Cable-Driven Parallel Robots: Proceedings of the 6th International Conference on Cable-Driven Parallel Robots 3031323211, 9783031323218

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Mechanisms and Machine Science

132

Series Editor Marco Ceccarelli , Department of Industrial Engineering, University of Rome Tor Vergata, Roma, Italy

Advisory Editors Sunil K. Agrawal, Department of Mechanical Engineering, Columbia University, New York, NY, USA Burkhard Corves, RWTH Aachen University, Aachen, Germany Victor Glazunov, Mechanical Engineering Research Institute, Moscow, Russia Alfonso Hernández, University of the Basque Country, Bilbao, Spain Tian Huang, Tianjin University, Tianjin, China Juan Carlos Jauregui Correa , Universidad Autonoma de Queretaro, Queretaro, Mexico Yukio Takeda, Tokyo Institute of Technology, Tokyo, Japan

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Stéphane Caro · Andreas Pott · Tobias Bruckmann Editors

Cable-Driven Parallel Robots Proceedings of the 6th International Conference on Cable-Driven Parallel Robots

Editors Stéphane Caro École Centrale Nantes, CNRS, LS2N Nantes Université Nantes, France

Andreas Pott University of Stuttgart Stuttgart, Germany

Tobias Bruckmann Chair of Mechatronics University of Duisburg-Essen Duisburg, Germany

ISSN 2211-0984 ISSN 2211-0992 (electronic) Mechanisms and Machine Science ISBN 978-3-031-32321-8 ISBN 978-3-031-32322-5 (eBook) https://doi.org/10.1007/978-3-031-32322-5 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

In 2012, leading experts from three continents gathered during the “First International Conference on Cable-Driven Parallel Robots (CableCon 2012)” in Stuttgart, Germany. This conference initiated a forum for the cable robot community. Due to the great success, the event was continued by the “Second International Conference on Cable-Driven Parallel Robots (CableCon 2014)” at the University Duisburg-Essen in 2014, establishing CableCon as a home for cable robot researchers. In 2017, the “Third International Conference on Cable-Driven Parallel Robots (CableCon 2017)” was organized by the Université Laval in Québec, Canada. This time, the conference went across the Atlantic and emphasized its role as an international event, where researchers from all over the world found a place to connect with the most recognized experts from the field. Back to Europe, the “Fourth International Conference on Cable-Driven Parallel Robots” took place in 2019 as part of the “15th IFToMM World Congress” in Krakow, Poland, a very special event which celebrated the 50th anniversary of IFToMM. Because of the COVID-19 pandemic, this fifth edition could not be organized as a physical event in Montpellier, France, as initially planned but was the first CableCon held online. All five previous Conferences on Cable-Driven Parallel Robots were organized under the patronage of International Federation for the Promotion of Mechanism and Machine Science (IFToMM), and this “Sixth International Conference on Cable-Driven Parallel Robots” is no exception. We are more than happy to return to a face-to-face event and to welcome the community in Nantes in 2023. Meanwhile, practical investigations on cable-driven parallel robots are attracting the focus of research teams around the world and interesting applications are presented. At the same time, fundamental research continues to provide new insights into the physical understanding of cable-driven parallel robots. This broad variety of research activities is reflected by the content of this book. The editors would like to thank the authors for their valuable contributions. Noteworthy, the strict schedule in the preparation of this book would not have been feasible without the support of the reviewers and the scientific committee. We would like to express our gratefulness to Springer for smoothly processing the material for this book. Stéphane Caro Tobias Bruckmann Andreas Pott

Organization

General Chairs Stéphane Caro Tobias Bruckmann Andreas Pott

CNRS, LS2N, Nantes, France University of Duisburg-Essen, Germany University of Stuttgart, Germany

Scientific Committee Sunil Agrawal Marc Arsenault Philippe Cardou Marco Carricato Clément Gosselin Marc Gouttefarde Darwin Lau Jean-Pierre Merlet Leila Notash Dieter Schramm Alexander Verl

Columbia University, New York, USA Laurentian University, Greater Sudbury, Ontario, Canada Université Laval, Quebec City, Quebec, Canada University of Bologna, Italy Université Laval, Quebec City, Quebec, Canada LIRMM, Univ. Montpellier, CNRS, France The Chinese University of Hong Kong Inria, Sophia-Antipolis, France Queen’s University, Kingston, Ontario, Canada University of Duisburg-Essen, Germany University of Stuttgart, Germany

Local Organizing Committee Karine Cantèle Camilo Charron Christine Chevallereau Fabien Claveau Vincenzo Di Paola Benoit Furet Solenn Gillouard Sophie Girault Alexandre Goldsztejn Sarah Le Marceau Métillon

LS2N, CNRS, France LS2N, PACCE Team, Université Rennes 2, France LS2N, ReV Team, CNRS, France LS2N, CODEx Team, IMT Atlantique, France LS2N, RoMaS and OGRE Teams, Centrale Nantes—University of Genoa, France LS2N, RoMaS Team, IUT Nantes, France LS2N, France LS2N, CNRS, France LS2N, OGRE Team, CNRS, France LS2N, France LS2N, RoMaS Team, CNRS, France

viii

Organization

Karim Moussa Elodie Paquet Tahir Rasheed Hugo Sorin Kévin Subrin Philipp Tempel Maxime Thieffry Bozhao Wang

LS2N, CODEx Team, IMT Atlantique and IRT Jules Verne, France LS2N, RoMaS Team, IUT Nantes, France LS2N, RoMaS Team, IUT Nantes, France LS2N, RoMaS Team, CNRS, France LS2N, RoMaS Team, IUT Nantes, France LS2N, ARMEN Team, France LS2N, CODEx Team, IMT Atlantique, France LS2N, RoMaS Team, Centrale Nantes, France

Contents

Kinematics Pose-Estimation Methods for Planar Underactuated Cable-Driven Parallel Robots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sara Gabaldo, Edoardo Idà, and Marco Carricato Manipulability Analysis of Cable-Driven Serial Chain Manipulators . . . . . . . . . . Sanjeevi Nakka and Vineet Vashista

3

16

Advances in the Use of Neural Network for Solving the Direct Kinematics of CDPR with Sagging Cables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jean-Pierre Merlet

30

Kinetostatic Modeling and Configuration Variation Analysis of Cable-Driven Parallel Robots on Spherical Surfaces . . . . . . . . . . . . . . . . . . . . . . Lei Jin, Tarek Taha, and Dongming Gan

40

Cable Modelling FEM-Based Dynamic Model for Cable-Driven Parallel Robots with Elasticity and Sagging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Karim Moussa, Eulalie Coevoet, Christian Duriez, Maxime Thieffry, Fabien Claveau, Philippe Chevrel, and Stéphane Caro

55

Absolute Nodal Coordinate Finite Element Formulation of Cables for Dynamic Modeling of CDPRs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chedli Bouzgarrou

69

Statics and Path of the Cables of a Cable-Driven Parallel Robot Wrapping on Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hao Xiong and Yuchen Xu

82

Workspace Model-Based Workspace Assessment of a Planar Cable-Driven Haptic Device . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Max B. Schäfer, Sophie Weiland, Lukas Worbs, Ing Tien Khaw, and Peter P. Pott

97

x

Contents

Payload Placement on Board a Cable-Driven Parallel Robot with Workspace Including Tilt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 Jean-Baptiste Izard Comparison Analysis of Tendon-Driven Manipulators Based on Their Wrench Feasible Workspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 Nicolas J. S. Testard, Christine Chevallereau, and Philippe Wenger On the Cable Actuation of End-Effector Degrees of Freedom in Cable-Driven Parallel Robots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 Jean-Baptiste Izard and Marc Gouttefarde Control and Dynamics A Practical Approach for the Hybrid Joint-Space Control of Overconstrained Cable-Driven Parallel Robots . . . . . . . . . . . . . . . . . . . . . . . . . . 149 Valentina Mattioni, Edoardo Idà, Marc Gouttefarde, and Marco Carricato Brief Review of Reinforcement Learning Control for Cable-Driven Parallel Robots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 Pegah Nomanfar and Leila Notash Force Control of a 1-DoF Cable Robot Using ANARX for Output Feedback Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 Marcus Hamann, Valentin Höpfner, and Christoph Ament Energy-Efficient Control of Cable Robots Exploiting Natural Dynamics and Task Knowledge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 Boris Deroo, Erwin Aertbeliën, Wilm Decré, and Herman Bruyninckx Modeling, Simulation, and Control of a “Sensorless” Cable-Driven Robot . . . . . 197 Atakan Durmaz, Özlem Albayrak, Perin Ünal, and M. Mert Ankaralı Validation of Emergency Strategies for Cable-Driven Parallel Robots After a Cable Failure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 Roland Boumann, Christoph Jeziorek, and Tobias Bruckmann Stability Analysis of Profile Following by a CDPR Using Distance and Vision Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 Thomas Rousseau, Nicolò Pedemonte, Stéphane Caro, and François Chaumette Comparison of Explicit and Implicit Numerical Integrations for a Tendon-Driven Robot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 Nicolas J. S. Testard, Christine Chevallereau, and Philippe Wenger

Contents

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Design A Design Method of Multi-link Cable Driven Robots Considering the Rigid Structure Design and Cable Routing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 Yaoxin Guo and Darwin Lau A Predictor-Corrector-Scheme for the Geometry Planning for In-Operation-Reconfiguration of Cable-Driven Parallel Robots . . . . . . . . . . . . 261 Felix Trautwein, Thomas Reichenbach, Andreas Pott, and Alexander Verl Variable Radius Drum Design for Cable-Driven Parallel Robots Based on Maximum Load Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 Jonas Bieber, David Bernstein, and Michael Beitelschmidt Reconfiguration and Performance Evaluation of TBot Cable-Driven Parallel Robot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 Jinhao Duan, Hanqing Liu, Zhaokun Zhang, Zhufeng Shao, Xiangjun Meng, and Jingang Lv Development Methodology of Cable-Driven Parallel Robots Intended for Functional Rehabilitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 Ferdaws Ennaiem, Juan Sandoval, and Med Amine Laribi Transmission Systems to Extend the Workspace of Planar Cable-Driven Parallel Robots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308 Foroogh Behroozi, Philippe Cardou, and Stéphane Caro Toward the Creation of a Hybrid 4-UPS CDPR . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 Paul W. Sanford, Juan Antonio Carretero, Andrew C. Mathis, and Stéphane Caro Effect of Antagonistic Cable Actuation on the Stiffness of Symmetric Four-Bar Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332 Vimalesh Muralidharan, Christine Chevallereau, Philippe Wenger, and Nicolas J. S. Testard Tensegrity-Inspired Joint Can Protect from Impacts by Isolating . . . . . . . . . . . . . . 344 Jonas Walter, Lukas Rothfischer, Richard Stierstorfer, Takeru Nemoto, Jörg Franke, and Sebastian Reitelshöfer Calibration and Accuracy Dynamic Parameter Identification for Cable-Driven Parallel Robots . . . . . . . . . . . 357 Tahir Rasheed, Loic Michel, Stéphane Caro, Jean-Pierre Barbot, and Yannick Aoustin

xii

Contents

Forward Kinematics and Online Self-calibration of Cable-Driven Parallel Robots with Covariance-Based Data Quality Assessment . . . . . . . . . . . . . . . . . . . . 369 Ryan J. Caverly, Keegan Bunker, Samir Patel, and Vinh L. Nguyen Elasto-Static Model and Accuracy Analysis of a Large Deployable Cable-Driven Parallel Robot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381 Zane Za¸ke, Nicolò Pedemonte, Boris Moriniere, Adolfo Suarez Roos, and Stéphane Caro Application A Warehousing Robot: From Concept to Reality . . . . . . . . . . . . . . . . . . . . . . . . . . . 397 Amir Khajepour, Sergio Torres Mendez, Mitchell Rushton, Hamed Jamshidianfar, Ronghuai Qi, Alireza Pazooki, Laaleh Durali, and Amir Soltani IPAnema Silent: A CDPR for Spatial Hearing Experiments . . . . . . . . . . . . . . . . . . 407 Christoph Martin, Marc Fabritius, Christian Lehnertz, Philipp Juraši´c, Johannes T. Stoll, Marc O. Ernst, Werner Kraus, and Andreas Pott Experimental Study on Thrustered Cable-Suspended Parallel Robot for Collaborative Task . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419 Kazuki Hayashi, Yusuke Sugahara, and Yukio Takeda The Robotic Seabed Cleaning Platform: An Underwater Cable-Driven Parallel Robot for Marine Litter Removal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430 Marc Gouttefarde, Mariola Rodriguez, Cyril Barrelet, Pierre-Elie Hervé, Vincent Creuze, Jose Gorrotxategi, Arkaitz Oyarzabal, David Culla, Damien Sallé, Olivier Tempier, Nicola Ferrari, Marc Chaumont, and Gérard Subsol A Cable-Based Haptic Interface with a Reconfigurable Structure . . . . . . . . . . . . . 442 Bastien Poitrimol and Hiroshi Igarashi Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455

Kinematics

Pose-Estimation Methods for Planar Underactuated Cable-Driven Parallel Robots Sara Gabaldo, Edoardo Id` a(B) , and Marco Carricato University of Bologna, Bologna, Italy {sara.gabaldo3,edoardo.ida2,marco.carricato}@unibo.it

Abstract. Planar underactuated cable-driven parallel robots (UACDPRs) employ less than three cables to control the pose of the end-effector (EE ). Consequently, the EE pose cannot be calculated only from cablelength geometrical constraint equations. In order to solve the direct kinematic problem, redundant measurements may be acquired, and the EE pose calculated by optimizing the resulting system of nonlinear equations. For the optimal pose estimation of planar UACDPRs, this paper presents three sensor fusion algorithms based on nonlinear weighted least squares equations, differing for the algorithm termination conditions. Additionally, different redundant measurement sets are experimentally compared. Keywords: Underactuated robots · cable-driven parallel robots es timation · redundant measurements · sensor fusion

1

· pose

Introduction

Cable-driven parallel robots (CDPRs) consist of several actuated winches on which cables are automatically coiled and uncoiled to displace an end-effector (EE ) in the working area [13]. Underactuated CDPRs (UACDPRs) are equipped with fewer cables than the number of degrees of freedom (DoFs) of the EE . They are mechanically simpler than their overconstrained counterparts, and their workspace is more accessible, but not all EE DoFs can be controlled. In addition, their EE is underconstrained and preserves some DoFs even when actuators are locked; in other words, if an external disturbance acts on the EE , the latter can oscillate as a multi-DoF parallel pendulum [10]. UACDPRs are usually assembled in a suspended configuration, so gravity is needed to keep the cables taut. Model-based control is commonly employed for operating CDPRs, and obtaining real-time solutions to kinematic problems is fundamental [20]. As for any parallel manipulator, solving the direct kinematics (DK) proves difficult because the loop-closure equations are nonlinear in the unknown (the EE pose). c The Author(s), under exclusive license to Springer Nature Switzerland AG 2023  S. Caro et al. (Eds.): CableCon 2023, MMS 132, pp. 3–15, 2023. https://doi.org/10.1007/978-3-031-32322-5_1

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Thus, a discrete set of solutions can generally be obtained. In addition, regardless of how a single solution is found, such a solution may be very sensitive to measurement errors on acquired data [6]. To speed up computation, ensure that a single solution to the problem is found, and limit solution errors due to measurement noises, redundant measurements can be used, and their constraint equations on the EE pose enforced (see [17] for a relevant application on CDPRs). Additional information obtained by adding sensors needs to be appropriately managed, since it is not to be taken for granted that it adds value [17]. To this end, sensor fusion strategies [5,7,14], or Extended Kalman Filters [15,18,21] have been employed on CDPRs. Solving the DK problem for UACDPRs is even more challenging [3] because the constraint equations are less than the DoFs of the EE , and cable-length geometrical constraints must be used alongside mechanical equilibrium equations (statics [1], or dynamics [12]). However, this approach requires knowledge of the EE inertial parameters [11], which may not be precisely known or may change over time. Even for UACDPRs, the natural alternative is to add sensors that provide additional data [9], and consequently extra geometrical constraint equations to formulate a determined or overdetermined system of equations in the EE pose. This paper introduces three DK sensor-fusion algorithms based on the method proposed in [7] for fully-constrained planar CDPRs: this method is thus adapted to planar UACDPRs. The method estimates the EE pose by optimizing an overdetermined or fully determined system of geometric residual equations; optimization is performed by an iterative nonlinear weighted least-square algorithm, which is solved by a standard Gauss-Newton method. The algorithms differ in how and why the iterations are terminated, and, to the knowledge of the authors, this paper originally proposes to modify one of the classical termination conditions of the Gauss-Newton method. Instead of stopping the method when the residual norm is below an assigned threshold, iterations are halted when each residual equation reaches an individually specified threshold. Lastly, experiments with different sensors are performed to acquire different data sets. Both theoretical and practical sensor limitations are commented on: encoders on the winch motors are used to estimate cable lengths, encoders on pulleys swivel angles measure cable inclinations [12], and vertical reference units (VRUs) measure the EE orientation [6]. The outline of the article is as follows. Section 2 briefly introduces the geometric model of the UACDPR, whereas the sensor fusion DK algorithms are detailed in Sect. 3. Tests on the algorithms, and on the use of different data sets, are reported in Sect. 4. Section 5 draws conclusions.

2

Geometric Model

A planar UACDPR consists of a 3-DoF platform moved by 2 actuated cables (Fig. 1). Oxy is an inertial frame, whereas P x y  is a mobile frame attached to the EE . Bi , the points where cables enter the workspace, and Ai , the points

Pose-Estimation Methods for UACDPR

5

where cables are fixed to the platform, are described by vectors bi and ai in the inertial frame. The orientation θ of the platform is described by the rotation  T matrix R(θ) and the pose of the EE is ζ = pT θ . Cables are considered massless and inextensible, and modeled as straight-line segments.

Fig. 1. Geometric model of a planar UACDPR

The distance between Ai and Bi is: P

ρi = ai −bi = p+R(θ)

ai −bi ;

i = 1, 2;

  cos(θ) − sin(θ) R(θ) = (1) sin(θ) cos(θ)

where P ai  is the (constant) position vector of Ai in P x y  . Cable orientation angles are: ρi (ζ) · j (2) ψi (ζ) = atan ρi (ζ) · i with i and j being unit vectors along the x and the y axis, respectively. The geometrical constraints imposed by cable lengths l1 and l2 on the EE pose are: ρi (ζ) − li = 0

3

i = 1, 2

(3)

Sensor Fusion DK Algorithms

In this Section, the sensor fusion DK method proposed in [7] for fully-constrained CDPRs is adapted to UACDPRs. The iterative method is then associated with

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different iteration termination conditions (ITCs), and three sensor fusion DK algorithms are defined. A residual is defined as the difference between a modeled pose-dependent variable and its measurement. In this paper, at any time instant, we consider three sets of measurements to be available, namely cable lengths zl = [l1 , l2 ]T , cable angles zψ = [ψ1 , ψ2 ]T , and platform orientation zθ = θ , where the superscript  means a measured value. For convenience, we also define the following arrays of pose-dependent variables (see Eqs. (2), (3)): hl (ζ) = [l1 (ζ), l2 (ζ)]T ,

hψ (ζ) = [ψ1 (ζ), ψ2 (ζ)]T ,

hθ (ζ) = θ(ζ)

(4)

Depending on which sets of measurements are used, residuals rj (ζ) = hj (ζ)−zj , with j = l, ψ, θ, can be defined. Since we consider planar UACDPR, at least two sets of residuals need to be considered for establishing a DK algorithm with a discrete set of solutions. In fact, there is no single residual with at least as many elements as the EE DoFs (i.e., 3). Even though the use of at least two residuals is a necessary condition for establishing a DK algorithm, it may not be sufficient for correct pose determination: their derivatives with respect to the EE pose may be linearly dependent, and thus the DK problem may still be underdetermined even with more equations than unknowns, or the solution may be very sensitive to measurement errors. In general, if we consider a residual r(ζ) ∈ Rw , with w ≥ 3, composed by a combination of two or more rj (ζ), the DK problem can be stated as min ζ

f (ζ)

with

f (ζ) =

1 T r (ζ)Q−1 r(ζ) 2

(5)

where Q ∈ Rw×w is the diagonal covariance matrix of r(ζ) so that the solution ζ is the best linear unbiased estimator (BLUE) of the EE pose [7]. To solve this nonlinear least-square problem, Gauss-Newton’s method may be applied to the gradient of f (ζ) with respect to ζ [2]. Let us denote said gradient by ∇f (ζ) ∈ R3 , and the Hessian of f (ζ) by ∇2 f (ζ) ∈ R3×3 . Given an initial approximation of the solution, ζ 0 , Gauss-Newton’s method inductively defines a sequence of approximations ζ k as [19]: ζ k+1 = ζ k − [∇2 f (ζ k )]−1 ∇f (ζ k )

(6)

If A ∈ Rq×q is a symmetric matrix non depending on the EE pose, c(ζ) ∈ Rq is vector depending on the EE pose, and α(ζ) = cT (ζ)Ac(ζ) is a symmetric quadratic form, the gradient of α(ζ) with respect to ζ is given by1 :   ∇α(ζ) = 2 ∇cT (ζ) Ac(ζ) (7) with ∇cT (ζ) ∈ R3×q . Accordingly, since Q−1 is diagonal and constant, and vector z is constant, the gradient of f (ζ) is computed as:     ∇f (ζ) = ∇rT (ζ) Q−1 r(ζ) = ∇hT (ζ) Q−1 r(ζ) = JT Q−1 r (8) 1

The proof is omitted for brevity sake, but is based on basic matrix product and derivative properties. See the appendix of [10] for similar calculations.

Pose-Estimation Methods for UACDPR

7

where ∇rT (ζ) = ∇hT (ζ) = JT ∈ R3×w , and the explicit dependence from ζ was dropped. The Gauss approximation is used for the evaluation of the Hessian of f (ζ), and the term dependent from the second order vector derivative of r(ζ), namely the gradient of J, is neglected2 :   T   (−1 (( T −1 T −1 ∇2 f (ζ) = (∇J (9) (((Q r + J Q (∇r)  J Q J Finally, if Jk and rk respectively denotes J and r calculated in ζ k , substituting Eqs. (8) and (9) in Eq. (6) yields: ζ k+1 = ζ k − [JTk Q−1 Jk ]−1 Jk T Q−1 rk

(10)

Classically, iterations are terminated when one of the following conditions (Iteration Termination Conditions, or ITCs) are met: 1. too many iterations have occurred; 2. the norm of the variation in the solution, namely Δζ k  = [JTk Q−1 Jk ]−1 Jk T Q−1 rk  is below a predefined threshold; 3. the norm of the gradient of f (ζ k ), namely the optimality condition ∇f (ζ k ) is below a near-zero predefined threshold; 4. the norm of rk , is below a near-zero predefined threshold. ITC 1 is a safeguard against ill-posed problems; if met, the algorithm has usually failed. If ITC 2 is met, the algorithm has usually stalled because the problem is ill-conditioned, and a solution may or may not have been found in practice. If ITC 3 is met, a local extremum of Eq. (5) is found, but this may not be the absolute minimum f . Finally, if ITC 4 is met, a global extremum of f , namely a solution to the DK problem, is found. In the authors’ opinion, there are several problems with the last ITC, which ensures that the DK solution is found. Since each element of rk may have different units and possibly different magnitudes, a condition on its norm is neither theoretically nor practically convenient. In addition, if some sensors are less accurate or their model less realistic, the residual associated with their measurement can be very high while other sensors are naturally associated with low residuals; then the algorithm may try to optimize a residual that is impossible to decrease, eventually incurring in ITCs 1–3, which are not the desired ones. Alternatively, one can try to assign an engineeringly meaningful near-zero threshold for ITC 4, but, for the above reasons, this is not straightforward. We then propose to modify condition 4 with: 5. the norm of each element of rk is below its own predefined threshold. Each element of r(ζ) is the difference between a measured and a modeled variable. Since measurements are affected by errors, it is reasonable to consider 2

Computing the second-order vector derivative of a vector results in a 3rd-order tensor; any computation between a 3-rd order tensor, matrices, and vectors is not straightforward, but an example of how to perform them can be found in the appendix of [10].

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a residual near the engineeringly acceptable value of said errors, if these are known. As an illustrative example, let us consider cable-angle measures. In a planar robot, encoders can directly measure pulley swivel angles [16]. Unfortunately, in practice, the geometric model of Eq. (2) is subject to various sources of error, since it is built under strong assumptions, such as (i) the cable and the pulley routing are co-planar, and (ii) the rotating swivel is frictionless. The robot may be experimentally characterized to determine the maximum error eψ , between modeled and measured cable angles, if a ground truth instrumentation is available (a laser tracker or a camera system to measure the EE pose accurately). Then, one can conservatively consider eψ (1 + p), with p an engineeringly acceptable percentage of the maximum error, as the threshold for each cable orientation residual. If the sensor and the model cannot agree more than eψ , it is not reasonable to iterate so that a lower residual threshold is met. Deeper considerations can be made to obtain configuration-dependent thresholds, which vary throughout the workspace, but this is out of the scope of this paper. Lastly, three sensor fusion DK algorithms are defined. For each of them, ITCs 1–3 are equal. The first algorithm, denoted as NormGN (Norm GaussNewton), is a classical version of the Gauss-Newton method, and results from Eq. 10 associated with ITC 1–4, where ITC 4 terminates iterations when the norm of the residual vector rk  is below an assigned threshold t. For NormGN, the rationale for defining ITC 4 threshold is the following: t = (1 + p)e

(11)

where p = 0.02 and e is the array of the maximum errors associated with the residuals under consideration. As an example, if we consider cable-length and cable-angle residuals, maximum cable-length error el = 0.003 m, and maximum cable-angle error eψ = 0.0523 rad (corresponding to 3◦ ), NormGN ITC 4 threshold is computed as  t = 1.02[el , el , eψ , eψ ]T  = 1.02 0.0032 + 0.0032 + 0.05232 + 0.05232 = 0.0756 (12) This example clearly shows the lack of physical ground at the basis of this threshold, since it sums meters and radians, and is severely biased by the measurement units that are selected. In this case, meters and radians are chosen in order to obtain a comparable order of magnitude in both terms, but, should different units be chosen, completely different results would be obtained. The second algorithm, denoted as StGN (Strict Gauss-Newton), differs from the previous one only by the ITC 4 threshold, which is set to be stricter than the previous one by 2 orders of magnitude, so that a higher accuracy may be expected. The third and last algorithm result from Eq. 10 along with ITC 1–3 and 5, where ITC 5 is set to terminate iterations when all the components of the residual array r(ζ k ) are below the components of an engineering array of thresholds t defined as t = (1 + p)e (13) This algorithm is denoted as EngGN (Engineering Gauss-Newton).

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4

9

Experimental Validation

The proposed sensor fusion DK algorithms were implemented in MATLAB to evaluate and compare their performances in terms of accuracy and computational running time. To this end, the prototype described in [16] was converted into an UACDPR by removing the lower cables (Fig. 2), and used to acquire the data sets (zj , with j = l, ψ, θ) necessary to run the algorithms. The prototype is defined by the following geometric and inertial parameters:

Fig. 2. Planar UACDPR used for experiments

P

a1  =



     −6.5 6.5 0.13 cm, P a2  = cm, P g = cm, 4 4 1.53     0 95 cm, b2 = cm, m = 1Kg b1 = 0 0

where P g is the (constant) position vector of the platform center of mass in P x y  . A trapezoidal velocity profile was assigned to robot cables, since a more complex cartesian trajectory planning was not needed [8]:     44.59 88.41 ls = cm, lf = cm (14) l(t) = ls − (lf − ls )u(t), 88.50 44.60

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where ls and lf are the cable lengths at the start and final equilibrium configurations, respectively ζ s = [15 cm, −50 cm, −36◦ ]T and ζ f = [80 cm, −50 cm, 36◦ ]T , and u(t) is the velocity trapezoidal motion law [8], with u(0) = 0 and u(T = 9.29 s) = 1. Actual cable lengths were estimated using incremental encoders ( 0.1◦ resolution) on each motor axis, and a kinematic model of the winches [12]. At the same time, cable angles were measured by incremental encoders ( 0.02◦ resolution) on each swivel pulley. Platform orientation data were acquired through an XSens MTi-630 AHRS sensor (pitch and roll accuracy 0.3◦ ), fixed on the EE . Based on a preliminary characterization of model-sensor accuracy, the maximum errors to be used in the DK algorithms were set to eψ = 0.0523 rad (namely 3◦ ) for cable angles, el = 0.003 m for cable lengths and eθ = 0.0087 rad (namely 0.5◦ ) for platform orientation; the acceptable percentage of the maximum error was set to p = 0.02 (2%). Radians and meters were chosen as measurement units so that the NormGN algorithm would produce nearly acceptable results, to further highlight that using the standard ITC 4 for the Gauss-Newton method is everything but straightforward and engineeringly meaningful. Encoders and VRU measurements were sampled at 100 Hz, and thus N = 292 sets of measurements z = [zl , zψ , zθ ]T were acquired along the assigned trajectory. Finally, ground-truth pose measurements were obtained using a computer vision system: an ArUco chessboard (ChArUco) was attached to the platform (see Fig. 2) and a GoPro HERO6 with 2.7K resolution and a 4 : 3 aspect ratio, shooting at 30 frames per second, was used to acquire images of the ChArUco. The images were then processed using an OpenCV Phyton library [4] to determine the actual evolution of the platform pose. The results were interpolated to be compared with the higher frequency samples of the sensor-fusion algorithm. Accuracy and running time performances were evaluated by considering the following: (i) different definitions of the residual vector r(ζ), i.e., assuming that different sets of sensors were available to collect data; (ii) different iteration termination conditions (ITCs) sets; (iii) different threshold values for some ITCs. For simplicity sake, the following acronyms refer to specific residual sets: – sl, which stands for swivel angles and cable lengths, represents r = [rTψ , rTl ]T ; – sli, which stands for swivel angles, cable lengths and inclinometer, represents r = [rTψ , rTl , rθ ]T ; – li, which stands for cable lengths and inclinometer, represents r = [rTl , rθ ]T – si, which stands for swivel angles and inclinometer, represents r = [rTψ , rθ ]T and the following assumptions were considered: – each algorithm tentative solution for the first pose in the trajectory was set to ζ 0 = [15 cm, −50 cm, −36◦ ], the theoretically assigned initial pose, while the remaining N − 1 tentative solutions were set to the solution computed in the previous point of the trajectory; – ITC 1 set to 100, ITC 2 set to 10−4 and ITC 3 set to 10−4 ; – each element of the diagonal matrix Q−1 was set to e−2 j , with j = l, ψ, θ depending on the corresponding elements of r in Eq. (5); this is standard practice when a more suitable estimation of the variance of r is not available [22].

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Fig. 3. Platform poses along the trajectory. Cam represents the trajectory reconstructed by the camera, while the acronyms represent the results obtained with different residuals.

The camera measurements (denoted as cam) and the pose components computed by the NormGN algorithm with the sl, sli, li, and si sets of residuals are shown in Fig. 3a. It can be noted that the DK solution is constant for a series of successive points of the trajectory. This happens because the ITC 4 is satisfied even before the algorithm performs the first iteration, and the tentative solution becomes the real solution. This phenomenon is perfectly reasonable: even if one (or more) components of the residual are way above their maximum error, other components of the residual are way below, and thus the norm of the residual may be below the assigned threshold. In Fig. 4, the blue bars show the pose-component average absolute errors for the NormGN across all residual sets, computed by subtracting the sensor fusion results from the camera-based poses through the trajectory. Residual sl performances are poor in determining

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platform orientation: the average error e¯θ is about 12◦ . In contrast, residual si performances are poor in the determination of platform position, yielding to average errors e¯x = 6 cm and e¯y = 3.5 cm. Residual li performances are slightly better than sli, especially in the determination of x pose-coordinate (¯ ex = 0.5 cm rather than e¯x = 1.5 cm average error): the poor accuracy of cable-orientation sensor drastically influences the algorithm result. Next, the threshold for the StGN was empirically set to t = 10−4 since a lower value resulted in a negligible accuracy improvement. Such an improvement can be observed in Fig. 4, where the pose-component average absolute errors are shown with a red bar. If residuals sli or li are considered, StGN is almost twice more accurate as NormGN. On the other hand, the average computing time of StGN is nearly double than NormGN (Table 1). Lastly, the camera measurements and the pose components computed by the EngGN algorithm with the sl, sli, li, and si sets of residuals are shown in Fig. 3b. The pose trend is smoother compared with NormGN since, in most points, the ITC 5 is not satisfied without at least one iteration of the algorithm. Thus, the solution changes continuously instead of discretely. The accuracy of the three algorithms is compared in Fig. 4 while Table 1 compares the average computing times. The experimental results clearly show that: – algorithm EngGN is more accurate (up to 67% accuracy increase) than NormGN (except for the determination of the orientation angle θ with the residual sl, where is 14% less accurate), while being comparable in terms of average computing time (EngGN is at worst 15% slower than NormGN, and never faster); – the accuracy of StGN and EngGN are comparable for any choice of residual, but the average computing time of EngGN is almost half that of StGN;

Fig. 4. Mean pose-variable errors along the trajectory

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Table 1. Table of average computational times of the algorithms. Set of residuals NormGN [ms] StGN[ms] EngGN[ms] sl

0.55

1.10

0.62

sli

0.65

0.99

0.75

li

0.45

0.68

0.45

si

0.38

0.65

0.41

– the accuracy results of NormGN and StGN are heavily influenced by the choice of the measurement units, and this choice is not straightforward and, in any case, lacks a physical meaning; – the use of EngGN allows for fast and accurate results, and, since it is physically sound, is also simple to implement: it is, thus, the best compromise among the three proposed algorithms; since the algorithm was run in Matlab, it is expected that a PLC implementation would be able to run within a small fraction of standard real-time cycle periods (0.5–2 ms).

5

Conclusions

In this work, the sensor-fusion DK method proposed in [7] was adapted to planar UACDPRs. To this end, the authors proposed 3 different algorithms differing in iteration termination conditions, and the numerical value of these condition thresholds. In particular, this paper originally proposed an engineeringly meaningful termination condition for the Gauss-Newton method used for sensor fusion, which was applied to one of the three algorithms. Experiments on a planar prototype with 2 cables were performed, and different sets of sensors were used in the sensor-fusion algorithms. It was shown that measurements of cable orientation tend to degrade the accuracy in determining platform poses, while measures of cable lengths and platform orientation significantly increase the accuracy of the results. Additionally, the original termination condition proposed in this paper allowed for (i) drastically reducing the sensor-fusion computation time while achieving comparable accuracy, and (ii) achieving a sound physical meaning, since its results are independent of the used measurement units. In the future, the originally proposed sensor-fusion DK method with engineering Gauss-Newton termination conditions will be applied to 6-DoF UACDPRs, and the optimal set of sensors and residual will be investigated. For these systems, orientation parametrization choice may hinder the effective application of the method, due to representation singularities and orientation measurement limitations (AHRS are very accurate for pitch and roll measurement, but not very effective for yam measurement). Additionally, 3-, 4-, or 5-cable UACDPRs may need different types of sensors, depending on their degree of underactuation.

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Acknowledgments. The authors acknowledge the support of the Italian Ministry of University and Research (MUR) through the PRIN 2020 grant “Extending Robotic Manipulation Capabilities by Cooperative Mobile and Flexible Multi-Robot Systems (Co-MiR)” (No. 2020CMEFPK).

References 1. Abbasnejad, G., Carricato, M.: Direct geometrico-static problem of underconstrained cable-driven parallel robots with n cables. IEEE Trans. Rob. 31(2), 468– 478 (2015) 2. Bell, B.M.: The iterated Kalman filter update as a Gauss-Newton method. IEEE Trans. Autom. Control 38(2), 294–297 (1993) 3. Berti, A., Merlet, J.P., Carricato, M.: Solving the direct geometrico-static problem of underconstrained cable-driven parallel robots by interval analysis. Int. J. Rob. Res. 35(6), 723–739 (2016) 4. Bradski, G.: The OpenCV Library. Dr. Dobb’s J. Softw. Tools (2000) 5. Fortin-Cˆ ot´e, A., Cardou, P., Campeau-Lecours, A.: Improving cable driven parallel robot accuracy through angular position sensors. In: IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), Daejeon, Korea (South), pp. 4350–4355 (2016) 6. Gabaldo, S., Id` a, E., Carricato, M.: Sensitivity of the direct kinematics of underactuated cable-driven parallel robots to redundant sensor-measurement errors. In: Altuzarra, O., Kecskem´ethy, A. (eds.) Advances in Robot Kinematics 2022. ARK 2022, Springer Proceedings in Advanced Robotics, vol. 24, pp. 131–138. Springer, Cham (2022) 7. Garant, X., Campeau-Lecours, A., Cardou, P., Gosselin, C.: Improving the forward kinematics of cable-driven parallel robots through cable angle sensors. In: Gosselin, C., Cardou, P., Bruckmann, T., Pott, A. (eds.) Cable-Driven Parallel Robots. MMS, vol. 53, pp. 167–179. Springer, Cham (2018) 8. Id` a, E., Briot, S., Carricato, M.: Robust trajectory planning of under-actuated cable-driven parallel robot with 3 cables. In: Lenarˇciˇc, J., Siciliano, B. (eds.) ARK 2020. SPAR, vol. 15, pp. 65–72. Springer, Cham (2021). https://doi.org/10.1007/ 978-3-030-50975-0 9 9. Id´ a, E., Merlet, J.-P., Carricato, M.: Automatic self-calibration of suspended underactuated cable-driven parallel robot using incremental measurements. In: CableDriven Parallel Robots. MMS, vol. 53, pp. 333–344. Springer, Cham (2019) 10. Id´ a, E., Briot, S., Carricato, M.: Natural oscillations of underactuated cable-driven parallel robots. IEEE Access 9, 71660–71672 (2021) 11. Id´ a, E., Briot, S., Carricato, M.: Identification of the inertial parameters of underactuated cable-driven parallel robots. Mech. Mach. Theory 167, 104504 (2022) 12. Id´ a, E., Bruckmann, T., Carricato, M.: Rest-to-rest trajectory planning for underactuated cable-driven parallel robots. IEEE Trans. Rob. 35(6), 1338–1351 (2019) 13. Id´ a, E., Mattioni, V.: Cable-driven parallel robot actuators: state of the art and novel servo-winch concept. Actuators 11(10), 290 (2022) 14. Korayem, M., Yousefzadeh, M., Kian, S.: Precise end-effector pose estimation in spatial cable-driven parallel robots with elastic cables using a data fusion method. Measurement 130, 177–190 (2018) 15. Le Nguyen, V., Caverly, R.J.: Cable-driven parallel robot pose estimation using extended kalman filtering with inertial payload measurements. IEEE Rob. Autom. Lett. 6(2), 3615–3622 (2021)

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16. Mattioni, V., Ida’, E., Carricato, M.: Design of a planar cable-driven parallel robot for non-contact tasks. Appl. Sci. 11(20), 9491 (2021) 17. Merlet, J.P.: An experimental investigation of extra measurements for solving the direct kinematics of cable-driven parallel robots. In: 2018 IEEE International Conference on Robotics and Automation (ICRA), Brisbane, QLD, Australia, pp. 6947– 6952 (2018) 18. Nguyen, V.L., Caverly, R.J.: CDPR forward kinematics with error covariance bounds for unconstrained end-effector attitude parameterizations. In: Gouttefarde, M., Bruckmann, T., Pott, A. (eds.) CableCon 2021. MMS, vol. 104, pp. 37–49. Springer, Cham (2021) 19. Nocedal, J., Wright, S.J.: Numerical Optimization, 2nd edn. Springer, Heidelberg (2006). https://doi.org/10.1007/978-0-387-40065-5 20. Pott, A.: An algorithm for real-time forward kinematics of cable-driven parallel robots. In: Lenarcic, J., Stanisic, M.M. (eds.) Advances in Robot Kinematics: Motion in Man and Machine. Springer, Dordrecht (2010). https://doi.org/10.1007/ 978-90-481-9262-5 57 21. Puri, N., Caverly, R.J.: Pose estimation of a cable-driven parallel robot using kalman filtering and forward kinematics error covariance bounds. In: Larochelle, P., McCarthy, J.M. (eds.) USCToMM MSR 2022. MMS, vol. 118, pp. 65–75. Springer, Cham (2022). https://doi.org/10.1007/978-3-030-99826-4 7 22. Sequeira, J., Tsourdos, A., Lazarus, S.B.: Robust covariance estimation for data fusion from multiple sensors. IEEE Trans. Instrument. Meas. 60(12), 3833–3844 (2011)

Manipulability Analysis of Cable-Driven Serial Chain Manipulators Sanjeevi Nakka and Vineet Vashista(B) IIT Gandhinagar, Palaj 382355, Gujarat, India [email protected]

Abstract. Understanding the manipulability of a manipulator is critical to interpreting manipulators’ motion generation abilities in the taskspace environment. A manipulator that can alter the manipulability characteristics provides an advantage in making it suitable for versatile applications. Recently, cable-driven serial chain manipulators, CDSMs, due to their promising features, such as low moving inertia, large payload handling capacity, and ability to alter the architecture, have emerged as an important robotic platform for various applications. These systems offer flexibility in architecture modulation, thus implying the possibility of modulating the manipulability. In this context, the current work focuses on the manipulability analysis of CDSMs by formulating the effect of cable routing on the capability of the manipulator to generate motion. In particular, two different planar cable routing architectures are considered. A quantitative measure for the CDSM’s manipulability is constituted to study the variations over the workspace. Further, from the application perspective, the potential of utilizing the cable routing alteration to improve the manipulability of cable-driven leg exoskeleton is presented.

Keywords: Cable-Driven Systems

1

· Manipulability · Exoskeletons

Introduction

Manipulators with serial-chain architecture have been widely used in the robotics community due to their unique properties, such as high repeatability, workspace, and ease of control. Historically, their usage has been mainly in industrial operations, such as pick-and-place, painting, welding, etc. [1]. Owing to their serialchain architecture, the design of these manipulators has been adapted as legs, arms, and fingers for humanoid robots to allow motion over different surfaces, interaction with the environment, the capability of carrying objects, and complex operations involving grasping and multi-arm collaboration [2]. Thus, their current usage is in a vast range of different applications ranging from industries to rehabilitation involving physical human-robot interaction [3]. Over the years, cable-driven serial-chain manipulators, CDSMs, where motoractuated cables apply external forces on the links to control the end-effector c The Author(s), under exclusive license to Springer Nature Switzerland AG 2023  S. Caro et al. (Eds.): CableCon 2023, MMS 132, pp. 16–29, 2023. https://doi.org/10.1007/978-3-031-32322-5_2

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motion, have emerged as important robotic platforms. Owing to the use of cables, a CDSMs offers flexibility in mounting actuators away from the joints on a separate frame, which adds the advantage of the reduced manipulator’s moving mass and inertia and improved force to weight capacity [4]. Further, CDSMs offer flexibility in varying actuators’ positions, cable attachment points, and cable routing, i.e., re-configuring the manipulator’s design and architecture, thus providing the potential to alter the system performance characteristics. These unique advantages have made CDSMs a preferred robotic manipulator for numerous applications, including robotics fingers [5], human movement rehabilitation [6], etc. As a cable can only apply pulling force and cannot push the body it is connected to, CDSMs require actuation redundancy and need to have non-zero positive cable tension during the operations [7]. This difference in actuation mode makes CDSMs exhibit different performance characteristics compared to torque driven serial-chain manipulators, TDSMs, where actuators are directly positioned at joints. Thereby, the performance analyses of TDSMs are not directly extendable to CDSMs. Accordingly, studies reporting the workspace analysis, stiffness analysis, cable tension optimization, and control of CDSMs have been conducted extensively [4,8,9]. Among the various performance measures of a manipulator, its manipulability plays a critical role in effectively controlling the manipulator’s position and orientation. Manipulability analyses have been extensively reported in the literature for different TDSM architectures and applications in robotics, such as trajectory planning, manipulation programming, redundancy treatment, optimal design, and improving robot dexterity [10–13]. However, manipulability analysis of CDSMs has not been explored markedly [14]. In TDSMs, the manipulator’s taskspace performance depends on the joint space actuation mapped through a jacobian matrix. In contrast, in CDSMs, cable-space acts as the actuation space and directly affects the manipulator’s joint space and taskspace performance. Further, there exists the potential of tuning cable space parameters to attain a desired manipulator’s performance in terms of its ability to generate motion and apply forces in arbitrary directions. Accordingly, the current work formulates the manipulability measure for CDSMs to study the effect of cable routing. In particular, a planar architecture of two degrees of freedom (DOFs) serial chain manipulator is considered with two different cable routing architectures for the analysis. The manipulability measures of the CDSMs are compared with TDSMs. Finally, the presented manipulability analysis has been applied to a cable-driven leg exoskeleton during the swing phase of walking, and insights into improving the exoskeleton performance by varying the cable routing are presented.

2

Manipulability Formulation

The concept of manipulability is interpreted as the ability to position and orientate the end-effector of a robotic arm in different directions. In particular, the

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Fig. 1. An illustration of the considered 2 DOFs serial chain manipulator architectures: (a) Torque-driven serial-chain manipulator (TDSM); (b) Cable-driven serialchain manipulator (CDSM) general architecture; and (c) CDSM co-shared architecture, where cables attached to distal link are routed through the proximal link.

manipulability can be established from the variations in the end-effector velocities across the workspace. Understanding the manipulator’s manipulability in taskspace is required for the design and control of robots during a particular task. 2.1

System Architecture

Figure 1 shows the schematics of planar two degrees of freedom two-link serial chain manipulator, where the rigid links are connected with revolute joints. A TDSM, where actuators are positioned at joints, is represented in Fig. 1a. Two CDSMs are shown in Figs. 1b and c, where motor actuated cables are connected to the links at some offsets to apply external forces on the links to generate the joint torques. As a cable can only apply a pulling force, actuator redundancy is required to control a CDSM [4]. For the presented analysis, four actuated cables, i.e., two actuators redundancy, are used to form two architectures referred as CDSM general and CDSM co-shared architectures in Figs. 1b and c, respectively. These architectures differ in how cables are routed from the motors to the links. For CDSM general architecture, each motor actuated cable is routed to apply a direct external force on only one link. In contrast, for CDSM co-shared architecture, cables attached to the distal link are routed through the proximal link such that these cables apply external forces to both the proximal and distal links. The link lengths are denoted as di , and the generalized position and force coordinates of the single DOF revolute joint are θi and τi , respectively. For the CDSMs in Fig. 1, the cables attachment point on each link is at a distance ai from the joint center and at an offset, hi from the link axis, and Mi denotes the corresponding motor position from O. TDSM: To describe the manipulability of a TDSM, noting that the actuators are directly positioned over the joint, the mapping between joint space to task space velocities has to be modeled. Let xi = fi (θ1 , θ2 , · · · , θn ) for i = 1, 2, · · · , r be a set of r equations relating the end-effector positions with the joint variables

Manipulability Analysis of Cable-Driven Serial Chain Manipulators

19

of a n DOFs manipulator. The time derivatives of xi can be written as function of θ˙i as follows, ⎤ ⎡ ⎤ ⎡ ∂f1 ∂f1 ∂f1 ⎡ ˙ ⎤ θ1 x˙ 1 ∂θ1 ∂θ2 · · · ∂θn ∂f2 ∂f2 ∂f2 ⎥ ⎢ ˙ ⎥ ⎢x˙ 2 ⎥ ⎢ · · · θ2 ⎥ ∂θ1 ∂θ2 ∂θn ⎥ ⎢ ⎥ ⎢ ⎥⎢ (1) ⎢ ⎢ .. ⎥ = ⎢ . ⎥ =⇒ X˙ = J θ˙ . . . .. · · · .. ⎥ ⎣.⎦ ⎢ ⎣ .. ⎦ ⎣ .. ⎦ ∂fr ∂fr x˙ r θ˙n · · · ∂fr ∂θ1 ∂θ2

∂θn

The X = [x1 , x2 , . . . , xr ]T ∈ Rr denotes r -dimensional position vector in task space and θ = [θ1 , θ2 , . . . , θn ]T ∈ Rn denotes an n-dimensional position vector in joint space. Essentially, Jacobian, J ∈ Rr×n , is a linear transformation matrix ˙ to the end-effector velocity vector, X, ˙ and that maps joint velocity vector, θ, serves as the means to describe the TDSM’s manipulability. CDSM General Architecture: In CDSMs, the joint actuation is the function of cable attachment and the cable actuation rates. For a CDSM with m cables, if lc = [l1 , l2 , · · · , lm ] ∈ Rm denotes the cable lengths, the relation between the ˙ and the cable actuation rates, l˙c , is given by Eqs. 2, 3 and joint velocities, θ, 4. Further, the mapping between joint space to task space velocities happens −→ similar to TDSM as presented in Eq. 1. In Eq. 2, Mi denotes ith motor position → from the origin, O, and − ri denotes ith cable attachment position on the link, shown in Fig. 1b. − → −→ → ri (θ) =⇒ ||li || = ||Mi − ri (θ)|| li = M i − − ⎡ ˙ ⎤ ⎡ ˆ ∂r1 ∂r1 −l1 . ∂θ1 −ˆl1 . ∂θ l1 2 ⎢ ⎢ l˙2 ⎥ ⎢ −ˆl2 . ∂r2 −ˆl2 . ∂r2 ∂θ1 ∂θ2 ⎢ ⎥ ⎢ ⎢ . ⎥=⎢ .. .. ⎣ .. ⎦ ⎣ . . ˙ m ˆlm . ∂rm lm −ˆlm . ∂r − ∂θ1 ∂θ2

⎤⎡ ⎤ ∂r1 · · · −ˆl1 . ∂θ θ˙1 n ∂r2 ⎥ ⎢ ˙ ⎥ ˆ · · · −l2 . ∂θn ⎥ ⎢ θ2 ⎥ ⎥⎢ . ⎥ .. ⎥⎣ . ⎦ ⎦ . ··· . ∂rm ˆ θ˙n · · · −lm . ∂θn

l˙c = −AT θ˙ ⎡ ∂ r1 −ˆl1 . ∂θ 1 ⎢ −ˆl ∂r2 ⎢ 2 ∂θ1 T − A = ⎢ ˆ ∂r3 ⎣ −l3 ∂θ1 ∂ r4 −ˆl4 . ∂θ 1

⎤ ∂ r1 −ˆl1 . ∂θ 2 ∂ r2 ⎥ −ˆl2 ∂θ ⎥ 2 ∂ r3 ⎥ ⎦ −ˆl3 ∂θ 2 ∂ r4 ˆ −l4 . ∂θ2

(2)

(3)

(4)

(5)

In Eq. 4, structure matrix, A, is a function of system geometry and captures the effect of cable routing and redundancy for a CDSM architecture. For the case of 2 DOF General CDSM shown in Fig. 1(b), structure matrix, A, can be represented as in Eq. 5. Further, from Eqs. 1 and 4, the end-effector velocities, ˙ can be mapped to the cable actuation rates, l˙c , as in Eq. 6. Thus, for CDSM, X, ˙ are a function of the manipulator’s configuration the end-effector velocities, X,

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represented by the Jacobian, J, and the cable routing represented by the structure matrix, A. Notably, a single transformation, Jc , is used in Eq. 6 to capture the effect of both of these terms on the CDSM’s manipulability. X˙ = −J(AT )−1 l˙c = −Jc l˙c

(6)

CDSM Co-shared Architecture: For the case of CDSM co-shared architecture, the co-sharing of cables alters the mapping between the joint rates and the cable actuation rates, i.e., structure matrix, A [15]. For the considered CDSM co-shared architecture with two co-shared cables, as in Fig. 1b, the changes in A are highlighted in Eq. 7. ⎡ ⎤ ∂ r1 ∂ r1 −ˆl1 . ∂θ −ˆl1 . ∂θ 1 2 ⎢−ˆl ∂r2 − (ˆl − ˆl ) ∂r1 −ˆl ∂r2 ⎥ ⎢ 22 1 21 22 ∂θ1 22 ∂θ2 ⎥ (7) − AT = ⎢ ˆ ∂θ ∂ r3 ˆl31 − ˆl32 ) ∂r1 −ˆl32 ∂r4 ⎥ ⎣−l32 ∂θ ⎦ − ( ∂θ1 ∂θ2 1 ∂ r4 ∂ r4 −ˆl4 . ∂θ −ˆl4 . ∂θ 1 2 2.2

Manipulability Modelling and Measure

TDSM: The transformation between the taskspace and the joint space in Eq. 1 is used to describe the manipulability. Manipulability ellipsoids and other measures based on manipulator Jacobian, J, have been reported in the literature [16,17]. Specifically, the preimage of the unit sphere in the joint space, θ˙T θ˙ = 1, under the mapping of Eq. 1, transforms to the velocity manipulability ellipsoid, as given in Eq. 8. Thus, the transformation JJ T defines a TDSM’s velocity and force manipulability characteristics. θ˙T θ˙ = 1 =⇒ X˙ T (JJ T )−1 X˙ = 1

(8)

In this work, we use a manipulability measure, w , that provides a quantitative measure of the manipulator’s ability to move in an arbitrary direction over the workspace [18,19]. From Eq. 9, the manipulability index, w , is derived from the transformation JJ T of a TDSM. For an invertible J, the matrix JJ T is positive semi-definite [19], which implies that the manipulability index has only positive real values, i.e., w ∈ R+ . Thus, for a configuration of the manipulator in the workspace, i.e., for a particular J, the w value describes the end-effector’s velocity and force transmission characteristics. Further, the magnitude of w value measures the manipulator’s proximity to singularity. Ideally, a zero value of w implies a singular configuration, and the smaller the w values, the closer the manipulator is to a singular configuration.  (9) TDSM: w = det(JJ T ) For the considered two DOFs planar TDSM, shown in Fig. 1a, w expression is computed and is given in Eq. 10. It is noted that w is dependent on the link

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lengths, d1 & d2 , and joint angle θ2 . From Eq. 10, the best posture of the TDM from the velocity and force manipulability capability perspective corresponds to the maximum w value, which is at θ2 = ±90◦ ; ∀ θ1 . Further, for link lengths such that d1 + d2 = constant, the manipulability measure, w , attains maximum value when d1 = d2 . w = (d1 d2 sin(θ2 ))2 = d1 d2 sin(θ2 ) (10) CDSM: The manipulator’s ability to generate motion in the taskspace is governed by two transformations. Firstly, from the taskspace to the joint space, which is represented by Jacobian, J, and secondly, from the joint space to the cable actuation space, which is represented by the cable routing A. Thus, unlike TDSMs, the transformation between the taskspace and the cable actuation space, as given in Eq. 6, is to be used to describe the manipulability of CDSMs. Accordingly, the quantitative manipulability measure, w , can be computed as given in Eq. 11 for CDSMs. For an invertible Jc , the matrix Jc JcT is positive semi-definite, which further implies only positive real w values, i.e., w ∈ R+ .  CDSM: w = det(J(AT )−1 (J(AT )−1 )T )   T −1 T (11) =⇒ w = det(J(AA ) J ) = det(Jc Jc T ) Due to the dependence on J and A, the expression of w for the considered CDSM general and co-shared architectures depends on the link lengths, joint angles, cable attachment points on the links, and actuators position. For a particular configuration of a manipulator and link lengths, J cannot be altered, but A can be effectively altered by changing the cable routing, i.e., the cable attachment points on the links and the actuator positions. Thus, with appropriate alterations in the cable routing, there exists the possibility of tuning the manipulability performance within the workspace for CDSMs to suit the desired task. Table 1. Motor Positions (in m) w.r.t O, in Fig. 1b and c for CDSM general and co-shared architectures Architecture M1

3

M2

M3

M4

General

[0.5,0] [1,0]

[−1,0]

[−0.5,0]

Co-Shared 1

[0.5,0] [1,0]

[−1,0]

[−0.5,0]

Co-Shared 2

[0.5,0] [1,−0.9] [−1,−0.9] [−0.5,0]

Manipulability Analysis

This section presents the manipulability analysis for the considered planar two DOFs TDSM and two architectures of the CDSM. Manipulability measure, w ,

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is evaluated, and contour plots corresponding to w values are plotted across the workspace. For the manipulability analysis, the link lengths for both TDSM and CDSMs are taken the same, di = 0.5 m. For the considered CDSM architectures, cable attachment points, ai and hi , are chosen as di /1.5 and di /3, respectively, and motor positions, Mi , are listed in Table 1.

Fig. 2. Manipulability measure, w , values for the CDSM general and Co-Shared architecture in (a) elbow up and (b) elbow down poses when the manipulator moves radially outwards at orientation angle φ = 15◦ and φ = 30◦ . The w values for the TDSM case are also plotted.

TSDM: Figs. 2a and b present the manipulability measure, w , values for the elbow up and elbow down poses of the TDSM when the manipulator moves radially outward w.r.t center O. It is observed that w increases as the end-effector progresses radially outwards, approaches a peak value, and finally reduces back to zero at boundaries. Essentially at the boundaries, as the manipulator is fully stretched, the Jacobian, J, becomes singular. The peak w value of 0.25 units is observed at a radial distance of 0.71 m and with θ2 = 90◦ . This is in accordance with Eq. 10 and denotes the TDSM’s best posture to generate the motion and force in the taskspace. Notably, the peak w value is dependent on chosen link lengths and will vary with varying link lengths, but the peak value is invariant of θ1 . From Figs. 2a and b, the w values are noted to be exactly the same for both elbow up and down poses along the considered radial trajectory. This is because w for TDSM depends on sin(θ2 ), and between the elbow up and down poses θ2 changes by 180◦ . Further, Fig. 3a shows the contour plots corresponding to the w values for the TDSM. A set of concentric circles are observed in the workspace, which implies that w values for the TSDM are invariant of the manipulator’s radial trajectory orientation. CDSM General Architecture: Figs. 2a and b plot the w values for the CDSM general architecture in both elbow up and down poses when the manipulator moves radially outwards. Two cases of radial trajectories with orientation angle φ = 15◦ and φ = 30◦ in the clockwise directions w.r.t x-axis are considered.

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Significant differences can be noted in the variations of w between the TDSM and CDSM. Unlike the same values of w for the TDSM’s elbow up and down poses, the w values are significantly different between the CDSM’s elbow up and down poses. Further, unlike the case of TDSM, the w for the CDSM is dependent on the radial trajectory orientation as noted by different w values for the two radial trajectories. In particular, the w values are higher for the CDSM elbow down pose compared to the elbow up pose. Thus, the manipulability measure, w , varies with the manipulator’s pose and the traversed trajectory over the workspace. Notably, for the CDSM, the best posture, i.e., the peak w value, is not unique but changes for each required trajectory. These observations for the CDSM explain the effect of structure matrix (A), i.e., cable routing, as it factors in w computation in Eq. 11. Further, larger values of w for CDSM implies better manipulability performance, i.e., the manipulator is far from singularity and has a better capability of generating end-effector velocities and forces.

Fig. 3. (a) Contours plots for the w values across the TDSM’s workspace (shaded region). (b,c) Contour plots for the w values across the wrench closure workspace (shaded region) of the CDSM general architecture for the elbow up and down poses. Significant variations in the w values are observed across the workspace for the two poses.

The contour plots corresponding to the w values across the elbow up and down poses of CDSM general architecture are plotted in Fig. 3b and c. The wrench closure workspace of the CDSM in the two poses, illustrated as a shaded region, reduces significantly compared to the TDSM’s workspace, which is shown as a shaded region in Fig. 3a. This reduction in the workspace is due to the unilateral force constraint of the cables. The contour plots for the CDSM general architecture do not form concentric circles as was the case for the TDSM but are symmetrically distributed about the Y-axis. Within the elbow up and down poses of the CDSM, significant variations are noted in the w values across the workspace (Fig. 3b and c). In general, the w values are higher for the elbow down pose. The w values for the CDSM are smaller at regions proximal and distal to the center O and increase in between, which is similar to the TDSM case. Apparently, the w values over the workspace are higher for the CDSM. For example, at the boundaries, it is noted that the poor manipulability performance of a TDSM improves significantly when it is cable-driven. Similarly, there are

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regions throughout the CDSM’s workspace where the manipulability measures improve. These differences in the case of the CDSM are essentially due to the incorporation of the structure matrix, A, in the computation of w .

Fig. 4. Contours plots for the w values across the wrench closure workspace (shaded region) of the CDM co-shared architecture for the (a) elbow up and (b) elbow down poses with motor positions Co-shared 1 as per Table 1. (c) and (d) showcase w contours for elbow and down poses Co-shared 2, motor positions as per Table 1.

CDSM Co-shared Architecture: The variations in the w values and the corresponding contour plots for the CDM co-shared architecture are plotted in Figs. 5 and 6 for the motor positions corresponding to co-shared 1 listed in Table 1. In general, these plots illustrate the effect of varying the cable routing within a CDSM. As noted from Fig. 1b and c, cables attached to the distal link are routed through the proximal link in the case of co-shared CDSM. This cosharing of cables alters the mapping between the cable actuation rate and the end-effector velocities, i.e., the structure matrix, A, Eq. 7, to alter the w values. In particular, the wrench closure workspace for the CDSM co-shared architecture improved compared to general architecture among the elbow up and down poses but remains lower than for the TDSM’s case. The gaps in Fig. 5 denote the infeasible workspace region. The w values also get redistributed across the workspace with smaller magnitudes at the proximal and distal regions. Notably, higher w values are reported for the CDM co-shared architecture as well. Thus, implying better manipulability in the case of CDSMs. Figures 4c and d present the contour plots of w for the elbow up and down poses of CDM co-shared architecture for motor positions corresponding to Coshared 2 listed in Table 1. Compared to the wrench closure workspace and con-

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tour plots in Figs. 4a and b, it is observed that the workspace remained similar but there is a significant re-distribution of w values just by altering the actuators positions for Co-shared 2. Thus, a redistribution of the manipulator’s manipulability across the workspace, i.e., variations in the best posture configurations and capability of generating the end-effector’s motion and force within the workspace. For example, for both the elbow up and down poses of Co-shared 1 and Co-shared 2 in Fig. 4, the proximal workspace regions to the center O for co-shared 2 have better manipulability performance as indicated by higher w values. Discussion: From the presented results, it is observed that there is a substantial difference in taskspace manipulability performance when cables are used to drive a serial-chain manipulator compared to the torque-driven method. It is observed that the manipulability performance of a CDSM varies based on the manipulator’s posture and the alignment of the trajectory within the workspace. Additionally, the optimal posture for a CDSM is not fixed and varies depending on the required trajectory. Further, it is also observed that altering the cable routing facilitates significant alterations in the manipulator’s manipulability performance in CDSMs. It is also shown that a CDSM having cable co-shared between the links can achieve diverse manipulability characteristics without affecting the tensionable workspace. Notably, works in the literature reported the usage of rigorous analysis on the tensionable workspace while altering system performance through altering system parameters [4,8,9,14]. The presented co-shared routing reportedly addresses this issue. Further, from the design and control viewpoint of a manipulator considering its widespread applications in industries and humanrobot interaction [5,6], the dependency of a CDSM’s performance on the cable routing presents a unique prospect of tuning its manipulability performance to make it suitable for a required task.

4

A Case of Cable-Driven Leg Exoskeleton

Leg exoskeletons have been developed for gait rehabilitation for their advantages in providing controlled repetitive motion, better quantification of motor recovery, and reduced labor need [20]. Among the works on leg exoskeletons, cable-driven based architectures are also being used for movement rehabilitation. Owing to the use of cables, these systems provide inherent advantages of being lightweight, flexible, and ease of altering cable routing [6]. An exoskeleton design for a particular rehabilitative intervention incorporates its suitability to a broad range of subjects’ anatomy and its ability to deliver required external forces while supporting varied requirements of human motion. Thus, the manipulability analysis can be an effective measure for an exoskeleton design. Considering the potential a CDSM provides in tuning the manipulability performance, the manipulability analysis of a cable-driven leg exoskeleton (CDLE) during the swing phase of the gait motion is presented. Specifically, the effect of cable routing has been analyzed and two CDLE architectures, general and co-shared, are considered as shown in Fig. 5.

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Fig. 5. Schematic of CDLE with two different cable routings, general and co-shared. Ti and li represent cable tension and length of the ith cable, respectively. Di , ri , and hi represent segment lengths, cable attachment points, and offset position on each segment. τi denotes the joint torque and Mi represents the motor positions w.r.t O.

Fig. 6. (a) Manipulability ellipse plotted at different points along the swing phase ankle trajectory to highlight the effect of the changes in manipulability characteristics of CDLE general and co-shared architectures. (b) Condition number, CN, variation for the chosen cases. Large values for the general CDLE case imply the anisotropic nature of the manipulability. (c) Manipulability measure, w , variation. Large values of w imply better manipulability. (d) Optimal Motor positions, M2 and M3 , for the coshared architecture during the swing phase of the gait cycle that facilitates reduction in CN.

CDLE Manipulability Analysis: Fig. 6a shows the human ankle trajectory (with swing phase highlighted) obtained from the human gait data set presented in [21]. Cable space to taskspace transformation matrix, Jc , is computed during the swing phase of the gait cycle for the cable space parameters reported in Sect. 3, and motor positions for both CDLE architectures correspond to general and co-shared 1 listed in Table 1. The manipulability ellipsoids are plotted using Jc across the swing phase ankle trajectory using transformation mapping presented in Eq. 8, refer Fig. 6a. The major axis of the ellipse indicates that the

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manipulator generates motion more effectively in that direction than in the direction of the minor axis. Compared to the general architecture, it can be observed that the major axis of the ellipsoid for the co-shared case is aligned with the ankle trajectory. The condition number (CN ), defined as the ratio of the major to minor radii, is plotted for both the CDLE architectures during the swing phase in Fig. 6b. This ratio, if it is equal to unity, implies isotropic manipulability, i.e., the manipulator can generate motion effectively in all directions. The CN values vary over the swing phase and are higher for the general architecture implying anisotropic manipulability. The manipulability measure, w , computed using JC as per Eq. 11 during the swing phase is also plotted in Fig. 6c for the two architectures. Smaller values of w are observed for both architectures at the latter part of the swing phase. CDLE Manipulability Tuning: The manipulability characteristics of the CDLE can be altered by tuning the cable routing. Such performance alteration can be useful for a CDLE design to deliver required external forces while supporting varied requirements of human motion. In this section, we demonstrate manipulability tuning of the co-shared architecture by optimizing the CN. An optimization problem is formulated in Eq. 12 to minimize CN for optimal coshared cable motor positions, M2 and M3 . The obtained optimal motor positions are plotted in Fig. 6d. min : CN = f (M )

(12)

s.t : M2x ∈ [0.12], M3x ∈ [−2, −0.1] Miy ∈ [−1, 2], i = 2, 3 Defining a new CDLE architecture using the mean values motor positions M2 and M3 in Fig. 6d. This architecture is called Optimized, and its manipulability measures are plotted in Fig. 6. As intended, the optimized motor positions resulted in noticeable changes in the manipulator characteristics during the swing phase. In particular, a significant reduction in the CN values is noted in Fig. 6b. This implies isotropic manipulability, i.e., the CDLE optimized architecture can follow the human leg motion while uniformly maintaining manipulability in all directions. Further, higher w values are also observed during the early part of swing in Fig. 6c, implying improved manipulability. These results indicate that a CDLE facilitates the modulation of its manipulability characteristics via cablerouting alterations. In summary, the current work demonstrates that CDSMs offer flexibility in architecture modulation to tune their manipulability performance. Further, in the context of movement training during a rehabilitation paradigm, the current work demonstrates the efficacy of utilizing architecture modulation in a CDLE to tune manipulability characteristics favorably. Such an approach can be taken to design novel gait intervention paradigms using CDLEs for effective rehabilitation during walking.

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Acknowledgment. This work was partly supported by the IHFC, TIH, IIT Delhi (Grant No. GP/2021/RR/014).

References 1. Kamrani, B., et al.: Optimal usage of robot manipulators. Robot Manipulators Trends Dev. 1-26 (2010) 2. Corves, B., Mannheim, T., Riedel, M.: Re-grasping: improving capability for multiarm-robot-system by dynamic reconfiguration. In: Jeschke, S., Liu, H., Schilberg, D. (eds.) ICIRA 2011. LNCS (LNAI), vol. 7101, pp. 132–141. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-25486-4 14 3. Veneman, J.F., et al.: Design and evaluation of the LOPES exoskeleton robot for interactive gait rehabilitation. IEEE Trans. Neural Syst. Rehabil. Eng. 15(3), 379– 386 (2007) 4. Rezazadeh, S., Behzadipour, S.: Workspace analysis of multibody cable-driven mechanisms, 021005 (2011) 5. Jung, S.-Y., et al.: Design of robotic hand with tendon-driven three fingers. In: 2007 International Conference on Control, Automation and Systems. IEEE (2007) 6. Hidayah, R., et al.: Gait adaptation using a cable-driven active leg exoskeleton (C-ALEX) with post-stroke participants. IEEE Trans. Neural Syst. Rehabil. Eng. 28(9), 1984–1993 (2020) 7. Ming, A., Higuchi, T.: Study on multiple degree-of-freedom positioning mechanism using wires. I: Concept, design and control. Int. J. Jpn Soc. Precis. Eng. 28(2), 131–138 (1994) 8. Du, J., Bao, H., Cui, C.: Stiffness and dexterous performances optimization of large workspace cable-driven parallel manipulators. Adv. Robot. 28(3), 187–196 (2014) 9. Lim, W.B., Yeo, S.H., Yang, G.: Optimization of tension distribution for cabledriven manipulators using tension-level index. IEEE/ASME Trans. Mechatron. 19(2), 676–683 (2013) 10. Merlet, J.-P.: Jacobian, manipulability, condition number, and accuracy of parallel robots. 199-206 (2006) 11. Dufour, K., Suleiman, W.: On maximizing manipulability index while solving a kinematics task. J. Intell. Rob. Syst. 100(1), 3–13 (2020) 12. Ukidve, C.S., McInroy, J.E., Jafari, F.: Using redundancy to optimize manipulability of Stewart platforms. IEEE/ASME Trans. Mechatron. 13(4), 475–479 (2008) 13. Zargarbashi, S.H.H., Waseem, K., Angeles, J.: The Jacobian condition number as a dexterity index in 6R machining robots. Robot. Comput.-Integr. Manuf. 28(6), 694–699 (2012) 14. Eden, J., et al.: Unilateral manipulability quality indices: generalized manipulability measures for unilaterally actuated robots. J. Mech. Design 141(9) (2019) 15. Sanjeevi, N.S.S., Vashista, V.: Stiffness modulation of a cable-driven serial-chain manipulator via cable routing alteration. J. Mech. Robot. 15(2), 021009 (2023) 16. Tsai, L.-W.: Robot Analysis: The Mechanics of Serial and Parallel Manipulators. John Wiley and Sons, Hoboken (1999) 17. Chiacchio, P., et al.: Global task space manipulability ellipsoids for multiple-arm systems. IEEE Trans. Rob. Autom. 7(5), 678–685 (1991) 18. Yoshikawa, T.: Manipulability of robotic mechanisms. Int. J. Rob. Res. 4(2), 3–9 (1985) 19. Emiris, D.M., Tourassis, V.D.: Singularity-robust decoupled control of dual-elbow manipulators. J. Intell. Rob. Syst. 8(2), 225–243 (1993)

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20. Pinto-Fernandez, D., et al.: Performance evaluation of lower limb exoskeletons: a systematic review. IEEE Trans. Neural Syst. Rehabil. Eng. 28(7), 1573–1583 (2020) 21. Winter, D.A.: Biomechanics and Motor Control of Human Movement. John Wiley and Sons, Hoboken (2009)

Advances in the Use of Neural Network for Solving the Direct Kinematics of CDPR with Sagging Cables Jean-Pierre Merlet(B) INRIA Sophia-Antipolis, 2004 Route des Lucioles, Valbonne, France [email protected]

Abstract. Direct kinematics (DK) is one of the most challenging problem for cable-driven parallel robot (CDPR) with sagging cables. Solving the DK in real-time is not an issue provided that a guess of the solution is available. But difficulties arise when all DK solutions have to be determined (e.g. in the design phase of the CDPR). Continuation and interval analysis have been proposed to find the solutions but they are computer intensive. A preliminary investigation on the use of classical neural networks (NN) for the DK has shown that they were performing poorly. We present in this paper several methodological improvements that allows to get on average 99.95% of the exact DK solutions in about 5 s. Still this result is not completely satisfactory and we present possible axis to obtain better results in terms of exact results and multiple solutions. Keywords: cable-driven parallel robot networks

1

· direct kinematics · neural

Introduction

In this paper we address the problem of finding all solutions of the DK for a 6 d.o.f. CDPR with sagging cable. The cable model is the classical Irvine textbook planar model [11] that has been experimentally proven to be valid for usual CDPR [19]. Provided that the length at rest L0 of the cable is known this model provides 2 non-algebraic constraint equations for each of the n cables with as unknowns the horizontal/vertical components Fx , Fz of the cable tension at its attachment point B on the platform and the planar coordinates xb , zb of B which can be derived from the 6 components of the platform pose parameters X. Further constraints are the 6 mechanical equilibrium equations. Hence we have 6 + 2n constraints equations for 6 + 2n unknowns (X, Fxi , Fzi , i ∈ [1, n]) and consequently we always get a square system that has usually multiple solutions. An important point is that there is no known method to predict how many DK solutions will be obtained for a given set of L0 . The DK may be used in the control law but this is not a problem as an appropriate Newton scheme Partly supported by ANR-18-CE10-0004 and ANR-19-P3IA-0002 grants. c The Author(s), under exclusive license to Springer Nature Switzerland AG 2023  S. Caro et al. (Eds.): CableCon 2023, MMS 132, pp. 30–39, 2023. https://doi.org/10.1007/978-3-031-32322-5_3

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usually allows one to get the solution in real-time. In this paper we are interested in the DK in the design phase where some specific CDPR properties have to calculated (e.g. the maximal cable tensions over a given workspace) very rapidly in order to assess the performances of a given geometry. In that case we have no a-priori knowledge of the CDPR state, meaning that for a given set of L0 we have to find all DK solutions as the considered property usually changes with the solution. If the property is established by sampling the L0 workspace, then a large number of DK problems will have to be solved. To the best of the author knowledge there are only 2 methods that have been developed to fully solve the DK: interval analysis [18] and an approach based on continuation1 [3] that first compute all DK solutions for rigid legs and then incrementally change the Young modulus E and linear density μ of the cable materiel from a high value for E and a small one for μ toward their known values [17]. Unfortunately both methods are extremely computer intensive (several hours for solving a DK problem) and therefore cannot be used in the design phase. Therefore it is interesting to investigate faster methods, such as neural networks, under the constraint that all solutions are obtained but allowing for possible errors in the X, Fx , Fz up to a limit that we have fixed arbitrarily to 5%.

2

First Trials with Neural Networks

There are several types of neural networks (NN) but one of the most commonly used is the multi-layer perceptron (MLP) [10]. It has an input layer, an output layer and in-between one or several hidden layers. A layer contains different neurons that receive as input a weighted sum of all the outputs of the neurons from the previous layer and map this input to the neuron output by using an activation function. A MLP require a training set with samples that maps the input (in our case the L0 ) to the desired output (here X, {Fx , Fz }). In the learning phase of the MLP a stochastic optimizer try to find the weights that minimize a statistical index (e.g. the Mean Squared Error (MSE)) on the errors between the MLP outputs and the desired one over the whole training set, this index being called the loss function. Being given an equations system F(L0 , X, {Fx , Fz }) = 0 there is theoretically a MLP that can approximate accurately a function G such that (X, {Fx , Fz }) = G(L0 ) but the parameters (number of layers, neurons, activation functions, . . . ) of this MLP are not known. In our case we also have a major issue as a MLP provides an output vector whose size is fixed: in our case the size should reflect the number of DK solutions which is basically unknown. Furthermore having a single MLP to obtain at the same time all the DK solutions seems highly improbable as explained later on. Another issue is that creating a MLP is a stochastic process so that reproducing a literature result is difficult: however the strategy proposed in this paper should overcome this problem. 1

Continuation basically amount to incrementally increase the continuation parameter(s) and to use the Newton method to compute the new solution at each step. But the amount of increase in the parameter(s) must be carefully selected to avoid skipping to another Newton solution.

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MLPs have been proposed for dealing with the DK of classical parallel robots [1,4,5,7,8,12,14,20] but with very mixed results and for approximating a single solution. More recently MLPs have been proposed for the kinematic analysis of CDPR with elastic cables [2](DK) or sagging cables [9](DK) and [6] (inverse kinematics). In this paper we will consider a complex case: the large 6 d.o.f. CDPR Cogiro with 8 cables. The geometry of this CDPR and the cable characteristic are presented in [13] while the platform mass is 1 kg. 2.1

Preliminaries: The Training Set

We started considering MLP in 2021 and we first designed an algorithm to provide arbitrarily large training set. Our exact methods allow us to calculate all DK solutions for a given set of L0 . We choose 8 poses distributed over the CDPR workspace and fix the L0 as the distance between the winch output point and the cable attachment point on the platform. We then solve exactly the DK n for the 8 sets L0j obtaining the set Sj = {Sj1 , Sj2 , . . . , Sj j } of nj DK solutions k for the j-th L0 set. For each Sj of a Sj we then select a random unit vector v in the L0 space and define a new L0 vector as L0 = L0j + λv where λ is a parameter starting at 0 that will be used for a continuation process: a small value, , is added incrementally to λ and the Newton scheme is used to obtain the DK solutions for the current λ, using as guess the ones obtained for the previous λ. Note that  has not a constant value: it is adjusted at each step to ensure that the Newton scheme converges to a solution that is coherent with the previous solution. We initially store Sj in the learning set and we will store a new set of DK solutions as soon as at least one of the L0 has changed by more than 5cm with respect to the previously stored DK solution. The continuation process stops as soon as soon as the Newton scheme does not converge for a very small . The process is then repeated with a new v until an arbitrarily number N of DK solutions has been obtained. A training set is therefore a set of files that have been obtained by using as starting point the solution Sjk of the set Sj . The initial choice for Sj together with the random choice of v allows for a good coverage of the L0 space. Furthermore we will see later on that during the process new training sets will be added. The number of samples in the training set may be huge as N is arbitrarily large. A training set with 144 files that represent 72144 DK solutions is available [15]. 2.2

Initial Results and Methods

As mentioned in the introduction we have to choose the geometry of the MLP, i.e. the number of hidden layers, the number of neurons in the layers, the learning rate and the activation function(s). There are no clear rules to choose these parameters but we noticed that the MLP training time was small (a few minutes) so that we use a systematic approach by creating all MLPs with 2 to 8 layers, 2 to 202 neurons with a step size of 10 and one activation functions in a set of 5 classical activation functions (namely ReLU, LeakyReLU, CELU, GELU, Softplus). Each combination was used for each file in the learning set. Initial results

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were very poor with 200–300% error on some of the unknowns for the 50 MLPs with the lowest final loss. A first improvement was obtained by using hybridization: each prediction of the MLP for a sample of the training set is used as initial guess for the Newton method. This has led to obtain about 5% of exact solutions with respect to all DK solutions in the whole training set. But we have noted that the number of Newton convergences over the training set does not automatically increase with the decrease of the loss. This may be explained as follows: around a given solution for some essential variables the difference between the prediction and their exact solution value must be very small for obtaining convergence while for the other non-essential variables this difference may be much larger before we get a divergence of the Newton scheme (note that the set of essential variables is not constant, its depends upon the solution). As a decrease of the loss may be obtained by a large decrease of the errors on the non-essential variables together with a small increase on the errors on essential variables we may thus obtain a lower number of Newton convergences although the loss has decreased. We also observe that during the loss optimization some MLPs were exhibiting small errors on some variables. This lead us to adopt a new strategy: – during the optimization we check the number of Newton convergences after each significant decrease of the loss and store the MLP having the highest number of convergences denoted as main MLPs with prediction Pm – we store the MLP having exhibited the lowest error on some variables. We then substitute their predictions on these variables in Pm and test Newton with these new predictions We then noted that for each file in the training set there was large differences in the exact Fx , Fz . We then define 12 different ranges for the Fx , Fz that cover all their possible values and distribute the samples in 12 new training sets called clusters. For each of the clusters we calculate new samples so that they have approximately 500 samples and we train MLPs with this new training sets. This has allowed us to discover about 30% of DK solutions over the whole training set but we discover more than one solution and never more than 2 solutions only in a few cases. Hence the above approach is still far from being satisfactory.

3

New Approach for Neural Networks

Hybridization appears to be working but the proposed clustering method was not very efficient and hence we have implemented a new clustering method. 3.1

New Clustering Approach

Let consider various sets L0j of cable lengths and their associated set of DK n solutions Sj = {Sj1 , Sj2 , . . . , Sj j }. We may represent the Sj in a planar graph: an horizontal line represents a given set L0j of L0 and we have nodes on this line that represent the DK solutions for this set. We define as level the height of the horizontal line, the bottom line having level 1 and the higher one level 8.

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Each Sjk includes a pose Xkj of the platform. Aspects may be defined in the product of the X space and the L0 space as the sets of all poses that may be connected by a continuous path in the L0 space, called kinematics branch. Aspects are separated by kinematic singularities and possibly by physical limits [16]. Aspects are much more coherent from a kinematics viewpoint than regrouping DK solutions according to their values for Fx , Fz but computing aspects is a demanding task. We therefore focus on DK solutions that can be connected by a linear interpolation between the L0 s from one level to the L0 s of another level. Furthermore we impose an arbitrary limit of 100 N for each of the Fx with the purpose of avoiding having very large changes in the Fx , Fz for very small changes in the L0 (typically this occurs when the height of the platform become close to the height of the winch output points). We consider two Sj , Sj1 , Sj2 at different levels, and define a potential kinematics branch as L0 = L0j1 + λ(L0j2 − L0j1 ) where λ is the branch parameter that lies in the range [0,1]. We then consider in turn all DK solution Sjk1 that will be the starting point of a continuation process with λ as parameter. Two cases may occur: – the continuation stops with λ = 1 so that DK solution Sjk1 is connected to the DK solution Sjk21 : we now have an edge in our graph that connect the nodes of Sjk1 and Sjk21 – the continuation encounters a singularity or one of the Fx exceed the limit value before λ reaches 1 so that Sjk1 is not connected to any element of Sj2 by a linear branch After having completed all the edge calculations we get the graph presented in Fig. 1. A seen in the figure we have a large number of connection between nodes but also nodes that have no connection. We select as kinematic branches the one emanating from the connected nodes of level 1 and going up to level n where n is at most 8. The continuation process for building the edge has

Fig. 1. The graph: the black squares (called nodes) on an horizontal line represent a DK solution for a given set of cable lengths. A line connect 2 nodes at different horizontal levels if a continuation process based on a linear interpolation on the cable lengths has led from the initial DK solution to the new one.

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allowed to determine multiples ordered pairs P n = (Ln0 , Yn ) where Y is a DK solution, the first point being the initial position. We select as sample H k any point such that |L0 (P k ) − L0 (P k−1 )| > d where d is a fixed threshold (these samples are the black circles on Fig. 2). After having completed the branch we create new samples from the H k , represented as white circles on the figure, by using a continuation process, the L0 moving along random unit vectors in the L0 space (we use the same selection strategy on the changes on the L0 to store the samples). The number of sample for each H k is calculated so that we end up with around 10 000 samples for each kinematic branch. For the graph nodes that are not connected we start directly selecting random unit vectors in the L0 space and use a continuation process with the node as starting point. Note that we may obtain other kinematic branches just by changing the order of the horizontal lines so that we end up with N kinematic branches.

Fig. 2. The sampling of a kinematic branch

3.2

MLP Training and Results

For creating a training set for a kinematic branch we select 1 over 8 of the samples obtained for the branch. Preliminary tests have shown that MLPs with 6 hidden layers, 70 or 80 neurons in the layers and using the activation functions LeakyReLU or CELU were providing the best results. For each branch we therefore create 4 MLPs that are exhibiting the largest number of Newton convergences, the training time for a MLP being about 30 mn. To test the efficiency of the training we check all the samples of the branch using the prediction of the 4 MLPs as input for the Newton method. We define the success rate as the percentage of samples for which we have obtained the exact DK solution (a success rate of 100% implies that for all samples we get the exact DK solution). Figure 3 shows the success rate for 32 kinematic branches. It may be seen that the success ranges between 70 and almost 100%: this is not perfect but much better than the result we have obtained up to now, although we have used only 4 MLPs for each test.

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Fig. 3. Success rate for all samples of 32 kinematic branches: 4 MLPs prediction are used for each sample.

The ultimate test will evidently be to check how many DK solutions are found for a given set of L0 . For that purpose we build a verification set as follows: for each of the 8 full DK solutions Sj , each one having nj solutions, we select a random unit vector v in the L0 space and move the L0 from L0j in that direction, applying a continuation process simultaneously on each of the DK solution until it fails for one of the solutions. During the process we store samples using the same L0 changes strategy, all the samples having nj solutions. We then select a new unit vector and start again until we have around 270 samples for each Sj . Note that although this process is somewhat similar to the one used for the training sets the random choice of v in the 8-dimensional L0 space ensures that the verification set is really different from any training sets. At the end we get a verification set with 2106 full DK solutions. We check this verification set for N = 43 kinematics branches using 172 MLPs. Here the success rate is defined as the percentage of DK solutions that are found relative to the expected one. We get a 66.66% success rate average with a maximum of 100% for 7 samples (all solutions are found), a minimum of 33.33 % for 2 samples and 35 samples having a success rate ≥ 90%. Figure 4 shows the success rate for all samples in the verification set. The average computation time for obtaining the DK solutions for a given set of L0 is 2.36 s. This time is however largely over-estimated as we mix a Pytorch model with a main procedure written in C. Although the success rate is not completely satisfactory these results may be improved incrementally in two manners: – by adding other kinematic branches. The first test with the verification set may show what branches lead to a low number of convergences and therefore should be complemented. For example with 456 MLPs we get an average success rate of 87.93%. The best result has been obtained for 1154 MLPs with a success rate of 99.955%. For the verification set with 2016 samples we get exactly all DK solutions for 1999 samples while the 17 remaining one have a success over 90%.

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Fig. 4. Success rate for each sample of the verification set for 43 branches

– a given sample on a kinematic branch is supposed to have the same number nj of DK solutions than its father Sj located at level lj . In most kinematic branches we move from level lj to another level ln that has nn DK solution. If nn > nj , then new DK solution(s) have to pop up during the continuation process and we indeed frequently noticed that new Newton solution(s) appear. We may thus define an intermediary level between lj and ln for constructing new kinematic branches. Thus our algorithm may learn from its failures to improve the process. Another way to evaluate the quality of the result is to look at the maximum of the cable tensions over the whole verification set, the exact value being 103.56 N. If we look only at the exact DK solutions obtained by our algorithm with 43 MLP we get a maximal tension of 102.28 N which represents an error of 1.2% while for 1154 MLPs we get the exact value. The maximal cable tension obtained from the 43 MLP predictions that lead to an exact solution is 212.83 N while with 1154 MLPs it is 158.94N. Hence it appears that using the exact solutions computed by our algorithm provides a reasonably accurate evaluation of the maximal tension while the MLPs prediction are largely over-estimating it.

4

Other Approaches

Although we get interesting results, using classical MLP may not be the only approach. Physics-informed neural networks (PINNs) are unsupervised, model based, neural network. We are currently trying to implement a PINN whose loss function will be the MSE of the 22 equation values and then we will have to manage the DK multiplicity of solutions.

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Another approach will be to use an autoencoder. The principle is to train the NN with an output which is encoded so that its size is smaller than the one from the original problem. Then the NN prediction is run through a decoder which translates the prediction to the output of the original problem. Autoencoding has been proposed mostly to reduce the learning time. In our case we are for example currently investigating MLPs that will provide a prediction only for X so that the end-point location of each cable is known. Then a numerical solver or a MLP may be used to to determine the single solution in Fx , Fz of the 2 Irvine equations for a given cable.

5

Conclusion

In summary NN offer another method to solve the DK problem although a blind use of their classical version leads to very poor result. As seen in this paper new methodologies have to be developed to start getting interesting results. Although the results we get are satisfactory thanks to the use of aspects we still have to make efforts for reducing the number of used MLPs. For that purpose we have noted than an aspect may include two different DK solutions for the same set of L0 . This induces trouble for the learning as the loss function cannot become 0. Therefore we are looking at refining the aspects by splitting them in characteristic varieties, regions in which the number of DK solutions is constant. Still with the current algorithm some properties such as the maximal cable tension can be estimated with an acceptable accuracy. A first drawback of the method is that we have assumed a constant platform mass. We plan to investigate the efficiency of the MLPs that have been trained for a mass of 1kg for managing other masses. If this efficiency is low continuation may be used but it remains to manage the issue of varying number of solutions for a given set of L0 as continuation does not allow to find new solutions. A second drawback for the design phase is that we sample the workspace for checking the property so that we may miss important changes in the property. However there are methods (e.g. the Kantorovitch theorem) that may allow to expand the point sample we have to a ball with bounded values for the unknowns and we plan to investigate this inflation approach.

References 1. Achili, R., et al.: A stable adaptive force/position controller for a C5 parallel robot: a neural network approach. Robotica 30(7), 1177–1187 (2012) 2. Ahouee, R., Moussavi, S., Hamedi, J.: Neuro-fuzzy intelligent control algorithm for cable-driven robots with elastic cables. In: 2nd International Conference on Cybernetics, Robotics and Control (2017) 3. Allgower, E.: Numerical Continuation Methods. Springer-Verlag, Berlin (1990) 4. Azar, W., Akbarimajd, A., Parvari, E.: Intelligent control method of a 6-DOF parallel robot used for rehabilitation treatment in lower limbs. Automatika 57(2), 466–476 (2016)

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5. Boudreau, R., Levesque, G., Darenfed, S.: Parallel manipulator kinematics learning using holographic neural network models. Robot. Comput.-Integr. Manufact. 14(1), 37–44 (1998) 6. Chawla, I., et al.: Neural network-based inverse kineto-static analysis of cabledriven parallel robot considering cable mass and elasticity. In: 5th International Conference on Cable-Driven Parallel Robots (CableCon). virtual, 7–9 July 2021 7. Dehghani, M., et al.: Neural network solutions for forward kinematics problem of HEXA parallel robot. In: American Control Conference. Washington, 11–13 June 2008 8. Geng, Z., Haynes, L.: Neural network for the forward kinematics problem of a Stewart platform. In: IEEE International Conference on Robotics and Automation, pp. 2650–2655. Sacramento, 11-14 April 1991) 9. Ghasemimi, A., Eghtesad, M., Farid, M.: Neural network solution for forward kinematics problem of cable robots. J. Intell. Robot. Syst. 60, 201–215 (2010) 10. Haykin, S.: Neural Networks: A Comprehensive Foundation. Prentice Hall PTR, Hoboken (1994) 11. Irvine, H.M.: Cable Structures. MIT Press, Cambridge (1981) 12. Kang, R., et al.: Learning the forward kinematics behavior of a hybrid robot employing artificial neural networks. Robotica 30(5), 847–855 (2012) 13. Lamaury, J., Gouttefarde, M.: Control of a large redundantly actuated cablesuspended parallel robot. In: IEEE International Conference on Robotics and Automation. Karlsruhe, 6–10 May 2013 14. Li, T., Li, Q., Payendeh, S.: NN-based solution of forward kinematics of 3DOF parallel spherical manipulator. In: IEEE International Conference on Intelligent Robots and Systems (IROS). Edmonton, 2–6 August 2005 15. Merlet, J.P.: Data base for the direct kinematics of cable-driven parallel robot (CDPR) with sagging cables. Technical report, INRIA (2021). https://hal.inria.fr/ hal-03540335v2 16. Merlet, J.P.: Computing cross-sections of the workspace of suspended cable-driven parallel robot with sagging cables having tension limitations. In: IEEE International Conference on Intelligent Robots and Systems (IROS). Madrid, 1–5 October 2018. https://hal.inria.fr/hal-01965229v1 17. Merlet, J.P.: Preliminaries of a new approach for the direct kinematics of suspended cable-driven parallel robot with deformable cables. In: Eucomes. Nantes, 20–23 September 2016. https://hal.inria.fr/hal-01419700v1 18. Merlet, J.P.: The forward kinematics of cable-driven parallel robots with sagging cables. In: 2nd International Conference on cable-driven parallel robots (CableCon), pp. 3–16. Duisburg, 24–27 August 2014. http://www-sop.inria.fr/coprin/ PDF/merlet cablecon2014.pdf 19. Riehl, N., et al.: Effects of non-negligible cable mass on the static behavior of large workspace cable-driven parallel mechanisms. In: IEEE International Conference on Robotics and Automation, pp. 2193–2198. Kobe, 14–16 May 2009 20. Yee, C., Lim, K.: Forward kinematics solution of Stewart platform using neural network. Neurocomputing 16(4), 333–349 (1997)

Kinetostatic Modeling and Configuration Variation Analysis of Cable-Driven Parallel Robots on Spherical Surfaces Lei Jin1 , Tarek Taha2 , and Dongming Gan1(B) 1 Purdue University, West Lafayette, IN 47907, USA

[email protected] 2 Robotics Lab, Dubai Future Labs, Dubai, UAE

[email protected]

Abstract. Cable-driven parallel robots (CDPRs) have good advantages to cover large part cleaning and manufacturing based on their lightweight designs and scalable workspace. This leads to CDPRs on curved surfaces with variable configurations due to the curvature of the cables. This paper presents a kinetostatic model to find the moving platform position with given set of cable lengths and vice versa, and to solve the cable tension force direction and magnitude for a 3anchor-point hemispheric CDPR considering the anchors’ quantity and arrangement. The problem is formulated by utilizing geodesic property on spheres and solving planar force balance equations. Corresponding anchor point is defined by the coverage of each set of anchors that counters the gravity. Equivalent positioning provides a solution for sets of non-ordinary-anchor-location-based reconfiguration which changes the maximum cable tension requirements and surface coverage. Simulations and calculations are conducted to illustrate the proposed model. Keywords: Cable-driven parallel robots · Curved surfaces · Kinematics · Statics · workspace · Reconfiguration

1 Introduction Cable-driven parallel robots, also known as CDPRs, are a kind of lightweight manipulators with excellent scalability in terms of size, payload, and dynamics capacities [1]. CDPRs are better than traditional parallel robots or human labor in terms of reduced mass in motion, larger workspace, high payload capacity, safe working environment for workers, and lower cable costs. Based on those, CDPRs have been applied in many applications including construction [2, 3], cranes [4–6], 3D printing [7], video recording [8], exoskeletons [9], and mobile manipulation [10]. They are also suitable for large part cleaning or manufacturing. For example, cable-driven window-cleaning robots are very common on skyscrapers. Some work used CDPRs to paint the outside of aircrafts as shown in Fig. 1 [11]. Related literature works mainly focused on planar or three-dimensional CDPR configurations. Shao [12] presented design and analysis of CDPRs on exterior walls of © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Caro et al. (Eds.): CableCon 2023, MMS 132, pp. 40–51, 2023. https://doi.org/10.1007/978-3-031-32322-5_4

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Fig. 1. a. 5-NIST RoboCrane painting an aeroplane 3-CableBot design with cable exit point fixed to a grid and b. Courtesy of the European FP7 Project CableBot [11]

buildings as an application of a 2D CDPR on flat surfaces. Paolo [13] studied CDPRs on flat planar surfaces. However, only few papers have mentioned cable-driven robots on curved surfaces. Bach and Yi [14] built a trajectory planning model on 3D curved surfaces. However, their system was still based on a traditional CDPR design with 6 dynamic anchors and 5 degrees-of-freedom (DOFs). In [15] a three-cable CDPR was proposed for inner surface cleaning of a large storage tank. General kinematic and static models were developed but no analysis on the curved surface geometry contact or coverage planning. Another CDPR for complex geometric surfaces was shown by [16] in which a four-cable CDPR was applied as the lifting mechanism for large turbine blade cleaning with an up-down function along the blade and the complex surface cleaning was mainly handled by the on-board cleaning heads. There has been relatively limited research on cable-driven parallel robots (CDPRs) configured on large curved surfaces, and corresponding modeling and analysis techniques have yet to be developed. In this work, we will address this gap by developing a model for CDPRs configured on a spherical surface. In our case, the cable will lay on the surface of a sphere, thus the direction of tension can be difficult to predict since the cable is no longer straight. This paper mainly focuses on three parts: the tension direction, the tension magnitude and workspace. We utilized geodesic properties to determine the direction of tension. Geodesics on spherical surfaces can be readily identified, as they invariably traverse the sphere’s center. Specifically, if the system under consideration is spherical and the end-effector is treated as a point mass, all anchor forces can be projected onto a tangent plane to the surface. This projection simplifies the force balance problem into a 2D plane. In this work, we focus on the maximum tension of the cable and the algorithm is based on the theory of superposition of dynamic control inspired by Chen [17] who used superposition to solve the disturbance problem. Workspace of the CDPR is defined by analyzing the maximum tension at each point. Cong [18] proposed a way to find the performance of CDPRs by looking at the Tension Factor. Excessive tension at a specific location can result in poor performance, whereas low tension implies that the system requires minimal force to counteract gravity, allowing for more force to counteract dynamic effects. In our study, we observed that relocating the anchor points on the spherical surface induced changes in the maximum tension, potentially enabling the

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optimization of the configuration setup. Equivalent positioning is also derived to simplify configuration variations when moving anchors by setting anchor to the equivalent anchor at base level. This paper is structured as follows: In Sect. 2, we introduce the geometric model of the hemispheric 3-anchor cable-driven parallel robot (CDPR). In Sect. 3, we address both the forward and inverse kinematics problems of the model. Subsequently, in Sect. 4, we discuss the statics of the system to determine the cable tension. Section 5 focuses on the variation of the model due to different anchor positions. Finally, in Sect. 6, we summarize the contributions of this work and outline potential avenues for future research.

2 Geometric Model 2.1 Hemi-Spherical Surface As in Fig. 2, the CDPR consists of three anchors distributed on the base of a hemispherical surface and an end-effector moving on the surface controlled by three cables. Different from CDPRs in 2D planes or 3D spaces, cables on curved surfaces are no longer straight lines. Instead, they will follow the shortest path from the moving platform to the anchor, which is the geodesic of these two points. The tension direction will be the tangent lines at this location. It can be inferred that all the tensions will be on the tangent plane at the moving platform. This prediction will result in solving force balance in the same plane. Our model has applied spherical coordinates for easier presentation with a base coordinate O-xyz system setting at the sphere center O. The angle starting from +x axis to +y axis is θ and starting from +z axis to x-y plane is ϕ. r is the distance from origin to the moving platform or anchors. As seen in Fig. 2, the horizontal plane in which three anchors (A, B, C) locate is called the base level. At each frame, P marks the position − → − → → − of the moving platform, and PA, PB, PC represents the tangent line pointing from the moving platform to the anchor direction. PA, PB, PC define the arc length from the moving platform to the anchor. A few assumptions are made to better solve the model and reduce the complexity of computation: 

(1) (2) (3) (4)





Both the anchors and the moving platform are treated as mass point. There is no gap between the cable and the surface. There is no friction on the surface. When there are multiple paths of the cable, i.e., the moving platform is on the opposite site of the hemi-sphere, the path is defined by the last frame that leads to this position.

Based on the hemi-sphere’s geometric property, changing the radius or base level has the same effect on the configuration setup. Figure 2’s configuration applies to all CDPRs on hemi-spherical surfaces (Fig. 3). However, in Fig. 3a, the scale difference may affect the model parameters, and modifying the base level shifts the anchor’s position relative to the origin. Nonetheless, the anchor’s position remains unchanged in both scenarios because the angles remain constant in spherical coordinates. Figure 3(b) shows a CDPR

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Fig. 2. Diagram for a hemi-spherical CDPR model with three anchors installed at the base level. Each colored line represents the cable (geodesics) of corresponding anchor.

configuration with unsymmetrical anchor positions. The kinematic and static models in the following sections will be suitable for all those variations.

Fig. 3. Diagram for a hemi-spherical model with three anchors installed at the base level where a. the base level changes to 1/3π and b. the anchors’ positions change to 4/3π, 3/2π, and 25/12π on the base plane.

3 Kinematic Model 3.1 Inverse Kinematics Given the moving platform location (r, θ, ϕ)|A,B,C parameters, to solve the three cable lengths is the inverse kinematics problem. Since the cables are on the sphere, the solution is the shorted curves between the anchors and the end-effector, which are also called the geodesics. On a sphere, a geodesic is an arc of a great circle through the two given points [19]. The equation of a geodesic on a sphere can be given as [20]: θ − θ0 = arccos(β cot(ϕ))

(1)

To intuitively understand the equation, β and θ 0 are parameters found by the positions of the moving platform and the anchor, and the equation represents the geodesic between

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the moving platforms and the respective anchor (r, θ, ϕ)|p , and in other words, the relationship of θ and ϕ angle. The geodesic on a sphere is simply the intersection of the sphere’s surface and the plane that contains the moving platform, the anchor, and the center of the sphere. The anchor vector is a vector originating from the origin and pointing towards each anchor, while the moving platform vector is a vector originating from the origin and pointing towards the moving platform. The normal plane refers to the tangent plane on the surface of a sphere, which is perpendicular to the sphere’s surface at the point of contact between the sphere and a plane. The plane can be given as C1x + C2y + C3z = 0 since it always goes through the origin as shown in Fig. 4(a), where A, B, and C can be found by the cross product of the anchor vector and the moving platform vector which is normal to the plane. In spherical coordinates, this equation can be transformed to C1 cos(θ ) + C2 sin(θ ) = −C3 cos(ϕ)

(2)

Thus, β and θ 0 are found by: θ0 = arctan( β = −

C2 ) C1

C3

(3) (4)

C12 + C22

Thus, for each anchor, with d being the straight-line distance between the anchor and the moving platform, the cable length d i (i = 1,2,3) can be calculated as: 1

di = 2arcsin( 2

d )π r

(5)

As part of the inverse kinematics, the direction of the cable tension also needs to be solved. Calculating the direction of cable tension becomes straightforward once it is established that the planes containing the moving platform, its corresponding anchor, and the center of the sphere are all perpendicular to the tangent plane at the location of the moving platform. If a plane (the geodesic plane) contains a line (the radius points from the origin to the moving platform) that is perpendicular to another plane (the tangent plane at the moving platform point), then these two planes will also be perpendicular to each other. Thus, all the projections of geodesics on the tangent plane will be straight lines starts from the moving platform to the anchors. Singularity occurs when the moving platform is moving along the equator and one of the anchors is at the opposite pole of the sphere. In this case there will be infinite geodesics of these two points since every geodesic is half the perimeter. Given that the geodesic is coplanar with the plane of the moving platform, the corresponding anchor, and the origin, the projection of geodesic on the tangent plane is a straight line as suggested in Fig. 4(b). The vector starting from the moving platform along the projection will be the force direction of the cable. A better solution will be − → projecting vector PA instead of arc PA to find the force direction, for example, because 

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Fig. 4. a. All geodesic planes will intersect with the origin. b. all planes are perpendicular to the tangent plane in which the moving platform locates. Thus, all force directions will be on the tangent plane.  − → − → the vector can be easily found by OA − OP. The F force direction of anchor A is given by:

(6) PA = PA − AA

(7)

PA F =  PA

(8)

PO is the unit vector of PO. AA is the line normal to the tangent plane. 3.2 Forward Kinematics Forward kinematics solves the moving platform location when all cable lengths are known. At least three cable lengths are needed to find out an accurate position. When one cable length is given, possible positions for the moving platform will be a circle on the sphere around the anchor, thus forming a plane of coverage cutting through the sphere. When three planes are defined, the cross section will be a point which is the moving platform location. For anchor point A, there is   −→ SA ∗r (9) OA = cos r −→ −→ −→ −→ Plane : OA i · x + OA j · y + OA k · z = OA2 (10) −→ where OA is the vector pointing from the origin to the plane of coverage in the direction of anchor A. A is the projection point from A to the plane of coverage defined in Eq. (10). −→ The dot products of i, j, and k (the unit vector in x, y, and z axis) with OA are the component magnitutes. sA is the arc length which is the given cable length of anchor A as input in forward kinematics, and sB and sC can be found out by the same procedure.

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−→ Since the plane is normal to the vector OA , the product of every vector to the plane −→ −→ and OA will be the square magnitude of OA . We can perform the same calculation for anchor B and C as well. The moiving platform’s coordinates can be solved by the following matrix by combining all three anchors and their known cable lengths: ⎛ −→ OA i ⎜ −→ ⎝ OB i −−→ OC i

−→ OA j −→ OB j −−→ OC j

−→ ⎞−1 ⎛ −→2 ⎞ ⎛ ⎞ OA k OA x −→ ⎟ ⎜ −→2 ⎟ ⎝ ⎠ = y OB k ⎠ ⎝ OB ⎠ −−→ −−→2 z OC k OC

(11)

Thus, the moving platform location P is the intersection of three planes of coverage, which is a point (x,y,z) on the hemi-sphere.

4 Statics In general, there will be more than two cables connecting the moving platforms, redundancy occurs in a system when there are more unknowns than equations, making it impossible to solve for all the variables. In this case, the redundancy arises because there are only two equations - the horizontal and vertical force balance equations - to solve for three unknown variables: the tension forces in each of the anchor cables and the force required to move the platform. As a result, the system is underdetermined and has infinite solutions. Like Hermus [21] analyzing the control of kinematically redundant moving platforms, our model will split the burden into force balance and dynamic control. Two of the cables will be selected by diagonal-pizza rule, which will be detailed later, to counteract the gravity, and all cables will be controlling the dynamics of the moving platforms. As stated above, all cable tensions will result in the same tangent plane, and since this paper only considers the moving platform as a mass point, the following force balance must be met: n

T k = F(stability control)

(12)

k=1



(T k , T k+1 ) + G = 0

(13)

where T k are the cable tension forces and G is the gravity of the end-effector. For static analysis in this paper, only Eq. (12) needs to be considered. The Diagonal-Pizza Rule selects two cables to counteract gravity, with boundary position being crucial for two-cable systems. Figure 5 shows that two cables cover identical force over each third of the sphere, with one cable having no tension at the boundary. Slicing the sphere into N parts, with N being the number of anchors, allows diagonal two anchors to balance forces in each slice, keeping the moving platform in their region and overall forces balanced. On the spherical surface, two cable forces and the tangent gravity projection will lie on the tangent plane as in Fig. 2, while the normal supporting force of the surface will cancel the normal force projection of the gravity. Thus, solving the three force balance

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Fig. 5. Every two anchors next to each other will form two boundaries across the apex of the hemisphere. The robot’s gravity will be cancelled by these two anchors in the shaded area between the boundaries.

equations in Eq. (12) will be equivalent to solve only the two force balance equations on a 2D plane, and it will provide the magnitudes of T k and T k+1 . A numerical example is given by setting the sphere radius to 5 and anchor 1 at position (5, 0.66π, 0.5π) and anchor 2 at position (5, 0.33π, 0.5π), Fig. 6a demonstrates the tension forces needed at each position of the hemi-sphere (the larger force of (T k and T k+1 ) is used to draw the dot of which the size shows the value of the force). It is obvious that the performance goes worse as approaching both the equator and the midpoint of each two anchors. It is because the corresponding cables are reaching the boundary of their workspace as in Fig. 6b. Any small increments around the boundary will cause the cable to swipe across the sphere and low latitude cases will result in small angle on vertical component force, thus the cable tension will be extremely high.

Fig. 6. a. Tension map of a three-anchor model. Tension increases as φ increases, and drastically enlarges around the base level with the maximum at the midpoint between two anchors. b. at low base level, any small increments will cause the top cable to swipe across the sphere, resulting in large vertical angle and vertical force to counteract the gravity.

5 Tension-Force-Oriented Configuration Variation The statics model assumes that all CDPR anchors are on the base level plane. Figure 6 shows that if the tension force exceeds cable capacity, the workspace is restricted. To

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cover the hemi-spherical surface fully, reconfiguration is necessary, as shown in Fig. 8, where lifting anchors reduces cable tension forces. Anchor lift reduces the angle between balance forces and cable forces needed to balance gravity force on the tangent plane for a given end-effector position. There is a same effect when moving anchor 1 to the equivalent anchor point on the equator as in Fig. 7 to reduce the angle between the two cables with θ.

Fig. 7. Moving up the anchor to the top has an equivalent anchor with decreased θ angle.

According to the Diagonal-Pizza rule, only two anchors need to be considered at a time. Thus, there are three reconfiguration ways to reduce the cable tension: narrowing the angle between the two anchors, moving one anchor up, and moving two anchors up. To illustrate the force change trend, a numerical example is shown here by giving radius equals to 5, θ angle between boundaries ranging from 2/3π to 1/6π, and ϕ angle decreasing from 1/2π to 0.1π. Based on those setup parameters, the three reconfiguration methods are applied and the average cable forces are plotted in Fig. 8 below when the moving platform is at different height levels, which are planes with different ϕ angle, on the hemi-sphere from the base level. Figure 8(a) shows that when moving one anchor from the equator level to 0.1π, the average tension is decreasing. Figure 8(b) shows that when moving two anchors from the equator level to 0.1π, the average tension is decreasing faster than moving single one. Figure 8(c) shows that when narrowing the angle between the two anchors from 2/3π to 1/6π, the average tension is decreasing. Figure 8(d) shows that when moving one anchor from the equator level to 0.1π, the cable of the moving anchor is taking more tension than the anchor stays at bottom. It can be inferred from Fig. 8(a), (b), and (c) that the maximum tension is decreasing along the height levels. Narrowing the angle has similar effect as moving one or two anchors up. However, moving single anchor has similar effect as moving two anchors, but the tensions of two cables will not stay the same. In other words, the anchor that has higher level will have higher tension. Since the string tension will not change along the cable, when the moving platform stays at the same position, the anchors can move along the cable without changing the tension and with the same force balancing on the moving platform tangent plane. This can lead to equivalent configuration. As illustrated in Fig. 9(a), the derived configuration A’B’C’ is called an equivalent configuration for an arbitrary configuration with actual

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Fig. 8. Average tension at different heights when a) only one anchor is moving up. b) both anchors are moving up. c) two anchors are moving closer. d) Cable tension comparison when only anchor 1 is moving up.

anchors set up at A, B, and C on the sphere since they have the same tension and angle to the moving platform.

Fig. 9. (a) Equivalent positioning will ease the calculation by move all the anchors to the base level and reduce the complexity in force directions. (b) The moving platform end-effector will be dangling when it is positioned below the base plane formed by the three anchor points.

When cable length is limited, installing all anchors on the same side of the moving platform allows for equivalent positions on the base level. Therefore, a single table with different angles of base-level anchor installation suffices for tension analysis in all scenarios. However, reconfiguring has drawbacks. Narrowing anchor angles reduces “slice” gaps, necessitating more anchors. Elevating anchors raises the setup’s base level, causing instability when the moving platform drops below it, as shown in Fig. 9(b).

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6 Conclusions and Future Work This work developed kinematic and static models for CDPRs on hemispheric surfaces using a diagonal pizza rule and geodesics property for force balance. Both forward and inverse kinematics were solved analytically based on spherical geometry. Superposition was used to simplify redundant static force calculations. Equivalent anchor positioning can simplify processes and reduce maximum tension. Future research can expand to non-ordinary curved surfaces with dynamics and path planning analysis [22].

References 1. Pott, A.: Cable-driven parallel robots theory and application. Springer International Publishing (2018) 2. Pinto, A.M., Moreira, E., Lima, J., Sousa, J.P., Costa, P.: A cable-driven robot for architectural constructions: a visual-guided approach for motion control and path-planning. Auton. Robot. 41(7), 1487–1499 (2016). https://doi.org/10.1007/s10514-016-9609-6 3. Iturralde, K., et al.: A cable driven parallel robot with a modular end effector for the installation of curtain wall modules. Proc. 37th Int. Symp. Autom. Robot. Constr. 1472–1479 (2020). https://doi.org/10.22260/isarc2020/0204 4. Seriani, S., Gallina, P., Wedler, A.: A modular cable robot for inspection and light manipulation on celestial bodies. Acta Astronaut. 123, 145–153 (2016). https://doi.org/10.1016/j.actaastro. 2016.03.020 5. Zi, B., Lin, J., Qian, S.: Localization, obstacle avoidance planning and control of a co-operative cable parallel robot for multiple mobile cranes. Robot. Comput. Integr. Manuf. 34, 105–123 (2015). https://doi.org/10.1016/j.rcim.2014.11.005 6. Duan, B.Y.: A new design project of the line feed structure for large spherical radio telescope and its nonlinear dynamic analysis. Mechatronics 9, 53–64 (1999). https://doi.org/10.1016/ s0957-4158(98)00028-2 7. Barnett, E., Gosselin, C.: Large-scale 3D printing with a cable-suspended robot. Addit. Manuf. 7, 27–44 (2015). https://doi.org/10.1016/j.addma.2015.05.001 8. Tanaka, M., Seguchi, Y., Shimada, S.: Kineto-statics of skycam-type wire transport system. In: Proceedings of USA-Japan Symposium on Flexible Automation, Crossing Bridges: Advances in Flexible Automation and Robotics, pp. 689–694 (1988) 9. Mao, Y., Jin, X., Dutta, G.G., Scholz, J.P., Agrawal, S.K.: Human movement training with a cable driven ARm EXoskeleton (CAREX). IEEE Trans. Neural Syst. Rehabil. Eng. 23, 84–92 (2015). https://doi.org/10.1109/TNSRE.2014.2329018 10. Jiang, Q., Kumar, V.: Determination and stability analysis of equilibrium configurations of objects suspended from multiple aerial robots. J. Mech. Robot. 4, 1–21 (2012). https://doi. org/10.1115/1.4005588 11. Gagliardini, L., Caro, S., Gouttefarde, M., Girin, A.: Discrete reconfiguration planning for cable-driven parallel robots. Mech. Mach. Theory 100, 313–337 (2016) 12. Shao, Z., Xie, G., Zhang, Z., Wang, L.: Design and analysis of the cable-driven parallel robot for cleaning exterior wall of buildings. Int. J. Adva. Robotic Sys. 18(1) (2021). https://doi. org/10.1177/1729881421990313 13. Williams, R., Gallina, P.: Planar Cable-Direct-Driven Robots, Part I: Kinematics and Statics. Proceedings of the ASME Design Engineering Technical Conference 2 (2001) 14. Bach, S., Yi, S.: Flexible motion control of a cable-driven robot on 3D curved surface. J. Ins. Cont. Robo. Sys. 26(x), 000–000 (20xx)

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15. Bu, W., Zhou, W., Fang, L., Chen, J., An, X., Huang, J.: A Novel Cable-Driven Parallel Robot for Inner Wall Cleaning of the Large Storage Tank. In: Tan, J. (ed.) ICMD 2019. MMS, vol. 77, pp. 28–40. Springer, Singapore (2020). https://doi.org/10.1007/978-981-32-9941-2_3 16. Lee, D.G., Oh, S., Son, H.: Il: Wire-driven parallel robotic system and its control for maintenance of offshore wind turbines. Proc. - IEEE Int. Conf. Robot. Autom. 2016-June, 902–908 (2016). https://doi.org/10.1109/ICRA.2016.7487221 17. Chen, D., Zhang, Y., Li, S.: Zeroing neural-dynamics approach and its robust and rapid solution for parallel robot manipulators against superposition of multiple disturbances. Neurocomputing 275 (2018) 18. Pham, C.B., Yeo, S.H., Yang, G., Chen, I.-M.: Workspace analysis of fully restrained cabledriven manipulators. Robotics and Autonomous Systems 57(9) (2009) 19. geodesic: Oxford Reference. Retrieved 30 Nov. 2022, from https://doi.org/10.1093/oi/author ity.20110803095848383 20. Hunt, R.E.: Geodesics on the Surface of a Sphere. Mathematical Tripos and Mathematical Methods II, University of Cambridge, Class Lecture (2007) 21. Hermus, J., Lachner, J., Verdi, D., Hogan, N.: Exploiting redundancy to facilitate physical interaction. IEEE Trans. Rob. 38(1), 599–615 (2022). https://doi.org/10.1109/TRO.2021.308 6632. Feb. 22. Wang, S., Wang, F., Du, X.: A method for solving the shortest path on curved surface based on psosa algorithm. J. Theoreti. Appl. Info. Technol. 46, 672–676 (2012)

Cable Modelling

FEM-Based Dynamic Model for Cable-Driven Parallel Robots with Elasticity and Sagging Karim Moussa1,3(B) , Eulalie Coevoet2 , Christian Duriez2 , Maxime Thieffry3 , Fabien Claveau3 , Philippe Chevrel3 , and St´ephane Caro3 1 2 3

Nantes Universit´e, IRT Jules Verne, LS2N, UMR 6004, 44000 Nantes, France [email protected] Universit´e de Lille, INRIA, CNRS, CRIStAL UMR CNRS 9189, Lille, France ´ IMT Atlantique, Nantes Universit´e, Ecole Centrale de Nantes, CNRS, LS2N, UMR 6004, 44000 Nantes, France

Abstract. Cable-Driven Parallel Robots (CDPRs) are a type of robot that is growing in popularity for different kinds of applications. However, the use of cables instead of rigid links makes the modelling of this robot a complex task, and therefore their trajectory planning and control are challenging. Assumptions such as inelastic, massless and non-sagging cables made when the CDPR is small are no longer valid when the robot becomes large. This paper presents a CDPR dynamic model taking into account cable elasticity and sagging, and its implementation within an open-source framework, named SOFA. Finally, the simulation results are compared to experiments conducted on a suspended CDPR. Keywords: Cable-Driven Parallel Robots · Dynamic Modeling Finite Element Method · Cable Model · Beam Theory

1

·

Introduction

Cable-Driven Parallel Robots (CDPRs, see Fig. 1) are used to perform operations in a very large workspace: from the assembly of solar panels [1] to flight simulators [2], through the development of a large radio telescope [3]. Simulating this kind of robots is of interest for carrying out potentially dangerous or time-consuming tests, such as the analysis and prediction of their behaviour, e.g. during their design and the synthesis of control laws, or to conduct tests at the safety limits, i.e. failure tests of actuator and cable breakage. It can also make it possible to estimate certain parameters or variables that cannot be measured on a real system. To do so, dynamic models that are both precise and computationally tractable are mandatory. This work was supported by both IRT Jules Verne in the framework of the PERFORM program and ROBOTEX 2.0 (Grants ROBOTEX ANR-10-EQPX-44-01 and TIRREX ANR-21-ESRE-0015). c The Author(s), under exclusive license to Springer Nature Switzerland AG 2023  S. Caro et al. (Eds.): CableCon 2023, MMS 132, pp. 55–68, 2023. https://doi.org/10.1007/978-3-031-32322-5_5

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Fig. 1. CDPR (CAROCA) located at IRT Jules Verne, from [4].

Several cable models are available in the literature as shown in Fig. 2. A first level of dynamic modeling is assuming massless, straight or non elastic cables [5, 6]. Some works consider elastic, but massless cables, therefore not undergoing cable sagging [7]. This may be of interest to model synthetic cables that have low mass density. In [8], the authors used the assumed mode approach to model a three degree-of-freedom (DOF) CDPR with sagging and non-elastic cables. The Irvine model [9] offers the possibility to model faithfully the cable geometry taking into account cable sag and elasticity, but only in the static case. Besides, it neglects the bending stiffness of the cable. It is therefore not directly suitable for dynamic control applications. Recently, a Rayleigh-Ritz cable model that takes into account the variation in cable mass and stiffness was studied in [10]. However, results are for now limited to a single DOF CDPR.

Fig. 2. Illustration of different cable models.

It should be noted that few CDPR simulators exist in the literature based on different cable models. In [11], a massless and inextensible cable model is implemented in Gazebo and ROS to simulate CDPR dynamic behaviour. Multiple nodes of Reissner beam are used in [12] to model cable elasticity, sag, shear and torsion in the XDE framework. However, some of the physical parameters of the cable such as the Young modulus need to be adapted in order to ensure

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simulation stability. The simulation time obtained in [12] is slow for testing and tuning controllers, namely, 10 h to simulate 10 s. CoppeliaSim offers elastic, but straight and massless cables [13]. Other frameworks like AGX Dynamics [14] and MapleSim [15] make it possible to consider elasticity and cable mass with a lumped parameter discretization. This work proposes a cable model suitable for CDPR control design. The proposed dynamic model considers both cable elasticity and sag. This dynamic model is implemented in SOFA framework [16], which was developed for finite element modeling and simulation of soft objects. A digital twin of CAROCA, a CDPR located at IRT Jules Verne, is developed in SOFA. The results from the simulation with SOFA are compared to the results from simulators developed with MaplesoftTM MapleSim software. The paper is organized as follows: Sect. 2 describes the cable model obtained based on the FEM model of beams in SOFA and its application to a suspended CDPR. Results from SOFA and MapleSim simulations are compared with experimental results in Sect. 3. Conclusions and future work are drawn in Sect. 4.

2

CDPR Dynamic Modeling

This section presents the dynamic model of CDPRs including both cable elasticity and sagging. The presented methodology is intended to be as generic as possible and is valid for all configurations (suspended and fully-constrained), but will be illustrated throughout the paper with a suspended CDPR. The method relies on the finite element method (FEM) implemented within the SOFA framework, a physics-based simulation platform that uses FEM to model, simulate, and control deformable objects. 2.1

FEM Model of Beams in SOFA

This method, first presented for SOFA in [17], relies on a representation based on three-dimensionnal Timoschenko beam theory [18] and a specific corotational formulation to account for large displacements [19,20]. It is implemented within the BeamAdapter1 plugin of the SOFA framework. The beam is discretized into N small elements, each of them corresponding to the one depicted in Fig. 3(a). The motion of the flexible beam is then decomposed into two parts, a rigid body motion and a deformation motion, with the assumption that the deformation remains small at the level of each elements. The equation of motion of an object according to Newton’s second law is given by: ˙ + H(q)λ (1) M¨ q = fe (q, q) where q ∈ Rn is the vector of generalized coordinates, M ∈ Rn×n is the inertia ˙ : Rn × Rn → Rn gathers external forces. matrix of the system and fe (q, q) n n×m is the Jacobian matrix that gives the direction of Finally, H(q) : R → R 1

https://github.com/sofa-framework/BeamAdapter.

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constraints forces, λ ∈ Rm the vector of Lagrange multipliers corresponding to the constraints and m is the number of algebric constraints. In the case of a beam element, q = (qTa , qTb )T where indices a and b represent the extremities of the beam element as shown in Fig. 3(a), qa and qb contain a positional vector ∈ R3 and a rotational term ∈ SO(3) defining the pose of each extremity. M is given in [18] and depends on the second moments of area Iy and Iz with respect to the axes y and z respectively, the cable cross section area A, the length of the beam element L, the second polar moment of area J, and the material density ρ.

Fig. 3. Cable discretization.

With the corotational assumption, the deformation in the cable elements is small locally and the external force can be calculated with the linear relation: fe (q) = K(u(q) − u(q))

(2)

where u(q) is the deformed configuration of q, u(q) is the rest configuration of q, both expressed in the local frame, and K is the stiffness matrix given in Eq. (3). ⎤

⎡

EA ⎢ L ⎢ 12EIz ⎢ 0 ⎢ L3 φy ⎢ ⎢ ⎢ 0 0 ⎢ ⎢ ⎢ ⎢ 0 0 ⎢ ⎢ ⎢ 0 0 ⎢ ⎢ ⎢ 6EI z ⎢ 0 ⎢ L2 φy K=⎢ ⎢ EA ⎢− 0 ⎢ L ⎢ ⎢ 0 − 12EIz ⎢ L3 φy ⎢ ⎢ ⎢ 0 0 ⎢ ⎢ ⎢ ⎢ 0 0 ⎢ ⎢ ⎢ 0 0 ⎢ ⎢ ⎣ 6EIz 0 L2 φy

12EIy L3 φz 0 6EIy − 2 L φz

GJ L 0

symmetric EIy Φpz Lφz

0

0

0

0

0

0

0

0

0

0

6EIy L2 φz

12EIy − 3 L φz

GJ 0 0 − L EIy Φnz 6EIy 0 − 2 L φz Lφz 0

0

0

EIz Φpy Lφy 0 6EIz − 2 L φy

EA L 12EIz 0 L3 φy

0

0

0

0

0

0

0

0

0

EIz Φny Lφy

6EIz 0 − 2 L φy

12EIy L3 φz 0 6EIy L2 φz 0

GJ L EIy Φpz 0 Lφz 0

0

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ EIz Φpy ⎦ Lφy

(3)

FEM-Based Model of CDPR

where Φpy = 3+φy , Φny = 3−φy , φy = 1+

59

12EIz , Φpz = 3+φz , Φnz = 3−φz , GAsy L2

12EIy E , E is the Young’s modulus, μ is the Poisson’s ,G= GAsz L2 2(1 + μ) ration, and Asy and Asz are the actual surfaces of the cable due to shear along the y and z directions. In CDPR use cases, the sheering can be neglected and therefore φy = φz = 1. The cables have a symmetrical cross-section and therefore Iy = Iz = I. Once the external force fe (q) is calculated in the local frame, it is transformed to the global frame using the rotation matrix of the local frame. The model presented in [18] is used to model beams. However, compared to beams, a cable can bend more and is unable to withstand compression force. This difference is mainly due to the second moment of area I that is smaller for a cable. The structure of the cable used for the CAROCA robot is shown in Fig. 3(b), it is a two layers concentric contra-helical cable made of steel where cable strands have opposite lay directions. As shown in [21], when the cable is straight all the wires are sticking together and act as a one homogeneous body that can bend around the central axis of the cable. The quadratic moment is at its maximum Imax . However, when the curvature k of the cable increases the wires start to slip and eventually at the most extreme case each wire bends around its own axis. The quadratic moment is then at its minimum Imin defined as: ns  Ismin ,i cos βs,i (4) Imin = Ic + φz = 1 +

i=1

where Ic is the quadratic moment of the steel core around its own axis, Ismin ,i is the minimum quadratic moment of strand i around its own axis calculated with Eq. (4) after replacing strand with wire, βs,i is the lay angle of strand i and ns is the number of strands. Then, Imax can be calculated: Imax = Imin +

a  ni i=1

2

Ai ri2 cos3 βl,i

(5)

where ni is the number of strands in layer i, Ai is the cross section area of each strand in layer i, ri is the radius of layer i, βl,i is the lay angle of the strands in layer i and a is the number of layers. Note that Imin and Imax are independent from the curvature of the cable and its tension force value. To simplify the model, a constant value for I ∈ [Imin , Imax ] will be chosen. Knowing the working region of the cable (e.g. assuming that the curvature will not reach significant values as much as k = 1 m−1 ), and based on the identified transition function between EImax and EImin in [21] shown in Fig. 4, the parameter α = 0.65 is chosen such that: I = Imin + α(Imax − Imin )

(6)

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Fig. 4. Theoretical evolution of bending stiffness EI as function of the curvature k for different tension T values, based on [21].

2.2

Application to CDPRs

In CDPRs, the cables are attached at one end to the platform using spherical joint-equivalent connectors and are rolled at the other end around the winches. The cables can be oriented by passing through pulleys to change the CDPR configuration from suspended to fully-constrained. In addition to cables, pulleys and winches have an influence on the full model of the CDPR. Their impact varies depending on their dimensions and the friction they induce. To get the full CDPR model, each cable is modeled by Eq. (1) where the constraints represent the connection between the cable and the platform, and its contact with the pulley and the winch. The spherical joint does not transmit torque, therefore only position constraints are applied. A 6-DOF, 2-DOF, and 1-DOF dynamic models for the platform, the pulley, and the winch, respectively are implemented. The cable/pulley, cable/winch and gearhead frictions are neglected. In the following section, the different simulations are compared with the experimental results.

3 3.1

Comparison with Existing Simulators Modeled Robot

For this first trial, the implemented model in SOFA described in Sect. 2.1 is compared to other state-of-the-art models (“Rope” and “Cable” described in Sect. 3.2) of the simulation software MapleSim, and to real-robot measurements from the CAROCA experimental platform (see Fig. 1). It is a suspended CDPR with 8 cables allowing 6-DOF for the platform. Steel cables (see Fig. 3(b)) are used, they can withstand significant tensions (10 kN), but can present significant sags given their mass. The working area of the robot is 7 m × 4 m × 3 m. The moving platform’s dimensions are 1.5 m × 1.5 m × 1 m, and it weighs 366 kg.

FEM-Based Model of CDPR

3.2

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MapleSim

The MapleSim [15] software provides some tools to model cable-driven robots including cable models, pulleys, and winches. This framework provides three possible models : rope, chain and cable. Rope is the simplest one where only the elasticity is considered, and cable is the most accurate where elasticity and sagging are considered. The rope model is made up of two massless nodes connected by a spring and a damper. The cable model relies on [22], where the cable is discretized into N equidistant nodes, being connected with each other by springs and dampers. Moreover, the cable mass is spread along the nodes. These first trials proposed in this paper are dedicated to these two models, they will serve as a reference to compare with SOFA’s implemented cable model’s results. 3.3

Definition of Scenario

The pick-and-place trajectory used to compare the simulators with respect to the experimental results is depicted in Figs. 5 and 6. Using an ideal cable model (massless and rigid), an inverse geometric model (IGM) and an inverse kinematic model (IKM) allow to find the desired joint space, i.e. the winches, angular position θd and velocity θ˙d , as shown in Fig. 7. Then, a PD controller is used to calculate the winch torques τw and track the reference trajectory. Controller gains in simulators are tuned slightly different than the robot as they must compensate for the friction [6].

Fig. 5. Reference Cartesian trajectory of the robot platform.

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Fig. 6. Viewing environment in SOFA with added winch numbering wi and the reference Cartesian trajectory of the platform from P1 to P4 .

Fig. 7. Trajectory generation and control architecture. Table 1. Tuning parameters associated to each cable model. Models

Parameters E μ (×1011 pa)

A ρ (kg m−3 ) Ca dr (×109 N s m−1 ) (×10−5 m2 )

α

BeamAdapter 1.022

0.28 2.1488

7850

–

–

0.65

Rope

1.022

–

2.1488

–

22.8

–

–

Cable

1.022

–

2.1488

7850

22.8

0.12 –

3.4

Simulation Results

The parameters needed to simulate the models (BeamAdapter in SOFA, rope and cable in MapleSim) and their values are shown in Table 1. Ca is the cable’s axial damping and dr is its bending damping ratio. As said before, the sheer is negligible in CDPR applications and so the precision of Poisson’s ratio’s value has no impact on the simulation.

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The first ten seconds of simulations have been cut on the results depicted in Figs. 8, 9, 10 and 11; they are associated to the initialisation of the robot, where the platform is kept at the initial target Cartesian position. This transitory behaviour before reaching the static equilibrium is not of interest. Then, the 30 s trajectory shown in Fig. 5 is applied. The measured cable tensions at the platform from both frameworks are compared to the experimental measures of the force sensors located between the cables and the moving-platform (MP). The resulting plots for cables 6 and 7 are shown in Fig. 8. The remaining six cables behave in a similar manner.

Fig. 8. Comparison between simulated and measured cable tensions.

It appears that the plots of both models in MapleSim are almost identical. This is probably due to the weight of the cable being insufficient to make a difference under the current cable tension between the massless (rope) and nonmassless (cable) models, and therefore the cables are almost straight. A small oscillation for the rope model is noticed, the PD controller gains might be high for this model. The plots of SOFA’s model match those of MapleSim. However, these three models have the following divergence with the experimental results. On one side, tensions in cables 2, 4, 6, and 8 match the experiment measurements. On the other side, they converge to a value different from the measured one in cables 1, 3, 5, and 7. This causes high Relative Root Mean Squared Errors (RRMSE) of some simulated cables tensions with respect to the the experiment (>20% )

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as can be seen in Fig. 9. The order of the cables and winches is given in Fig. 6. The SOFA model has slightly lower RRMSE mean value compared to MapleSim. As a first explanation, because of the redundancy of actuation, multiple set of cables tensions can result in the same platform position. A common distribution algorithm should be implemented in the three simulators and the experimental setup to handle this aspect.

Fig. 9. RRMSE (%) for each cable tension.

As for the winches, the angular position of winch 6 is shown in Fig. 10. The plots of the remaining 7 being similar.

Fig. 10. Angular position of winch 6.

Contrary to the cables tensions, the plots of the three simulated models match the experimental measurements. This is due to the controller being implemented in the joint space and not considering the cable tension. This can be seen in Fig. 11 where all RRMSE are very small ( 3, the manipulators are kinematically redundant since 3 dofs would be enough to control the endeffector motion in the plane. Moreover, n > 3 means that the manipulators are also under-actuated. Indeed, since the cables can only pull, one more cable than the number of modules should exist to control all the modules [10]. 2.1

Tensegrity Modules

We want to define a planar tensegrity manipulators inspired by the bird neck upon stacking several basic mechanisms or modules. These modules play the role of intervertebral joints. Each module consists of articulated bars and springs and are operated by cables. Springs and cables play the role of muscles and tendons. Tendon forces must be positive and are bounded by the actuators maximal torques. Since only planar motions are involved, the relative movement between two vertebrae is mainly a rotation. Both revolute joints and anti-parallelogram

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joints (referred to as X-joint) can be used to produce planar motions between two vertebrae [11]. The former generate a pure rotation about a fixed point while the latter have a variable center of rotation. An important feature of the X-joint is its ability to increase stiffness under an increase in the antagonistic tendon forces, contrary to the revolute joint [12]. We thus decide to use X-joints (Fig. 1, left). The ratio between the length L of the crossed bar and the length b of the base or upper bar influences the kinetostatic performance of the X-joint but should also take into account the maximal and minimal elongation of the springs [13]. Different numbers of modules, with or without offsets will be considered in this work. Figure 1, right shows a manipulator built with two X-joints and offsets. 2.2

Stack of Modules

The manipulators at hand are composed of a stack of modules. The stack can be build with or without offsets. Offsets can be viewed as the possibility to adjust the dimension of the vertebrae or, equivalently, the maximal reach of the manipulator, independently of the X-joints ratio L/b. Let define the joint configuration of the manipulator by q = [q1 , q2 , ..., qn ], where n is the number of modules. Let X = [xn , yn , γn ] define the pose of the end-effector (EE), i.e. the coordinates of the center of the upper bar of the last module n and its orientation angle. We have: ⎧  n n q ) L2 − b2 cos2 ( q2i ) − i=1 sin(γi )ho ⎨ xn = − i=1 sin(γi−1 + 2i  n n y = i=1 cos(γi−1 + q2i ) L2 − b2 cos2 ( q2i ) − i=1 cos(γi )ho ⎩ n n γn = γ0 + i=1 qi

(1)

where ho is the offset height (Fig. 1, right), γ0 is the orientation angle of the base bar of the first module and γi = γi−1 + qi . 2.3

Tendon Routing

For a planar manipulator, three dofs are sufficient to control its EE pose. We use nf =4 tendons with remote motors on the base, regardless of the number of modules, in order to reduce inertia, complexity and costs. We also operate our manipulator with a long tendon connected to all the modules on the left side and 3 shorter tendons grouping together sub-groups of modules (see Fig. 2). This choice results from a simplified implementation of the muscle organization of the bird neck [1,14]. We will also consider fully actuated manipulators, namely, manipulators with 3 modules actuated by a long tendon connecting the 3 modules on the left and 3 short tendons on the right. These short tendons actuate each of the modules independently, like in the prototype analyzed in [15]. Each tendon can be routed in different ways on each of the modules (Fig. 3): – tendon j placed on the left or on the right of module i, along the spring (Fig. 3, left): when pulling this tendon, the associated motor modifies the module orientation so as to reduce the tendon length on this side;

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Fig. 2. Example of cable routing with 6 modules: one long cable on the left and three shorter cables on the right.

Fig. 3. Tendon routing situations.

– tendon j run along the bars of module i (Fig. 3, right): this routing allows reaching the modules located above module i while nullifying tendon impact on this module. We define an actuation matrix A of size (n × nf ) as follows: each column j associated with tendon j describes how this tendon passes along module i. Each entry A(i, j) can take on three possible values: 1 if the tendon passes on the left, 2 if the tendon passes on the right, 3 if the tendon j does not act on module i. The unwounded length of tendon j is denoted by lj . The tendon length lj can be expressed as follows [15]: lj =

ljc

+

n  i=1

A(i,j)

where ljc is a constant value. lj

A(i,j)

lj

(qi )

(2)

depends on the tendon routing:

⎧ 1  2 q q q ⎨ lj = L − b2 cos2 ( 2i ) + 2 h cos( 2i ) − 2r sin( 2i ) q q 2 i i l = L2 − b2 cos2 ( 2 ) + 2 h cos( 2 ) + 2r sin( q2i ) ⎩ j3 lj = 0

(3)

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3 3.1

125

Wrench-Feasible Workspace Static Model

The static model is of primary importance for the study of tensegrity manipulators. For a manipulator built with several modules in series, the static model allows determining its configuration as functions of the tendon forces. The potential energy U of the manipulator can be written as: U = Ug + Uk +

4 

F j lj ,

(4)

i=1

where Ug (resp. Uk ) is the contribution of gravity (resp. of all springs), lj are the tendon lengths and Fj are the tendon forces. Each term Fj lj accounts for the potential energy associated with the work done by the cable forces. The equilibrium condition of the manipulator is: dU =0 dq

(5)

dl T k Let G = dqg + dU and F = dq and Z(q) = − dq , where l = [l1 , l2 , l3 , l4 ] T [F1 ...F4 ] . The above equation can written as: dU

G(q) = Z(q)F The associated linearized model writes:  dG(q) dZ(q)F − δq = Z(q)δF dq dq

(6)

(7)

An equilibrium is stable if its stiffness matrix is definite positive: K= 3.2

d2 U >0 dq2

(8)

WFW Calculation

Tendons impose positive forces and these forces are bounded by the actuators. The set of poses that satisfy the abovementioned constraints and in which the robot can balance a bounded set of external wrenches is called the wrenchfeasible workspace (WFW) [16]. In our case, the external wrenches are contributed by the springs and the gravity effects only. In the literature, the WFW is most often calculated for cable-driven parallel robots, see [16] and references herein. These methods cannot be adapted easily to serial tendon-driven manipulators. Continuation methods have been employed to compute the WFW of a 2-dof tensegrity manipulator [17]. The time taken for such computations has not been presented. A brute-force scanning technique has been followed in [13], where a 2-dimensional (D) scanning was performed in the joint space of a 2-DoF

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manipulator to determine the WFW. The limitation of such a technique is that a high scanning resolution is required to obtain the boundary points with sufficient accuracy, which is untractable for manipulators with more than 3 modules. Interval analysis has been used to compute manipulator workspaces [18] or cabledriven parallel robots [16] with garanteed results but the high computational cost limits its practical implementation to manipulators with few dofs. Joint Space Scanning. A simple approach is to scan the joint space as proposed in [1]. The linearized model (7) is solved to determine a vector of forces F satisfying (7). When the number of tendons nf is greater than the number of modules n, there exist infinitely many solutions and one can select the solution with minimal norm such that 0 < Fj < Fmax . This method cannot be used in our case for two major reasons. First, the computational cost increases exponentially with the number of modules and second, it does not work for under-actuated manipulators (i.e. when n > nf ). For those latter, indeed, the static model admits solutions only if rank([G(q) Z(q)]) = rank(Z(q)). Force Space Scanning. Scanning the force space is more realistic since we have only 4 forces, whatever the number of modules. For each F, it is then necessary to solve Eq. (6) to find the equilibrium configuration qe . We then calculate the EE coordinates via Eq. (1). As observed in [2,13], several equilibrium solutions qe can be obtained under two conditions: (i) the X-joints can reach a configuration close to their flat singularities and (ii) the gravity effects are dominant. In our case, the spring effects are very high as compared to gravity and the joint ranges are limited, so that there is only one feasible solution to Eq. (6). The equilibrium solution qe is obtained iteratively, starting from the equilibrium configuration at rest. For a given set of input forces Fe , we seek for the solution qe that minimizes ||G(qe ) − Z(qe )Fe ||, using a Newton-Raphson approach. Starting from a configuration qep , this method consists in writing the linearized model (7) in the neighborhood of this configuration. We compute the variation that tends to nullify ||G(qe ) − Z(qe )Fe ||. The joint solution is thus updated as follows:  −1 dG(qep ) dZ(qep ) − Fe qe = qep − (G(qep ) − Z(qep )Fe ) (9) dq dq The solution is updated until: ||G(qe ) − Z(qe )Fe || < 

(10)

where  is a decision parameter, which we take here as  = 1.e − 6. The matrix that needs to be inverted in (9) is in fact the stiffness matrix K defined in Eq. (8). Convergence is thus guaranteed whenever the equilibrium configurations are stable. On the other hand, the springs are selected such that the equilibrium configuration at rest is stable (see further). We thus expect that a stable equilibrium solution is always found. In fact, we have been able to verify that it is indeed

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the case in all tested examples. Convergence speed depends on the starting configuration qep , which must be close to qe . We start the force space scanning at F = 0, namely, at the (stable) equilibrium configuration at rest. Convergence proves quite fast, after only 2 or 3 iterations in all tested examples. The WFW is then built in the 3D (xn , yn , γn ) space upon calculating the EE pose associated with each equilibrium configuration. The force and workspace sampling data used are as follows. The maximal forces are fixed to 140 N. We limit the number of tested forces to 20 for each tendon. Nature often relies on frugality. Therefore, bio-inspiration leads us to minimize energy. Accordingly, our objective is to move with minimum actuation forces. This goal can be achieved by trying to remain close to the equilibrium configuration at rest. Since we are interested to poses that can be reached with small forces, we choose a non-regular sampling to explore more values for low forces. The tested forces are [0, 1, 2, 3, 5, 7, 9, 12, 16, 20, 25, 30, 35, 40, ...125, 130, 135, 140] for each tendon. To plot the WFW, we define a regular grid along the xn , yn and γn coordinates. The grid is built in a box defined by ±1.1h along xn and yn and γ0 ± nqmax along γn , where h is the manipulator height in its vertical straight configuration, qmax is the maximal bending angle of the X-joints and γ0 is the orientation of the base bar of the first module. A WFW cell is declared reachable as soon as one pose of the EE belongs to it.

4

WFW Comparative Analysis

The goal of this section is to compare manipulators with different numbers of modules, with or without offsets. For more realistic comparisons, we impose similar features to all the manipulators studied: – all modules in a given manipulator are identical with symmetric joint ranges ±qmax ; – all manipulators have the same height h in their straight vertical configuration; – for all manipulators, the sum of the rotation ranges of all their joints Δq = 2nqmax is the same; – all manipulators have the same width b; – all manipulators have the same mass, assumed equally distributed in all the modules; – all manipulators have the same EE pose in their rest configuration; – stiffness in the rest configuration is greater than a given minimal value to ensure stability. A first objective is to study the effect of offsets and module height (defined by L/b) on the WFW size, for completely actuated manipulators, i.e. with three modules. The bird neck includes a large number of vertebrae. However, it is not clear if manipulators with a large number of modules would be a right choice, as the actuation system of our manipulators is a highly simplified implementation of the complex muscle organization of the bird neck. A second objective is thus to study the effect of the number of modules on the WFW size.

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Manipulator Data

Manipulator Compared. For given√module dimensions L and b, the module height in its zero orientation is hm = b2 + L2 . We start with modules similar to those used in the prototype analyzed in [15]. Their geometric parameters are b = 0.05 m and L = 0.1 m. The corresponding module height is 0.0866 m. We fix the manipulator height of all manipulators to 6 times this module height: h = 0.516m. The joint range decreases with the number of modules in order to keep the same sum of joint ranges Δq = 2nqmax (see above). We consider 7 manipulators, described in Table 1. Four of them have 3 modules with offset of values ho = 0, ho = hm , ho = 2hm , and ho = 3hm , respectively. Besides, we take three offset-free manipulators with different numbers of modules, namely, 6, 9 and 12. In all tested cases, the configuration at rest is defined as follows. The orientation angle of the manipulator base and the one of the EE bar are fixed to π/4 and −π/4, respectively, and the EE position is fixed at xm = 0 and ym = 0.9h. The choice of these data allows the manipulators to feature a C-shape equilibrium configuration at rest. Although the rest configuration of the bird neck features a S-shape [14], this shape is difficult to reach for manipulators with 3 modules only. In a C-shape configuration, the manipulator is not in a singularity and it can thus move more easily in all directions, like in a S-shape configuration. Finally, the sum of joint ranges is fixed to Δq = 3pi and, to have a stable configuration, we impose a minimal stiffness at rest. This stiffness at rest can be obtained with suitable spring constants (see below). Spring Selection. An essential element for the dimensioning of our manipulators is the choice of springs. The springs make it possible to define the equilibrium configuration at rest. The stiffer the springs, the higher forces are needed to move the manipulator and, for the same maximal forces, the more the WFW is reduced. On the other hand, the role of the springs is to ensure the stability of the manipulator. It is particularly important that the equilibrium configuration at rest be stable. This allows the manipulator to remain in this configuration under small perturbations and without any actuation. The springs are thus chosen on the basis of the following 2 requirements: – impose a C-shape configuration at rest in the prescribed EE pose; – ensure stability at rest. We impose a positive stiffness via the smallest eigenvalue of the stiffness matrix K, which must be greater than a prescribed minimal value Km = 1 Nm/rad. We want to limit the spring stiffnesses, while satisfying the above constraints. The difference in stiffness between the 2 opposite springs in a module will modify the equilibrium configuration at rest while the average value of the springs will contribute to the X-joint stiffness. For those manipulators with 3 modules, the configuration is fully defined by the EE pose. For the other manipulators, we first determine the equilibrium configuration qe which makes it possible to reach the desired EE position xdn , ynd , γnd while minimizing the norm of the joint configuration vector:

Comparison Analysis of Tendon-Driven Manipulators

qe = minq ||q|| s.t.[xn (q), yn (q), γn (q)] = [xdn , ynd , γnd ]

129

(11)

We then calculate the spring constants as follows: [kl kr ] = min||k

l + kr || G(qe ) = 0 s.t. min(eig(K(qe ) ≥ Km

(12)

The spring constants for the 7 manipulators compared are given in Table 1. Table 1. Robot data. n

offset qmax Spring constants [N/m]

3

0

π/2

kl = [1816 1339 874] kr = [3860 3216 1512]

3

hm

π/2

kl = [1592 1454 593] kr = [4346 3001 2278]

3

2hm

π/2

kl = [1283 1563 452] kr = [4979 2821 3303]

3

3hm

π/2

kl = [1042 1710 369] kr = [5843 2762 4835]

6

0

π/4

kl = [2859 2780 2401 1825 1190 676] kr = [5258 4692 3851 2832 1813 1027]

9

0

π/6

kl = [3804 3826 3637 3268 2761 2171 1562 1003 593]

12 0

π/8

kr = [6608 6175 5579 4837 3987 3087 2207 1426 858] kl = [4708 4770 4679 4444 4085 3624 3090 2513 1929 1374 891 547] kr = [7947 7586 7111 6526 5842 5080 4268 3440 2630 1879 1234 772]

4.2

Example: A Manipulators with 6 Modules

In this section, we describe the case of a 6-X manipulator without offsets in order to provide some complementary information. Its data are given in Table 1, manipulator 5. Figure 4 depicts the 3D WFW. Its shape looks like a twisted banana. This shows that the position and orientation coordinates are highly coupled. Moreover, the banana is rather flat, which shows that the EE orientation range is limited at every position. Therefore, the manipulator turns out to be more appropriate to positioning tasks. For now on, accordingly, the WFW will be analyzed in terms of point-reachable workspace, namely, as the set of points associated with at least one feasible EE orientation [19]. Figure 5 shows the resulting 2D WFW. The WFW was calculated for the minimal forces: the orientation was then fixed by the minimal forces at each point. Colors indicate the norm of the force vector. 4.3

Comparison Results

Figure 6 shows the WFW of all compared manipulators. The WFW plots are arranged in a table with two columns and 4 rows. In each of the 4 rows, the

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Fig. 4. Representation of the 3D WFW in (xn , yn , γn ). The orientation of the effector varies significantly with the position of the effector but for a given pose, the orientation range is limited.

Fig. 5. WFW projected onto (xn , yn ). The minimal forces have been considered at each point and colors indicate the norm of the force vector (dark blue: ||F|| ≤ 10 N, blue: 10 < ||F|| ≤ 50 N, green: 50 N < ||F|| ≤ 100 N, yellow:100 N < ||F||). At the EE poses [−0.45, 0] and [0.24, 0.2], the manipulator is shown with its EE orientation corresponding to minimal forces.

manipulators have the same module ratio L/h. In the left columns, all the manipulators have 3 modules. The first one (resp. second, third, fourth) has no offsets (resp. offsets of height hm , 2hm and 3hm ). The presence and height of offsets is difficult to analyze since the modules ratios L/b have different values for different offset numbers or heights. The second manipulator has clearly the largest WFW. This was expected since the module ratio of this manipulator satisfies the ideal ratio L/b = 2 [13]. Manipulators in the second row have no offsets. The first WFW of this row is the same as in the previous row for better visual

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Fig. 6. Influence of the presence of offsets (first column) and of the number of modules (second column) on the WFW shape and size. The same WFW has been reproduced in the first row. The presence of offsets tends to reduce the WFW size, while a greater number of modules tends to increase the WFW size.

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comparison. Clearly, the more the number of modules, the larger the WFW. It is worth noting that it is still the case for the manipulators with 9 and 12 modules, although the module ratio L/b is smaller than the ideal value. It is also interesting to compare each WFW of the first row with its neighbor in the second row (starting from the second row as the first row displays the same WFW), since their joint ratios are similar. The same conclusions as above can be drawn, namely, the WFW is larger when the number of modules is greater.

5

Conclusion

A family of planar manipulators built upon stacking a series of tensegrity Xjoints has been analyzed in this paper. The manipulators are actuated with four tendons, regardless of the number of modules. The main goal of this work was to study the influence of offsets and number of modules on the WFW size. The comparison analysis was conducted on the basis of equal manipulator height, width and mass and of equal maximal actuation forces. The manipulators spring constants were determined so that the configuration at rest features a C-shape at a given EE pose with a minimal stiffness to ensure stability. We have shown that the more the number of modules, the larger the WFW. Besides, the effect of offsets did not prove so significant. In fact, the module ratio turned out to be of more importance for manipulators with 3 modules. The WFW was calculated upon scanning the force space. Another possibility could be to scan the workspace. This will be the object of future work. Moreover, we will also study the influence of obstacles. Kinematically redundant and under-actuated manipulators should have a better ability to adapt to cluttered environments by shaping around obstacles.

References 1. Fasquelle, B., Furet, M., Khanna, P., Chablat, D., Chevallereau, C., Wenger, P.: A bio-inspired 3-DOF light-weight manipulator with tensegrity X-joints. In: 2020 IEEE Int. Conf. on Robotics and Automation, pp. 5054–5060, IEEE (2020) 2. van Riesen, A., Furet, M., Chevallereau, C., Wenger, P.: Dynamic analysis and control of an antagonistically actuated tensegrity mechanism. In: Arakelian, V., Wenger, P. (eds.) ROMANSY 22 – Robot Design, Dynamics and Control. CICMS, vol. 584, pp. 481–490. Springer, Cham (2019). https://doi.org/10.1007/978-3-31978963-7 60 3. Trivedi, D., Rahn, C.D., Kier, W.M., Walker, I.D.: Soft robotics: biological inspiration, state of the art, and future research. Appl. Bionics Biomech. 5(3), 99–117 (2008) 4. Hannan, M., Walker, I.: Analysis and initial experiments for a novel elephant’s trunk robot. In: Proceedings 2000 IEEE/RSJ Int. Conf. on Intelligent Robots and Systems, vol. 1, pp. 330–337 (2000) 5. Guan, Q., Sun, J., Liu, Y., Wereley, N.M., Leng, J.: Novel bending and helical extensile/contractile pneumatic artificial muscles inspired by elephant trunk. Soft Rob. 7(5), 597–614 (2020). PMID: 32130078

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6. Liu, Y., Ge, Z., Yang, S., Walker, I.D., Ju, Z.: Elephant’s trunk robot: an extremely versatile under-actuated continuum robot driven by a single motor. J. Mech. Robot. 11(07), 051008 (2019) 7. Laschi, C., Cianchetti, M., Mazzolai, B., Margheri, L., Follador, M., Dario, P.: Soft robot arm inspired by the octopus. Adv. Robot. 26(7), 709–727 (2012) 8. Buckingham, R.: Snake arm robots. Ind. Robot. 29(3), 242–245 (2002) 9. Porez, M., Boyer, F., Ijspeert, A.J.: Improved Lighthill fish swimming model for bio-inspired robots: modeling, computational aspects and experimental comparisons. Int. J. Robot. Res. 33(10), 1322–1341 (2014) 10. Carricato, M., Merlet, J.-P.: Stability analysis of underconstrained cable-driven parallel robots. IEEE Trans. Rob. 29(1), 288–296 (2012) 11. Furet, M., et al.: Estimating motion between avian vertebrae by contact modeling of joint surfaces. Comput. Methods Biomech. Biomed. Eng. 25(2), 123–131 (2022) 12. Muralidharan, V., Wenger, P.: Optimal design and comparative study of two antagonistically actuated tensegrity joints. Mech. Mach. Theory 159, 104249 (2021) 13. Furet, M., Wenger, P.: Kinetostatic analysis and actuation strategy of a planar tensegrity 2–x manipulator. ASME J. Mech. Robot. 11(6), 060904 (2019) 14. Terray, L., et al.: Modularity of the neck in birds (Aves). Evol. Biol. 47(2), 97–110 (2020). https://doi.org/10.1007/s11692-020-09495-w 15. Fasquelle, B., et al.: Identification and control of a 3-X cable-driven manipulator inspired from the bird neck. ASME J. Mech. Robot. 14(1), 1–25 (2021) 16. Gouttefarde, M., Daney, D., Merlet, J.-P.: Interval-analysis-based determination of the wrench-feasible workspace of parallel cable-driven robots. IEEE Trans. Rob. 27(1), 1–13 (2011) 17. Boehler, Q., Charpentier, I., Vedrines, M.S., Renaud, P.: Definition and computation of tensegrity mechanism workspace. J. Mech. Robot. 7(4), 044502 (2015) 18. Chablat, D., Wenger, P., Majou, F., Merlet, J.P.: An interval analysis based study for the design and the comparison of three-degrees-of-freedom parallel kinematic machines. Int. J. Robot. Res. 23(6), 615–624 (2004) 19. Vijaykumar, R., Waldron, K., Tsai, M.: Geometric optimization of serial chain manipulator structures for working volume and dexterity. Int. J. Robot. Res. 5(2), 91–103 (1986)

On the Cable Actuation of End-Effector Degrees of Freedom in Cable-Driven Parallel Robots Jean-Baptiste Izard1(B) and Marc Gouttefarde2 1 Alted, Clapiers, France

[email protected] 2 LIRMM, Univ Montpellier, CNRS, Montpellier, France

Abstract. By routing the cables that support and drive the mobile platform of a Cable-Driven Parallel Robot (CDPR) to the moving parts of an end-effector (EE) installed on said mobile platform, it is possible with combined motions of the winches of the CDPR to drive this EE, such as the opening and closing of a gripper. Compared to CDPRs without internal EE, the cable tension distribution is then modified. In this paper, the static equilibrium of a CDPR platform with an internal EE driven by the supporting cables is analyzed, showing that an additional platform wrench needs to be balanced when force is applied on the EE. Both the magnitude of the wrench on the EE degrees of freedom and the way the cables generate motion at the EE influence this additional platform wrench. Considering a cable configuration and an external wrench applied on the EE, several EE designs are proposed where the EE motion is driven by one or two cables leading to different results in terms of workspace and wrench feasibility. Keywords: Platform internal end-effector motion · static equilibrium · wrench-feasible workspace analysis · available wrench set analysis

1 Introduction Cable-Driven Parallel Robots (CDPRs) and Delta-type parallel robots share one design issue in industrial applications: Creating additional large-amplitude Degrees Of Freedom (DOFs) at the mobile platform leads in both cases to significant complexity. In the case of the Delta robot, the dynamic capabilities rely partly on having a lightweight mobile platform, but integrating an actuator in the platform adds weight. In addition, electric and pneumatic cables can be damaged by the numerous bending cycles and flutters caused by the rapid movements of the platform. In the case of CDPRs, most applications require motion at the end-effector (EE) mounted on the mobile platform: at least one DOF is generally required, frequently two, for instance generating a large rotation together with opening and closing a gripper. Space into the mobile platform and payload capabilities may not be an issue. However, when the workspace dimensions are large, it is complex to realize an electrical and/or pneumatic harness, connecting the platform to the robot base structure, which is reliable, cost-effective and does not hinder the platform movements.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Caro et al. (Eds.): CableCon 2023, MMS 132, pp. 134–145, 2023. https://doi.org/10.1007/978-3-031-32322-5_11

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In Delta robots, the problem has been circumvented by using one or more UPU kinematic chains in parallel to the three Delta kinematic chains. A notable design, proposed for the PAR4 and used in the Quattro robot marketed by Omron [1], consists in adding a fourth R-PAR actuated kinematic chain, identical to the original 3 Delta chains, and using the resulting actuation redundancy to generate a rotation created by means of an articulated mobile platform. This design has limited impact on the dynamics and workspace size of the robot. Other similar concepts have since emerged to add a DOF to the platform of a Delta-like robot, e.g. [2, 3], but none of them provides more than one DOF and all of them are based on rigid-link kinematic chains. Rotational limitations in CDPRs have also led to design specific configurations that are able to generate a large rotation within the mobile platform. In [4], a rotational DOF within the platform is directly actuated by cables to generate rotations of large amplitudes. Another new type of spatial CDPR providing unlimited rotations about an axis is presented in [5]. In this particular design, eight cables are wrapped around a cylindrical mobile platform. In [6], a planar CDPR with infinite platform rotation is introduced. It uses a mechanism embedded into the mobile platform actuated from the CDPR base by a cable loop and does not suffer from parasitic motions. Moreover, full unlimited 3-DOF rotations are obtained in [7, 8] by using an embedded spherical wrist implemented with omni-wheels. On the other hand, 6-DOF CDPRs are often redundantly actuated, typically using eight cables, in order to achieve large workspaces and/or to fully constrain the mobile platform. This actuation redundancy may also be used to drive additional motions inside the mobile platform. There are many ways to route cables inside a CDPR platform using pulleys and eyelets which direct them towards mobile elements to which they are attached [9]. This creates internal mobility within the platform, driven by the cables, but this also has an impact on the cable tensions. It is crucial for the operation of the CDPR that the introduction of this internal mobility does not hinder the static equilibrium of the platform. The first contribution of this paper is the mathematical analysis of a CDPR with n > 6 degrees of freedom and m ≥ n cables in order to determine the impact of introducing internal EE motions to an existing CDPR design, in particular in how it impacts the wrench applied to the platform. Several ways to drive internal EE DOFs constructed on the base of an 8-cable suspended CDPR configuration are then proposed and analyzed to compare their performance in terms of wrench capabilities and workspace. This analysis considers one internal EE DOF but could be generalized to more DOFs in future works. From this perspective of wrench modification, designs suggested in [4, 7, 8] are discussed.

2 Wrench Equations Singularities set aside, a set of n actuators can fully drive up to n DOFs. A CDPR is redundantly actuated when there are more actuators than mobile platform DOFs: at a given platform pose, there exists infinitely many cable tension combinations that can balance a given wrench at the mobile platform. The same mechanism with less actuators than mobile platform DOFs is underactuated: at a given pose, only a limited

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set of wrenches can be balanced by cable tensions. Isostatic mechanisms have as many actuators as DOFs, and one combination of cable tensions corresponds to one platform wrench (and vice-versa). The vector of cable forces τ is mapped into the wrench f through the wrench matrix W with n rows and m columns. The CDPR is fully actuated at the necessary condition that the rank of W is equal to n. With forces and velocities being dually linked, W can also be used to calculate the vector ˙l of cable length time derivatives (cable velocities) in function of the platform twist t. f = Wτ ; ˙l = −W T t

(1)

The column vector wi of W, of dimension n, represents the wrench generated by cable i = 1..m at the level of the mobile platform for a unit cable force. In the case of a CDPR with massless cables, vectors wi have a well-known expression given in Eq. (2), with bi the vector representing the position vector of cable i attachment point to the mobile platform in the platform reference frame, and ui the unitary vector directed along the cable segment from the platform to the base.     ui W = w1 w2 · · · wm ; wi = (2) bi × ui Another description of W is given in Eq. (3) where the row vectors vj , j = 1..n, correspond to the rows of W; T  (3) W = vT1 vT2 · · · vTn Each vj corresponds to the cable velocities that generate a movement on one of the translational or rotational degrees of freedom j of the mobile platform. Let us consider a CDPR with m cables to which we add a supplementary cable m+1. The cable tension vector τ therefore gains an element, the cable tension τm+1 . Moreover, the wrench matrix W gains a column wm+1 representing the wrench generated by cable m + 1 for a unit cable force. All platform DOFs being potentially impacted by this new cable, all components of wm+1 are likely to be non-zero. On the other hand, adding an internal EE motion in the mobile platform leads to adding an element to the wrench f , corresponding to the force or moment fn+1 to be generated on the EE DOF, and adding a row vm+1 to W built from the combination of cable velocities that generate a movement along this new DOF. Let us now consider a CDPR with m cables, n degrees of freedom and k ≤ m − n internal EE DOFs driven by the cables. The wrench f is then composed of the wrench f n applied on the platform and of a set of forces/moments f k , k = 1 . . . m−n on the platform internal EE DOFs. The wrench matrix W can be decomposed in two submatrices: W n×m , the wrench matrix of the CDPR if there were no internal EE motion, as described in Eq. (2), and W k×m , the wrench matrix of the EE DOFs considered individually. These two matrices are considered to have full rank. Equation (1) then becomes:     fn W n×m = τ (4) fk W k×m

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3 Evaluation of Static Equilibrium This section presents an analysis of the Constant-Orientation Wrench-Feasible Workspace (COWFW) and Available Wrench Set (COAWS) of various designs of CDPRs having internal EE DOFs. An example CDPR configuration is crafted taking inspiration from the CoGiRo CDPR configuration [10]. It sits on a footprint measuring 12 × 8 × 6 m, with a box-shaped mobile platform frame measuring 1 × 1.4 × 1.5 m. The platform reference point is at the center of the pattern drawn by the cable fixing points on the lower side of the platform. The empty platform weighs m0 = 150 kg. The coordinates of the center of mass of the empty platform are (0, 0, 0.75) m. The resulting 6D wrench from gravity is denoted f 6 . The cable diameter is 4 mm, weighing 65 g per meter. Winches are designed with a safe working load of 2500 N (security factor of 4). Table 1. Example configuration drawing and fixing point coordinates in m. Cable # Drawing points

Fixing points

1

2

3

4

5

6

7

8

X

−5.5

−6

6

5.5

5.5

6

−6

−5.5

Y

−4

−3.5

−3.5

−4

4

3.5

3.5

4

Z

6

6

6

6

6

6

6

6

X

0.5

−0.5

0.6

−0.5

−0.5

0.5

−0.6

0.5

Y

−0.8

0.7

0.7

−0.7

0.8

−0.7

−0.7

0.7

Z

0

1.5

0

1.5

0

1.5

0

1.5

The purpose of the CDPR is to carry a payload fEE = 500 N that comes in addition to the platform weight. A vertical translational EE DOF is considered: the purpose of the EE is to move the payload up and down. We consider that fEE is applied vertically right below the platform reference point, therefore an additional platform wrench f EE = (0, 0, fEE , 0, 0, 0)T adds up to f 6 when the load is carried. Different designs derived from this configuration will be evaluated in terms of COWFW and COAWS in Sects. 3.3 to 3.6. Details on how the COWFW and COAWS are calculated are given in Sects. 3.1 and 3.2, respectively. 3.1 Wrench-Feasible Workspace Analysis The COWFW is the set of positions of the CDPR mobile platform where a given external wrench or set of wrenches can be balanced by a set of cable tensions within a feasible tension set. The latter is defined as non-negative lower and upper bounds on the cable tensions. The lower bounds are defined for each cable as the positive cable tension required to have a length to sag ratio of 5 [11]. The upper bound is defined by the safe working load of the winch. The method detailed in [10] is used to test if a given pose belongs to the COWFW. Validity of a pose is also conditioned to cables not colliding with the platform. With the CoGiRo CDPR cable configuration, no collision occurs between the cables for any pose at reference orientation within the footprint.

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The search of the COWFW boundary is performed by testing a set of directions covering the space around the reference pose which is to be tested valid beforehand. Each direction is tested with a bisection algorithm and outputs a boundary position. All boundary positions are arranged to shape a boundary surface. 3.2 Available Wrench Set Analysis The COAWS can be defined as the set of wrenches that can be balanced by a cable tension set, with values within a feasible tension set, for all platform positions within a prescribed set, the platform orientation being fixed at the reference orientation. The tension feasible set is defined as in Sect. 3.1. At each position, the COAWS is calculated using the hyperplane shifting method detailed in [12]. The COAWS for the given set of positions is the intersection of the COAWS of all individual positions. In the following analysis, the prescribed set of positions is a meshing of the constant orientation workspace representing 2/3 of the horizontal footprint dimensions and 1/2 of the vertical footprint dimension. This workspace corresponds to the prescribed one in the optimization of the CoGiRo CDPR configuration [10], with the exclusion of rotations. Once the COAWS is built, the intersection of the COAWS with a subspace of the wrench space, where some wrench components are fixed, can be computed. In the following, all platform wrench components are set at 0 except for the vertical force. The wrench coordinate fZ along the vertical force direction is thereafter transformed into admissible additional mass to the platform madd through the following equation: madd = −

fZ − m0 g0

(5)

where g0 is the acceleration of gravity equal to 9.81 m/s, and madd indicates the additional weight that the platform can be carry on top of platform empty weight. In cases where the EE is driven by a combination of cables, another variable in the wrench is the force applied on the EE embedded in the platform. In the 2D space with coordinates in madd and fEE , the COAWS is a convex polygon. Moreover, we consider that the force fEE drives the added mass madd with a speed reduction ratio of 1/η. A given value of madd carried by fEE requires η to be equal to madd g0 /fEE . 3.3 EE DOF Driven Independently In a first design, the EE is driven independently from the cables. This is the reference design to which the designs discussed in the next sections will be compared. The effects of physically installing the EE actuators on board of the platform (transmission of energy and signal to the platform) are ignored. Figure 1 shows the corresponding COWFW, which size is evaluated at 9.6 × 6.3 × 4.0 m. As for the COAWS, the only variable left is madd , valid from 0 to 261 kg.

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Fig. 1. COWFW of the example configuration with a translational EE DOF driven separately against fEE = 500 N. Coordinates are shown in m.

3.4 Directly Wiring a Cable to the EE DOF A direct solution to drive an internal EE DOF with a cable of a CDPR would be to draw one cable on the platform through an eyelet and then route this cable into the platform to actuate the internal EE DOF. Let us consider the example configuration defined at the beginning of Sect. 3. Instead of being fixed to the platform, cable 1 is run on an eyelet and routed to the payload through a block and pulley mechanism with η cable leads routed on the block. This corresponds to a reduction ratio of 1/η. With w1 ..w8 the columns of the wrench matrix of the configuration at the considered position without the driving of the EE DOF, Eq. (4) writes:       w1 w2 ...w8 ui f 6 + f EE = τ ; wi = , i = 1...8 (6) fEE η 01×7 bi × ui This solves into the following equation: τ1 =

fEE ; f 6 + f EE − w1 fEE /η = [w2 ..w8 ]τ 2..8 η

(7)

This equation corresponds to setting τ1 equal to fEE and thereafter solving the static equilibrium of the CDPR using only cables 2 to 8, with the gravity wrench f 6 being incremented by −w1 fEE /η, which is the effect of the cable 1 with tension τ1 . Considering all positions in the prescribed set leads to an empty COAWS, i.e. it is not possible to carry a payload of any magnitude this way in all positions (Fig. 2). Four subsets of positions are then determined, which lead to different COAWS that do not intersect (Fig. 3).

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Fig. 2. COWFW of the example configuration where the translational EE is driven by cable 1 (blue cable) against fEE = 500 N with = η1. Coordinates are shown in m.

Fig. 3. AWS of the design featuring cable 1 driving the translational EE DOF. The AWS is shown on the left, and corresponding position sets are shown with the same colors on the right, where coordinates are shown in m.

3.5 Doubling a Cable to Drive a Drum The designs proposed in [7, 9] are meant to modify a CDPR cable configuration by selecting a cable which is doubled. Both cables from this doubling are driven by individual winches. Inside the platform, these cables are routed to a drum where they are wound in opposite directions. Running the winches of these two cables in opposite directions therefore actuate the internal EE DOF. Let us select cable 1 of a configuration and double it. The two corresponding cables are numbered 0 and 1, increasing the total number of cables to m + 1. The fixing and drawing points of cables 0 and 1 are built from the former positions of the fixing and drawing point of cable 1: both are moved by a position vector denoted d 01 in opposite directions in order to keep the unit vectors u0 and u1 directing cables 0 and 1 equal.   ⎡ ⎤   u1 u1 w2 ..wm ⎦ f 6 + f EE = ⎣ (b1 + d 01 ) × u1 τ (8) (b1 − d 01 ) × u1 fEE η −η 01×m−1

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Equation (8) shows the wrench matrix for this configuration. Changing coordinates for τ , replacing τ0 and τ1 by τ1+ = τ0 + τ1 and τ1− = τ0 − τ1 respectively, leads to: 

f 6 + f EE fEE



⎡

03 = ⎣ d 01 × u1 2η



⎤ w1 w2 ..wm ⎦

τ1− τ1+ τ2 ..τm

T

(9)

0 01×m−1

Which solves into the following set of equations:   03 − τ1 = fEE /2η; f 6 +f EE − fEE /2η = [w1 ..wm ][ τ1+ τ 2.m ]T d 01 × u1   

(10)

f6+

This set of equations corresponds to the static equilibrium of the CDPR without the internal EE, where the tension in former cable 1 is replaced by the sum τ1+ of the tensions in cables 0 and 1, and to which an extra wrench noted f + 6 , with only torque values, adds to gravity wrench f 6 + f EE . The magnitude of f + 6 is proportional to fEE and to the norm of the cross product d 01 × u1 .

Fig. 4. COWFW with the translational EE driven by a combination of cable 0 and cable 1 obtained by doubling original cable 1 against fEE = 500 N with η = 1. Coordinates are in m.

Fig. 5. COAWS of the example configuration with cable 1 being doubled to drive a translational EE DOF. Limiting positions are shown on the right as blue crosses. Black dots show non-limiting feasible positions which are part of the prescribed set of positions.

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In order to evaluate this design, the example configuration is modified by doubling

T cable 1 with d01 = 0 0 0.1m . fEE is once again set equal to 500 N and η to 1. The COWFW and COAWS are shown in Fig. 4 and Fig. 5. In the case of the robot presented in [7, 8] with 6 degrees of freedom and 8 cables, cables are assembled by pairs which are routed towards 4 actuated drums installed in the platform. Middle points of fixing point pairs shape a square. fEE is replaced by a  T vector of torques fEEi , i = 1..4, actuated on the drums, with η matching the radius of the drums. Cable tensions are denoted τij , i = 1...4 and j = 1...2, τi+ = τi1 + τi2 and τi− = τi1 − τi2 , Eq. (4) becomes the following equation:  ⎡ ⎤    + 03  f 6 + f EE w i ⎣ d i × ui ⎦ τi−  T = (11) τi fEE1 .. fEE4 i=1..4 2η 01×m−1 which solves into Eq. (12), corresponding to an underactuated CDPR with 4 cables attached to the platform at the vertices of a square, with the addition of a perturbation wrench f + 6 with non-zero values on torque directions proportional to values fEEi , which is applied in addition to f 6 : ⎧ τi1 − τi2 = 2fEEi /η, i = 1..4 ⎪ ⎪ ⎪ ⎪   03  ⎨ fEEi /η = [w1 ..w4 ] τi+ i=1..4 2 f 6 +f EE − (12) d i × ui ⎪ i=1..4 ⎪ ⎪    ⎪ ⎩ + f6

With the CDPR being underactuated, this perturbating wrench leads to a perturbation of the position. This would explain why, in [8], the platform shows erratic movements when EE motion is performed while the platform is suspended. Friction in the drums require torque from the EE DOFs, which turn into additional wrench on the platform. As the CDPR static equilibrium is equivalent to that an underactuated cable configuration, this wrench generates the shift in position and orientation visible in the video in [8]. 3.6 Running a Pair of Cables Towards a Drum Carrying the Load The design proposed in [4] features a platform built with a vertical central rod to which 6 cables are attached in 2 clusters of 3 cables drawn upwards and downwards, which makes for an overconstrained CDPR with 5 DOF, excluding the rotation of the central rod. The central rod is then fixed by a drum coaxial to it to which two cables are attached, drawn from the base in a mostly horizontal direction, in order to pull on the drum in antagonistic directions. Similar constructions are also proposed in [9]. We therefore study in this section the design where two cables from the initial configuration are taken through an eyelet to an EE with a reduction ratio 1/η. The number of cables is unchanged. The wrench matrix is augmented with a row vector with non-zero values for the cables driving the EE and 0 for the other cables. We choose to + − = τ1 + τ2 and τ12 = τ1 − τ2 , Eq. (4) leads to: drive the EE with cables 1 and 2. With τ12    

T w1 − w2 w1 + w2 w3 ..wm − + f 6 + f EE = (13) τ12 τ12 τ3 ..τm fEE 2η 0 01×m−2

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which solves into Eq. (14), which corresponds to the static equilibrium of a modification of the original configuration where a cable runs from drawing point 1 to fixing point 1, then through eyelets and pulleys to fixing point 2, and then to drawing point 2; and to which an additional wrench f + 6 is applied, proportional to fEE : ⎧ − EE τ12 = f2η ⎪ ⎪ ⎪ ⎨

T + fEE f 6 +f EE −(w1 −w2 ) = [(w1 +w2 ) w1 ..wm ] τ12 τ3 ..τm (14) 2η ⎪ ⎪    ⎪ ⎩ f6+

The corresponding COWFW (fEE = 500 N, η = 1) and COAWS are drawn in Fig. 6 and Fig. 7.

Fig. 6. COWFW of the example configuration where the translational EE is driven by a combination of cable 1 (blue) and cable 2 (red) against fEE of 500 N with η = 1. Coordinates are in m.

Fig. 7. COAWSs of the configuration with cable 1 and 2 driving antagonistically a translational EE DOF. Limiting positions are shown on the right as blue crosses. Black dots show non-limiting feasible positions which are part of the prescribed set of positions.

3.7 Discussion – Comparison of Designs A series of cable-driven configurations of an EE embedded in a CDPR mobile platform (internal EE), built from a CDPR configuration derived from the CoGiRo configuration

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detailed in Table 1, have been studied with respect to their workspace with a payload set at fEE = 500 N and payload capability within a designated workspace of 8 m × 5.3 m × 3 m. These results are compiled in Table 2. Table 2. Comparison of designs EE drive

Section

Workspace size with Payload capability fEE = 500 N (m)

Comments

Independent

3.3

9.6 × 6.3 × 4.0

261 kg

Effects of umbilical are neglected

One cable, directly acting

3.4

(non-prismatic)

(217 to 261 kg)

Does not comply with the designated workspace

One doubled cable, 3.5 both acting antagonistically

Z = 0: 6.1 × .9 (offset on XY) Z = 4: 9.6 × 6.2

261 kg, with |fEE | < Requires an 388 N additional winch with respect to original configuration

Two cables, acting antagonistically

(non-prismatic – dramatic effect at low Z)

245 kg, with 39 N < Important impact fEE nd cables, which are always under tension. nd is the number of EE DoFs, and the degree of redundancy [11] is defined as μ = n − nd . Cables are coiled

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Fig. 1. Geometric model of a cable transmission.

and uncoiled by servo-controlled winches, characterized by a nominally constant transmission ratio. Their length varies proportionally to the actuator rotations if elasticity and hysteresis are neglected [6]. In this work, cables are considered ideal unilateral constraints, thus no mass and elasticity effects are taken into account. According to Fig. 1, Oxyz is an inertial frame, whereas P x y  z  is a mobile frame attached to the EE center of mass. The EE pose is described by the position vector p of P , and a rotation matrix R = R(), which describes the orientation of the mobile platform with respect to the base of the mechanism.  is a minimal set of orientation parameters, i.e., T  three Euler’s angles. The EE pose is defined as ζ = pT T . Cables are attached to the platform, and the fixed base at distal and proximal anchor points Ai and Bi , respectively, for i = 1, ..., n. ai and ai are the position vectors of Ai with respect to O and P in the inertial frame (Fig. 1). The constant position vector of Ai in the mobile frame is denoted by ai,P . bi denotes the position of B i with respect to O. The latter is a constant vector if the cable exit point is an eyelet; otherwise, it depends on the cable transmission model [5], and, ultimately, the EE pose. The i-th cable is modeled as the line segment between points Ai and Bi and, thus, the i-th cable vector can be expressed as [14]: ρi = ai − bi = p + R ai,P − bi

(1)

Considering li as the i-th cable length, the constraint imposed by the i-th cable, and the unit vector of the i-th cable, pointing from the base towards the platform, are: ρ ti = i (2) ρTi ρi − li2 = 0, li The static equilibrium of the mobile platform can be formulated as follows:   ˜i − JT τ − W = 0, (3) Ji = ti T −tTi a

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where τ ∈ Rn is the array of cable tensions, W ∈ Rnd is the external wrench acting on the platform, JT ∈ Rnd ×n , usually referred to as the structure matrix, is the transpose of the inverse kinematics Jacobian matrix, and Ji is the i-th row of J. The symbol ∼ over a vector denotes its skew-symmetric representation. For OCDPRs (μ > 0), the solution of Eq. 3 is underdetermined if all bodies are considered rigid, and infinitely many solutions exist for a given wrench and EE pose. The solution to this problem consists in computing a tension distribution according to a given criterion [12]. If a hybrid joint-space controller is used, tension errors in force-controlled cables influence the tension in length-controlled cables, and, ultimately, the FD and the robot performances. Such influence can be analyzed by computing the force-distribution sensitivity to cable-tension error. Assuming that the tension of the last μ cables, denoted by τ c , is to be controlled, the array τ of cable tensions and the structure matrix can be partitioned as:     τ (4) JT = Jd Jc τ  d , τc where Jd ∈ Rnd ×nd and Jc ∈ Rnd ×μ , and τ d ∈ Rnd denotes the tension of the length-controlled cables. The force-distribution sensitivity σ is defined as the maximum tension variation in the position-controlled cables generated by a unit tension variation (or error) of the force-controlled cables, namely [10]: σ

max

τ c ∞ =1

Δτ ∞ =  − J−1 d Jc ∞

(5)

The FD sensitivity index can be computed for every set j of force-controlled cables, thus obtaining C values, σj , for j = 1, . . . , C. C is the maximum number of ways of selecting μ cables out of n, without repetition. After computing all possible C values of the FD sensitivity, the minimum value can be identified as:   n! n  (6) j = 1, . . . , C, C= = σ = min σj , μ (n − μ)!μ! The index σ  points out the cable set that, if force-controlled, propagates the least tension control errors in the other cables. That cable set is denoted by j  .

3

HRPCable Workspace Characterization

In this section, the constant-orientation wrench-feasible WS of an 8-cable OCDPR prototype at LIRMM is characterized with respect to its forcedistribution sensitivity to cable-tension error. The prototype, known as HRPCable, was used as an experimental set-up during the EU project Hephaestus [16]. It can be installed in a suspended or overconstrained configuration. For this work, the robot is configured as an OCDPR (Fig. 2).

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Fig. 2. HRPCable prototype at LIRMM (CNRS - University of Montpellier). Table 1. Geometrical properties of the HPRCable OCDPR. i

1 ⎤ ⎡ −4.1948 ⎥ ⎢ ⎥ bi [m] ⎢ ⎣−1.7756⎦ 3.0136 ⎤ ⎡ −0.2492 ⎥ ⎢  ⎢ ai [m] ⎣ 0.2021 ⎥ ⎦ 0.1291

2 ⎡

⎤ −3.8532 ⎥ ⎢ ⎢−1.3288⎥ ⎦ ⎣ 0.2429 ⎤ ⎡ −0.2483 ⎥ ⎢ ⎢ 0.2021 ⎥ ⎦ ⎣ −0.2112

3 ⎤ ⎡ −3.7834 ⎥ ⎢ ⎢ 1.3632 ⎥ ⎦ ⎣ 0.2441 ⎤ ⎡ −0.2488 ⎥ ⎢ ⎢−0.2021⎥ ⎦ ⎣ −0.2876

4 ⎤ ⎡ −4.0968 ⎥ ⎢ ⎢ 1.8087 ⎥ ⎦ ⎣ 3.0151 ⎤ ⎡ −0.1909 ⎥ ⎢ ⎢−0.2743⎥ ⎦ ⎣ 0.2230

5 ⎤ ⎡ 4.1016 ⎥ ⎢ ⎢1.7823⎥ ⎦ ⎣ 3.0206 ⎤ ⎡ 0.2483 ⎥ ⎢ ⎢−0.2021⎥ ⎦ ⎣ 0.2112

6 ⎡

⎤ 3.7872 ⎥ ⎢ ⎢1.3318⎥ ⎦ ⎣ 0.2484 ⎤ ⎡ 0.1887 ⎥ ⎢ ⎢−0.2749⎥ ⎦ ⎣ −0.2987

7 ⎡

3.7722

8 ⎤ ⎡

⎥ ⎢ ⎢−1.3400⎥ ⎦ ⎣ 0.2488 ⎤ ⎡ 0.2503 ⎥ ⎢ ⎢ 0.2019 ⎥ ⎦ ⎣ −0.2090

⎤ 4.0825 ⎥ ⎢ ⎢−1.7894⎥ ⎦ ⎣ 3.0190 ⎤ ⎡ 0.2482 ⎥ ⎢ ⎢0.2021⎥ ⎦ ⎣ 0.1291

In general, an OCDPR is in a wrench-feasible pose ζ if at least one FD (τ ∈ Rn ) exists, for a given wrench W ∈ Rnd , such that the EE equilibrium in Eq. (3) is satisfied with bounded tensions: ∃τ :

τmin  τ  τmax ,

−JT τ − W = 0

(7)

where τmin and τmax are the tension limits, and the symbol  denotes elementwise inequality between a scalar and a vector quantity. The inertial frame Oxyz is located in the center of the base, and the moving frame P x y  z  is at the center of the mobile platform, coinciding with its center of mass. The only external load applied to the robot EE is gravity, and its mass is m = 23 kg. The geometrical properties of the robot are summarised in Table 1, and the tension limits are set to τmin = 30 N, and τmax = 2000 N. The robot has two degrees of redundancy (μ = 2), and thus 2 cables have to be force-controlled, whereas the others may be length-controlled when a hybrid joint-space strategy is adopted for its control. According to Eq.(6), the number of possible cable combinations is C = 28. The minimum FD sensitivity σ  in the constant-orientation wrench-feasible WS (with R = I3 , where I3 is the identity matrix of order 3) is shown in Fig. 3,

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Fig. 3. Variation of the minimum FD sensitivity σ  throughout the wrench-feasible WS of HRPCable OCDPR.

Fig. 4. Wrench-feasible WS of HRPCable characterized by 28 distinct constant j  zones, where j  identifies the cable pair to be force-controlled to have the lowest error in the tension distribution in each zone. Each color is associated with a certain j  pair, and a detailed description of coloring is omitted since it would not add descriptive value.

whereas Fig. 4 highlights the volumes where the optimal cable set j  remains constant. Each color is associated with a certain j  pair. The 28 distinct constantj  zones cannot be easily pointed out, and their boundaries are not linear and not easily identifiable. The analysis of the FD-sensitivity variation for different choices of forcecontrolled cable pairs is conducted to show how a poor choice of force-controlled cable pair can have negative consequences on robot control. To quantitatively assess how specific cable pairs can be an issue when force-controlled, two thresholds for the sensitivity values are selected: σ = 2N is a practically acceptable value of the minimum sensitivity σ  [10], whereas σ = 10N is a value where control performances are expected to be appreciably degraded. As an example, if we choose cable pair 2-4 as the force-controlled one (Fig. 5), only in 4% of

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(a) Variation of the FD sensitivity when the cable pair 2-4 is force-controlled.

155

(b) σ values percentage.

Fig. 5. Variation of the FD sensitivity σ for the cable pair 2-4 of HRPCable robot.

(a) Variation of the FD sensitivity when the cable pair 7-8 is force-controlled.

(b) σ values percentage.

Fig. 6. Variation of the FD sensitivity σ for the cable pair 7-8 of HRPCable robot, which gives the best results above all the possible combinations.

the WS , the sensitivity value σ is lower or equal to 2N, 30% of the workspace is characterized by 2N < σ < 10N, and 66% by σ ≥ 10N (Fig. 5b), with peak values higher than 1000N (Fig. 5a)1 . On the contrary, considering the pair 7-8 (Fig. 6), 68% of the WS poses have sensitivity values lower than 2N, while the others are between 2N and 4.41N, which is the absolute maximum value of σ in this case (Fig. 6a). This means that by choosing the pair 7-8 to be force-controlled, the tension error throughout the whole WS would be, at most, around 4.4 times the error in the force-controlled cables.

1

Void zones in the top right corner of Fig. 5a are due to the contour plot nature of the illustration: the FD sensitivity change negligibly in said volume with respect to the rest of the workspace, and contour lines are not depicted.

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A Practical Cable-Selection Strategy

As highlighted in Sect. 3, the constant-j  map (see Fig. 4) derived from the FDsensitivity analysis is not straightforward to interpret for control purposes. In fact, the constant-j  zones are numerous (C = 28) and they have a highly irregular shape: switching between force- and length-controlled cables while crossing the borders of constant-j  zones could be very frequent, impractical, and possibly leading to various kinds of control problems, such as control signal discontinuities, vibrations, or unjustifiably large control efforts. A solution to these problems can be found, such as smoothing the control signal with additional low-pass filters to be activated during switching. We propose an alternative that does not require the addition of such control modules, which ultimately may limit the performances due to the induced control lag. A heuristic method to keep the tension error near the theoretical smallest possible value during the EE motion throughout the WS is introduced. A single cable set can be chosen to be force-controlled in the whole WS . The analysis of several cases showed that to select this cable set, one should evaluate how many points over the totality of the WS have an acceptable value of σ. We define a value of σ as acceptable, if it is less than or equal to the maximum value of the  ) in the WS . σ  is by definition, the minimum FD sensitivity σ  (called σmax minimum value of σ for every j-th cable set (Eq.(6). As an example, for the  is equal to 2N. HRPCable σmax Considering this criterion, the force-controlled cable set can be chosen by comparing the percentage of WS configurations whose sensitivity is within the  ) for each cable combination. The best candidate to acceptable limit (σ ≤ σmax be force-controlled would be the cable set having the highest percentage of WS configurations within the acceptable sensitivity value. For example, comparing all the cable combinations for the HRPCable robot (the complete analysis is not here reported for the sake of brevity), the pair 7-8 is chosen, as the percentage of  = 2N is the highest (68%). If this FD sensitivity values lower or equal to σmax rationale is followed to keep the tension error as low as possible, during the EE motion, one should take into account that in certain WS zones the cable-tension error could be excessively amplified where σ has its peaks (e.g., the bottom right corner of the WS , in Fig. 6a), which could limit the robot WS . In other words, the EE should work far from the peaks to avoid tension-error amplification, unless these peaks have a reasonably small value. Furthermore, for a given force-controlled cable set, the percentage of WS configurations whose sensitivity is within the acceptable limit could be enlarged, optimizing the robot geometry. In fact, from the analysis of the 8-cable cases, it was noticed that a lack of symmetry in the robot structure makes some cable sets more suitable to be force-controlled with respect to others. In a symmetric case, such as the 8-cable IPAnema 3 OCDPR reported in [10], there is no pair significantly better than the others. On the contrary, in the HRPCable case, whose cable attachment points are not symmetric to avoid cable interference (see Table 1 and Fig. 2b), the results obtained with the pair 7-8 are clearly better than the results obtained with other pairs, and could be further optimized, for

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Fig. 7. HRPCable prototype.

example by allowing the EE to rotate as demonstrated in [8]. Pair 7-8 not only has the highest percentage of configurations with an acceptable σ among all cable combinations, but it also shows the lowest maximum value of σ in the entire WS (4.41N). Taking this observation into account, it should be possible to optimize the cable anchor point positions in order to enlarge the volume with σ < 2N for this cable set, thus completely removing the need for switching tension- and length-controlled cables2 . In general, the analysis reported in this Section highlights that some robot geometries are more effective at being hybrid-controlled than others, and considering this control option at design phase could be more effective than an a-posteriori analysis.

5

Experimental Validation

The practical hybrid-input control strategy proposed in this paper is tested on the HRPCable prototype (Fig. 7). Geometrical and inertial parameters of the robot are summarized in Sect. 3, and gravity is the only external wrench acting on the EE . The trajectory planning, the computation of the tension distribution, and the hybrid-input control strategy run on a Beckhoff real-time IPC at 0.5 kHz rate. At the servo-drive level, for ease of implementation, position set-points are always assigned. The þk control update on the þj motor, pj,k , is obtained as follows: ∀k, for j = 1, . . . , 8 (8) pj,k = pj,k−1 + Δpj,k , where Δpj,k is computed differently, according to the cable being length- of tension-controlled. In the former case, Δpj,k is calculated through inverse kinematics, whereas in the latter case Δpj,k = vj,k Δt, with Δt = 2 ms and vj,k is 2

This option will be further analyzed in future studies, but preliminary simulations showed its theoretical feasibility.

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Fig. 8. Experimental tension distribution errors.

Fig. 9. FD error 2-norm when cable pairs 7-8, 2-4, and 2-7 are tension controlled.

assigned, as in Eq. 1 of [15], as the nominal cable velocity to follow the prescribed path. While the EE orientation is kept such that R = I3×3 , the same trajectory of the EE reference point is tracked by first controlling the tensions of the cable pair 7-8, which is the best overall, and then by controlling the tensions of the pair 2-4, the worst overall. In both cases, the EE is moved from the home position, pH = [0, 0, 0]T m, to the final position pF = [−0.2, 0.2, −0.05]T m in 25 s. Then, the robot is halted for 10 s, and finally goes back to the home position in 25 s. Please note that the HRPCable prototype is intended for quasi-static operations, and faster trajectories could not be tested. The tension distribution is computed by applying a minimal 2-norm algorithm with τmin = 30N and τmax = 500N, that is [3]: min τ 2 τ

with

τmin  τ  τmax , −JT τ − W = 0

(9)

Tension error norms, denoted by ei  = τi − τi , with i = 1, . . . , 8, and computed by subtracting the results of Eq. (9) τi to the experimentally obtained val-

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ues τi , are shown in Fig. 8; the first trajectory from pH to pF is executed from t = 25 s to t = 50 s, while the second one from pF to pH is executed from t = 60 s to t = 85 s. Errors are generally fairly large due to the robot’s need for maintenance and re-calibration, and the limited time available for experiments was not sufficient for statistical analysis of more trajectories. Nonetheless, Fig. 8a shows that tension-controlling cable pair 7-8 performs generally better, with lower maximum tension errors in several cables, with respect to tension-controlling cable pair 2-4 (Fig. 8b). This result is further confirmed in Fig. 9 by looking at the 2-norm of the FD distribution errors, denoted by exy = τ xy − τ xy  with xy indicating the experiment where the x and y cables were tension-controlled, τ is the result of Eq. (9), and τ  is the array of the experimental tensions: tension-controlling cable pair 7-8 always results in a smaller 2-norm error than tension-controlling cable pair 2-4. Figure 9 shows an additional result, which was expected in this study: there may be a better cable pair to be locally tension controlled (cable pair 2-7 for the selected trajectory), and the larger error in cable pair 7-8 is the trade-off for not needing to switch the tension-controlled cables throughout the workspace.

6

Conclusions and Outlook

This paper proposed a practical approach for the selection of tension-controlled cables when a hybrid joint-space control technique is used in overconstrained cable-driven parallel robots. The proposed approach is based on the computation of the force-distribution sensitivity to cable-tension errors (FD sensitivity). In general, selecting tension-controlled cable pairs through a FD sensitivity analysis would require switching the selected cables throughout the workspace, which may be inconvenient. Thus, in this paper, a “best overall” selection strategy that would allow to tension-control always the same cable pair was introduced. This strategy is based on evaluating which tension-controlled cable pair has the largest workspace volume with a force distribution sensitivity below an acceptable threshold. The method was successfully validated through experiments, also showing that the best cable pair overall may not be locally better than other choices. Future studies will focus on optimizing a manipulator EE geometry so that the best cable pair overall is also locally better than any other cable, a result that was hinted at by the analysis proposed in this paper. Acknowledgments. The authors greatly acknowledge support of the European Union through FEDER grant n◦ 49793. This work has also been partially supported by ROBOTEX 2.0 (Grants ROBOTEX ANR-10-EQPX-44-01 and TIRREX ANR-21ESRE-0015) funded by the French program Investissements d’avenir.

References 1. Bouchard, S., Gosselin, C.: A simple control strategy for overconstrained parallel cable mechanisms. In: Proceedings of the 20th Canadian Congress of Applied Mechanics (CANCAM 2005), Montreal, Quebec, Canada (2005)

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2. Bruckmann, T., Mikelsons, L., Hiller, M., Schramm, D.: A new force calculation algorithm for tendon-based parallel manipulators. In: 2007 IEEE/ASME International Conference on Advanced Intelligent Mechatronics, Zurich, Switzerland, pp. 1–6 (2007) 3. Gouttefarde, M., Lamaury, J., Reichert, C., Bruckmann, T.: A versatile tension distribution algorithm for n-DOF parallel robots driven by n + 2 cables. IEEE Trans. Rob. 31(6), 1444–1457 (2015) 4. Gouttefarde, M., Collard, J.F., Riehl, N., Baradat, C.: Geometry selection of a redundantly actuated cable-suspended parallel robot. IEEE Trans. Rob. 31(2), 501–510 (2015) 5. Id´ a, E., Merlet, J.-P., Carricato, M.: Automatic self-calibration of suspended underactuated cable-driven parallel robot using incremental measurements. In: CableCon 2019. MMS, vol. 74, pp. 333–344. Springer, Cham (2019). https://doi.org/10.1007/ 978-3-030-20751-9 28 6. Id` a, E., Mattioni, V.: Cable-driven parallel robot actuators: state of the art and novel servo-winch concept. Actuators 11(10) (2022) 7. Kraus, W., Miermeister, P., Schmidt, V., Pott, A.: Hybrid position-force control of a cable-driven parallel robot with experimental evaluation. Mech. Sci. 6(2), 119– 125 (2015) 8. Mattioni, V., Id` a, E., Carricato, M.: Force-distribution sensitivity to cable-tension errors: a preliminary investigation. In: Gouttefarde, M., Bruckmann, T., Pott, A. (eds.) CableCon 2021. MMS, vol. 104, pp. 129–141. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-75789-2 11 9. Mattioni, V., Ida’, E., Carricato, M.: Design of a planar cable-driven parallel robot for non-contact tasks. Appl. Sci. 11(20), 9491 (2021) 10. Mattioni, V., Id` a, E., Carricato, M.: Force-distribution sensitivity to cable-tension errors in overconstrained cable-driven parallel robots. Mech. Mach. Theory 175, 104940 (2022) 11. Ming, A., Higuchi, T.: Study on multiple degree-of-freedom positioning mechanism using wires. I: concept, design and control. Int. J. Japan Soc. Precis. Eng. 28(2), 131–138 (1994) 12. M¨ uller, K., Reichert, C., Bruckmann, T.: Analysis of a real-time capable cable force computation method. In: Pott, A., Bruckmann, T. (eds.) Cable-Driven Parallel Robots. MMS, vol. 32, pp. 227–238. Springer, Cham (2015). https://doi.org/10. 1007/978-3-319-09489-2 16 13. Oh, S.R., Agrawal, S.K.: Cable suspended planar robots with redundant cables: controllers with positive tensions. IEEE Trans. Rob. 21(3), 457–465 (2005) 14. Pott, A.: Cable-Driven Parallel Robots. STAR, vol. 120. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-76138-1 15. Santos, J.C., Gouttefarde, M.: A simple and efficient non-model based cable tension control. In: Gouttefarde, M., Bruckmann, T., Pott, A. (eds.) CableCon 2021. MMS, vol. 104, pp. 297–308. Springer, Cham (2021). https://doi.org/10.1007/978-3-03075789-2 24 16. Taghavi, M., Iturralde, K., Bock, T.: Cable-driven parallel robot for curtain wall modules automatic installation. In: Teizer, J. (ed.) Proceedings of the 35th International Symposium on Automation and Robotics in Construction (ISARC), Taipei, pp. 396–403 (2018)

Brief Review of Reinforcement Learning Control for Cable-Driven Parallel Robots Pegah Nomanfar(B)

and Leila Notash

Queen’s University, Kingston, ON K7L 3N6, Canada {19pn9,leila.notash}@queensu.ca

Abstract. Reinforcement Learning (RL) is a branch of machine learning applied to many applications, such as mechatronics and robotics. RL allows more challenges to be resolved in robotics due to the high capacity for problem definition and active environmental interaction. Another notable property of RL is the capacity to include a variety of mathematical models, which the best-fitted model can be employed for solving any challenge. This characteristic of RL is highly beneficial for cable-driven parallel robots (CDPR) with applications in industry, construction and rehabilitation, in which the robot’s task space contains uncertainties. Recently many researchers have employed RL for various control tasks in CDPRs. Since RL has been applied to CDPRs’ different control challenges and satisfying results have been achieved, the future of RL in CDPR applications seems promising. In this paper, the applications of RL in CDPR control will be studied, and the most reported RL methods for these robots in the literature will be discussed. The promising future research subject will be described as well. Keywords: Cable-driven parallel robots · Reinforcement learning Machine learning · Reinforcement learning control · Review paper

1

·

Introduction

Parallel mechanisms that utilize cables as actuators are referred to as cabledriven parallel robots (CDPR). These robots are renowned for their simple structure, high speed and acceleration, maneuverability, and large workspace [1–3]. These characteristics of the CDPRs make them candidates for many industrial applications, such as material handling and assembly processes and rehabilitation devices, refer to Fig. 1. Also, large-scale flexible wings that are attached with cables to satellites can be modelled as a cable-driven parallel mechanism in order to be controlled for vibration suspension with small force [4,5]. Although the benefits of CDPR are notable in different aspects, there are still challenges in controlling them. One major challenge in CDPR relates to cable tension since the cables can only be pulled but not pushed, which means the tension in cables should always be non-negative. One solution to this problem is to ensure that the c The Author(s), under exclusive license to Springer Nature Switzerland AG 2023  S. Caro et al. (Eds.): CableCon 2023, MMS 132, pp. 161–172, 2023. https://doi.org/10.1007/978-3-031-32322-5_13

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number of cables is larger than their degrees of freedom (DOF). Other challenges can exist in kinematic and dynamic modelling, vibration control and stability analysis of CDPRs. Especially in pose control, different methods can be found in the literature, such as experimental proportional-integral-derivative (PID) controller and robust PID for rehabilitation CDPR, as well as sliding mode control to compensate for the uncertainties in parameters, disturbing effects or changes in system parameters [2]. Most of the proposed controllers for CDPRs are classical PID and model-based state-space. A precise classical controller design approach is highly challenging due to environmental uncertainties and dynamic robot complexity. One reason is that CDPR robots need to employ the tension distribution algorithm for the model-based controller to ensure the positive tension distribution and uniqueness of the solution, which may be preferred to be performed offline based on the control parameters for highly accurate tasks. All these challenges in the control realm have required an accurate dynamic model of the CDPR with limited complexity and modelled uncertainties in the robot workspace. Therefore, achieving a dynamic model of the robot that can address all the uncertainties, disturbances, and complexities is time-consuming and challenging to design.

Fig. 1. Sample applications of CDPRs: (a) Large-scale flexible wing in satellites [5], (b) construction [6], (c) rehabilitation [7].

Some of the noted challenges in the previous paragraph have been addressed through machine learning (ML), which is using mathematical models of the data in computer systems to assist the robot learning process. This process occurs by employing an artificial neural network (ANN). ANNs consist of input and output and a group of neurons between input and output that apply a specific function to the input and send them to output. This series of neurons that connect input and output is called layers. A deep neural network (DNN) is made of multiple layers. Each layer receives input from the previous layer, performs an operation on the input, and passes the output to the next layer. Nowadays, many researchers find ML a desirable solution for many challenges in robotics and mechatronics. In ML, robot learning is generally improved by iteration in simulation or repeating tasks without being explicitly programmed for different tasks. Many studies have proved that machine learning and control design can cooperate for better and more precise results in an uncertain environment without any demand

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for an accurate dynamic model of the robot or estimation of uncertainties. Reinforcement learning (RL) is a branch of ML that allows more challenges to be resolved in robotics since it facilitates for active interactions with the environment. RL has been developed in many research areas recently, it has been used for solving some of the well-known challenges in robotics in the past decade. However, the RL control method in CDPRs is an emerging technique, and the lack of survey papers about RL in this area can be observed. The goals of this review paper are as follows: 1) Review existing literature on RL in CDPRs and classification of applications based on RL methods. 2) State-of-art for RL in CDPRs and future research topics. To achieve these goals, first, RL methods in CDPRs will be reviewed in the next section. A summary of essential concepts in RL and the mathematical model of the mentioned methods in the literature will be reported. The RL methods based on their applications will be classified in Sect. 2 and more details about the learning control methods of the CDPR will be presented. The state-of-art for RL in CDPRs will be discussed in the final section. 1.1

Literature Review of Reinforcement Learning in Cable-Driven Robots

In recent years, RL has mostly been used in robotics for different purposes, including complex tasks such as obstacle avoidance and path planning [2]. Deep reinforcement learning (DRL), which is a combination of RL with deep learning (DL), implies DNN has been implemented with a nominal model of CDPRs and a basic control law for fully constrained CDPRs in [8]. The goal of this research was to improve the controller to adjust to the uncertainties in the robots, such as cable elasticity and mechanical friction. To achieve this goal, a DRL framework of a high-level controller, which identifies whether a system is stable or unstable, called a Lyapunov-based DRL framework, was simulated with experimental results [8]. Also, DRL with the same control method as [8] for flexible structures of the solar panels in satellites was simulated in [4]. The main goal was controlling and suspending the vibration in the structural model based on the CDPRs due to the suspension of the structure with cables [4]. In [5], CDPR was employed to model flexible structures in the satellites to surpass and reject the vibration with a small and controllable output force [4]. The dynamic model of the CDPRs with four cables was achieved using the finite element method. Three types of controllers were proposed to achieve this goal: Fuzzy PID, deep reinforcement learning (DRL), and Lyapunov-based DRL [4]. In [9], CDPR was proposed as a suspension system in wind tunnels with high attack angles, and RL was used to suspend the vibration. The aerodynamic characteristics of the wind were considered in the nonlinear controller consisting of the RL technique and the classical control method. The simulated methods were a deep deterministic policy gradient (DDPG) and computed-torque controller (CTC). In [2],

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the position control of 3 DOF planner CDPRs was simulated using the DDPG method with no cable tension model, and its performance was compared to the PD control method. Optimizing cable tension distribution in a 3 DOF CDPR for rehabilitation assistance was simulated using a deep Q-network (DQN), a combination of Q-learning and DRL [7]. Q-learning is a technique in RL that describes the quality of action for the system’s state. In this research, the tension in three cables was controlled using the PID method, and for one cable, DQN was applied [7]. Another application of RL and CDPRs in rehabilitation has been reported in [10] to realize continuous mode adaptation between passive and active working modes in interaction with human users for upper body rehabilitation training. The result was achieved from both simulation and experiment with optimal trajectory control, and RL was employed for the online calculation to adjust the device to the patient’s needs [10]. Another research considered RL for the kinematic configuration of CDPRs in situations where the attachment points of the cables to the moving platform shift [11]. Moreover, in [6], the eight-cable CDPR with 6 DOF was considered in construction sites for heavy loads. To maintain the tension in cables, a PID controller and the Q-learning method of RL were used to balance the highly unstable load by finding the desired platform position and orientation, as well as its rotational speed in simulation [6]. As noted, CDPR using RL has found its way to many applications. The majority of the research goals are for position control and vibration control. Using RL in CDPR, is not required to employ a cable tension distribution algorithm. The robot can learn cable tensions in a trial-and-error process, and a neural network-based controller can be established with low computational cost. On the other hand, the need for more experiments and implementation of RL for CDPRs in the real world can be noted. In the next section, the essential concepts of the RL to investigate the frequently used methods in the literature will be discussed.

2

Reinforcement Learning

The past decade was a progressive era for using artificial intelligence and learning methods in robotics, such as unsupervised and supervised learning, reinforcement learning and their applications in robotics [12]. In unsupervised learning, the goal is to find the hidden patterns and structures for raw data and label them according to the needs and purpose of the application. In supervised learning, the goal is to label the output data called “test data” based on the input data’s labels called “trained data”. RL is one of the subsets of machine learning that searches for scenarios that result in maximum accumulative rewards. The reward is a signal from the environment, usually including the robot task space. The reward signal helps to find the desirable scenarios, defined by high positive rewards, and undesirable scenarios with negative rewards. This interaction with the environment in RL is highly beneficial for extracting long-term solutions or different tasks but with the same task space, especially in an uncertain environment that might alter the dynamic model. In these cases, RL can learn the best scenarios

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and solutions to pursue from rewards. Therefore, a robot can be trained to act more purposefully and effectively in environments that contain uncertainties by learning from interaction with the environment [12,13]. RL has many benefits since it can provide conditions to seamlessly transfer the solutions in the simulated environment that can be executed with a physical robot. This means the control strategy enables robots to learn the desired goal, such as efficiency in position control from the reference motion and imitate the movement with high accuracy. In robot control with RL, the feedback from the RL controller, which is in the form of a scaler (reward), is used to find the detailed explicit solution to a problem. Moreover, in RL, there is no need for the exact dynamic model of the robot, which is hard to achieve. Also, the dynamic model of CDPR usually contains unknown uncertainties and complexity. RL can predict multi-scenarios of robots for specific tasks and their results, and the agent can be trained to avoid failure and damaging scenarios. As well, the system can have an acceptable performance by using the average-accuracy dynamic model, and there would be no demand for high-level control design [12,14]. In reinforcement learning, the system’s state defines a comprehensive prediction of future occurrences, and the action is the system’s command. The agent has the most significant task, exploring the possible strategies and receiving feedback on the outcome of the choices. The controller must find a policy for the long-term maximum sum of the rewards. The major components in RL and their tasks can be described as follows: the decision-maker unit, known as the “agent, learns how to interact with the unit called the “environment” from its action and experience. The agent receives the state and reward from the environment as the input, and by processing through the agent unit using the neural network algorithm called “policy,” a decision for a proper “action” is taken and sent as the output of the agent to the environment as an input. In this process, the agent continuously interacts with the environment, and the goal is to execute the actions that result in maximum rewards. It should be noted that the environment concept in RL varies from what is known in robotics, which is the space where the robot tasks take place. Depending on the application, the environment in RL could be only the task space or include both robot and the task space, such as in bipedal robots, where both robot and its task space are included in the environment [6]. In Fig. 2, the agent and environment interaction is illustrated, where St is the set of states s, at time step t. The action At at t is decided by the agent and conducted in the environment. The change in environment state will lead to a new state St+1 and new reward Rt+1 , which become the current state St and current reward Rt feedback from the environment. Two famous approaches in designing RL are model-free (MFRL) and modelbased (MBRL), also known as direct and indirect methods. In the model-free approaches, there is no dynamic model or model that characterizes the environment, and the optimal actions are derived by trial and error using the physical system. In the model-based method, the dynamic model of the robot exists and adds its characteristics to the environment, which helps with other components in

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Fig. 2. Agent and environment interaction.

RL control, such as reward derivation and optimal actions [13]. In the model-free methods, there is no need for knowledge of modelling the transition dynamics, which eases the implementation process. The disadvantages of MFRL are slow learning convergence, being highly time-consuming and the high possible risk of damage to the robot [13]. It should be noted that MFRL is highly beneficial for applications with the high complexity of dynamic models or environment properties where learning the dynamic system model is challenging [16]. The MBRL has received more attention in recent years as it can converge to the optimal solution faster with the few interactions between the agent and the environment. MBRL is more “sample-efficient,” but the learning rate and the general performance of this method are highly dependent on model accuracy. On the other hand, learning the environment properties empowers the agent to figure out different tasks for the same environment. Moreover, the clarity of decisions, which are taken in MBRL, allows the outcome to be checked for different actions and their rewards that can be used for similar tasks or similar environments [14, 16]. In this paper, the approach for solving the noted challenges in the literature review is from the perspectives of model-based or model-free approaches. To appreciate the implemented methods in the literature, the basic concepts and algorithms of reinforcement learning will be reviewed in the following subsection. 2.1

Review of Reinforcement Learning Methods in CDPR

A robotic task such as navigation or manipulation can be formalized as the Markov Decision Process (MDP), in which the interaction between the environment and the agent is conducted through the sequence of observation, action and reward signals. The robot’s state is s, and s is the next state. The robot’s state can be described continuously or discretely. The reward function is r, and the action a is derived from the policy function, π(.), which maps the state to action: π : s → a. In the stochastic model, the policy function depends on a random variable , which is a domain defined by the set of all possible outcomes of an event, and the state-to-action mapping is defined as π(a | s, ), which is the probability distribution of states over actions. Therefore, the interaction between the robot and environment, modelled as MDP, is a tuple that can be found in

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[S, A, , P (s | s, a), R, γ] where S is the set of possible states, A is the set of actions, P (s | s, a) is the probability of transition at a future state, s , which depends on the applied actions a. Also, R, is the set of possible rewards, can be described as R(s, s , a), which represents the reward an agent can achieve by taking action a from state s and considering the next state as s . The discount factor of the reward function, γ, varies between zero and one. At the time interval t, the action At is chosen based on the received state from the environment St . The reward from the following change in environment state is Rt+1 . It is important to follow the policy that maximizes the sum of the future rewards or “return” called Gt , which can be defined as follow: Gt = Rt+1 + γRt+2 + γ 2 Rt+3 + ... =

∞ 

γ k Rt+k+1

(1)

k=0

Since the main goal is to maximize the total amount of reward received in the long term, there are two methods in order to maximize the return from the environment: value-based and policy-based functions. Value-Based. RL is a method based on estimating the expected returns that bring the highest total rewards. The value-based common approaches are statevalue-function V π (s), which is the expected return by starting the state s. The action-value function Qπ (s, a), the expected return from state s, follows policy π by taking action a. The optimal value function Q∗ (s, a) is the maximum value of action after execution. The optimal policy function π ∗ (s, a) is the policy with the optimal value function [15,17]. This method is highly suitble for continuous action state space. Policy-Based. RL is the technique that works directly on policies and does not work on value estimation, which is described above. Policy-based RL allows for parametrizing policies and searching for parameters that maximize the return functions. There are two common approaches: the stochastic policy π(s | a), in which the actions are extracted from the probability distribution, and the deterministic policy μ(s), in which actions are deterministically selected [15,17]. The policy-based method can be applied to both continuous and discrete state space. The most reported RL control of CDPRs in the value-based approach is Qlearning and DQN (a deep neural network form of the Q-learning method), and in the policy-based approach is DDPG. The traditional Q-learning method is useful for finite states and actions. In the traditional Q-learning method, a table is used where the rows are for states, and the columns are for actions. Each cell has the estimated Q-value, which measures the overall expectation rewards for each state-action pair. The Q-values in the table are updated by each agent and environment interaction: Q(st , at ) ← Q(st , at ) + α(Rt+1 + γmaxa (st+1 , a ) − Q(st , at ))

(2)

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Equation (2) is the recursive RL procedure, α is the learning rate and a is the target action with the highest Q-value from the next state and is used in updating the Q-value of the current action.

Fig. 3. Actor-critic method.

Traditional Q-learning is not appropriate for enormous state and action space since it can face high computational costs and a lack of memory phenomena for the processor system. Therefore, for an infinite number of states and action space, employing layers of the neurons in the network is essential, which results in DQN. The DQN method is a Q-learning but uses ANN instead of the table. This method can solve more complicated and high-dimensional problems where the state space is continuous with large state space and action space. DDPG is another most used method in the RL control of CDPRs. This method is policybased and can be applied to high-dimensional problems and continuous action capable of accepting states and actions with infinite space. This method includes DQN and actor-critic algorithms where actors observe the current state and determine an action with the highest accumulative rewards in the long term. The critic estimates the expected value of long-term reward by taking states’ observations and actions as input. Actor-critic interaction can be seen in Fig. 3. There are four NN in DDPG algorithms: actor current network and actor target network are the actor’s network, critic current network and critic target network are the critic’s network. The updating process for policy parameters is executed in the actor’s current network, and in the actor target network, the optimal action is chosen. In the current critic network, value network parameters

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regarding Q-value are updated, and in the critic target network, the target Qvalue is calculated. 2.2

Learning Control for CDPR

In most studies, the combination of ML and basic control law has been referred to as learning control. Researchers have proposed many techniques for solving problems based on the mentioned combination. This combination can include the classical control, such as PID, or a high-level controller, such as the Lyapunov function. Also, different learning methods using ML, such as unsupervised learning, supervised learning and RL can be utilized. Therefore, the control system can operate in various scenarios in learning control. The control parameters are updated based on the requirements to guarantee performance accuracy for future runs. The learning controller allows for using the nominal model of the robot and a basic controller, plus an ML algorithm. In this scenario, the system can learn the control parameters in a closed-loop with the ML algorithm by receiving the system’s state. The nominal dynamic equation of CDPR can be seen as follows: T

M(x)¨ x + C(x,x) ˙ x˙ + G(x) + (J) f = u

(3)

where M(x), C(x,x), ˙ G(x) and f represent the inertia matrix, Coriolis and centrifugal force matrix and gravity and friction term can be expanded to actuators and cables friction. In addition, the state of the CDPR is a vector containing the motion of its moving platform’s orientation and position,i.e., The position vector x, velocity vector x˙ and acceleration vector x ¨. Also, u, and J are the Jacobian matrix and wrench vector, which is the moving platform force vector. The Jacobian matrix calculate the cables tensions, which is u = JT τ , and τ is the tension vector. The system’s state in most RL studies in robotics is the same as the environment state and plays a vital role in reward function. Whether the RL algorithm is MFRL or MBRL, with system states, the connection between the control algorithm and RL occurs. It is noteworthy that Eq. (3) can include variables based on the needs and problem definition and become more than a nominal equation. These variables can be gear ratio, nonlinear factors, or introducing parameter uncertainties due to error and shift in the motion parameters. This high-level control study is usually recommended for cases with high precision. In CDPR, the mobile platform is connected to a fixed base by cables. This means solving the control problem for CDPR is more complicated than other types of robots, such as serial robots or parallel robots. Using cables instead of joints and links leads to considering factors such as cable tension, which usually can be neglected in other types of robots or problems with fewer accuracy requirements. Therefore, the learning control for CDPR with all the mentioned parameters is highly beneficial. In this scenario, the control signal u is the combination of the classical control signal ua and learning control signal ur , which is the control signal to be learned by the RL algorithm. This relation can be defined as u = ua + ur . A summary of RL control method of CDPRs in literature is reported in Table 1.

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P. Nomanfar and L. Notash Table 1. Summary of RL Control Methods for CDPRs in Literature.

Ref Application

Method

Reward Function MFRL/MBRL Experimental Work

[2]

Position control of CDPR

DDPG

Platform pose error, platform velocity and angular velocity error

MFRL

No

[4]

Vibration Fuzzy PIDDRL, suspension of Lyapunov DRL CDPR, Satellites large flexible structure wings

Position error, and velocity error, of platform, cable tension

MFRL

No

[5]

Vibration DDPG suspension of CDPR, Satellites large flexible structure wings

Position and velocity of platform, position error

MFRL

No

[6]

Balancing of highly unstable load using CDPR, construction

Ball position and MFRL speed, platform pose

No

[7]

Optimizing cable PID and deep tension, Q-learning rehabilitation assisting

Cable tensions and rate of acceleration (cable jerks)

MFRL

No

[8]

Compensating negative effects of uncertainties CDPR

Lyapunov DRL

Platform pose error, velocity and angular velocity error

MFRL

Yes

[9]

Vibration suspension of CDPR in the wind tunnel

DDPG

Orientation error MFRL of platform, angular error

PID and Q-learning

No

RL framework has the capability to be used with controllers such as simple PID, computed-torque control or high-level control fuzzy method and the Lyapunov-based controller. The reason is that the reward function in RL brings the possibility of considering important variables for research goals. One of the widely used variables is the robot’s states which have been proposed as the reward function. Also, Lyapunov-based control methods have been used to ensure the system’s stability, whether it is MFRL or MBRL. The most reported RL methods used in RL control of CDPRs are Q-learning and the deep deterministic policy gradient (DDPG), known as MFRL. Moreover, it is important to check for the dynamic model of the robot in the environment of the RL model. If it exists, then it will be categorized as model-based. Table 1 contains a summary of RL methods in the literature. As reported in Table 1, system states in robotics are mostly the position and velocity of moving platforms, which are attached to cables. The pertinent pose error and velocity error of moving platforms are the reward signals to agents for position control and for

Brief Review of Reinforcement Learning Control

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applications that demand highly accurate actions. Also, for vibration suspension applications, the reward signal may contain the cables’ tension force as well.

3

Concluding Remarks

In this paper, CDPRs with RL control were studied based on the applications of CDPR and RL methods employed to solve the challenges. The goal of this paper was to understand the state-of-art of RL in CDPR as well as their applications. The future of RL in this field is very promising, and RL can be used for solving different challenges in different applications. RL can define the goals and find a proper mathematical model that best fits the nature of the data. RL frameworks allow for the proper dynamic model of the environment that can later be utilized for different investigations and transfer learning. The model-based RL is highly recommended for applications that require precise action, such as rehabilitation and human-machine interaction. Also, model free is highly recommended for applications where there is no demand for high accuracy or the dynamic model is difficult to learn. Suggested future research can be MBRL in CDPR, experimental RL control of CDPR, as well as using the simulation results for virtual reality (VR) applications in industry, construction or rehabilitation. Another promising research topic is using multi-agent RL for CDPR to learn coordination strategies between multiple robots, enabling them to work together more efficiently to accomplish complex tasks. Also, safety-aware RL for CDPR robots is a topic with high potential since CDPR can be dangerous if not controlled properly. Safety-aware RL approaches can ensure that robots operate within safe limits, even in unpredictable or changing environments. Overall, the future of reinforcement learning research for CDPR is bright, with many exciting opportunities for developing new algorithms that enable these robots to operate more effectively in a wide range of applications.

References 1. Qian, S., Zi, B., Shang, W.W., Xu, Q.S.: A review on cable-driven parallel robots. Chin. J. Mech. Eng. 31(4), 66 (2018) 2. Sancak, C., Yamac, F., Tik, M.: Position control of a planar cable-driven parallel robot using reinforcement learning. Robotica 40(10), 3378–3395 (2022) 3. Notash, L.: Artificial neural network prediction of deflection maps for cable-driven robots. In: Proceedings of the ASME Design Engineering Technical Conference and Computers and Information in Engineering Conference IDETC/CIE2020, 17– 19 August 2020 (2020) 4. Sun, H., Tang, X., Hou, S., Wang, X.: Vibration suppression for large-scale flexible structures based on cable-driven parallel robots. JVC/J. Vib. Control 27(21–22), 2536–2547 (2021) 5. Sun, H., Tang, X., Wei, J.: Vibration suppression for large-scale flexible structures using deep reinforcement learning based on cable-driven parallel robots. In: ASME International Mechanical Engineering Congress and Exposition, Proceedings (IMECE), 7A-2020 (2021)

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6. Grimshaw, A., Oyekan, J.: Applying deep reinforcement learning to cable driven parallel robots for balancing unstable loads: a ball case study. Front. Rob. AI 7, 212 (2021) 7. Xie, C., Zhou, J., Song, R., Xu, T.: Deep reinforcement learning based cable tension distribution optimization for cable-driven rehabilitation robot. In: 2021 6th IEEE International Conference on Advanced Robotics and Mechatronics (ICARM), Chongqing, China, pp. 318–322 (2021) 8. Lu, Y., Wu, C., Yao, W., Sun, G., Liu, J., Wu, L.: Deep reinforcement learning control of fully constrained cable-driven parallel robots. IEEE Trans. Ind. Electron. 70, 7194–7204 (2022) 9. Wang, W., Wang, X., Shen, C., Lin, Q.: Reinforcement learning-based composite controller for cable-driven parallel suspension system at high angles of attack. IEEE Access 10, 36373–36384 (2022) 10. Yang, R., Zheng, J., Song, R.: Continuous mode adaptation for cable-driven rehabilitation robot using reinforcement learning. Front. Neurorob. 16, 1068706 (2022) 11. Aref, M.M., Mattila, J.: Automated calibration of planar cable-driven parallel manipulators by reinforcement learning in joint-space. In: 6th RSI International Conference on Robotics and Mechatronics (ICROM), Tehran, Iran, pp. 172–177 (2018) 12. Zhang, T., Mo, H.: Reinforcement learning for robot research: a comprehensive review and open issues. Int. J. Adv. Rob. Syst. (2021) 13. Polydoros, A.S., Nalpantidis, L.: Survey of model-based reinforcement learning: applications on robotics. J. Intell. Rob. Syst. 86(2), 153–173 (2017). https://doi. org/10.1007/s10846-017-0468-y 14. Sutton, R.S., Barto, A.G.: Introduction to Reinforcement Learning, 2nd edn. MIT press, Cambridge (2018) 15. Pal, C.V., Leon, F.: Brief survey of model-based reinforcement learning techniques. In: 2020 24th International Conference on System Theory, Control and Computing (ICSTCC), Sinaia, Romania, pp. 92–97 (2020) 16. Tai, L., Zhang, J., Liu, M., Boedecker, J., Burgard, W.: A survey of deep network solutions for learning control in robotics: from reinforcement to imitation (2016) 17. Arulkumaran, K., Deisenroth, M.P., Brundage, M., Bharath, A.A.: Deep reinforcement learning: a brief survey. IEEE Signal Process. Mag. 34(6), 26–38 (2017)

Force Control of a 1-DoF Cable Robot Using ANARX for Output Feedback Linearization Marcus Hamann(B) , Valentin H¨ opfner , and Christoph Ament Faculty of Applied Computer Science, University of Augsburg, 86159 Augsburg, Germany {marcus.hamann,valentin.hoepfner,christoph.ament}@uni-a.de

Abstract. This paper presents a data-based approach to force control. For this purpose, representative data of the robot in operation is recorded to model it afterwards by methods of time series modeling. Specifically, a form of NARX (Nonlinear AutoRegressive eXogenous) models is used, so-called Additive NARX or ANARX. In this way, a state space representation can be generated from the resulting neural network. Based on this conventional model, a controller can be subsequently designed. This approach is holistic and is thus not only applicable for cable robots, but for any class of robots. The resulting controller is evaluated on a physical system and compared with a conventional PID controller.

Keywords: cable robot

1

· force control · data-based control · ANARX

Introduction

Cable robots have advantages over serial robots in terms of scalability, dynamics and payload-weight ratio. However, they also have disadvantages. Cable robots can only transmit limited and tractive forces. This is a special feature for cable robots, as it affects the workspace, safety and dynamics. Therefore, it is important that a valid cable force distribution is provided at all times, even within real-time applications. In [13], the development of a real-time capable force calculation algorithm for cable robots was adressed. Subsequently, there were other approaches that improved computational efficiency [11] or addressed the influence of sagging [12] or elastic [10] cables on the cable force distribution. In [19], a closed-form force distribution approach was presented and improved in [18]. These approaches addressed the compliance with defined cable force limitations. At the same time, position and force control of the platform has to be realized. This depends on the configuration of the robot, i.e. size or location and number of cables. Pose control can pursue several goals. These include optimal tracking control or the realization of predefined dynamics of the closed-loop system. There are numerous approaches available for position control, for example based on c The Author(s), under exclusive license to Springer Nature Switzerland AG 2023  S. Caro et al. (Eds.): CableCon 2023, MMS 132, pp. 173–183, 2023. https://doi.org/10.1007/978-3-031-32322-5_14

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exact linearization [3,4], Lyaponov-based PD controller [1,7] or flatness based feed forward controller [23,24]. The goal of force control is the fast realization of required cable forces while keeping the kinetic limitations. In [9] an admittance control, and in [21] an impedance control is presented for this purpose. In [20], a hybrid position-force controller approach is presented that ensures both cable tension and positioning accuracy despite kinematic inaccuracies. NARX models in general can be used to derive an output equation from a given time series. ANARX models are a special class of NARX models resulting from a restructuring of the underlying neural network. They have the advantage that a state space model and therefore also a control law can be determined from the model equations. In [17], a dynamic output feedback linearization is presented. The control signals are calculated by an ANARX model. The approach can be applied to control a nonlinear dynamic MIMO system. In [25], an approach based on state feedback linearization to control nonlinear SISO and MIMO systems using NN-SANARX models is presented. The objective of this work is to train an ANARX model from a force trajectory and subsequently design a force controller. The design of the control law is based on a state space, which can be automatically derived from the structure of the underlying neural network. The advantage of this approach is that no physical model has to be developed. Only a (preferably) short but representative time series of the force trajectory is to be used. Figure 1 shows the current planar configuration of the cable robot at the University of Augsburg. The cable robot consists of four modules, each with an motor, pulleys, a load cell and the cable. The modules of the robot are fixed inside a cube with edge length 2.5 m. In addition, the measuring system, a laser tracker, can be seen in the foreground of the figure. It also consists of four single modules and measures the pose of the platform. The red lines indicate the laser beams. The measuring system will not be the focus of the following work. To demonstrate our results, we extracted a single module of the cable robot, since it already contains a major part of the nonlinearities. This module is depicted in Fig. 2. In the following Sect. 2 the controller design is presented. This is closely linked to the modeling of the system. Subsequently, in Sect. 3 the experiment on the physical system is presented and discussed in Sect. 4. Finally, Sect. 5 concludes with a summary.

2 2.1

Controller Design Modeling with ANARX

For time series modeling, a variety of methods are available. One way is through NARX (Nonlinear AutoRegressive eXogenous) models, which are capable of modeling time series and performing single- or multi-step predictions.

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Pulley Motor

500 mm Load Cell

400 mm

Cable

Fig. 1. Planar Configuration of the Cable Robot of the University of Augsburg

Fig. 2. Single Module Consisting of an Motor, Pulleys, a Load Cell and the Cable

Each predicted output value of a time series is calculated from past values of the same time series as well as from the current and past values of the exogenous input. The output of a NARX model can be noted as y(k) = f (y(k − 1), . . . , y(k − N ), u(k − 1), . . . , u(k − M )) + ε(k) ,

(1)

with output y(k) at time k, input u(k) at time k and an error term ε(k) that contains all characteristics that cannot be modeled. The function f is an arbitrary nonlinear function. This allows a NARX model to be interpreted as a neural network. In this case, the function f corresponds to the input/output behavior of a neural network. Figure 3 shows one interpretation of a NARX model as a neural network with one input, one hidden and one output layer. Here, the function f corresponds to a fully connected feedforward neural network. NARX models implemented in the form of neural networks belong to the class of black box models. Only the input/output behavior is modeled. The internal states in general do not correspond to any physical variables anymore. ANARX (Additive Nonlinear AutoRegressive eXogenous) models are intended to eliminate this drawback by reformulating Eq. (1). The inputs and outputs of the network from Fig. 3 are sorted according to their lags and each pair forms the input of a separate nonlinear function. For the case of an equal number of input and output lags, the output equation results in y(k) = f1 (y(k − 1), u(k − 1)) + · · · + fN (y(k − N ), u(k − N )) + ε(k) ,

(2)

with nonlinear functions f1 , . . . , fN . If these functions are interpreted as neural networks, the network from Fig. 4 can be generated. To express that the nonlinear

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Fig. 3. Conventional NARX time series model with N lags of the output and M lags of the input

functions f1 , . . . , fN are represented by neural networks, these systems are also called NN-ANARX [17]. The approximation capability of this network is not

y(k − 1) u(k − 1) y(k) y(k − N ) u(k − N )

Fig. 4. Additive NARX time series model with N lags of the input and output

affected compared to the network from Fig. 3. However, this reformulation has the decisive advantage that a state space can be derived from it. For an ANARX model y(k + 1) = f1 (y(k), u(k)) + f2 (y(k − 1), u(k − 1)) + . . . + fN (y(k − N + 1), u(k − N + 1))

(3)

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of order N , the following state space y(k + 1) = x1 (k + 1) = x2 (k) + f1 (y(k), u(k)) x2 (k + 1) = x3 (k) + f2 (y(k), u(k)) .. . xN −1 (k + 1) = xN (k) + fN −1 (y(k), u(k))

(4)

xN (k + 1) = fN (y(k), u(k)), with the artificial states x1 , . . . xN can be derived. By substituting the states from Eq. 4 into Eq. 3, the equivalence of the state space representation and the output equation can be easily shown. A detailed mathematical derivation is given in [8]. 2.2

Training

In order to train the network, representative data must be recorded – this means data that covers the workspace as good as possible. One approach is to record arbitrary steps within the reachable working range for a certain time. The model can be trained open loop or closed loop. In case of open-loop training, the actual output measurements are used for prediction of the output of the next time step. In case of closed-loop training, multiple time steps and not just one time step are predicted, where the current prediction depends on the N last predictions [14]. For this purpose, Backpropagation-Through-Time is applied. This makes the training more stable and robust and reduces the risk of overfitting. However, the training process takes significantly longer compared to open-loop training, because parallelization is not possible. In our case, the network was pre-trained open-loop and then re-trained closed-loop. For optimization, ADAM is used [6]. The software on the developer PC is written in Python 3.9.10 [22] using the framework PyTorch [15]. The training can be executed in simulation or on the physical system. The developer PC works with a sampling time of 0.0025 s. The PLC works with a sampling time of 0.0025 s. The PLC is programmed using MATLAB Simulink (The MathWorks, Inc., MA, USA). 2.3

Controller

The formulation given by Eq. 4 allows to implement various control approaches. The approach presented in the following originates from [16]. The predicted output of the system is given by y(k + 1) = x2 (k) + f1 (y(k), u(k))

(5)

according to Eq. 4. Since the output is supposed to follow a reference r(k + 1), the optimization problem 0 = x2 (k) + f1 (y(k), u(k)) − r(k + 1)

(6)

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can be formulated. From this, the optimal input (or the manipulated variable) u(k) is determined by solving Eq. 6 using line search. That means the manipulated variable u(k) is calculated in each time step k based on the measured variable y(k) and the reference r(k + 1) using the model. However, this also results in some challenges: – The controller is generated on the basis of an ANARX model. Therefore, the quality of the controller depends strongly on the quality of the model. – If the physical system is not able to follow the reference, this results in unrealizable values for u(k). – The optimization problem can be complex and possibly not solvable in real time. – It is not guaranteed that a (unique) solution exists. It can be shown, that invertibility of f1 with respect to u is a sufficient condition. In order to overcome the last two issues, a linear function f1 1 can be applied. The other functions f2 . . . fN can be arbitrary nonlinear functions. In the case of a SISO system, Eq. 6 therefore becomes     a1 0 = x2 (k) + u(k) y(k) + b − r(k + 1) . (7) a2 The control variable u(k) =

r(k + 1) − x2 (k) − y(k) · a2 − b a1

(8)

can be generated after training by rearranging Eq. 7. Figure 5 illustrates schematically the resulting network. These models are called Simplified NN-ANARX (NN-SANARX) [16]. In our experiments, the linear network for the first lag did not negatively affect the model quality. The resulting controller is significantly dependent on the quality of the model. However, a perfect model is not to be expected. Therefore, the control loop can be extended by an integral controller to prevent a permanent control error. Figure 6 shows the block diagram of the control loop including the cable robot (CR), ANARX model, ANARX controller and integral controller. The input of the ANARX model corresponds to the output of the cable robot – a force. The output of the ANARX model corresponds to the artificial state x2 (k), which is required for the control law. All other variables of the control law – the reference r(k + 1) as well as the factors a1 , a2 and b - are already known and constant in this case. In addition, an integral controller has been used in this block diagram to provide stationary accuracy, since the ANARX controller does not necessarily guarantee it.

3

Experiment

In the following, using the physical system from Fig. 2 as an example, an ANARX model is first generated in order to subsequently use it for control. 1

Specific implementation of the linear transformation in PyTorch: https://pytorch. org/docs/stable/generated/torch.nn.Linear.html.

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y(k − 1) u(k − 1) y(k − 2) u(k − 2) y(k) y(k − N ) u(k − M )

Fig. 5. SANARX time series model with N lags of the output and M lags of the input, the first lag is processed linear uI (k)

r(k + 1)

u(k) =

r(k+1)−x2 (k)−y(k)·a2 −b a1

uA (k) x2 (k)

CR

y(k)

ANARX Model

Fig. 6. Control loop including the cable robot (CR), ANARX model, ANARX controller and integral controller

3.1

Training

First of all, the underlying model must be learned using the ANARX structure. For this purpose, a 30-s series of measurements with arbitrary step changes within the working range (0 to 20 N) was recorded at 4 kHz on the physical system. It became apparent that the learning time can be significantly reduced if the resulting signal is downsampled to – in our case – 400 Hz. During the training, a number of hyperparameters has been varied. These include batch size (in the range from 1 to 1000), number of hidden layers (in the range from 2 to 4), hidden layer size (in the range from 3 to 100), activation function (ReLU or tanh) and bias (with or without). In addition, the number of input and output lags is relevant. These were also part of the parameter

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sweep. It turned out that in this case more than four input delays and more than ten output delays should be used. For the experiment, five input delays and 14 output delays were used. The force measurement by the load cell shows a distinct creep behavior. That means that the recorded force constantly changes its value slightly after reaching a setpoint value. This behavior can be modeled if not only the force but also the position signal is used as input of the network, since the creep behavior can be measured using the position signal. However, the position signal cannot be used because it is potentially also required for position control. Therefore, the force remains the only input to the network, resulting in a steady-state error remaining in the model. Figure 7 shows the trajectory Flc of the load cell (black) as well as the ANARX-modeled trajectory Fpr (red). Figure 8 shows a zoomed part of the trajectory. As can be seen, a steady-state error remains. In addition, it can be seen that the model shows minor inaccuracies when dropping to zero. However, this scenario was not part of the learning dataset and thus represents extrapolation behavior. Nevertheless, the dynamics can be followed quite well in general. 20 F 30

lc

F pr

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Fig. 7. Trajectory Flc of the load cell (black) and the ANARX-modeled trajectory Fpr (red) – validation time series.

3.2

Force in N

Force in N

40

F lc

15

F pr

10 5 0 -5 43.5

44

44.5

45

time in s

Fig. 8. Trajectory Flc of the load cell (black) and the ANARX-modeled trajectory Fpr (red) – snippet of validation time series.

Controller Performance

The following figures show the performance of the resulting controller compared to a conventional PID controller tuned by an expert. Figure 9 shows the result of an aperiodic reference variable (black). It shows the physical system controlled by the ANARX controller (red), the PID controller (blue) and an extension of the ANARX controller using an additional integral controller (green). Figure 10 shows the result of an periodic reference variable (black). It also shows the physical system controlled by the ANARX controller (red), the PID controller (blue) and an extension of the ANARX controller using an additional integral controller (green).

u in N m

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Fig. 9. The physical system controlled by the ANARX controller (red), the PID controller (blue) and an extension of the ANARX controller (green) – aperiodic reference

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Fig. 10. The physical system controlled by the ANARX controller (red), the PID controller (blue) and an extension of the ANARX controller (green) – periodic reference

Discussion

The controller design approach described in this paper appears to be very promising. However, there are some challenges to overcome along the way. At first, representative data of the physical system has to be recorded. The question arises how much data, in which quantity and in which workspace has to be gathered. In our case, a sequence of step changes of different initial and final step values over a period of less than a minute was sufficient to generate a model appropriate for the application. Subsequently, a model is generated from the data. The model depends on a number of hyperparameters. The optimal choice of these parameters is usually made empirically. In addition, the learning time of a single network of several minutes to a few hours should not be underestimated. Figure 7 an Fig. 8 show that even with a small amount of data - in this case 30 s - a very good model can be generated. The dynamics of the real system are modeled very well. The advantage of the approach described here is, that a state space can be extracted from the neural network. From this, a control law can be generated without the necessity of developing a physical model in advance. In our case, once the parameters are determined, the control law is constant. Figure 9 and 10 show that very good control results can be generated from this already. The control variable follows the reference variable closely. However, a permanent control error remains. In our case, however, this can be eliminated easily by an integral controller. The control results are also better than those achieved by an expert tuned PID controller. The challenge of using a PID controller is that it is a linear approach, which tends to oscillate for the given system and a different parameter choice (e.g., higher P gain) in closed loop control.

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The trajectories of the manipulated variable also show that less energy is required for the data-based control, either with or without an additional integral controller. This can be determined by comparing the integrals of the manipulated variables of all three cases.

5

Conclusion

The presented work demonstrated a data-based approach to controller design. For this purpose, data was collected from a physical system in order to model it by a special class of NARX models. From this model, a state space can be derived. The state space representation can subsequently be used for controller design. The control has been evaluated on a physical system and compared with a conventional PID controller. The results are very promising. The approach seems to be very effective, but requires some parameter exploration in advance. Moreover, the method is not yet as stable as we aspire to. That means that a suitable controller cannot always be generated. The step from the data to the model is still subject to some heuristics, which we try to eliminate. Furthermore, a git repository was developed, which contains numerous features [5]. These include – creation of neural network based NARX, ANARX and SANARX models in SISO and MISO configuration, – open-loop training, – closed-loop prediction and training (based on a variant of backpropagationtrough-time), – conversion of NN-ANARX models to state space representation, – computation of optimal control input for NN-SANARX models (SISO case), – export of all models (including state space representation) as ONNX to facilitate its inference on arbitrary hardware [2].

References 1. Alp, A., Agrawal, S.: Cable suspended robots: feedback controllers with positive inputs, pp. 815–820. American Automatic Control Council (2002). https://doi.org/ 10.1109/ACC.2002.1024915 2. Bai, J., Lu, F., Zhang, K., et al.: Onnx: open neural network exchange (2019) 3. Hamann, M., Ament, C.: Model-based control of a planar 3-dof cable robot using exact linearization (2021). https://doi.org/10.1007/978-3-030-75789-2 21 4. Hamann, M., Winter, D., Ament, C.: Model-based control of a pendulum by a 3-dof cable robot using exact linearization. IFAC-PapersOnLine 53, 9053–9060 (2020). https://doi.org/10.1016/j.ifacol.2020.12.2130 5. H¨ opfner, V.: Nn-anarx (2023) 6. Kingma, D.P., Ba, J.: Adam: a method for stochastic optimization (2014) 7. Kino, H., Yahiro, T., Takemura, F., Morizono, T.: Robust pd control using adaptive compensation for completely restrained parallel-wire driven robots: translational systems using the minimum number of wires under zero-gravity condition. IEEE Trans. Rob. 23, 803–812 (2007). https://doi.org/10.1109/TRO.2007.900633

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8. Kotta, U., Sadegh, N.: Two approaches for state space realization of narma models: bridging the gap. Math. Comput. Model. Dyn. Syst. 8, 21–32 (2002). https://doi. org/10.1076/mcmd.8.1.21.8340 9. Kraus, W., Mangold, A., Ho, W.Y., Pott, A.: Erratum to: Haptic interaction with a cable-driven parallel robot using admittance control (2015). https://doi.org/10. 1007/978-3-319-09489-2 23 10. Kraus, W., Miermeister, P., Pott, A.: Investigation of the influence of elastic cables on the force distribution of a parallel cable-driven robot (2013). https://doi.org/ 10.1007/978-3-642-31988-4 7 11. Lamaury, J., Gouttefarde, M.: A tension distribution method with improved computational efficiency (2013). https://doi.org/10.1007/978-3-642-31988-4 5 12. Li, H., Zhang, X., Yao, R., Sun, J., Pan, G., Zhu, W.: Optimal force distribution based on slack rope model in the incompletely constrained cable-driven parallel mechanism of fast telescope (2013). https://doi.org/10.1007/978-3-642-31988-4 6 13. Mikelsons, L., Bruckmann, T., Hiller, M., Schramm, D.: A real-time capable force calculation algorithm for redundant tendon-based parallel manipulators. pp. 3869– 3874. IEEE (2008). https://doi.org/10.1109/ROBOT.2008.4543805 14. Nelles, O.: Nonlinear System Identification. Springer, Cham (2020). https://doi. org/10.1007/978-3-030-47439-3 15. Paszke, A., et al.: Pytorch: an imperative style, high-performance deep learning library (2019). http://papers.neurips.cc/paper/9015-pytorch-an-imperative-stylehigh-performance-deep-learning-library.pdf 16. Petlenkov, E.: Nn-anarx structure based dynamic output feedback linearization for control of nonlinear mimo systems, pp. 1–6. IEEE (2007). https://doi.org/10. 1109/MED.2007.4433965 17. Petlenkov, E., Belikov, J.: Nn-anarx structure for control of nonlinear siso and mimo systems: neural networks based approach, pp. 138–144. IEEE (2006). https://doi.org/10.1109/CHICC.2006.4347185 18. Pott, A.: An improved force distribution algorithm for over-constrained cabledriven parallel robots (2014). https://doi.org/10.1007/978-94-007-7214-4 16 19. Pott, A., Bruckmann, T., Mikelsons, L.: Closed-form force distribution for parallel wire robots (2009). https://doi.org/10.1007/978-3-642-01947-0 4 20. Reichenbach, T., Rausch, K., Trautwein, F., Pott, A., Verl, A.: Velocity based hybrid position-force control of cable robots and experimental workspace analysis (2021). https://doi.org/10.1007/978-3-030-75789-2 19 21. Reichert, C., M¨ uller, K., Bruckmann, T.: Robust internal force-based impedance control for cable-driven parallel robots (2015). https://doi.org/10.1007/978-3-31909489-2 10 22. Rossum, G.V., Drake, F.L.: Python 3 Reference Manual. CreateSpace (2009) 23. Stoltmann, M., Froitzheim, P., Fuchs, N., Fl¨ ugge, W., Woernle, C.: Linearised feedforward control of a four-chain crane manipulator (2019). https://doi.org/10. 1007/978-3-030-20751-9 20 24. Stoltmann, M., Froitzheim, P., Fuchs, N., Woernle, C.: Flatness-based feedforward control of a crane manipulator with four load chains (2019). https://doi.org/10. 1007/978-3-319-98020-1 8 25. Vassiljeva, K., Petlenkov, E., Belikov, J.: State-space control of nonlinear systems identified by anarx and neural network based sanarx models, pp. 1–8. IEEE (2010). https://doi.org/10.1109/IJCNN.2010.5596581

Energy-Efficient Control of Cable Robots Exploiting Natural Dynamics and Task Knowledge Boris Deroo1(B) , Erwin Aertbeli¨en1 , Wilm Decr´e1 , and Herman Bruyninckx1,2 1 Department of Mechanical Engineering, KU Leuven, Leuven, Belgium {boris.deroo,erwin.aertbelien,wilm.decre,herman.bruyninckx}@kuleuven.be 2 Mechanical Engineering, TU Eindhoven, Eindhoven, The Netherlands

Abstract. This paper focusses on the energy-efficient control of a cabledriven robot for tasks that only require precise positioning at few points in their motion, and where that accuracy can be obtained through contacts. This includes the majority of pick-and-place operations. Knowledge about the task is directly taken into account when specifying the control execution. The natural dynamics of the system can be exploited when there is a tolerance on the position of the trajectory. Brakes are actively used to replace standstill torques, and as passive actuation. This is executed with a hybrid discrete-continuous controller. A discrete controller is used to specify and coordinate between subtasks, and based on the requirements of these specific subtasks, specific, robust, continuous controllers are constructed. This approach allows for less stiff and thus saver, and cheaper hardware to be used. For a planar pick-andplace operation, it was found that this results in energy savings of more than 30%. However, when the payload moves with the natural dynamics, there is less control of the followed trajectory and its timing compared to a traditional trajectory-based execution. Also, the presented approach implies a fundamentally different way to specify and execute tasks. Keywords: task-specific control · cable-driven parallel robot brake control · pick-and-place · natural constraints

1

· passive

Introduction

Traditionally the execution of robot manipulator tasks is not focussed on limiting the energy consumption, but rather on speed and precision. This often results in robots that are more precise, and thus stiffer, heavier, and more expensive, than strictly necessary for the task. In addition, increasing energy costs and a growing All authors gratefully acknowledge the support by the Research Foundation-Flanders (FWO) project ELYSA (FWOSBO37), and Flanders Innovation & Entrepreneurship Agency (VLAIO). c The Author(s), under exclusive license to Springer Nature Switzerland AG 2023  S. Caro et al. (Eds.): CableCon 2023, MMS 132, pp. 184–196, 2023. https://doi.org/10.1007/978-3-031-32322-5_15

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need for more sustainability push for more energy-efficient solutions. Practical applications of tasks that can also be accomplished with less accurate robots are numerous in industry, e.g. palletising, truck unloading, box stacking, etc. These types of pick-and-place tasks only require high precision at the start and end of the execution, but not for the gross of the motion. These tasks are the main focus of this paper and will henceforth be referred to as ‘transportation tasks’. This paper demonstrates that by adapting the definition of a task in a smart way, by utilising knowledge about the system and task directly in the control execution, a more energy-efficient control can be achieved using simple control strategies that do not require accurate modelling nor a high computational load. To illustrate these strategies, a robot manipulator was built to manipulate relatively high payloads of 10 to 100 kg (Fig. 1). For the majority of the task execution, high precision is not necessary, and natural constraints [1] can be used to achieve the required precision. By using cables as actuation, the moving mass of the robot is minimal, and lowering the inherent energy consumption.

Fig. 1. Cable-driven parallel robot that was used in this work. The actuated cables are highlighted with blue lines. All motors are equipped with a brake. (Colour figure online)

Research covering non-stiff manipulators for such applications typically attempts to mimic the general purpose robot requirements and applications, actively trying to eliminate vibrations induced by its natural dynamics [2]. This work argues that a better way is to compete on specific application use cases by making them more energy efficient, and not attempt to replace industrial robots altogether. The energy consumption can be minimised by optimising the executed trajectory, by optimising the execution time [3], or the followed path in space [4]. However, the robot is still constrained to follow a point-to-point (PTP) trajectory that might differ from the natural dynamics.

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Attaching springs to a SCARA robot in order to store energy in between cycles of a task has shown to significantly increase the energy efficiency [5]. The natural dynamics of a robot with series elastic actuators (SEA) were also used to achieve end effector velocities that were much higher than a stiff robot [6], limiting the required power of the motors. While these works show promising results, the concept of moving with the natural dynamics has not been explored extensively in robotic manipulation. To hold a constant position during control, usually a standstill torque is generated in the actuators, continuously consuming power. Instead brakes could be used to achieve the same effect. Commercial robots are already equipped with normally-on brakes, consuming a constant power to deactivate the brakes. Thus, utilising the brakes in the control has a double positive effect on the energy consumption. The downside being that it introduces a time delay to (de)activate them, and the user usually cannot individually control the brakes of an industrial robot. Brakes have already been used to passively control robots that experience an external force [7,8]. Braking a joint implies a geometric constraint, discretely changing the natural dynamics of the system. This constraint can be easily incorporated in the kinematic control. A common challenge in cable robots is the redundancy resolution [9,10]. As will be explained in Sect. 3.1, the method described in this work will solve this by selectively disabling the motors. The task context associated with the application determines the requirements and constraints throughout the task execution. For transportation tasks, these requirements and constraints are subject to change during the operation. For example, the placement or insertion of a payload typically requires a higher precision than the transportation in free space. This knowledge can be taken into account to split the task into multiple simple subtasks with different requirements. For these subtasks, specific controllers and mechatronics can be developed that focus on their robust execution, and their specific requirements. Monitors can be used to track the trends of the continuous execution, and coordinate the discrete switching between the controllers. Thus, for flexible robots it makes sense to focus on such tasks that do not have strict precision requirements, or where this precision does not need to originate directly from the control. Summarising, the contributions of this paper are the following: – Exploiting the natural dynamics of the system for higher energy-efficiency task execution, while still fulfilling the required position precision of the task. – Using insights about the task and the mechanical system to develop simple, robust continuous controllers with realtime task execution monitors that feed into a discrete task execution. – Achieving the required task precision by making use of natural constraints of the environment, or artificially induced constraints on the cable lengths by means of braking.

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187

Task Specification

As specified earlier, the methodology focusses on a transportation task. The goal is to place a payload next to a previously placed payload, or mechanical constraint (Fig. 2). The concrete execution will be explained in Sect. 3.2.

Fig. 2. Different subtasks to be executed. The arrows indicate the motion in each subtask. The numbers indicate when a monitor triggers that indicates the end of the current subtask. Cables depicted in green are actuated, in orange can passively move, in red are constrained by a brake. (Colour figure online)

2.1

Assumptions and Knowledge

The task specification of the use case investigated in this work makes following assumptions: – The payload is not fragile and the environment can be used to mechanically dampen vibrations, without damaging the payload or the environment. – The world model information, such as the position of objects already placed in the workspace, is known to the controller. – Motor brakes can be individually activated and controlled. – The top motor is positioned over the previously placed payload, such that the current payload can be swung over it. – Grasping the payload is out of the scope of this paper The control approach makes use of the following knowledge in the task specification: – The payload has a significant mass, thus gravitational force can be used to ensure cable tension. – The execution can be split up in a lifting, transporting, dropping and (optional) fine-positioning state.

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Energy Measures

The total electric power consumed by a single motor is given by: Pel = Pmech + Pmech,loss + Pel,loss = V Ia ,

(1)

with Pmech the mechanical power, Pmech,loss the mechanical losses due to friction in the motor and transmission, Pel,loss the electrical losses of the motor, V the voltage applied to the motor, and Ia the armature current. The electrical losses are typically dominated by the copper losses Pcu,loss , determined by the motor resistance Ra and armature current Ia : Pcu,loss = Ra Ia2 .

(2)

The energy consumption can be found by integration of the power over time t:  T Eel = Pel dt. (3) 0

2.3

Approach

This paper describes three concepts with the aim of a robust, energy-efficient robot control. While these concepts are rather straightforward and intuitive, they are not yet utilised in mainstream manipulator control. It is not argued that all of these principles need to be used, but if the (sub)task allows it, using any of these principles can have a positive impact on the energy-efficiency. Contextual Prior Knowledge: When humans perform a transportation task, they do not perform strict PTP motions such as traditional industrial robots. Instead, movements with a certain tolerance on the position are performed. This allows the natural dynamics of the system to be exploited, as will be explained in the following subsection. Typically, the tolerances come from knowledge about both the environment and the task context. For example, the spatial constraints, fragility of the payload, if a certain part of the task requires a higher precision, etc. It is clear that this knowledge precedes the task execution and determines how the human will perform the task. The execution is generally done in multiple states, e.g., picking up the payload, moving and placing near the target position, making small adjustments when necessary. This knowledge is used to split up the task in multiple subtasks and identify the different requirements. Robust controllers and monitors are then developed to perform and coordinate between these subtasks. Examples of such requirements are crane like operations such as: lifting the load to a certain height, transporting it without colliding, and lowering the load until contact is made. The task also does not require high control precision throughout, but only for the initial grasping and final placement. In addition, this does not need to come only from the control. Geometric constraints such as the environment or a previously placed payload can be used to achieve this accuracy by sliding against them. This is further explained in Sect. 3.2.

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By using this knowledge, lower-cost (and often also lower-weight) hardware can be used, so that a more robust, energy efficient execution can be developed. Thus, for a repetitive task, the cost of designing and implementing a task-specific controller is not necessarily higher than a generic, less energy-efficient controller. Exploiting Natural Dynamics: In this work, the natural dynamics of the system are used to inject as little energy as possible, resulting in energy-efficient motions. However, precise control of the timing is lost when the system freely follows its natural dynamics. Due to the layout of the used cable robot (Fig. 1), when the end effector is in a fully constrained position, releasing the power of one (or more) of the motors, will result in a pendulum-like swing around the cables that are still powered, or braked. This swing is used in the control strategy to cover the horizontal distance while consuming a minimal amount of energy. Active Use of Brakes: Based on the context, certain subtasks may occur where a joint does not need to move. Instead of producing a constant standstill torque, it can also be opted to brake the joint. Another case occurs when the demanded motion is in line with external forces such as gravity. In case of a continuous brake, the brake force can be directly controlled to achieve a certain resulting force. With a discrete brake, a tolerance region can be determined between which the brake switches on-and-off to achieve a similar effect. Section 3.1 utilises this concept to drop the payload without driving the motor. The brakes can also be used to stop the natural dynamics, if necessary.

3

Control Strategy

The controller is of the hybrid continuous-discrete type. The task is split up in subtasks which are executed with specific continuous controllers. Monitors are used to trigger transitions in the discrete control, implemented as a finite state machine (FSM). 3.1

Continuous Control

At the lowest level, each of the motors is either controlled by a velocity PID, or a current controller. The latter offering more opportunities for energy savings, at the cost of a higher control design effort. Depending on the subtask and the joint, inverse kinematics are used to construct a Cartesian velocity controller, or a joint current controller. The layout of the cable robot is illustrated in Fig. 3.

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Cartesian Controller: The top motor act as a hoist, and is situated such that it can always deliver the majority of the gravity compensating force, and hence can take the role of a hoist. Depending on which side of the vertical plane (Fig. 3) the payload is situated, only the side motor that is in the same half plane will be able to deliver a force that can counteract gravity. The other one is not used for the manipulation. This reduces the redundancy to zero, meaning that a unique inverse kinematic solution exists, with two complementary discrete modes. The control input of the robot is the velocity of the non-redundant motors which is related to the rate of change of their cable length d˙i = ni ri θ˙i . With ni , ri , and θ˙i respectively the gear ratio, the drum radius, and the velocity of motor i. In the following, the symbol xj and xj signify parameter x of respectively the driven and non-driven side motor. The Jacobian can easily be derived from the kinematics. The resulting inverse kinematic equations are given by: ⎡ ⎤−1 −sin(αj ) sin(α1 )     ⎢ sin(α1 − αj ) sin(α1 − αj ) ⎥ d˙1 Vx ⎢ ⎥ =⎣ , −cos(α1 ) ⎦ cos(αj ) Vy d˙j sin(α1 − αj ) sin(α1 − αj )

(4)

when the movement is in the ‘push’ direction of the driven side motor (l˙i > 0), gravity is used as the driving force of the motion. To ensure that the non-driven side cable does not slack, a current, just slightly larger than the static friction,  is maintained in the non-driven side motor when the motion is along the −d˙j direction. Otherwise gravity ensures cable tension, and the motor is not powered.

Fig. 3. Kinematics of a planar, redundant cable robot. In the depicted scenario, the right actuator (depicted in blue) is not being driven. (Colour figure online)

Current Controller: The current controller is used when one of the actuators is braked. Braking an actuator implies a geometrical constraint, such that the end effector has to be on a spherical surface with a radius that is determined by the cable length. This reduces the mobility of the end effector to 1 DoF in a plane, where the other joints can be used to move along the circular constraint. This results in simple kinematics (assuming the braked cable remains tensioned). This method is used when the desired motion is in the direction of gravity, and thus the control can occur passively. E.g., when dropping the payload the side motor brake is controlled to avoid a holding torque (Fig. 4).

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Fig. 4. Dropping the payload by passive control. Red circle segments indicate the side motor is braked. Orange segments indicate that the side motor is passive, allowing the cable to move freely due to gravity. (Colour figure online)

3.2

Discrete Control

The aforementioned continuous control is implemented for each of the specific subtasks by means of a higher level discrete controller. Figure 2 illustrates the different subtasks. Each circled number represents when the end criteria monitor of one of the discrete states should trigger, which serves as a signal for the discrete controller to go to the following state. The high level control consists of the states illustrated in Fig. 2. 1 indicates the system is operational and the first subLifting State: Monitor  task can be executed. The load is lifted upwards with a feed-forward Cartesian velocity input. When the end effector has reached a certain height hd , monitor 2 triggers, signalling the discrete controller to go to the ‘swing state’. hd is  such that the payload does not collide with previously placed payloads, nor the ground during the swing state. This height can be determined by, e.g., a camera system and is assumed to be known in this paper. During this state, energy is injected into the payload and stored as gravitational energy. Swing State: Releasing the side motor power causes the payload to execute a free fall. However, the top motor brake is activated, constraining the motion along a circle, resulting in a pendulum-like motion. During this motion, the direction of the end position is monitored. After half a period, when the end 3 triggers. This causes the brake of the other effector reaches its apex, monitor  side motor to activate, holding the payload steady. A small amount of energy is consumed by a single side motor to maintain cable tension. By keeping the brake of the top motor active, no holding torque (and thus no energy) is required. Drop State: After the swing, the load is dropped to the ground in the vicinity of the target position by applying braking actions (Fig. 4). During the dropping motion, the current of the top motor is monitored. A sudden discrepancy of this value can be used to detect an impact force. This is used to detect the collision 4 Which in turn when the payload touches the ground, triggering monitor . switches to the fine-positioning state.

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Fine-Positioning State: A sequence of actions is performed to place the payload at the target position. First the load is slightly lifted upwards until the cable length l2 is such that the payload can not collide with the ground. This 5 Which causes the top motor to brake and the side motor to triggers monitor . release power, causing a free swing motion, at very low speed until the payload 6 Afterwards collides with the previously placed payload, triggering monitor . the payload is moved straight down with a Cartesian controller, until it collides 7 terminating the fine-positioning and the with the ground triggering monitor , task execution. To lift the payload a small amount of energy is injected, which is also true for the final drop since this is done with active actuation.

4

Experimental Set-up

These concepts were validated on a parallel cable-driven robot (Fig. 1). The system is able to manipulate a payload of 14 kg in the vertical plane, with a workspace of 1 × 1 m through a 3 DoF redundant actuation. A 750 W Beckhoff AM8032 motor with a gear ratio of 70, controlled by a Beckhoff AX5203 drive, is positioned at the top such that it mostly compensates the gravitational force of the payload. It has a normally-on electromagnetic brake, allowing discrete braking actions. A power of Pb,top = 11 W is required to deactivate the brake. The cable is clamped and wound around a 4.2 cm radius drum that is connected at the shaft output. The payload is connected at both sides to a 188 W BLDC motor (Fig. 5). Both motors are controlled through a VESC drive, an open-source hardware project [11]. The motors have an internal gear ratio of 8, and are connected by a timing belt and pulley system with an additional gear ratio of 30/16 to a rotary shaft with a radius of 8 mm. The cables are directly wound around this shaft, and guided along pulleys towards the side of the payload (Fig. 1 and 5). The motors are equipped with normally-off electromagnetic brakes, instead of normally-on due to long supplier lead times on the latter. However, in an industrial application the robot would be equipped with normally-on brakes. Thus, the brakes will be treated as normally-on during the energy consumption analysis. A power of Pb,lef t = Pb,right = 8 W is necessary to activate the brakes.

Fig. 5. BLDC motor, brake and timing belt transmission used for the side actuation.

All the used cables are made from Dyneema SK78 with a diameter of 1 mm. These cables are lightweight, have a low working stretch (< 1%), and can carry

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up to 1.95 kN. The current Ia and voltage V supplied to the motors are directly obtained from the drives. The control feedback relies on the position of the end effector. In most cable robots this can be derived from the encoder positions, but since cable tension cannot be ensured during the free swing of the payload, a camera with a sensor size of 1920 × 1280p is used. This allows direct end position tracking of the end effector at a rate of 150 Hz with a resolution of ±1 mm.

5

Experimental Results

The proposed approach was experimentally compared against a sequence of PTP motions, each with a trapezoidal velocity profile. The trajectories corresponding to both methods are shown in Fig. 6. The PTP-trajectory performs a vertical lift up, followed by a horizontal transportation, a vertical drop slightly above the ground, and finally a horizontal and vertical movement that finishes the placement. Parts of the trajectory that are similar to the end criteria of the proposed method are marked with circled numbers in the figure The PTP-trajectory does not make use of a contact with the 3 to . 5 ground to detect when the drop has ended, it goes straight from  It should be noted that the side motors are not intended for servo applications, causing jittery motions at low speeds. This caused a chattering effect when the PTP controller was used, since an exact position needs to be followed, 1 and ). 2 The proposed controller does resulting in higher jerks (e.g. between  not rely on position tracking, and as such does not induce this chattering.

Fig. 6. Executed trajectory of the proposed and the PTP controller.

The consumed energy is directly related to the total execution time. The lift and drop state are executed at the same speed, since these are similar and interchangeable parts in both trajectories. The other parts of the PTP-trajectory are scaled such that the same total execution time is achieved, and the maximum motor velocity is not exceeded. The energy consumption of each motor is given by eq. (3), and is depicted in Fig. 7. Table 1 gives the consumed energy of each motor during each subtask, and the cumulative energy consumed by all motors. 1 → 2 energy is injected to overcome gravity. During  2 → 3 the During  biggest difference in energy consumption occurs. This is mainly because the top

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Fig. 7. Energy consumption corresponding to the trajectories of Fig. 6, for each motor, as well as for the whole system. Table 1. Energy consumption during each subtask, and the cumulative total. Subtask energy consumption [J] 1 → 2  2 → 3  3 → 4  4 → 5  5 → 6  6 → 7  Top

PTP 183.87 Swing 199.06

162.81 7.16

94.76 76.89

Left

PTP 62.85 Swing 71.22

18.08 0.17

21.59 0

Right

PTP 5.90 Swing 6.27

47.77 14.98

27.95 0.01

489.81 298.85

616.02 375.74

Total PTP 263.35 (Cumulative) Swing 276.54

27.97

26.46 9.10

18.80 11.15

0

24.05 0

24.45 0

3.56

6.40 0.15

3.99 0.71

407.27

651.34 416.52

674.54 428.38

motor, which consumes the most power, is braked in the proposed method. In the proposed method no force is generated in the left motor from this point onward. Either gravity produces tension in the corresponding cable for the rest of the motion, or the motion occurs in the half plane where this motor is disabled. In this particular experimental setup, the weight of the payload cannot overcome the friction of the top motor, due to its high gear ratio. Thus, it still needs 3 →  4 of the proposed to be powered in order to drop the payload during  method. However, by using the brakes of the right motor to control the drop, instead of actuating the motor, no power is consumed by that joint compared to the PTP method. For the PTP controller, the brakes are continuously energised and consume a constant power. In the proposed control method, the brakes are only energised when it is necessary (Fig. 8). The consumed energy is calculated by multiplication of the on-time of the brake, with its power consumption (Table 2).

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Fig. 8. Time during which each brake is active in the proposed method. The left brake is omitted from this figure since it is always off. Table 2. Brake energy consumed for the PTP- and the proposed controller. Brake energy consumption [J] Left Right Top Total PTP

104.37 104.37 170.79 379.54

Swing 103.73 94.03

97.60

295.36

As indicated in Tables 1 and 2, the total energy consumed by the robot with the proposed approach is 723.74 J compared to 1054.08 J of the PTP approach, which is a gain of more than 30%.

6

Discussion and Conclusion

In this paper a novel energy-efficient control method was introduced. This method is built upon simple principles, and only uses the kinematics of the system. It does not rely on complex models and does not need extensive quantitative identification. The dynamic effects in the motors and cables could be neglected. However, more effort is required to specify the task in the control execution. While this might be beneficial for repetitive tasks with high cycles, for task with low cycles, the added effort might not outweigh the energy gains. Based on experimental results of a transportation task, the total energy consumption was 31% lower with the proposed method compared to a conventional PTP controller, on exactly the same hardware and software setup. The current implementation is not yet optimised, so it is expected that the energy can still be lowered if the execution is optimised. This would however require knowledge of the dynamics of the system, and thus a more in-depth identification. The principles of this method can be used for any task that have similar tolerances on path following and timing as the (subtasks) of the described use case. The validation set-up was built to a large extend with low-cost hardware that was readily available. The jittery behaviour of the side motors (especially at low speeds) currently limit the achievable tracking behaviour. But even with this non-ideal hardware, the task could be reliably executed.

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In future work additional actuation will be added such that the top motor can be moved horizontally, and the swing can occur wherever in the workspace, giving more flexibility in the execution.

References ´ 1. Eppner, C., Deimel, R., Alvarez-Ruiz, J., Maertens, M., Brock, O.: Exploitation of environmental constraints in human and robotic grasping. Int. J. Rob. Res. 34(7), 1021–1038 (2015) 2. Malzahn, J.: Modeling and Control of Multi-Elastic-Link Robots under Gravity. Ph.D. thesis, Technische Universit ¨ at Dortmund (2014) 3. Pellicciari, M., Berselli, G., Leali, F., Vergnano, A.: A method for reducing the energy consumption of pick-and-place industrial robots. Mechatronics 23(3), 326– 334 (2013) 4. Paes, K., Dewulf, W., Vander Elst, K., Kellens, K., Slaets, P.: Energy efficient trajectories for an industrial ABB robot. Procedia CIRP 15, 105–110 (2014) 5. Goya, H., Matsusaka, K., Uemura, M., Nishioka, Y., Kawamura, S.: Realization of high-energy efficient pick-and-place tasks of SCARA robots by resonance. In: 2012 IEEE/RSJ International Conference on Intelligent Robots and Systems, pp. 2730–2735 (2012) 6. Haddadin, S., Huber, F., Albu-Sc¨ affer, A.: Optimal control for exploiting the natural dynamics of variable stiffness robots. In: 2012 IEEE International Conference on Robotics and Automation, pp. 3347–3354 (2012) 7. Hirata, Y., Suzuki, K., Kosuge, K.: Improvement in the performance of passive motion support system with wires based on analysis of brake control. In: 2011 IEEE International Conference on Robotics and Automation, pp. 4272–4277 (2011) 8. Andreetto, M., Divan, S., Fontanelli, D., Palopoli, L.: Passive robotic walker path following with bang-bang hybrid control paradigm. In: 2016 IEEE International Conference on Intelligent Robots and Systems, pp. 1054–1060 (2016) 9. Lamaury, J., Gouttefarde, M., Chemori, A., Herve, P.E.: Dual-space adaptive control of redundantly actuated cable-driven parallel robots. In: 2013 IEEE International Conference on Intelligent Robots and Systems, pp. 4879–4886 (2013) 10. Oh, S.R., Agrawal, S.: A control lyapunov approach for feedback control of cablesuspended robots. In: Proceedings 2007 IEEE International Conference on Robotics and Automation, pp. 4544–4549 (2007) 11. VESC-project. https://vesc-project.com/. Accessed 3 Jan 2023

Modeling, Simulation, and Control of a “Sensorless” Cable-Driven Robot ¨ ¨ 3 , Atakan Durmaz1,2(B) , Ozlem Albayrak3 , Perin Unal 1,2 and M. Mert Ankaralı 1

Electrical and Electronics Engineering Department, Middle East Technical University, Ankara, Turkey {atakand,mertan}@metu.edu.tr 2 Robotics and Artificial Intelligence Technologies Application and Research Center (ROMER), Middle East Technical University, Ankara, Turkey 3 TEKNOPAR Industrial Automation, Ankara, Turkey {albayrak,punal}@teknopar.com

Abstract. This paper focuses on the modeling, simulation, control, and experimental validation of a planar cable-driven robot system that operates on the vertical (x-z) plane. Cable-driven robots have recently gained significant attention due to their successful commercial applications, such as spider cameras in large areas like stadiums. In this study, we model the robotic system as a planar dynamical system driven by visco-elastic tension elements (i.e., cables) attached between the four corners of the rectangular end-effector and corners of the workspace where the electric motors are connected. We assume that limited sensory information is available to the controller, such that we can only measure motor torque, and no direct information is available from the end-effector. This sensorless measurement assumption poses significant control challenges. We propose a novel control approach that adopts a parallel feedforward velocity and feedback force/torque control topology. The control inputs of the system are assumed to be the reference motor velocities, as we utilize industrial servo controllers with built-in velocity control capabilities. We model the motor dynamics as a first-order low-pass filter to account for the phase lag between the reference and actual motor commands. We first simulate the closed-loop system and test the effectiveness of the control policy under different unknown system parameters such as stiffness, damping, and motor lag. We then experimentally verify the topology on an actual experimental setup. We believe that these results are promising for future cable-driven robotic applications, especially for systems with limited sensory equipment. Keywords: Cable-driven mechanism · sensorless control · series-elastic-actuation · parallel feedforward and feedback control

1

Introduction

Cable-driven robots are a type of parallel robots that have multiple actuators merged in one end effector. Although other parallel robots utilize rigid bodc The Author(s), under exclusive license to Springer Nature Switzerland AG 2023  S. Caro et al. (Eds.): CableCon 2023, MMS 132, pp. 197–208, 2023. https://doi.org/10.1007/978-3-031-32322-5_16

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ies meeting at the end effector, cable-operated robots use cables controlled by actuators at every corner [6]. Compared to the parallel robots with actuators connected to the rigid bodies meeting at the end effector, cable-driven robots have a higher ratio of the workspace area to the area of the mechanism since cables can be wound in the winch system. Therefore, cable-driven robots present a more convenient solution to perform similar tasks [3]. Cable-driven robots have become popular with commercial applications; for example, cable-operated robots are being used as spider cameras to record events, such as football matches, concerts, and performing arts. Since there is a requirement for wide-angle shooting to record all activity, there is a high risk of using a drone over a group of people in a closed showplace. These led us to use cableoperated robots. Our aim in this study is to design and control a cable-driven robot via minimal sensory information and instrumentation, where the main future focus is construction applications [2,5]. For example, we envision that a cable-driven solution can be developed such that the robot can apply plaster, paint a large flat wall, etc. Utilizing a cable-driven robot for such tasks could make it possible to benefit from the same workforce to perform other tasks. Table 1. The mechanism layout (Corners are defined as the axis of rotation of the winches, in meters). Part

Corner 1

Corner 2

Corner 3

Corner 4

Body

(1.05,0.58)

(−1.05,0.58)

(−1.05,−0.58)

(1.05,−0.58)

End effector (0.08,0.095) (−0.08,0.095) (−0.08,−0.095) (0.08,−0.095)

Different methods exist used to analyze and control cable-driven mechatronic systems [11]. However, in the vast majority of the studies and applications realtime (or quasi-real-time) position and pose information of the end effector is available through different sensors or cameras [1,7], hence easing the closedloop control problem. However, obtaining such information brings additional costs and maintenance requirements and decreases robustness to the failures due to increased instrumentation and complexity. In that respect, the core goal of this paper is to develop a control algorithm that can regulate the position and pose of the end-effector using minimal sensory information. We only rely on measurements on the motors (speed and torque) and assume that we don’t have any direct sensory information from the end effector. To achieve such a challenging task, we developed a parallel feed-forward and feed-back control structure that can still effectively track the reference trajectories. Previously, we published a very limited version of this study (in Turkish) that lacks the intensive modeling and simulation analysis conducted in this paper, which only focuses on some empirical observations [9] (Fig. 1).

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Fig. 1. Experimental cable-driven planar robot adopted in the study.

2 2.1

Materials and Method Description of the Overall System

Mechanical Infrastructure. In the mechanism, 3 mm diameter kevlar cables by ISO 10325 standards are used. The layout of the mechanism is presented in Table 1. Throughout this study, the origin point is defined as the center of the mechanism by conventional wisdom. The radius of the winch belonging to the circle around which the cables are rotated is 0.0195 m. In addition, the radius of the cables is 1.5 mm and the gaps on the threads are selected as 1.6 mm, which protects the cables against abrasion. In the first tests, due to the compression force in the radial direction as a result of the tensile force, we started the experiments with the cables with a radius of 2 mm to have a negligible compression effect. However, we changed the cable radius to 1.5 mm due to the safety issues that may occur due to the abrasion in the cables. Since our method takes into account the compression force in the cables, we anticipate that this change will not affect the results. Hardware Infrastructure. We used Eight Siemens 1fk7042-2af71-1qa0 servo motors connected to Siemens Simotion D445-2 to control the mechanism. Since we have more than six motors in the mechanism, we required to use Siemens S120 driver for four rear servo motors. We partitioned the front and rear motors between the Siemens Simotion D445-2 and Siemens S120 driver, to prevent any delay that may occur between the engines located on the same plane. On the other hand, we observed that there was a negligible delay between the front and rear planes; however, this did not make any difference in our intended results.

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Software Infrastructure. We used Siemens Simotion Scout v5.1 IDE and ST(Structured Text) programming language, which can perform all the tasks we aimed, for programming. 2.2

Modeling and Assumptions

Fig. 2. Illustration of the cable-driven planar mechanism. The distance between the front and the rear planes is 1 m, and they are controlled in the same way to hold the end effector parallel to the ground. In future work, different yaw and pitch angles can be achieved to increase the covered area for the corresponding task.

In this section, we describe the modeling of the series elastic cable-driven planar mechanism, its kinematics, dynamics, and associated simplifications & assumptions. The model, illustrated in Fig. 2, consists of a rigid body with inertia I, mass m, length lb , and height hB , to which four elastic cables are attached from its four corners. The position and orientation of the body are represented by a bodyfixed frame B with respect to an inertial world frame W, located at the center of the workspace which has a length and height of lw , and hw respectively. The corner positions are also defined with respect to world frame, where origin is the center of the end effector, with ln /2 and hn /2 denoting x and y coordinates, respectively. We express the Cartesian position and orientation of the body, i.e., T B, with respect to W by W PB = [xB yB ] , and α respectively. In this  context, W we can define the rotation matrix, RB , and its derivative, Dα W RB to model some of the kinematic transformations between W and B:

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Dα

W

 RB (α) =

cos α − sin α sin α cos α

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   − sin α − cos α RB (α) = cos α − sin α

(1)

One can express the position of body corners, as illustrated in Fig. 2, with respect to B as: B

 P c1 =

ln 2 hn 2

 ,

B

 P c2 =

− l2n

 ,

hn 2

B

 P c3 =

− l2n − h2n



& B P c4 =



ln 2 − h2n

 .

(2)

where as following rigid body transformation computes the location of each corner i with respect to the inertial frame, W: W

Pci = W RB (α)B Pci + W PB

(3)

We also need to derive the kinematic transformations that map the body velocities to the corners’ velocities in W. Both the translational velocity, W P˙B and the rotational velocity, α, ˙ of the body affect the corner velocities. Following transformation for each corner i, performs this mapping: W

P˙ ci = Dα

W

 RB (α) B Pci α˙ + W P˙B

(4)

Four motors located at the corners of the workspace drive and control the motion of the mechanism. The positions of motor drive locations with respect to the inertial frame, W, take the following form W

 Pm1 =

lw 2 hw 2

 ,

W

 Pm2 =

− l2w

 ,

hw 2

W



Pm3

− l2w = − h2w

 &

W

 Pm4 =

lw 2 − h2w

 .

(5) To derive the equations of motion of the dynamics and develop control algorithms, we also need the cable length and rate of change of cable length, i.e. ρi &ρ˙ i , for each connection. The cable length for each cable can be computed via: ρi = ||W Pmi − W Pci ||2

(6)

To compute the rate of change of cable lengths and derive the Jacobean matrix, we first find unit norm vectors, vi ı ∈ {1, 2, 3, 4}, which represents the direction of forces applied by each cable: W

P m i − W P ci (7) ρi We illustrate these vectors in Fig. 2 with dark blue arrows. Based on these vectors, simple inner product projection yields the rate of change of cable length of a single cable: (8) ρ˙ i = −vi , W P˙ci  = −viT W P˙ci vi =

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As we stated before the cables can only apply tension force, thus the force vector acting on the body generated by cable i can be represented by Fi = fi vi where fi ∈ R+ .

(9)

In this context, we can derive the translational and rotational dynamics of the system as   4  0 W ¨ m PB = vi fi + −mg i=1 (10) 4  W  W Iα ¨= RB (α) Pci × vi fi i=1

We can also re-write the dynamics in matrix form by constructing the Jacobian matrix ⎡ ⎤ ⎡ ⎤  W    f1 0 ⎢ ⎥ ¨ · · · ⎢ f2 ⎥ ⎣ ···  vi m PB  + −mg ⎦ = · · · W RB (α)W Pci × vi · · · ⎣ f3 ⎦ Iα ¨ 0 f4 (11) ⎡ ⎤  W  0 m P¨B = Jf + ⎣ −mg ⎦ Iα ¨ 0 To finalize the derivation of the dynamical system of equations, we need to find the cable forces. As we stated before, we assume that motors drive the system in a series elastic framework. In this context, the motors pull the cables from the corners of the workspace by manipulating the rest-length of the cable and its rate of change. In this framework, the individual motor dynamics produces two outputs, ρoi and ρ˙ oi (i.e. rest length and its rate of change), and feeds them to the plant dynamics. Unlike some other studies that assume that cables’ visco-elastic characteristics remain constant, we follow a more realistic and parsimonious modeling approach and assume that both the spring constant and damping coefficient depends on the effective cable length ρi [10]. Thus, instantaneous spring and damping coefficients are computed using the following relations: ki =

κ ρi

,

di

=

β ρi

(12)

Based on these state-dependent coefficients we can compute the force on a cable fi = min{ki (ρi − ρoi ) + di (ρ˙ i − ρ˙ oi ) , 0}

(13)

Note that cables cannot apply compression forces. Thus, if the computed effective force on a cable ends up being negative (which means that it is starting to apply compression force) the min argument in (13) assigns fi = 0.

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We further assume that the motors are controlled via high but limited bandwidth motion controllers, such that motors track a reference velocity signal, ρ˙ ∗oi (t). In this regard, we model this tracking behavior with an LTI transfer function that has a first-order low-pass filter structure. Figure 3 illustrates the block-diagram structure of the motor dynamics. Note that this series elastic modeling method is only used at the simulation, whereas the actual controller uses calculated and measured force values corresponding to each motor in the feedback loop.

Fig. 3. Block diagram structure of the motor control dynamics.

3

Control Policy

In our study, we assume that direct measurement from the body/end-effector is not available. Thus we use the “sensorless” control term to define our methodology. In cable-driven systems, to get measurement feedback from the body, the most common approach is to utilize external (mostly expensive) vision-based tracking systems that can measure/estimate the state variables of the body (pose and velocities) in real time. A different approach is to instrument some types of equipment and sensors on the body (such as line encoders, IMUs) and communicate with the base control station via wireless protocols. The most apparent disadvantage of both approaches is the extra cost, which can be quite substantial if the requirements on sampling-time, communication delay, etc. are critical and strict. On the other hand, additional measurement and instrumentation increase design, manufacturing, and maintenance complexities. By going with a sensorless approach, we technically eliminate these problems while also making the control problem much harder. In this study, we adopted a parallel feedforward and feedback control policy that can track a given reference trajectory on cartesian body coordinates. 3.1

Feedforward Velocity Control

To develop the feedforward/open-loop component of the control policy, we make some “assumptions” and develop a purely kinematics-based controller. First of all, we ignore the lag associated with the motor control dynamics and consider that ρ˙ oi (t) ≈ ρ˙ oi (t)∗ , where ρ˙ oi (t) is the actual rate of change of cable length, ρ˙ oi (t)∗ is the control input that will be produced with our controller. In this phase, we also ignore the compliance in the cable; in other words, we treat the cable as an almost rigid component. Under these assumptions, we can assume that ρ˙ i (t) ≈ ρ˙ oi (t) ≈ ρ˙ ∗oi (t) (14)

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Technically we assume that motor velocity can directly/kinematically control the rate of change of cable length. Under this simplification, we can develop a kinematic motion controller by computing the desired cable velocities based on the reference trajectory defined on the body coordinates. Let W PB∗ (t) be a smooth (i.e. differentiable) and continuous reference trajectory and W PB∗ (0) =W PB (0). To track this time-dependent trajectory body velocities should satisfy: d W ∗  (15) P˙B (t) = PB (t) , ∀t dt Note that in our study, we want the body to preserve its original orientation at all times, i.e. W

α∗ (t) = 0 & α˙ ∗ (t) = 0, ∀t

(16)

Since we already assumed that we can have direct instantaneous control on the cable velocities, we only compute desired motor velocities, i.e. ρ˙ ∗i (t) based on desired body velocity variables, W P˙B∗ (t) and α˙ ∗ (t) using the kinematic relations derived in Sect. 2.2. We can also use the Jacobean matrix that we derived in (11) to project cable forces on the body forces to transform the body velocity variables to cable length change rates. In this context, we can compute the control inputs associated with the feed-forward control as ⎡ ∗ ⎤ ρ˙ o1 (t) W ∗  ⎢ ρ˙ ∗o (t) ⎥ P˙B (t) ⎢ ∗2 ⎥ = −J T (17) ⎣ ρ˙ o (t) ⎦ 0 3 ∗ ρ˙ o4 (t) The negative sign in (17) comes from the fact that we treat the tension force as positive which has a shortening effect on cable leg lengths. The feedforward controller feeds this control input vector to the next action the control policy. 3.2

Feedback Torque Control

To lay the foundations of the feedforward velocity control part of the algorithm, we performed some preliminary assumptions. Specifically, we neglected both the low-pass filter motor dynamics and compliance in the cables. However, these assumptions as well as other possible dynamical uncertainties and effects will lead to some discrepancies between the actual body trajectories and desired ones. Obviously, we need to adopt feedback control-based corrections for these kinds of deviations. Normally, the most obvious choice would be to obtain feedback from directly the outputs, i.e. the state-variables (pose and velocities) of the body. However, in our application we do not have any direct measurement from the body, thus we name our approach as “senseless”. Even though we drive the cables based on the series elastic velocity control approach, we can still measure the cable forces based on the current/torque measurements of the individual motors. To apply a feedback control based on these measurements, we need to compare them with respect to some reference/desired

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force values. We compute these reference force trajectories based on two important constraints. First of all, we make sure that the reference force vector satisfies the static force balance on the body. In this context at a given time t, the reference force vector should satisfy the following equation ⎡ ⎤ 0 ⎣ mg ⎦ = J ∗ f ∗ , f ∗ ∈ R4 (18) 0 Note that to compute J ∗ , one needs to use the reference trajectory. The equation in (18) has 4 degrees of freedom since f ∗ ∈R4 , but only three equalities need to be satisfied. In this sense, it is an over-determined linear problem, and there exist infinitely many solutions. We need to use this extra degree of freedom to ensure that computed forces are always positive since cables can only apply tension (positive force). Moreover, practically, it is preferred that cable forces are always above some minimum threshold at all times in cable-driven systems. In this context, we have the following linear constraint min{f1∗ , f2∗ , f3∗ f4∗ } ≥ fmin > 0

(19)

This converts the equation into a linear programming problem, which we solve using standard optimization techniques.

Fig. 4. Block diagram structure of the whole system.

Our algorithm then computes the error between desired and measured force vectors, i.e., ef = f ∗ − f , multiply the error vector with a scalar gain Kf (i.e., P control action) and supplies this feedback correction component to the addition block. The addition block sums the outputs of feedforward and feedback controllers and feeds them to the motor dynamics. Figure 4 illustrates the blockdiagram structure of the whole system. Note that there are no jumps in the torque and force distributions along a trajectory due to the low pass characteristics of the servo motors we selected to be able to perform torque control slowly. The reason for selecting the P controller is due to the memoryless structure of the Siemens Simotion platform we used.

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Results and Discussion

During the experiments, we aimed to minimize position errors and optimize the control of the end effector by using only torque measurements. We observed that there is an increase in errors as the end effector moves longer distances. However, the errors always stay below 10%, which is an acceptable level for applications in the construction field. Also, since we need to ensure a safe work environment, we selected the cables to work under a maximum tensile force of 1000N safely, and we limited the motor torques at 10 Nm. These limitations allowed us to work confidently from −40 to +40 cm on the x-axis and from −30 cm to +30 cm on the y-axis.

Fig. 5. Torque measurements from each motor.

Actual position(m) for the desired point at (0.25 m, 0.00m)

0.25

Actual x position Actual y position

0.2

Position(m)

0.15

0.1

0.05

0

-0.05 0

50

100

150

200

250

300

Time(ms)

Fig. 6. Experimental measurements of the position(m) vs time(ms).

In all of our experiments, the robot moved to start from the point (0.0 m, 0.0 m) to the desired points (xd , yd ). In Fig. 6, the result of an experiment with the aimed point (0.25 m, 0.0 m) can be seen. As shown in Fig. 5 above, the torque values during the experiment stay below 2.6 Nm, which satisfies safe conditions, as we have stated previously. Corresponding simulation results can also be seen in Fig. 7 below, as well as simulation results for (xd , yd ) = (0.25 m, 0.25 m). Since the verification on the actual system is too time-consuming not to overload the paper

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with too many unnecessary figures, we limit the results to two sufficient examples to show the harmony between the simulations and the actual response. 8 motors behave in two ways with groups of 4 motors. Since the robot moves in the + xaxis, 4 motors numbered 1 and 4 in each plane reacted with the maximum torque of 2.6 Nm. The other four motors responded with smaller maximum torque values. The minus sign on the y-axis in Fig. 5 shows the clockwise direction; therefore, we focus only on the magnitudes instead of the signed values.

Motor Torques for (x d,y d) =(0.25m,0.00m)

2

Desired vs actual positions for (xd,y d)=(0.25m,0.00m)

0.3 Motor 1 Motor 2 Motor 3 Motor 4

1.8 1.6

0.25 Desired y position Actual y position Desired x position Actual x position

0.2 1.4 0.15

1.2 1

0.1

0.8 0.05

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0.2 -0.05 0

0 0

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1.2

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0.4

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0.8

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1.4

(a) Simulated torque values. (b) Simulated position values. Motor Torques for (x d,y d)= (0.25 m, 0.25 m)

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Desired vs actual positions for (xd,y d)=(0.25m,0.25m)

0.3 Motor 1 Motor 2 Motor 3 Motor 4

0.25 Desired y position Desired x position Actual y position Actual x position

0.2 2 0.15 1.5

0.1

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1.2

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0.8

1

1.2

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1.4

(c) Simulated torque values. (d) Simulated position values.

Fig. 7. Simulated torque (a) and position (b) values for (xd , yd ) = (0.25 m, 0.00 m), and Simulated torque (c) and position (d) values for (xd , yd ) = (0.25 m, 0.25 m).

Also, the result of this experiment, as the actual positions on the x and y axes are shown in Fig. 6. The actual position values shown in Fig. 6 are obtained by OpenCV library, and filtering techniques with a camera located opposite to the mechanism for measurement purposes only [4]. The result, according to our observation for this example was (0.24 m, 0.01 m). As you can check, the error in the x-axis is 4%, less than 10%, as we said before. On the other hand, there is an error of 0.01 m in the y-axis, which is in the measurement error range (± 1 mm); therefore, this error is practically insignificant in this sense. In this research, we worked on the control of a cable-operated robot and tried to optimize the control of the robot to minimize errors. In comparison to the readily existing planar cable-driven robots, we worked on 8 motors instead of 4 motors and obtained a better control strategy and lower errors. If we look at the results of the study, the effects of our solution can be seen by the comparison of the actual positions on the x and y axes [8]. Figure 6 shows how effective

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our solution is compared to other systems with four motors and other control strategies by avoiding overshoot and, at the same time reducing the error below 10% range [8]. Finally, the system is tested at different desired points, and all results are consistent with the results submitted here, which shows the efficiency of the suggested solution. Acknowledgement. This work was supported by The Scientific and Technological Research Council of Turkey (TUBITAK) through Project 9170021.

References 1. Borgstrom, P.H., et al.: NIMS-PL: a cable-driven robot with selfcalibration capabilities. IEEE Trans. Robot. 25(5), 1005–1015 (2009) 2. Bruckmann, T., Boumann, R.: Simulation and optimization of automated masonry construction using cable robots. Adv. Eng. Inform. 50, 101388 (2021) 3. Gosselin, C.: Cable-driven parallel mechanisms: state of the art and perspectives. Mech. Eng. Rev. 1(1), DSM0004 (2014) 4. Huang, R., Pedoeem, J., Chen, C.: YOLO-LITE: a real-time object detection algorithm optimized for non-GPU computers. In: 2018 IEEE International Conference on Big Data (big Data), pp. 2503–2510. IEEE (2018) 5. Iturralde, K., et al.: A cable driven parallel robot with a modular end effector for the installation of curtain wall modules. In: ISARC. Proceedings of the International Symposium on Automation and Robotics in Construction, vol. 37, pp. 1472–1479. IAARC Publications (2020) 6. Jin, X.J., et al.: Four-cable-driven parallel robot, pp. 879–883 (2013) 7. Jin, X.J., et al.: Geometric parameter calibration for a cable-driven parallel robot based on a single one-dimensional laser distance sensor measurement and experimental modeling. Sensors 18(7), 2392 (2018) 8. Khosravi, M.A., Taghirad, H.D.: Robust PID control of fully-constrained cable driven parallel robots. Mechatronics 24(2), 87–97 (2014) ˙ et al.: Sens¨ 9. Kunduz, I., ors¨ uz Seri Elastik Tekni˘ gi ile Bir Kablo S¨ ur¨ uml¨ u D¨ uzlemsel Mekanizmanın Kontrol¨ u. Avrupa Bilim ve Teknoloji Dergisi 41, 324–330 (2022) 10. Tempel, P., Schmidt, A., Haasdonk, B., Pott, A.: Application of the rigid finite element method to the simulation of cable-driven parallel robots. In: Zeghloul, S., Romdhane, L., Laribi, M.A. (eds.) Computational Kinematics. MMS, vol. 50, pp. 198–205. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-60867-9 23 11. Zi, B., et al.: Dynamic modeling and active control of a cable-suspended parallel robot. Mechatronics 18(1), 1–12 (2008)

Validation of Emergency Strategies for Cable-Driven Parallel Robots After a Cable Failure Roland Boumann(B) , Christoph Jeziorek , and Tobias Bruckmann Universit¨ at Duisburg-Essen, Forsthausweg 2, 47057 Duisburg, Germany {roland.boumann,christoph.jeziorek,tobias.bruckmann}@uni-due.de

Abstract. This work describes the experimental validation of emergency strategies for cable-driven parallel robots after a cable failure. In previous work of the authors, two strategies have been introduced and verified within simulation. One strategy is based on the minimization of the robots kinetic energy, while the other strategy utilizes potential fields to guide the end effector to a safe pose. In order to validate those strategies on a prototype, a decoupling device is developed which induces a cable failure at the end effector of the SEGESTA prototype. Moreover, a very basic failure detection algorithm is proposed. Using the decoupling device, experiments are conducted. The robot detects the failure on itself, switches to emergency mode and applies one of the strategies. As both strategies safeguard the end effector, they are successfully validated. Keywords: cable-driven parallel robot · cable break · cable failure · force distribution · emergency strategies · nonlinear model prediction potential fields · failure detection algorithm · decoupling device

1

·

Introduction

Cable-driven parallel robots (CDPRs) commonly employ an end effector with a number of cables attached, coiled on digitally controlled winches. Since the industrial usage increases, safety requirements must be considered. For well-known components like cables or pulleys, guidelines and standards exist. Nevertheless, a small risk of a cable failure remains [7]. Several publications deal with strategies after a cable failure, failure modes and error detection. [14] gives first thoughts on conceivable faults and fault tolerance. The consequences of defect or slack cables leading to diverging cable forces are investigated in [10]. [2] proposes an emergency strategy by planning a dynamic trajectory along an elliptical path, while the platform can be outside the workspace after a cable failure (hereinafter referred to as post-failure workspace). In [11], algorithms for dynamical trajectory planning back into the post-failure workspace after cable failure are introduced for a cable driven camera system. An algorithm for planning a feasible straight line trajectory back into the postfailure workspace is proposed by Boschetti. It is extended considering drivetrain c The Author(s), under exclusive license to Springer Nature Switzerland AG 2023  S. Caro et al. (Eds.): CableCon 2023, MMS 132, pp. 209–220, 2023. https://doi.org/10.1007/978-3-031-32322-5_17

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models and successfully tested on a prototype using 3 degrees-of-freedom and 4 cables [3]. Here, the cable failure is represented by rapidly unwinding a cable. In [16], safety concepts for CDPRs are outlined, focusing on emergency braking and workspace monitoring. [13] propose strategies to identify and recover from failure in planar CDPRs using reconfiguration. In preliminary work [5] of the authors, two strategies for handling cable failure have been proposed based either on minimization of the systems kinetic energy or on potential fields. Contrary to the emergency strategies described in the literature, the author’s strategies seek to safeguard the end effector into the postfailure workspace after cable failure without predefining a trajectory. They have been verified in simulation for various numbers of cables, degrees-of freedom and thus redundancy. The strategy of kinetic energy minimization was extended by active reconfiguration in [7]. Moreover, post-failure workspace recovery through reconfiguration was demonstrated. This strategy was verified in simulation using spatial multibody models [8], investigating real-time capability of the algorithm. Furthermore, the malfunction of a conventional control algorithm in the case of cable failure was analyzed. Proposing the Nearest-Corner Method [4], realtime capability of the potential field method was demonstrated in simulation. Summing up, the methods proposed by the authors have never been validated on a prototype, which is the scope of this paper. Furthermore, to the authors’ best knowledge, no mechanism was proposed yet to decouple a cable from an end effector, enabling realistic testing. Indeed, most strategies were only tested in simulation or by unwinding a cable. Lastly, investigations on detection of a cable failure in a prototype are mostly absent. This paper is structured as follows: First an introduction on CDPRs and existing emergency strategies is given, highlighting the contribution of this work. Modeling fundamentals are explained in Sect. 2. The authors’ two emergency strategies – namely minimization of kinetic energy and potential fields – are introduced in Sect. 3. In Sect. 4 a decoupling device developed by the authors in order to simulate a cable failure on a real prototype is presented, along with a very primal failure detection algorithm (FDA). The post-failure workspace of the SEGESTA prototype is examined in Sect. 5. Within a test scenario, both emergency strategies are successfully validated on SEGESTA. Finally, a summary and an outlook are given.

2

Modeling Fundamentals

The robot’s platform, which is also the end effector, has a degree of redundancy of r = m − n with n degrees-of-freedom (DOF) and m cables. The coordinate system 6 P B - is fixed at the platform, which is referenced in the inertial system 6 -. B The orientation Φ of the platform with respect to 6 - is described by the rotation matrix B RP using yaw-pitch-roll angles. B rP is the position. Using both row vectors, the pose can be defined by B xP = [B rP , Φ]T .As depicted in Fig. 1, the base vectors are B bi , which can be denoted in B = B b1 . . . B bi . The vectors B bd,i describe the points where the cables enter the workspace. They can be

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derived using inverse pulley kinematics [12]. With the vectors P pi denoted in P = [P p1 . . . P pi ], where the cables are attached to the platform, the cable vectors B li can be determined. The inverse kinematics yields: B

li = B bd,i − (B rP + B RP P pi ),    B pBi

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(1)

For simplicity, the top-left index is omitted in the following for all variables referenced in 6 B -. Each cable’s direction is described by the unit vector νi with a tension fi in cable direction. The force exerted at the platform by each cable is li fi = fi · = fi · νi , 1 ≤ i ≤ m. (2) li 2 All cable tensions can be brought together in the cable force vector f ∈ Rm×1 . The external forces fE and torques τE acting onto the platform (except from the cable forces) are described by wE , including e.g. gravitational forces. Using the structure matrix AT , which can be derived from the robot’s Jacobian, the static force equilibrium at the platform is set up as ⎡ ⎤   f1 −fE νm ν1 . . . ⎢ .. ⎥ (3) = − wE = = AT f . −τE p1 × ν1 . . . pm × νm ⎣ . ⎦ fm The platform’s inertia tensor is B θP which can be derived using Steiner-Huygens relation, while its mass is mP . The vector B P - to the center of P rS points from 6 gravity. Setting up Newton-Euler equations [7] expressed in 6 B - leads to ¨ P + K(xP , x˙ P ) + Q(xP , x˙ P ) = AT f = −w. M (xP )x

(4)

¨ P are time derivatives of the end effector pose. The mass matrix of the x˙ P and x platform is M (xP ), while K(xP , x˙ P ) contains Coriolis and centrifugal forces

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and torques. Q(xP , x˙ P ) holds all remaining force and torques including disturbances, friction and gravitational forces. The cables are assumed to be nonsagging and massless. A minimum tension fmin needs to be maintained to keep the cables tensed, as they can only pull but never push. Considering mechanical load and motor capabilities of the robot’s components, a maximum tension of fmax should not be exceeded. For a desired trajectory, different wrenches w result and Eq. (4) needs to be solved for feasible cable forces. Several well known methods exist to find solutions to this problem [12]. Note, that modeling of the drivetrain system, friction and disturbances is neglected in this work. All values are given with three decimal places. Trailing zeros are omitted to save space.

3

Emergency Strategies

As introduced in Sect. 1, cable failure during the operation of CDPRs may have severe outcomes ranging from uncontrolled robot movements to collisions or loss of payload. Consequently, having emergency strategies is crucial. As displayed in former work of the authors, the post-failure workspace can be drastically smaller, increasing the chance of the end effector getting outside of it [7]. In this case, no static force equilibrium can be found and the platform will necessarily start to move. Moreover, common control approaches are likely to fail [8]. In the following, the failure of one single cable is assumed for simplicity, thus m − 1 cables still exist which can influence the platform. Without the broken cable, the remaining vector of cable forces gets f ∗ ∈ R(m−1)×1 , with the force ∗ ∗ , fmax ∈ R(m−1)×1 . Furthermore, the structure matrix becomes boundaries fmin T∗ n×m−1 , i.e. without the column of the broken cable. In former work of A ∈R the authors, two strategies to deal with a cable failure emergency scenario have been introduced and verified in simulation, see Sect. 1. They are briefly described in the following and validated on a prototype within this work. Minimization of Kinetic Energy. The main idea of the first strategy is to minimize the platforms kinetic energy after a cable failure, which is proportional to the square of its velocity, assuming a rigid body. If the velocity can be minimized, the system is led to a full stop. If it is able to stop and remains standing still in force equilibrium, it will then automatically be in the post-failure workspace. One major advantage of this method is that the final position does not need to be predefined. For this purpose, a nonlinear model-predictive control (NMPC) [1] is introduced, based on the given model in Sect. 2. The system is described using the nonlinear discrete state equations   x(k + 1) = f x(k), u(k) y(k + 1) = C x(k + 1).

(5)

  f x(k), u(k) is the nonlinear system function based on Eq. (4). The input vector u = f ∗ (k) contains the remaining cable forces. The state vector of platform ˙ rP (k), Φ(k) ]T and the output vecvelocity and position is x(k) = [ r˙P (k), Φ(k), T ˙ tor is y(k) = [ r˙P (k), Φ(k) ] . The output matrix C = [ I 6 06×6 ] for n = 6 is

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defined with the identity matrix I6 . The NMPC should now predict the systems state into the future, in order to find cable forces which minimize the velocity, contained in y. The state vector in the next time step x(k + 1) is delivered by the nonlinear system function using numerical integration with a prediction step size of Δtc . Using the Euler-Cromer method this yields ¨ P (k)Δtc x˙ P (k + 1) = x˙ P (k) + x xP (k + 1) = xP (k) + x˙ P (k + 1)Δtc .

(6)

¨ P , resulting from the chosen cable forces, can be The systems acceleration x derived from Eq. (4). Using the prediction horizon np and the control horizon nc a cost function J is set up which is minimized at each time step generating set point cable forces for the system. J=

np  

T   y(k + i) − yr (k + i) R y(k + i) − yr (k + i) + . . .

i=1 nc  

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i=i

(7) The optimization criteria can be weighted using the parameters R and r1 . The first criterion causes the system to follow the desired output yr , while the second criterion keeps the cable forces continuous. Taking into account the cable force limits, the nonlinear optimization problem is constrained and J needs to be ∗ ∗ ≤ f ∗ ≤ fmax . Equation (7) will be zero as soon minimized considering fmin as the robot comes to a full stop and the cable forces remain constant. The implementation is described in Sect. 5. Potential Fields. Potential fields can be used in path planning for robots avoiding also obstacles while the robot approaches its goal pose [15]. In the following, the approach is briefly described. The main idea is to choose a feasible goal pose in the post-failure workspace of the system and guide the end effector there, while avoiding collisions using potential fields. The robot’s end effector in Cartesian space is subject to virtual forces F (rP ) generated through a potential field U (rP ), formed by a number of attractive Uatt and repulsive fields Urep . The force is defined by the negative gradient of the field. The attractive field is used to guide the end effector into the desired position, while the repulsive fields can be utilized to avoid obstacles or regions in the workspace. Based on the distance ρf (rP ) = rP − rP,final  to the desired position rP,final , the attractive force Fatt can be determined as the negative gradient of Uatt : ⎧ : ρf (rP ) ≤ d ⎨−ζr (rP − rP,final ) Fatt (rP ) = −∇Uatt (rP ) = (8) dζr (rP − rP,final ) ⎩− : ρf (rP ) > d. ρf (rP )

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A combination between a quadratic and a conic potential field is chosen in Eq. (8). The quadratic field is continuously differentiatable and increases monotonically with the distance to its origin. Since with greater distance ρf the virtual forces increase drastically for the quadratic field, it is combined with a conic one at the distance d to the origin. Note, that at the border ρf = d, both potentials are equal allowing for a smooth transition. The field’s influence can be scaled using the scaling factor matrix ζr = diag(ζx , ζy , ζy ). For a number of different obstacles o ∈ R, the same number of repulsive fields can be employed leading to forces Frep,o (rP ). As within this work, no repulsive fields are employed, they are not further detailed here. Additional to the attractive forces, an attractive torque τatt (Φ) is generated, dependent on Φ and the desired final orientation Φfinal . Here, only a quadratic potential is used, analogous to Eq. (8), employing the scaling factor matrix ζΦ = diag(ζϕ , ζθ , ζφ ). Moreover, virtual damping wd onto the platform is introduced, proportional to the translational and rotational velocity as described in [6], using the matrices D1 and D2 which are diagonal matrices providing one scaling factor for each DOF. Considering all virtual forces and torques including external ones, this yields   o fE Fatt (rP ) + i=1 Frep,i (rP ) wE = (9) + wd + τE τatt (Φ) If now the cables generate this wrench including all virtual forces, the end effector will move according to the chosen potential fields.

4

Decoupling Device and Failure Detection Algorithm

To conduct realistic experiments on the SEGESTA prototype [9], a device is required which decouples a cable from the end effector with reliability and reproducibility in order to induce a cable failure. For this purpose an electromechanical lock mounted onto the platform is used, see Fig. 2. A loop is tied with a bowline at the cable to be released. The loop is connected to the lock while passing through the rounded cable guidance. This prevents damage at the cable and enables reliable operation without the cable getting stuck. To eject the cable, the latch is released by an electro magnet and ejected with spring assistance. Thus, the decoupling also works with minimum cable tension. The lock can withstand forces far above the maximum cable force of SEGESTA, enabling normal operation of the robot. The device allows for the decoupling of each cable at the platform since it can be mounted at different positions. Moreover, the mechanism provides fast and easy reassembling for efficient test series. A trigger signal is integrated into the SPS and the device is galvanically isolated using an optocoupler. Carrying out numerous tests when evaluating the emergency methods, the cable decoupling worked flawlessly. To determine the event of a cable failure, a very primal failure detection algorithm (FDA) is proposed. A combination of three measured signals is used. First, a measured cable force below a threshold near zero could indicate a cable failure, but also cable slackness. Second, from the current motor angle and the

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Fig. 2. (a) CAD representation of decoupling device mounted onto the bottom of SEGESTA. The cable (red) is connected through a hole on the end effector (gray), passed around the cable deflector (green) and locked with the latch (silver) of the lock (black). (b) The SEGESTA Prototype with m = 8 cables and n = 6 DOF. Pulleys attached at B =

0.73 −0.73 0.70 −0.74 0.73 −0.73 0.74 −0.74 −0.64 −0.64 −0.68 −0.68 0.64 0.64 0.68 0.68 0.07 0.10 1.01 1.01 0.11 0.06 1.01 1.01

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inverse kinematics, the cable length error is determined. If the cable is too short and has low tension at the same time, then it cannot be slack. Third, from the current motor velocity it is determined, if the motor is winding up the cable. If now the cable is not tensed, too short and gets wound up, the FDA assumes that the cable is disconnected from the platform and switches to emergency mode.

5

Experiments

To conduct experiments investigating the proposed emergency methods, the SEGESTA prototype is used [9]. The force boundaries are set to fmin = 5 N changes the platforms paramand fmax = 50 N. Since the decoupling device   45 −45 35 −35 45 −45 35 −35 0 0 −35 −35 0 0 35 35 mm, eters, it yields: mP = 0.262 kg, P = 35 −35 −35  35 35 −35 −35  35 6.13 0.06 −0.56 T P 0.06 6.05 0.41 · 10−4 kg m2 . Using P rS = [−4.4, 3.3, −30.5] mm and ΘS = −0.56 0.41 2.42 the given parameters, the static equilibrium workspace SEW of the prototype for platform orientation aligned with 6 B - can be determined [8], see Fig. 3. Certainly, the workspace extends if the platform is allowed to change its orientation [6]. It can be observed that the post-failure workspace is drastically smaller, promoting the chance of the platform being outside of it at the time of a cable break. A red cross indicates the starting position for the experiments (rP = [−0.15, 0.15, 0.65]T m, Φ = [0, 0, 0]T rad), which is inside of the prefailure but outside of the post-failure workspace. Before the experiment, the platform is moved to this position in normal mode. Afterwards, the trigger signal for the decoupling device is manually toggled while the platform is at rest. The SPS is not allowed to process this information. It needs to detect the cable decoupling of cable 8 using the proposed FDA and afterwards switches to one of the proposed emergency strategies. Position and velocity of the platform are determined using forward kinematics [12].

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Minimization of Kinetic Energy. The method according to Sect. 3 was implemented using MATLAB’s fmincon function with a SQP algorithm after code generation for the BECKHOFF PLC of SEGESTA. The iterations were limited to 50 to restrict the maximum computation time allowing suboptimal solutions in the control loop [1]. The experiments could be performed with 2 kHz cycle time, thus computation time issues of the method are resolved. The weights are set R = diag(0.5, 1, 1.2, 0.125, 0.125, 0.01) and r1 = 0.12. Furthermore np = nc = 1, Δtc = 0.5 ms and yr = 0 were chosen. The computed forces of the algorithm fdes are converted to motor torques and used as set-point for the motor controller. Figure 4 shows the experiment. After initial acceleration due to the cable break, the desired velocity of 0 is reached after 1.4 s and the platform is brought to a static stable pose at rP = [−0.048, 0.274, 0.403]T m and Φ = [−1.09, −0.74, −0.75]T rad with constant cable forces. Hence, the emergency strategy is proven to work. Avoidance of platform tilting could be improved by varying R or adding a position control loop. The platform neither collides with the robot frame, nor with the ground. The maximum velocity occurring is −0.92 m/s in z-direction and the maximum angular velocity is 14.23 rad/s about the z-axis, which peaks for a short time when cable 3 has an impulse in force. The choice of the weighting parameter r1 leads to a continuous progression of fdes . Figure 5 shows a close up of fdes and fmea to display the FDA. After the trigger signal T , the device decouples the cable which then slides out of the guidance. As soon as the cable force drops below the threshold, the FDA can finally detect the failure D. Immediately after detection, the controller switches to emergency mode and commands fdes . The transition is continuous, since the previously commanded forces are used as initial value for the NMPC. In this experiment, the detection has a delay of ∼25 ms after the trigger and throughout the test series, the detection worked reliably without error. The measured forces show that due to the cable failure, especially cable 3 gets slack, leading to an impulse when getting back in tension. The same effect has been observed in

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[8]. Moreover, fdes and fmea are fairly different, which is due to unmodeled effects like friction or drivetrain inertia. Hence, severe differences appear as expected. Furthermore, no control loop of the cable forces is used, which is recommended for future work. However, the method is able to successfully rescue the platform, even though very simple models have been used. Potential Fields. An attractive potential field according to Sect. 3 is placed at the goal pose rP,final = [0.175, 0, 0.5]T m, which is well within the post-failure workspace. Now d = 0.25 m and ζx = ζy = ζz = 950. Φfinal = 0 rad are

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chosen with ζΦ = diag(0.4, 0.4, 0). Virtual damping is parametrized with D1 = diag(25, 25, 25) and D2 = diag(0.48, 0.48, 0). No repulsive field is set. Since the desired wrench may vary drastically, the robot might get outside of its WFW and Eq. (3) might not be solvable within given force limits. Thus, within this work, the Nearest-Corner Method as introduced in [4] is employed to solve for approximated forces within the limits in real-time. The Puncture Method [9] is set as standard method and the exponential weight to p = 16. With a small error of [−0.022, −0.005, 0.007]T m and [−0.05, −0.03, 0.02]T rad, the goal pose is reached leading to a stop of the system and a safe guidance back into the post-failure workspace after ∼ 1 s, see Fig. 6. The residual error remains since the potential field does not have an integral controller part, which might be introduced in future work. fdes stays well within the given boundaries and the algorithm makes extensive use of extreme force distributions in the initial phase. Compared to the first method, the set forces are remarkably higher since the robot is pulled quickly into the goal pose by the algorithm. The same accounts for the resulting Cartesian and rotational velocity. This effect might be reduced by weakening the field’s potential or by reducing p, which would lead to cable ∗ ∗ ∗ +(fmax −fmin )/2 [4]. Here as well, some cables get slack and forces closer to fmin experience an impulse in cable force when getting back in tension. This could be improved by a control loop closure on the cable forces. When the robot re-enters the workspace, the Nearest-Corner method delivers a continuous transition of the forces to the Puncture Method solution.

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Summarizing, both methods are able to successfully rescue the platform after cable decoupling and the detected cable failure. Since the robot is brought back into the post-failure workspace, it may continue operation there with a reduced set of cables or wait for maintenance.

6

Conclusion and Outlook

In this work, two emergency strategies after cable failure, based on former work of the authors, have been presented and implemented at the SEGESTA prototype. A cable decoupling device was developed to test the algorithms under realistic conditions. The device can disconnect a cable at the end effector while the robot is in operation. Moreover, a very primal failure detection algorithm was introduced, which automatically detects the failed cable and switches to one of the emergency strategies. Both strategies, which were simulatively verified in previous publications, have been proven to work in real-time on the prototype SEGESTA under realistic conditions using the decoupling device. To the authors’ best knowledge, this is the first demonstration of an emergency method with a cable decoupling on a prototype. After rescue of the platform into the postfailure workspace, the robot may continue operation with a reduced set of cables. The experiments showed that both algorithms are very sensitive to parametrization. Consequently, a deep investigation on how the methods can be generalized is necessary. For future work, the decoupling device allows for broader testing throughout the whole workspace of SEGESTA, also in dynamical state, in order to improve the proposed emergency methods or to try different approaches. The operating principle of the decoupling device might also be upscaled for tests on larger prototypes. To avoid incorrect fault detection, the very primal failure detection algorithm needs to be extensively tested. The presented emergency strategies could be extended to consider e.g. position control within the NMPC approach, testing of collision avoidance with obstacles in the workspace for both strategies, a generalized parametrization, a cable force control loop and a consideration of the cables within collision avoidance. Acknowledgments. This work was supported as part of the fundings “Langfristige experimentelle Untersuchung und Demonstration von automatisiertem Mauern und 3D-Druck mit Seilrobotern” and “Auf dem Weg zur digitalen Bauausf¨ uhrung: Automatisierung des Rohbaus mit Seilroboter-Technik” by the Ministry of Regional Identity, Communities and Local Government, Building and Digitalization of the Land of North Rhine-Westphalia.

References 1. Adamy, J.: Regelungen f¨ ur lineare und nichtlineare Regelstrecken. In: Nichtlineare Systeme und Regelungen, pp. 439–524. Springer, Heidelberg (2018). https://doi. org/10.1007/978-3-662-55685-6 6

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2. Berti, A., Gouttefarde, M., Carricato, M.: Dynamic recovery of cable-suspended parallel robots after a cable failure. In: Lenarˇciˇc, J., Merlet, J.-P. (eds.) Advances in Robot Kinematics 2016. SPAR, vol. 4, pp. 331–339. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-56802-7 35 3. Boschetti, G., Minto, R., Trevisani, A.: Experimental investigation of a cable robot recovery strategy. Robotics 10(1), 35 (2021). https://doi.org/10.3390/ robotics10010035 4. Boumann, R., Bruckmann, T.: Real-time cable force calculation beyond the wrench-feasible workspace. Robotics 9(2), 41 (2020). https://doi.org/10.3390/ robotics9020041 5. Boumann, R., Bruckmann, T.: Development of emergency strategies for cabledriven parallel robots after a cable break. In: CableCon 2019. MMS, vol. 74, pp. 269–280. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-20751-9 23 6. Boumann, R., Bruckmann, T.: Simulation von Seilrissen und Erweiterung von Abfangstrategien f¨ ur Seilroboter. In: Bertram, T., Corves, B., Gr¨ aßler, I., Janschek, K. (eds.) Fachtagung Mechatronik: Paderborn, 27–28 March 2019, pp. 49– 54. Paderborn, Germany (2019) 7. Boumann, R., Bruckmann, T.: An emergency strategy for cable failure in reconfigurable cable robots. In: Gouttefarde, M., Bruckmann, T., Pott, A. (eds.) CableCon 2021. MMS, vol. 104, pp. 217–229. Springer, Cham (2021). https://doi.org/ 10.1007/978-3-030-75789-2 18 8. Boumann, R., Bruckmann, T.: Simulation and model-based verification of an emergency strategy for cable failure in cable robots. Actuators 11(2), 56 (2022). https:// doi.org/10.3390/act11020056 9. M¨ uller, K., Reichert, C., Bruckmann, T.: Analysis of a real-time capable cable force computation method. In: Pott, A., Bruckmann, T. (eds.) Cable-Driven Parallel Robots. MMS, vol. 32, pp. 227–238. Springer, Cham (2015). https://doi.org/10. 1007/978-3-319-09489-2 16 10. Notash, L.: Wrench recovery for wire-actuated parallel manipulators. In: Padois, V., Bidaud, P., Khatib, O. (eds.) Romansy 19 - Robot Design, Dynamics and Control, pp. 201–208. Springer, Vienna (2013). https://doi.org/10.1007/978-3-70911379-0 25 11. Passarini, C., Zanotto, D., Boschetti, G.: Dynamic trajectory planning for failure recovery in cable-suspended camera systems. J. Mech. Robot. 11(2), 021001 (2019). https://doi.org/10.1115/1.4041942 12. Pott, A.: Cable-Driven Parallel Robots. STAR, vol. 120. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-76138-1 13. Raman, A., Walker, I., Krovi, V., Schmid, M.: A failure identification and recovery framework for a planar reconfigurable cable driven parallel robot. IFACPapersOnLine 55(37), 369–375 (2022). https://doi.org/10.1016/j.ifacol.2022.11. 211 14. Roberts, R.G., Graham, T., Lippitt, T.: On the inverse kinematics, statics, and fault tolerance of cable-suspended robots. J. Robot. Syst. 15(10), 581–597 (1998) 15. Spong, M.W., Hutchinson, S., Vidyasagar, M.: Robot Modeling and Control, pp. 75–118. Wiley, New York (2005) 16. Winter, D.L., Ament, C.: Development of safety concepts for cable-driven parallel robots. In: Gouttefarde, M., Bruckmann, T., Pott, A. (eds.) CableCon 2021. MMS, vol. 104, pp. 360–371. Springer, Cham (2021). https://doi.org/10.1007/978-3-03075789-2 29

Stability Analysis of Profile Following by a CDPR Using Distance and Vision Sensors Thomas Rousseau1,2(B) , Nicol` o Pedemonte2 , St´ephane Caro1 , and Fran¸cois Chaumette3 1

Nantes Universit´e, Ecole Centrale Nantes, CNRS, LS2N, UMR 6004, Nantes, France {thomas.rousseau,Stephane.Caro}@ls2n.fr 2 IRT Jules Verne, Bouguenais, France [email protected] 3 Inria, Univ Rennes, CNRS, IRISA, Rennes, France [email protected]

Abstract. Cable-Driven Parallel Robots (CDPRs) form a class of robots well-adapted to large workspaces since they replace rigid links by cables. However, they lack in positioning accuracy. In a previous work, a control law has been proposed to enable a CDPR to perform a profile-following task, based on the data measured by two different and redundant sensors that are fused using the Gradient Projection Method (GPM). However, its robustness had not been assessed yet. This paper analyzes the stability of such a control law in function of the systematic errors on the parameters of the system. Numerical simulations show that the characteristics of the system ensure the stability of the control law in the robot workspace even in the presence of significant errors, provided the initial angle between the surface to follow and the moving-platform is smaller than 11◦ .

Keywords: Stability Analysis Parallel Robot

1

· Sensor-based control · Cable-Driven

Introduction

Cable-driven parallel robots (CDPRs) are robots where the end-effector, named the moving-platform (MP), is actuated by a set of cables instead of rigid links. The length of the cables is usually controlled by a drum-pulley system, actuated by motors fixed to the main frame. These robots are able to carry loads in large workspaces and could thus help in large industrial assembly tasks. However, their accuracy, especially for positioning with regards to a given object, needs to be This work is supported by IRT Jules Verne in the framework of the PERFORM program. c The Author(s), under exclusive license to Springer Nature Switzerland AG 2023  S. Caro et al. (Eds.): CableCon 2023, MMS 132, pp. 221–233, 2023. https://doi.org/10.1007/978-3-031-32322-5_18

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improved before they can find a place on production lines. While many research works brought significant improvements to CDPR models [1–3], sensor-based control, and especially vision-based control [4–6], can also enable such robots to reach a better precision, without needing the tedious computations required for the Forward Kinematics. However, vision-based control has some important limitations: embedded cameras suffer from a limited field of view, while external cameras provide a global overview of the workspace but at the cost of worse accuracy. A solution to these shortcomings was proposed in [7] where measurements from both distance sensors and an external camera are combined using the Gradient Projection Method (GPM) [8,9] to prioritize tasks. The robustness of this control law was not considered in [7]. While the stability analysis for N-tasks problems was investigated in [10], the proposed stability criterion is only valid in the ideal case while a specific study based on the quality of the estimations and possible sensor biases is required in our case. The robustness of CDPR control for one pure visual task has been tackled in [5]. A novel workspace named Control Stability Workspace (CSW), enclosing the stable poses for given perturbation bounds, was also introduced in [11]. In this paper, the robustness of the GPM control for a profile following task is assessed. First, a stability criterion is derived from the control law of the system. Then, the impact on stability of all the identified error sources is investigated. Finally, a combined analysis is run in the robot operating conditions to confirm its robustness. Since experimental results have already been presented in [7], only simulation results are discussed in this paper. A view of the experimental platform used in this paper, ACROBOT, a six degrees of freedom (DoF) suspended CDPR, is shown in Fig. 1.

Fig. 1. ACROBOT, a CDPR located at IRT Jules Verne

This paper is organized as follows: Sect. 2 presents the two sensors and their kinematics modeling. The control law is recalled from [7] in Sect. 3. Section 4 focuses on the design of the stability criterion and Sect. 5 presents the numerical results of the stability analysis. Conclusions are drawn in Sect. 6.

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Fig. 2. Parametrization of the three distance sensors embedded on the MP and facing the target surface. The external camera observes the MP.

2

Task Definition

The CDPR considered in this paper is subject to a control law enabling its MP to perform a profile following task. This task can be divided into two subtasks: maintaining the MP parallel to the followed surface at a constant distance, denoted T1 , and moving from a starting pose to a target pose along the profile, denoted T2 . The control of the robot stems from sensor-based control and combines measurements provided by two types of sensors: an array of three distance sensors embedded on the MP, and an external camera located in front of the robot. The boarded sensors and the surface to follow are shown in Fig. 2. 2.1

Sensors

Distance Sensors. Three ultrasonic distance sensors are placed on the bottom of the MP, facing downwards along the z-axis of the MP frame Fp , as shown in Fig. 2. Since all sensors are coplanar and their directions are parallel, for the MP to be locally parallel to the surface, the three sensors must return the same distance. The vector of distances measured by the sensors is denoted as s = [s1 , s2 , s3 ]T , si being the distance measured by the ith sensor Si . The time derivative of the measured distance vector, noted s˙ , is then expressed as a function of the MP twist vp : s˙ = p L1 vp (1) T

with vp = [νp T , ωp T ] where νp is the MP linear velocity vector and ωp is its angular velocity vector, expressed in Fp . p L1 is the interaction matrix of the distance sensors [12,13]. Using a constant value of the interaction matrix corresponding to the desired configuration is a usual choice in sensor-based control [13]. The interaction matrix p L1 thus only depends on the respective positions (xi , yi ) of the distance sensors expressed in Fp , and can be evaluated for the particular case s = s∗ , where s∗ = [s∗ , s∗ , s∗ ]T and with s∗ = 0.2 m the common reference distance so that the MP is locally parallel to the followed surface:

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p

L1|s=s∗

⎤ ⎡ 0 0 −1 −y1 x1 0 = ⎣0 0 −1 −y2 x2 0⎦ 0 0 −1 −y3 x3 0

(2)

External Camera. The images acquired by the external camera are processed by an algorithm recognizing several AprilTags, placed on the MP as well as on the starting and target points of the trajectory. The relative pose of the reference frames attached to these April Tags is then estimated with regard to the camera frame Fc . Then, this pose is expressed in the world frame Fw , and, as for PoseBased Visual Servoing (PBVS), the visual features are selected as: w  tp (3) sv = θu where w tp is the position vector of the MP expressed in Fw and θu is the axis∗ angle representation of the rotation matrix p Rp . The target pose is given by: w ∗  tp s∗v = (4) 03 where w t∗p is the position of the target AprilTag in Fw . The interaction matrix L2 of these visual features can be found in [13]:  I −[w tp ]× w L2 = 3 (5) 03 Lω

w

with [w tp ]× the skew-symmetric matrix of w tp . Lω can be approximated by I3 , the (3 × 3) identity matrix. To express this matrix in Fp , the adjoint matrix w Vp is used [11]: w Rp [w tp ]× w Rp w Vp = (6) w 03 Rp The change of reference frame is then performed: p

L2 = w L2

w

Vp

(7)

This interaction matrix only depends on the estimated position and orientation of the MP in Fw . 2.2

Fusion Using the Gradient Projection Method

The fusion between the data from the external camera and the three distance sensors is performed with the GPM. In this method, a first main task is considered as a priority, while only the part of the second task compatible with the realization of the first one is retained [9].

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Main Task. The main task T1 is selected from the distance sensor measurements, such that the task error e1 is given by: e1 = s − s∗

(8)

The next step is to determine the Jacobian of the main task. As described above in Sect. 2.1, the time variation of the main task is related to the MP twist vp by (1). As the inputs of the low-level controller of the CDPR are the velocities l˙ of the m cable lengths, they are related to vp by: l˙ = A vp

(9)

where the forward Jacobian matrix A of the CDPR is given by [5]: 

um u1 . . . ui . . . A= b1 × u1 . . . bi × ui . . . bm × um

T (10)

where ui are the cable direction unit vectors, pointing from the exit points (pulleys) of the CDPR to the anchor points placed on the MP, and where bi are the position vectors of the anchors points, known from the platform design. All these vectors are expressed in Fp . ui vectors are calculated geometrically from the MP pose expressed in Fw using the straight and inelastic cable model [5]. Equations (1) and (9) can be combined to obtain: e˙ 1 = J1 l˙

(11)

where the Jacobian J1 of the main task is a (3 × m) matrix, given by J1 = L1 A+

(12)

with A+ the Moore-Penrose pseudo-inverse of A. Secondary Task. T2 consists in minimizing the error between the current and the desired MP poses, expressed in Fw , and noted as follows: e2 = sv − s∗v

(13)

From (7) and (9), its Jacobian is a (6 × m) matrix given by: J2 = w L2

w

Vp A+

(14)

Similarly to (11), e˙ 2 satisfies the following relationship: e˙ 2 = J2 l˙

3

(15)

Control Strategy

In practice, the true Jacobians J1 and J2 can be subject to approximations, which is the case for the form given in (2), noise measurements, and calibration

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errors. The Jacobians used in the control scheme are thus different from the

in the following. The control law, pictured in Fig. 3 true ones and are denoted J and performing the fusion described above, is detailed in [7]. It relies on the

1 on the kernel of the primary task given by: projection operator P

+ J

1 = Im − J

P 1 1

(16)

The control law is expressed in terms of e1 and e2 , with the gain ratio rλ = λ2 /λ1

+ e1 − λ1 (J2 P

1 )+ (rλ e2 − J2 J

+ e1 ) l˙ = −λ1 J 1 1

(17)



+ e1 and matrix J

1 ˜ 2 = J2 P ˜2 = rλ e2 − J2 J To simplify the notations, the term e 1 are defined such that the control law has the simple form: +

+ e1 + J ˜ e l˙ = −λ1 (J 2 ˜2 ) 1

(18)

Fig. 3. Control scheme of the CDPR with the fusion of the distance and pose errors.

4

Stability Criterion

The stability of the control scheme can be classicaly assessed using a Lyapunov stability analysis. Since the main task is of dimension three, the projection of the secondary task on the kernel of the main task causes the secondary task to lose three degrees of freedom. Hence, the stability of the control law cannot be assessed directly by considering ||e1 ||2 + ||e2 ||2 as candidate Lyapunov function. 4.1

Secondary Task Error Redefinition

To solve the analysis problem, a three-dimensional secondary task e2r can be defined by projecting the six-dimensional secondary task e2 on the null space of  e1 , expressed in the MP frame. First, the projected error e2 is considered: +



1 e2 , with p P

1 = I6 − p e2 = p P L1|s=s∗ p L1|s=s∗

(19)

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Considering only the three non-zero terms of e2 , the reduced error vector e2r can be defined as: ⎤ ⎡w xp −w x∗p e2r = ⎣ w yp −w yp∗ ⎦ (20) θuz The stacked vectors e1 and e2r hence form a six-dimensional error. The corresponding stability problem can be assessed since the dimension of the variables matches the number of DoF of the MP. The modified interaction matrix p L2r is then obtained by selecting the rows corresponding to the retained features ˜ 2r are computed similarly by and the corresponding modified matrices J2r and J substituting L2 by L2r in their definition. A modified control law follows: +

+ e1 + J ˜ e l˙ = −λ1 (J 2r ˜2r ) 1

4.2

(21)

Lyapunov Stability Criterion

By considering the reduced secondary task, the Lyapunov candidate function L, strictly positive and continuously differentiable, is defined as: L=

1 1 ||e1 ||2 + ||e2r ||2 2 2

(22)

Its time derivative is nothing but: L˙ = eT1 e˙ 1 + eT2r e˙ 2r = eT1 J1 l˙ + eT2r J2r l˙

(23)

Assuming the low level robot controller is able to perfectly apply the cable velocity vector computed by the control law (21), we obtain:  + +



T

+ e1 − eT J1 J

+ ˜ J ˜ ˙ L = −λ1 eT1 J1 J 1 2r 2r J1 e1 + rλ e1 J1 J2r e2r 1 (24) + +



+ + T T T



˜ ˜ +e2r J2r J1 e1 + rλ e2r J2r J2r e2r − e2r J2r J2r J2r J1 e1 The previous equation can be expressed in a matrix form such that the Lyapunov candidate function derivative L˙ is expressed as:  e T T ˙ L = −λ1 [e1 e2r ] Π 1 (25) e2r which leads to the stability criterion Π > 0 with: ⎤ ⎡ + +



+



˜ ˜ (I − ) J r J J J J J λ 1 2r ⎦ 2r 2r 1 Π=⎣ 1 m + +



+



˜ ˜ J (I − J J )J r J J 2r

m

2r 2r

1

λ 2r 2r

(26)

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In the ideal case, that is when the Jacobians used in the control scheme

1 = J1 and J ˜ 2 , it follows that J1 J ˜ + = 03 ˜2 = J correspond to the real ones, i.e., J 2r and the expression of matrix Π becomes:  I3 03 Π= (27) + ˜+ ˜+ J2r J+ 1 − J2r J2r J2r J1 rλ J2r J2r As expected, the resulting expression in the ideal case matches with the relationship already established in [10].

5

Stability Analysis

The sign of the Lyapunov function derivative L˙ depends on the sign of the criterion Π. If the criterion is positive definite, the derivative is negative for all input vectors, thus guaranteeing the stability of the control law. This sign depends on the quality of the estimation of a series of parameters. These are the position error of the exit points of the pulleys and the anchor points of the tp , the cables on the MP δai and δbi , the vision-based pose estimate error δ w

s,d , positionning error of the distance sensors in the platform reference frame δ L the relative orientation of the surface to follow with regard to the platform frame, given by the vectors nTi , as well as the gain ratio rλ . Since there is an infinity of combinations of poses and parameter values, the path P validated experimentally in [7] on the ACROBOT is chosen for this study. An analysis parameter by parameter is run to determine which ones have an important impact on the stability and which ones can be neglected. Then, a combined analysis using known parameter values is conducted to provide us with the expected stability domain of the robot during its standard behaviour. The analytical sign analysis of Π is rather complex, due the multiple interactions between the parameters. A numerical analysis is thus proposed. For this analysis, an initial angle between the MP and the tangent plane to the surface followed of 15◦ was retained. CDPR Geometrical Parameters. The precision of the CDPR geometry

and hence on Π. Considerparameters ai and bi values has an impact on A ing only errors in the estimation of these parameters along the path P, the stability margin versus both ai and bi parameters is shown Fig. 4. The control law remains stable up to an error of 0.08 m for ai and to 0.10 m for bi . The stability isocontours, i.e., the lines where the lowest eigenvalue remains constant, closely follow the altitude profile of the path, hinting at a significant dependency on this parameter regarding the stability of the control law. For this experiment system, the accuracy of these parameters is respectively 0.01 m for δai and 0.008 m for δbi . Hence, the stability of the robot is not threatened by these errors. Vision-Based Pose Estimation. The pose estimation impacts the stability since it is involved in the estimation of the robot Jacobian A and p L2 , necessary

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Fig. 4. Stability of the robot along P versus the CDPR geometrical parameters.

Fig. 5. Stability of the robot along P versus the position estimation error δ w tp

Fig. 6. Stability of the robot along P versus the norm of the sensor position error

to compute the estimates of J1 and J2r . The impact of the position measurement bias resulting of a low image resolution is shown Fig. 5. However, this bias is not uniformly distributed along the three directions, and the estimated position error is larger along the depth of the image observed by the camera. The norm of the position error plotted in Fig. 5 was thus distributed accordingly. The impact of the orientation error about the three directions of Fp was computed along P, and is represented for a central position in the robot cell in Fig. 7. In both cases, the accuracy of the pose-estimation, 70 mm for the position and 3◦ for the orientation at worst, does not overshoot the stability limits. However, the resulting margin is lower than the one obtained for the CDPR geometrical parameters errors and pose -estimation errors have a larger stability impact. Distance Sensors Positioning. The position of the distance sensors with regards to the MP-frame is measured and these measured values xi and yi are

s,d (2). The impact of a measurement error on the used for computing matrix L

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(a) Orientation error δθux along P

(b) Orientation error δθuy along P

(c) Orientation error δθuz along P

(d) Stability isosurface of the robot at position [0, 0, 0.2]

Fig. 7. Stability of the robot versus the three components of the orientation estimation error δθu

stability of the control law is shown Fig. 6. From this figure, it is clear that the quality of such measurements significantly impacts the stability since the maximum norm of the error ensuring stability for these parameters is twice lower than for the parameters studied above. However, an adequate accuracy on these parameters can easily be achieved with a robust design of the MP or following a characterization of this MP. For ACROBOT, this accuracy is estimated to 1 mm. Gain Ratio. Eventually, the gain ratio between the main task and the secondary task is also involved in the computation of the stability criterion Π2 . The corresponding plot is not shown due to lack of space but the result obtained shows that this parameter cannot prevent the stability of the control law. However, the values below rλ = 0.25 reduce the stability margin. Furthermore, no difference in stability has been noticed all along the path P for this parameter.

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Combination of Parameters. To analyze the combination of these factors, another analysis is conducted, combining all the error sources. A series of 20.000 trajectories was run, selecting randomly the errors applied. Each of these parameters was assigned an error box whose side corresponds to its respective estimated accuracy. For each trajectory and for each parameter discussed above, a random vertex of the corresponding error box is selected. The distance measurements returned by the distance sensors are generated randomly with a Gaussian centered on μ = 0.2 m and σ = 0.025. The distance difference between the three sensors generates an angle α between the normal nS and the normal to the tangent plane nT , equivalent to the angle between the plane xp Op yp and the tangent plane. The results of this analysis with regard to the angle α are presented Fig. 8.

Fig. 8. Stability margin of the tested configurations versus orientation angle α. In blue: stable configurations and in red: unstable configurations. (Color figure online)

When observing the minimum eigenvalues of the stability criterion versus the angle α, all configurations are stable when α ≤ 11◦ . However, after this threshold, the proportion of stable configurations decreases when the angle increases. No stable configurations were found when α ≥ 33◦ . This implies that an initial configuration of the system with α lower than 11◦ and that, while executing the two tasks, the tracking error with regard to the main task remains below 11◦ is a sufficient condition for the convergence of the system. On the tested trajectory P, the orientation of the tracking error does not exceed 8◦ , and the stability of the control law with the measured parameter errors is thus ensured.

6

Conclusion

This paper proposed a stability analysis of a sensor-based control law for profile following using a CDPR. First, a stability criterion was derived, then a numerical

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parameter by parameter analysis was conducted, highlighting the most critical parameters that ensure the stability of the system. The most critical parameters are the coordinates of the distance sensor in the MP frame as well as the bias on the vision-based pose estimation. A combined analysis taking into account the most relevant parameters for various configurations of the robot was also presented. This analysis shows that all these configurations for the biases of the relevant parameters of the prototype ACROBOT are stable when the angle between the MP and the tangent plane to the target surface is smaller than 11◦ . A sufficient condition for the stability of the system on the initial configuration of the robot and on the tracking error hence results from this analysis. This implies that the response time of the main task must remain short so that the moving MP during the trajectory following does not overshoot the stability angle threshold. While these results are specific to the ACROBOT prototype, the method is transferable. Stability domains of other CDPRs using the same control scheme can be computed similarly if the relevant parameters are given.

References 1. Pott, A.: Influence of pulley kinematics on cable-driven parallel robots. In: Lenarcic, J., Husty, M. (eds.) Latest Advances in Robot Kinematics, pp. 197–204. Springer, Dordrecht (2012). https://doi.org/10.1007/978-94-007-4620-6 25 2. Mishra, U.A., Caro, S., Mishra, U.A., Caro, S.: Forward kinematics for suspended under-actuated cable-driven parallel robots with elastic cables: a neural network approach. J. Mech. Robot. 14(4), 041008 (2022) 3. Schmidt, V.L.: Modeling Techniques and Reliable Real-Time Implementation of Kinematics for Cable-Driven Parallel Robots Using Polymer Fiber Cables. Fraunhofer Verlag, Stuttgart (2017) 4. Dallej, T., Gouttefarde, M., Andreff, N., Michelin, M., Martinet, P.: Towards vision-based control of cable-driven parallel robots. In: 2011 IEEE/RSJ International Conference on Intelligent Robots and Systems, pp. 2855–2860. IEEE (2011) 5. Zake, Z., Chaumette, F., Pedemonte, N., Caro, S.: Vision-based control and stability analysis of a cable-driven parallel robot. IEEE Robot. Autom. Lett. 4(2), 1029–1036 (2019) 6. Ramadour, R., Chaumette, F., Merlet, J.P.: Grasping objects with a cable-driven parallel robot designed for transfer operation by visual servoing. In: 2014 IEEE International Conference on Robotics and Automation (ICRA), pp. 4463–4468. IEEE (2014) 7. Rousseau, T., Pedemonte, N., Caro, S., Chaumette, F.: Constant distance and orientation following of an unknown surface with a cable-driven parallel robot. In: 2023 IEEE International Conference on Robotics and Automation (ICRA). IEEE (2023) 8. Nakamura, Y., Hanafusa, H., Yoshikawa, T.: Task-priority based redundancy control of robot manipulators. Int. J. Robot. Res. 6(2), 3–15 (1987) 9. Siciliano, B.: Kinematic control of redundant robot manipulators: a tutorial. J. Intell. Rob. Syst. 3(3), 201–212 (1990) 10. Antonelli, G.: Stability analysis for prioritized closed-loop inverse kinematic algorithms for redundant robotic systems. IEEE Trans. Rob. 25(5), 985–994 (2009)

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11. Zake, Z., Chaumette, F., Pedemonte, N., Caro, S.: Control stability workspace for a cable-driven parallel robot controlled by visual servoing. In: Gouttefarde, M., Bruckmann, T., Pott, A. (eds.) CableCon 2021. MMS, vol. 104, pp. 284–296. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-75789-2 23 12. Samson, C., Espiau, B., Borgne, M.L.: Robot Control: The Task Function Approach. Oxford University Press, Oxford (1991) 13. Chaumette, F., Hutchinson, S., Corke, P.: Visual servoing. In: Siciliano, B., Khatib, O. (eds.) Springer Handbook of Robotics, pp. 841–866. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-32552-1 34

Comparison of Explicit and Implicit Numerical Integrations for a Tendon-Driven Robot Nicolas J. S. Testard(B) , Christine Chevallereau, and Philippe Wenger Nantes Universit´e, Centrale Nantes, CNRS, LS2N, 44000 Nantes, France {nicolas.testard,christine.chevallereau,philippe.wenger}@ls2n.fr

Abstract. A tendon-driven robot inspired from the bird neck is studied in this paper. The objective is to propose numerical integrations for this robot. The classical explicit and implicit Euler integration schemes can be used on this robot. When the elasticity in the tendons is not considered, these schemes are stable and give similar numerical integration results but the explicit implementation is faster. By considering the dynamics of the robot joints on one hand and the dynamics of the motors on the other hand, we propose two Euler integration schemes, explicit and implicit, that take into account the elasticity in the tendons. These schemes rely on the dynamics of the robot joints on the one hand, and on the dynamics of the motors on the other hand and it links them by the tendon elastic model. We observe that with the tendon elasticity model, the explicit integration becomes unstable, depending on the time step, while the implicit one stays stable. The numerical integration that uses this implicit scheme with tendon elasticity allows one to obtain more precise results, while the model of the system is more realistic. Moreover, the evolution of the tendon elongation during a desired motion can be obtained while they are computed in the integration scheme. Keywords: Dynamics Euler · Tendon-driven

1

· Numerical integration · Explicit and Implicit

Introduction

Robots are more and more complex in their design and the numerical integrations of their dynamic model must be more and more efficient and accurate. Two different methods exist for the numerical integration of the dynamic equations, namely, explicit and implicit [6]. A tendon-driven robot inspired from the bird neck was first proposed in [8]. This paper proposes a study of dynamic numerical integration for this robot and a similar one. First, we study the model with no tendon elasticity and we compare the explicit and implicit integration. We then propose an integration of the dynamic equations in the presence of tendon elasticity. We show how to compute the explicit and implicit Euler integration scheme for both models and what results can be expected for these numerical integrations of our robot models. c The Author(s), under exclusive license to Springer Nature Switzerland AG 2023  S. Caro et al. (Eds.): CableCon 2023, MMS 132, pp. 234–245, 2023. https://doi.org/10.1007/978-3-031-32322-5_19

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As indicated in [10], the stability of the explicit Euler integration can depend on their time step. Furthermore, they indicate that the higher the highest natural frequency of the structure is, the smaller the time step will have to be. In Sect. 3, in fact, this integration scheme such that it stays stable can be observed for these robots under study in the absence of tendon elasticity. When tendon elasticity is ignored, the dynamic equation of these robots can also be integrated with an implicit Euler method as shown in [4] and [12]. We will show that with this method, the numerical integration is also always stable. However, tendon elasticity must be considered to have more realistic results. R R [13] or Maple [11] exist that model tendon elasticSoftwares like Matlab ity and use explicit and implicit integration but these softwares are not free. R proposes several explicit integration schemes like Runge-Kutta methMatlab ods with different orders or the Heun’s method. It also proposes an implicit R graphical programming environment. FurtherEuler scheme in the Simulink R /Multibody library in more, the tendon model is only present in the Simscape R R  proposes Simulink which is an additional paying feature. Similarly, Maple R  several explicit Runge-Kutta integrations and the MapleSim environment can use the implicit Euler or the Rosenbrock scheme. In contrast, SOFA ([1]) is a free framework that also models tendon elasticity and integrates the dynamic equations with a explicit and implicit scheme. Explicit Euler and Runge-Kutta 4 solvers are used among the explicit schemes and the implicit conjugate-gradient based Euler solver or the Newmark implicit solver are used among the implicit ones. However, this framework can be difficult to handle for those who are not familiar with it and the implicit integration uses more complicated schemes than the implicit Euler. Accordingly, Sect. 4 proposes two integration methods, implicit and explicit, that take into account tendon elasticity. The proposed schemes integrate both the dynamics of the robot joints and the dynamics of the motors that pull the tendons, these two dynamics being linked by the tendon model. It will be observed that the implicit Euler integration scheme is still always stable. At the opposite, the explicit integration becomes unstable if the time step is not small enough. This limit time step for the explicit integration decreases with the complexity of the robot. All the computations presented R and can be done with any other in this article have been done with Matlab programming software.

2

Robot Model

The robot is made of a stack of anti-parallelogramm joints or X-joints [15]. This joint is actuated through 2 tendons that pull on each side shown in Fig. 1. The tendons are pulled by motors and springs are added in parallel of the tendons to obtain a stable configuration at rest. Different robots can be designed by arranging several X-joints in series. [8] proposed a robot with Nj = 3 joints and Nt = 4 tendons, we also proposed to study a robot with Nj = 4 joints and Nt = 5 tendons. One tendon pulls all the joints on the left while the Nt -1 other ones pull the different joints on the right (see Fig. 2).

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t2c

αi

t1c θi L φi

tic

Rd

ψi b

Fig. 1. X-joint (left) and motor pulling a tendon (right)

(a)

(b)

Fig. 2. Tendon-driven robot with 3 joints (a) and with 4 joints (b)

We first present the model with tendon elasticity. Let α = [α1 , α2 , ..., αNj ] define the vector of joint configurations of the robot. Let θ = [θ1 , θ2 , ..., θNt ] define the vector of motor positions. The dynamic model with tendon elasticity can be written as [3,7]:  Msα α ¨ + cs (α, ˙ α) + g(α) = Ztc − Γ rα (α, α) ˙ (1) m¨ ˙ Mθ θ = Γ m − Γ rθ (θ, θ) + Btc where: – Mm θ = diag(Iθ ) is the inertia matrix of the motors in the motor space (with Iθ the inertia of the motor around its axis); – Γ m are the motor torques; ˙ are the friction in the motors; – Γ rθ (θ, θ) – B = diag( Rrgd ) is the link between the tendon lengths and the motor positions (Rd is the radius of the drum and rg the gears ratio); – tc are the tension in the tendons;

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– Msα is the inertia matrix of the structure such that the kinetic energy is ˙  Msα α, ˙ T = 12 α – cs (α, ˙ α) is the Coriolis effects; – g(α) corresponds to the forces associated to the potential energy (gravity and springs); ˙ ˙ with no elasticity); – Z = −J θ B (such that θ = Jθ α – Γ rα (α, α) ˙ is the friction in the structure. As shown in [2,14], there are different tendon models. We will use the one proposed in [9]: tc = kc xc + dc x˙ c (2) where xc is the tendon elongation that can be expressed as xc = B(f θ (α) − θ), where f θ (α) is the vector of the motor positions computed from the joint angles when there is no elasticity. kc defines the link between the tension and the elongation in statics and dc is a damping coefficient. In the absence of tendon elasticity, this system of equations can be written as: ¨ = Γ sys (α, α) ˙ + J (3) Mα α θ Γm where:   – Γ sys (α, α) ˙ = −c(α, ˙ α) − g(α) + Z B−1 Γ rθ − Γ rα m – Mα = Msα + J θ Mθ Jθ m˙ – c(α, ˙ α) = cs (α, ˙ α) + J ˙ θ Mθ Jθ α The diagonal and top bars of the X-joints are made with aluminium alloy with dimensions 0.1 m and 0.05 m, respectively. Their mass is 64 g and 32 g, respectively. The radius of the motor drum is Rd = 0.02 m and the gear ratio a dry friction in the motors of the form Γ rθ = is rg= 25. We consider  2 ˙ diag arctan cs θ Γ s as proposed in [9] where Γ s = 0.008 N.m for all motors π

and cs = 0.5. also consider a viscous friction in the pivots of the X-joints  We 2 2   ∂φj ∂ψ Γ rα,j = fv 2 ∂αj + 2 ∂αjj α˙ j with fv = 0.001 N.m/(rad/s). In the tendon model, we take kc = 105 N/m and dc = 100 N/m/(rad/s). Each X-joint is equipped with the same springs on each side. On each side, the spring constants for the 3-modules (resp. 4-modules) robot are [600,600,200] N/m (resp. [800,600,200,200] N/m,), from bottom to top. The free length is 46 mm for all springs.

3 3.1

Numerical Integrations Without Elasticity Explicit Numerical Integration Without Elasticity

The simplest way to integrate the system dynamics is to first compute the joint accelerations at step i with the dynamic equation. The joint velocities are then integrated at step i + 1 with these accelerations. The joint orientations at step

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i + 1 are finally integrated with the velocities at step i. The scheme of this simple explicit numerical integration is represented by the following Eq. (4): ⎧   ⎪ Γ sys (αi , α ¨i = M−1 ˙ i ) + J α ⎨α θ (αi )Γ m (4) α ˙ i+1 = α ˙i +α ¨ i dt ⎪ ⎩ ˙ i dt αi+1 = αi + α where dt is the time step and αi = α(i.dt). 3.2

Implicit Numerical Integration Without Elasticity

For the implicit integration, the velocities (resp. orientations) at step i + 1 are computed with the accelerations (resp. velocities) at step i + 1 and not at step i, as described by Eq. (5): ⎧ ⎪ ¨ i+1 = Γ sys (αi+1 , α ˙ i+1 ) + J ⎨Mα α θ (αi+1 )Γ m (5) α ˙ i+1 =α ˙i +α ¨ i+1 dt ⎪ ⎩ = αi + α ˙ i+1 dt αi+1 However, the accelerations at step i + 1 also depend on the velocities and orientations at step i + 1 that we want to compute. Therefore, we linearize the above equation as in [5]. First, we define dα ˙i = α ˙ i+1 − α ˙i = α ¨ i+1 dt and dαi = αi+1 − αi = α ˙ i+1 dt. The first order linearization gives: Mα dα ˙ i = Γ sys (αi , α ˙ i )dt + Kdαi dt + Ddα ˙ i dt + J θ (αi )Γ m dt

(6)

∂Γ sys (αi , α ˙ i ) ∂J ˙ i) ∂Γ sys (αi , α θ (αi )Γ m + and D = . where K = ∂α ∂α ∂α ˙ To compute dα ˙ i from this equation, we express dαi as a function of dα ˙ i by: dαi = α ˙ i+1 dt = (α ˙ i + dα ˙ i )dt We then obtain:   Mα − Kdt2 − Ddt dα ˙ i = Γsys (αi , α ˙ i )dt + Kα ˙ i dt2 + J θ (αi )Γ m dt Thus, ⎧ ⎪ ˙i ⎨dα α ˙ i+1 ⎪ ⎩ αi+1

(7)

(8)

the implicit integration scheme is defined by:  −1   Γ sys (αi , α = Mα − Kdt2 − Ddt ˙ i )dt + Kα ˙ i dt2 + J θ (αi )Γ m dt =α ˙ i + dα ˙i = αi + α ˙ i+1 dt (9)

Comparison of Explicit and Implicit Numerical Integrations

3.3

239

Comparison Between Explicit and Implicit Numerical Integration

For the integration of the equation, the dry friction in the motors will be considered as constant at step i. For the robots, we can use a computed torque control on the joints as presented in [9]. An example of motion for the robot with 3 and 4 joints is observed in Fig. 3 with a time step of dt = 2 ms. We can observe some difference in the evolution of the angles that are negligible w.r.t. the amplitude of the motion. Moreover, the time step can be increased and decreased and the 2 numerical integrations will remain stable.

Fig. 3. Numerical integration of the robots (on the left) and comparison between the implicit and explicit numerical integration results (on the right)

Table 1. Computation time for a motion of 6.3 s with dt=2 ms 3 joints 4 joints explicit integration ≈ 4 s

≈ 5s

implicit integration ≈ 18 s

≈ 30 s

Table 1 presents the computation time for these motions. It can be observed that the explicit integration is faster than the implicit one and that the computation time increases with the number of joints.

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Numerical Integrations with Tendon Elasticity Explicit Numerical Integration with Tendon Elasticity

We now consider that the tendons are elastic. In this case, the evolution of motor positions is not defined by the joint angles only. Thus, the motor positions need to be integrated as well. Accordingly, the explicit integration scheme of the system with tendon elasticity is defined in the system of equations (10). ⎧⎧   −1 ⎪ ⎪ ⎪ θ¨i Γ sys,θ (αi , α = (Mm ˙ i , θ i , θ˙ i ) + Γ m + Btc,i ⎪ θ ) ⎨ ⎪ ⎪ ⎪ ⎪ θ˙ ˙ ¨ ⎪ i+1 = θ i + θ i dt ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ˙ ⎪ ⎪ ⎧θ i+1 = θ i + θ i dt ⎪  ⎪ ⎪ −1 ⎪ ⎪ ¨i Γ sys,α (αi , α = (Msα ) ˙ i , θ i , θ˙ i ) + Z(αi )(tc,i ) ⎨⎪ ⎨α (10) α ˙ i+1 = α ˙i +α ¨ i dt ⎪ ⎪ ⎪ ⎪⎪ ⎩ ⎪ ⎪ = αi + α ˙ i dt α ⎪ ⎪ ⎧ i+1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨tc,i = kc xc,i + dc x˙ c,i ⎪ ⎪ ⎪ xc,i = B(f θ (αi ) − θ i ) ⎪ ⎪ ⎪ ⎩⎪ ⎩ x˙ c,i = B(Jθ (αi )α˙ i − θ˙ i ) where:

4.2

Γ sys,α (α, α) ˙ = −cs (α, ˙ α) − g(α) − Γ rα (α, α) ˙ ˙ = −Γ rθ (θ, θ) ˙ Γ sys,θ (θ, θ)

(11)

Implicit Numerical Integration with Tendon Elasticity

Similarly, the implicit integration of the system with tendon elasticity is defined by the system of equations (12). ⎧⎧ m ¨ ⎪ ˙ i+1 , θ i+1 , θ˙ i+1 ) + Γ m + Btc,i+1 ⎪ ⎨Mθ θ i+1 = Γ sys,θ (αi+1 , α ⎪ ⎪ ⎪ ˙ ˙ ¨ ⎪ θ i+1 = θ i + θ i+1 dt ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ = θ i + θ˙ i+1 dt ⎪ ⎪ ⎧θ i+1 ⎪ s ⎪ ⎪ ¨ i+1 = Γ sys,α (αi+1 , α ˙ i+1 , θ i+1 , θ˙ i+1 ) + Z(αi+1 )(tc,i+1 ) ⎨⎪ ⎨Mα α (12) =α ˙i +α ¨ i+1 dt α ˙ i+1 ⎪ ⎪ ⎪ ⎩ ⎪ = α + α ˙ dt α ⎪ i i+1 ⎪ ⎧ i+1 ⎪ ⎪ ⎪⎪ t = k x + dc x˙ c,i+1 ⎪ c,i+1 c c,i+1 ⎨ ⎪ ⎪ ⎪ ⎪ xc,i+1 = B(f θ (αi+1 ) − θ i+1 ) ⎪ ⎪ ⎩⎪ ⎩ ˙ − θ˙ i+1 ) x˙ c,i+1 = B(Jθ (αi+1 )αi+1 By defining: dθ˙ i = θ˙ i+1 − θ˙ i = θ¨i dt,

dθ i = θ i+1 − θ i = θ˙ i+1 dt = (dθ˙ i + θ˙ i )dt

(13)

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241

we can develop at the first order: dxc,i = xc,i+1 − xc,i = B(Jθ (αi )dαi − dθ i )   ˙i ∂Jθ (αi )α dαi − dθ˙ i ˙i + dx˙ c,i = x˙ c,i+1 − x˙ c,i = B Jθ (αi )dα ∂α

(14)

Thus, we can obtain: tc,i+1 = tc,i + where:

∂tc ∂tc ∂tc ∂tc ˙ dαi + dα ˙i + dθ i + dθ i ∂α ∂α ˙ ∂θ ∂ θ˙

∂tc ˙i ∂Jθ (αi )α = kc BJθ (αi ) + dc B , ∂α ∂α ∂tc ∂tc = −kc B, = −dc B ∂θ ∂ θ˙

∂tc = dc BJθ (αi ) ∂α ˙

(15)

(16)

Therefore, by applying the same computation as in Sect. 3.2, the linearization of the equations leads to: 

    2 (Mm (−Kθα dt2 − Dθα dt) Γθ dθ˙ i θ − Kθθ dt − Dθθ dt) = (17) Γα (−Kαθ dt2 − Dαθ dt) (Msα − Kαα dt2 − Dαα dt) dα ˙i

where: ∂Γ sys,θ ∂tc (θ i , θ˙ i ) + B , ∂θ ∂θ ∂tc ∂tc , Dθα = B =B ∂α ∂α ˙

Kθθ = Kθα

Dθθ =

∂Γ sys,θ ∂tc (θ i , θ˙ i ) + B ∂ θ˙ ∂ θ˙

∂Γ sys,α ∂Z(αi )tc,i ∂tc (αi , α + Z(αi ) ˙ i) + ∂α ∂α ∂α ∂Γ sys,α ∂tc ∂tc (αi , α , Kαθ = Z(αi ) = ˙ i ) + Z(αi ) ∂α ˙ ∂α ˙ ∂θ ∂tc = Z(αi ) ∂ θ˙

(18)

Kαα = Dαα Dαθ

    Γ θ = Γ sys,θ (θ i , θ˙ i ) + Γ m + Btc,i dt + Kθθ θ˙ i + Kθα α ˙ i dt2   Γ α = (Γ sys,α (αi , α ˙ i ) + Z(αi )tc,i ) dt + Kαθ θ˙ i + Kαα α ˙ i dt2

(19)

(20)

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Thus, the implementation of the implicit integration taking into account tendon elasticity is: ⎧ 

−1  ⎪ ⎪ dθ˙ i (Mm Γθ − Kθθ dt2 − Dθθ dt) (−Kθα dt2 − Dθα dt) ⎪ θ ⎪ = ⎪ 2 s 2 ⎪ d α ˙ dt − D dt) (M − K dt − D dt) Γ (−K ⎪ i αθ αθ αα αα α α ⎪ ⎪ ⎪ ⎪ ˙ i+1 ˙ i + dθ˙ i θ = θ ⎪ ⎪ ⎪ ⎪ ⎪ = θ i + θ˙ i+1 dt ⎨θ i+1 =α ˙ i + dα ˙i α ˙ i+1 ⎪ ⎪ ⎪ ⎪αi+1 = αi + α ˙ i+1 dt ⎪ ⎪ ⎪ ⎪ ⎪ = B(f (α x c,i+1 i+1 ) − θ i+1 ) θ ⎪ ⎪ ⎪ ⎪ = B(Jθ (αi+1 )α ˙ i+1 − θ˙ i+1 ) x˙ c,i+1 ⎪ ⎪ ⎪ ⎩ = kc xc,i+1 + dc x˙ c,i+1 tc,i+1 (21) 4.3

Comparison Between the Numerical Integrations with and Without Elasticity

We apply the same control on the robots as in the Sect. 3.3. We obtain motions similar to the one obtained with implicit integration without elasticity for the implicit integration with dt = 2 ms (Fig. 3) for the robot with 3 joints and the one with 4 joints. For the explicit integration, we observe that if the time step dt is greater than a certain value, approximately 0.4 ms, the numerical integration does not converge for the robot with 3 joints. Thus, a condition on the time step for the stability appears when tendon elasticity is taken into account. For the robot with 4 joints, the numerical integration does not converge even with a time step of dt = 0.025 ms.

Fig. 4. Comparison of the numerical integration of the models with or without elasticity for the robot with 3 joints for the implicit (dt=2ms) and explicit (dt=0.4ms) methods

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The difference between the numerical integration results with and without elasticity for 3 joints is presented in Fig. 4. We observe that the motion differences are similar between implicit and explicit and that these differences are negligible as compared with the amplitude of the motion. With these numerical integrations, moreover, it is also possible to compute the tendon elongation as presented in the Fig. 5.

Fig. 5. Elongation of the tendons with the implicit (dt=2ms) and explicit (dt=0.4ms) integration for the robot with 3 joints

Table 2 presents the comparison of the computation time for the model with and without elasticity. We observe that the computation time does not increase significantly for the explicit and implicit integration. Table 3 compares the performance of the 2 numerical integrations. It can be observed that when the elasticity of the tendons is taken into account, the implicit integration can becomes necessary for more complex robots. However, these results can change depending on the value of stiffness and friction in the robot. Table 2. Comparison of the computation time between models with or without elasticity for a motion of 6.3 s for 3 joints without elasticy with elasticity explicit integration (dt=0.4 ms) ≈ 17 s

≈ 20 s

≈ 18 s

≈ 22 s

implicit integration (dt=2 ms)

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Table 3. Comparison of the explicit and implicit integrations with elasticity for a motion of 6.3 s

limit time step computation time for 6.3 s integrated 3 joints explicit integration ≈ 0.4 ms implicit integration none

≈ 20 s (dt=0.4 ms) ≈ 22 s (dt=2 ms)

4 joints explicit integration < 0.025 ms implicit integration none

> 700s ≈ 35 s (dt=2 ms)

5

Conclusion

Numerical integration of tendon-driven robots can be made with explicit or implicit integration. When tendon elasticity is not considered, both approaches are stable for the numerical integration. However, for a given time step, the explicit integration is faster than the implicit integration. Thus, it is better to use explicit integration when there is no elasticity of the tendons. We have proposed an explicit and implicit integration that takes into account tendon elasticity and that integrates the dynamics of the joints and the dynamics of the motors. These numerical integrations give similar results on the robot joints evolution as the numerical integrations without elasticity, while they also allow computing the tendon elongation without increasing the computation time. We have observed that the implicit integration is still always stable with tendon elasticity. However, the explicit integration becomes unstable if the time step is not small enough. This limit time step becomes smaller when the robot becomes more complex. Accordingly, the minimal computation time that can be obtained with the explicit integration becomes much higher than the computation time that can be obtained with the implicit integration. Thus, when tendon elasticity is considered, the implicit integration is better for numerical integrations. In future work, we will take into account external efforts that can be applied on the robot, such as contacts against obstacles.

References 1. Allard, J., et al.: SOFA - an open source framework for medical simulation 2. Baklouti, S., Courteille, E., Caro, S., Dkhil, M.: Dynamic and oscillatory motions of cable-driven parallel robots based on a nonlinear cable tension model. J. Mech. Robot. 9(6) (2017) 3. Chalon, M., Friedl, W., Reinecke, J., Wimboeck, T., Albu-Schaeffer, A.: Impedance control of a non-linearly coupled tendon driven thumb. In: 2011 IEEE/RSJ International Conference on Intelligent Robots and Systems, pp. 4215–4221 (2011). ISSN: 2153–0866 4. Coevoet, E., et al.: Software toolkit for modeling, simulation, and control of soft robots. Adv. Robot. 31(22), 1208–1224 (2017) 5. Coevoet, E., Escande, A., Duriez, C.: Optimization-based inverse model of soft robots with contact handling. IEEE Rob. Autom. Lett. 2(3), 1413–1419 (2017)

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6. Efimov, D., Polyakov, A., Levant, A., Perruquetti, W.: Discretization of asymptotically stable homogeneous systems by explicit and implicit euler methods. In: 2016 IEEE 55th Conference on Decision and Control (CDC), pp. 5545–5550 (2016) 7. Fasquelle, B., Furet, M., Chevallereau, C., Wenger, P.: Dynamic modeling and control of a tensegrity manipulator mimicking a bird neck. In: Advances in Mechanism and Machine Science Proceedings of the 15th IFToMM World Congress on Mechanism and Machine Science, pp. 2087–2097 (2019) 8. Fasquelle, B., Furet, M., Khanna, P., Chablat, D., Chevallereau, C., Wenger, P.: A bio-inspired 3-DOF light-weight manipulator with tensegrity X-joints. In: 2020 IEEE International Conference on Robotics and Automation (ICRA), pp. 5054– 5060 (2020). ISSN: 2577–087X 9. Fasquelle, B., et al.: Identification and control of a 3-X cable-driven manipulator inspired from the bird’s neck. J. Mech. Robot. 14(1) (2022) 10. Gui, Y., Wang, J.T., Jin, F., Chen, C., Zhou, M.X.: Development of a family of explicit algorithms for structural dynamics with unconditional stability. Nonlinear Dyn. 77(4), 1157–1170 (2014) 11. Hrebicek, J., Rezac, M.: Modelling with maple and maplesim. In: Louca, L.S., Chrysanthou, Y., Oplatkova, Z., Al-Begain, K. (eds.) ECMS 2008 Proceedings, pp. 60–66. ECMS (2008) 12. Jourdes, F., Valentin, B., Allard, J., Duriez, C., Seeliger, B.: Visual haptic feedback for training of robotic suturing. Frontiers in robotics and AI 9, 800232 (2022) 13. Michelin, M., Baradat, C., Nguyen, D.Q., Gouttefarde, M.: Simulation and control with XDE and Matlab/Simulink of a cable-driven parallel robot (CoGiRo). In: Pott, A., Bruckmann, T. (eds.) Cable-Driven Parallel Robots. MMS, vol. 32, pp. 71–83. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-09489-2 6 14. Ottaviano, E., Castelli, G.: A study on the effects of cable mass and elasticity in cable-based parallel manipulators. In: Parenti Castelli, V., Schiehlen, W. (eds.) ROMANSY 18 Robot Design, Dynamics and Control. CICMS, vol. 524, pp. 149– 156. Springer, Vienna (2010). https://doi.org/10.1007/978-3-7091-0277-0 17 15. Snelson, K.D.: Continuous tension, discontinuous compression structures. US Patent 3169611 (1965)

Design

A Design Method of Multi-link Cable Driven Robots Considering the Rigid Structure Design and Cable Routing Yaoxin Guo(B) and Darwin Lau The Chinese University of Hong Kong, Sha Tin, Hong Kong [email protected], [email protected]

Abstract. Multi-link cable driven robots (MCDRs) have attracted much attention in recent years and have been widely used in complex working environments due to their inherent high compliance and flexibility. Although lots of work have studied on the cable routing arrangement of MCDRs, because of the highly coupled relationship between the rigid structure and cable routing, to improve the overall performance of the robot, it is necessary but challenging to take the rigid structure into account. In this paper, we propose a co-design method for MCDRs, which is based on configuration planning with target reachability and structure design with cable routing arrangement according to wrench feasibility. The complete set of design variables can be determined by applying the proposed co-design method for both rigid-links and the cable routing arrangement. Furthermore, the example results for a planar MCDR demonstrate the rationality and applicability of the proposed method.

1

Introduction

Multi-link cable driven robots (MCDRs) are a type of multi-joints serial robot actuated by cables [1]. This type of robot has attracted increasingly more attention in recent years owing to its many advantages, such as high compliance and dexterity, low self-weight compared with traditional serial rigid robots, and large rotational workspace. Due to these advantages, MCDRs have allowed a range of exciting applications, such as the hazardous material industry [2] and in-pipe inspection robot [3]. For MCDRs, both the cable routing arrangement, that means the number of cables and the cable attachment of each cable on every rigid-link, and the dimensions of rigid-links are vital to the robot’s performance [4]. Most importantly, all the design variables that can fully define the whole MCDR significantly influence the robot’s Jacobian matrix [5], and other robot properties such as the workspace, required maximum cable forces and stiffness [6]. However, determining the complete set of design variables is challenging due to the highly non-linear and complex relationship between the performance of MCDRs and the design variables. c The Author(s), under exclusive license to Springer Nature Switzerland AG 2023  S. Caro et al. (Eds.): CableCon 2023, MMS 132, pp. 249–260, 2023. https://doi.org/10.1007/978-3-031-32322-5_20

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Fig. 1. General Model of MCDRs.

Most existing MCDR designs only focus on studying cable routing schemes, and many different routing schemes have been proposed. The most intuitive one is a directly driven routing scheme in that the cables directly connect one rigid-link to the base [7], thus, the number of cables is relatively large. Then, the co-sharing routing scheme is put forward, which can be divided into two categories as external co-sharing routing scheme and internal co-sharing routing scheme [8]. The difference between these two schemes is internal scheme will have the cable routing inside one specific rigid-link. Thus, different cable routing schemes can be considered for different required performances, such as tension reduction [9] and workspace amplification [10]. In addition to the optimal cable routing scheme, the design of rigid-links also has a crucial impact on the performance of MCDRs. For example, based on specific tasks, if the number of rigid-links is over-required, it will not only increase the redundancy and complexity of the entire MCDR, but also increase the self-weight and inertia, which will weaken the advantages of using this type of robot. Thus, the complete design method of MCDRs, which includes both the cable routing arrangement and the design of rigid-links, is needed to be studied in depth. However, the complete design problem is challenging due to the highly coupled relationship between rigid-links and cable routing schemes. In this paper, a novel co-design method of configuration planning and structure design is proposed to determine the entire set of design variables for MCDRs that are both for rigid-links and cable routing arrangement, which is a taskspecific design method that considers reachability and wrench feasibility. Finally, a planar MCDR is used as an example to illustrate the rationality and applicability of the proposed method.

2

Background of MCDRs

Figure 1 shows a general multi-link cable driven robot (MCDR) that actuated by m cables with i = 1, ..., m, and is comprised of N rigid-links that j = 1, ..., N . Then, the attachment point (both fixed point and go-through point) of i-th cable on k-th segment (total k¯ ≥ N ) is denoted by Aik . The kinematics of the cable vector can be expressed as lik = Aoik (q) − Aoi(k−1) (q)

(1)

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T T where q = [q T 1 , ..., q N ] is the whole-body configuration, which is depends on the corresponding joint type. Moreover, Aoik (q) represents the corresponding cable attachment point in global frame, and lik is the cable vector through k-th segment of i-th cable. Then, the length of i-th cable li and the corresponding l˙ i are expressed as

li =

¯ k 

||lik ||

j=1

l˙ i =

¯ k  k=1

l˙ ik =

¯ k 

(2) ˆlik · l˙ik

k=1

Hence, differential kinematics of MCDRs can be concluded in the generalized ˙ with J(q) is the Jacobian matrix. Denoting Ji as coordinate that l˙ = J(q) · q, i-th row of the Jacobian matrix, which can be obtained by solving Eqs. (1) and ˙ Based on the kineto-static duality of robotics, the static (2) that l˙ i = Ji · q. equilibrium equation of MCDRs is J T (q) · f = τ (q)

(3)

where f = [f1 , ..., fm ]T is the cable force vector such that f  f  f¯ , and τ (q) is the generalized external force vector. It is well known that the overall performance of MCDRs can be analyzed through the analysis of the Jacobian matrix. However, the Jacobian matrix of MCDRs depends on not only the configuration q but also structure parameters, such as the number of cables and the length of each rigid-link. Thus, it is crucial to consider the performance of the MCDR during the design phase.

3

Problem Formulation

In order to design a MCDR according to specific task requirements, the set of design variables considered in this work is defined as D = {N, Lj , m, Aij , f¯i }, where N is the total number of rigid-links and m is the total number of cables, while Lj is the length of rigid-link j, j = 1, ..., N , also Aik and f¯i are the cable attachment point on each segment k of cable i, i = 1, ..., m and the corresponding cable force limit, respectively. The first two kinds of variables are rigid-link parameters, while the latter three can decide the cable arrangement and guide motor selection. The whole set of design variables is necessary to be determined before actual processing. The proposed design method is based on specific task requirements, and the design inputs are: – S, T are the desired start and end position of the tip, respectively; – wd is the required wrench vector on the tip of the MCDR; and – O represents obstacles that the MCDR needs to avoid during work processes.

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Fig. 2. Proposed Co-design Concept for MCDRs.

As shown in Fig. 2, the proposed co-design method of MCDRs with configuration planning and structure design includes two steps. 1. Configuration planning based on reachability: To achieve the desired tippositions (S and T ) while avoiding the obstacles (O), N and each Lj must satisfy some conditions. In this work, the rapidly exploring random tree (RRT) method is used to decide N and Lj , j = 1, ..., N , which will be discussed in Sect. 4.1 and Sect. 4.2. Then, to enable smooth transitions of the tip from S to T , a set of whole-body configurations Q will be derived in Sect. 4.3. Moreover, the first step is mainly focus on the design of rigid-links. 2. Cable routing selection based on wrench feasibility: Denoting the cable routing arrangements of MCDRs by {m, Aik , f¯i }, and continuing on the result of the first step, the optimal cable routing scheme can be determined through cable force attributes according to the desired wrench wd at each configuration q ∈ Q. The second step will be explained in Sect. 5. By applying the proposed design method, the output will be the entire set of design variables D.

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4

253

Configuration Planning Based on Reachability

It is well known that rigid-link structure plays a dominant role in the reachability of MCDRs for specific targets of the tip, and reachability represents one of the conditions that MCDRs must match in practical projects. Thus, in this section, the design variables of rigid-links N and Lj , j = 1, ..., N will be decided first to achieve the desired tip positions. Furthermore, through the tip-transition from S to T , a set of whole-body configuration Q will yield. 4.1

RRT Algorithm

The proposed design method uses the RRT planning algorithm with a fixed search step for finding the necessary number of rigid-links N of the MCDR based on reachability while avoiding obstacles. While, as shown in Fig. 3, nodes in the RRT concept represent the corresponding joints of MCDRs in this work, which is related to N , and the fixed search step can be treated as the length Lj of the j-th rigid-link. Consequently, the line-segment between two nodes is shown as the corresponding rigid-link. The basic concept of the RRT algorithm with referring to [11], the initial search tree T is generated first according to the known obstacles, and V is the selection set of feasible samples. Then the nearest node (joint of the MCDR) xnearest ∈ V to the samples xsample ∈ T can be selected. Thus, by applying the fixed search step (rigid-link length) L, the new node can be assigned as xnew = xnearest + L ·

xsample − xnearest ||xsample − xnearest ||

(4)

It is necessary to check the collision situation with obstacles and line-segment xnearest xnew , and if it is collision free, the new node xnew can be added into V which is connected to the corresponding xnearest . Furthermore, if the distance between xnew and Ttarget (like the desired tipposition S and T in this work) is no more that the fixed search step L that |xnew Ttarget | ≤ L, the search process will end. The output result will be the node set V with the information of node-connections with fixed search step.

Fig. 3. Illustration of RRT Algorithm with Fixed Search Step.

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Adjustment of the Length of Rigid-Links

Because the fixed search step (rigid-link length) is considered during the RRT search process, except the length of the last rigid-link (|xnew Ttarget | ≤ L), all other rigid-links are of equal length to the fixed L that L1 = L2 = ... = LN −1 = L. However, for the length of the last rigid-link LN , it may be different for different desired tip-position, thus, the adjustment process of LN needs to be considered. By applying the RRT with a fixed search step based on the start/end tippositions S and T , the length of the last rigid-link is denoted as Ls = |xsN −1 S| and Lt = |xtN −1 T |, separately. In order to make the designed MCDR has lower self-weight and lower inertia, by regulating the whole-body configuration slightly, it is possible to make the length of the last rigid-link as that LN = min (Ls , Lt )

(5)

Thus, the number of rigid-links of the MCDR with the corresponding rigidlink length (N, Lj , j = 1, ..., N ) are decided based on reachability. 4.3

Whole-Body Configuration Planning

Furthermore, as shown in Fig. 3, the corresponding whole-body configuration is known simultaneously to achieve the desired tip position with the designed parameters of rigid-links. For the start tip-position, the obtained configuration T T T is denoted as q S = [q T S1 , ..., q SN ] , while, q is for the end tip-position. To make a smooth transition of all N joints to meet the tip-position change from S to T , in this work, we do whole-body configuration planning by discretely interpolating each joint angle. Thus, to interpolate for k¯ steps, we have k q k = q S + ¯ · (q T − q S ), k = 1, ..., k¯ − 1 k

(6)

Then, a set of whole-body configurations to make the tip transfer from S to ¯ T will be decided as Q = {q S , q 1 , ..., q k−1 , q T }, and it will be used for the future configuration-based design of the cable routing.

5

Cable Routing Selection Based on Wrench Feasibility

The cable routing arrangement of MCDRs includes the number of cables m, the cable attachment point Aik which is for i-th cable on k-th segment with i = 1, ..., m, and the corresponding cable force limit f¯i . It is known that the cable routing arrangement has a decisive effect on the overall performance of MCDRs through the influence of the Jacobian matrix. Besides reachability, wrench feasibility based on the desired external wrench is another condition MCDRs must meet in practical projects. Thus, according to the static equilibrium in Eq. (3), it is vital to design the cable routing arrangement to make the MCDR wrench feasible.

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Heuristic-Based Cable Routing Selection for MCDRs

Drawing on the heuristic-based design idea of cable driven parallel robots (CDPRs) as in [12], the heuristic-based selection of optimal cable routing scheme is used in this work. First, the heuristic cable routing scheme is not a definite cable routing arrangement but a general scheme to produce the cable routing arrangement for different requirements, which is explained in detail in [12]. Moreover, a library of heuristic cable routing schemes for MCDRs should be generated in advance based on previous experience and analysis. Thus, the task-specific preliminary set of heuristic cable routing schemes can be picked out by designers after the rough selection process. Then, different cable routing arrangements can be derived by applying the preliminary set of heuristic cable routing schemes. After that, the final optimal cable routing arrangement will be determined through the evaluation simulations and results analysis. 5.2

Optimal Cable Routing Arrangement

While achieving the whole-body configuration as planned in Sect. 4.3, to satisfy the wrench requirement for each time-step, in this work, the evaluation for selecting the optimal cable routing arrangement is the performance of cable forces based on the required wrench vector wd . First, the minimum norm of cable force vector is denoted as one performance indicator that min ||f ||2 s.t. J T (q) · f = τ

(7)

f 0 where τ is the projection of the desired wrench vector wd from task space onto generalized configuration space. In addition, since it takes into account the smooth transition of all joints of the MCDR, the change in cable forces during the movement should be the other evaluate indicator. For the change in t and t + 1 time-step, that min ||f t+1 − f t ||2 s.t. J T (q t+1 ) · f t+1 = J T (q t ) · f t = τ f t+1 , f t  0

(8)

q t+1 , q t ∈ Q Furthermore, the upper limit of cable force can be decided as f¯i ≥ max{fi (q t ), ∀q t ∈ Q}

(9)

Finally, the optimal cable routing arrangement (m, Aik , f¯i , i = 1, ..., m) can be determined through the evaluation simulations and results analysis process. When the whole design of the proposed design method for MCDRs is finished, the output results are the complete set of design variables that D = {N, Lj , m, Aik , f¯i }, with j = 1, ..., N, i = 1, ..., m.

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Example and Results

This section uses a planar MCDR as an example to demonstrate the proposed co-design method for MCDRs. The task requirements (S, T , O, and τ which is projected from wd ) are shown in Fig. 4. In the end, the set of design variables D will be given with the result analysis. The units of length and force are cm and N , respectively. For brevity, the units are omitted below.

Fig. 4. Design Inputs for the Example.

6.1

Configuration Planning Based on Reachability

The first step of the proposed design method is to find a fair number of rigid-links with the corresponding length for achieving the desired tip positions. By denoting the fixed search step as L = 9, the result is shown in Fig. 5. The necessary number of rigid-links is the same as N1 = N2 = 3, then we can determine the designed number of rigid-links as N = 3. For the length of each rigid-link, as we explained before, except for the last rigid-link length, the others are the same as the fixed search step that L1 = L2 = L = 9. However, the length of the last rigid-link differs from LS3 = 7.495 and LT3 = 7.927. Thus, by applying Eq. (5), the design last rigid-link length is given as L3 = min (LS3 , LT3 ). In order to make the manufacturing easier, we take the approximation that L3 = 7.5. After adjusting the length of rigid-links, the whole-body configurations q S and q T that are related to the desired start/end tip-positions are also decided, which is shown in Fig. 6(a). By following Eq. (6) to interpolate k¯ = 5 steps between q S and q T , as in Fig. 6(b), the set of whole-body configurations can be obtained as ⎡ ⎤ −37.11 −33.52 −29.93 −26.34 −22.75 −19.16 Q = ⎣ 88.59 79.98 71.37 62.76 54.16 45.55 ⎦ −5.56 −20.47 −35.39 −50.30 −65.21 −80.13

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Fig. 5. Result of the RRT-planning Step with L = 9.

Fig. 6. (a) Designed Length in (cm) of Rigid-links based on Desired Poses Reachability with Corresponding Whole-body Configurations in degress (◦ ). (b) Planning Result of the Whole-body Configuration.

After the first step of the proposed design method, the parameters of designed rigid-links (N = 3, L1 = L2 = 9 and L3 = 7.5) for the MCDR and the set of whole-body configurations Q are decided. 6.2

Cable Routing Selection Based on Wrench Feasibility

As discussed in Sect. 5.1 and [12], the generation of the library of heuristic cable routing schemes for MCDRs and selection of the preliminary set of heuristic cable routing schemes are highly depend on the experience of designers. Thus, those parts will not be expanded in detail in this example, however, Fig. 7 shows the different resulting cable routing arrangements by applying different preliminary cable routing schemes. Figure 7(a) shows the directly driven routing scheme, which represents each joint is driven independently by two cables, and total 6 cables are needed.

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Fig. 7. A Simple Illustration of Different Cable Routing Arrangements as in (a)–(c) with Red-circles Means Where Cables Go Through, and (d) is the Designed Dimensions from the Previous Steps.

Figure 7(b) and (c) are related to external co-sharing routing scheme and internal co-sharing routing scheme, respectively, and both need 4 cables. By applying different cable routing schemes on a specific task, different cable routing arrangements containing m and Aik are decided as in Fig. 7. Based on the configurations in Q, to make the designed MCDR satisfies wrench feasibility for the desired external wrench wd on the tip, the optimal cable routing arrangement can be determined through two performance indicators defined in Sect. 5.2. First, the required maximum cable force is calculated by Eq. (7) and Eq. (9), and the results of different cable routing arrangements can be seen in Fig. 8. It shows that wrench infeasible configurations exist for directly driven routing scheme. Then, although internal co-sharing routing scheme has the smaller

Fig. 8. Maximum Cable Force for Each Desired Pose in Q based on Different Cable Routing Arrangements.

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Fig. 9. (a) Combined Wrench Feasible and Collision Free Workspace of the Designed MCDR with the Joint Limits. The Colorbar Shows the Maximum Cable Force. (b) Hardware Setup of the Designed MCDR.

required cable force to meet wrench feasibility, external co-sharing routing scheme is smoother in the changes of cable force. Then, the second performance indicator expressed in Eq. (8) needs to be considered together due to the maximum cable force (13.94 N ) that is required for external co-sharing routing scheme is acceptable. The smoother change in cable force is the most important in practical applications, thus, the optimal cable arrangement is decided as the arrangement derived by external co-sharing routing scheme. Finally, the complete set of design variables D is derived that N = 3, L1 = L2 = 9, L3 = 7.5, m = 4, and in local frame of each rigid-link that cable 1: A11 = [0, −3]T A12 = [8, −2.5]T f¯1 = 13.94 cable 2: A21 = [0, 3]T A22 = [8, 2.5]T A23 = [8, 2.5]T f¯2 = 1.07 cable 3: A31 = [0, 2]T A32 = [8, 2]T A33 = [8, 2]T A34 = [6, 2]T f¯3 = 9.59 cable 4: A41 = [0, −2]T A42 = [8, −2]T A43 = [8, −2]T A44 = [6, −2]T f¯4 = 8.82 The designed MCDR is shown in Fig. 9(a), and the hardware setup is in Fig. 9(b). It shows that the designed MCDR has the capability to meet the reachability and wrench feasibility by following the expected configurations.

7

Conclusion and Future Works

In this work, we put forward a new concept for MCDRs: reachability guides the design of rigid-links for MCDRs, and wrench feasibility will determine the optimal cable routing arrangement by considering two evaluate indicators about cable forces. Furthermore, all design variables can be derived through the proposed co-design method. For future works, some crucial points will be considered, such as the cable interference with both obstacles and rigid-links, the configuration singularity and friction analysis in motion.

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Acknowledgements. The work was supported by the Research Grants Council (General Research Fund Reference No. 14203921) and the Innovation and Technology Commission (Innovation and Technology Support Programme Reference No. ITS/347/19FP).

References 1. Mustafa, S.K., Agrawal, S.K.: On the force-closure analysis of n-DOF cable- driven open chains based on reciprocal screw theory. IEEE Trans. Robot. 28(1), 22–31 (2011) 2. Buckingham, R., Graham, A.: Dexterous manipulators for nuclear inspection and maintenance–case study. In: 2010 1st International Conference on Applied Robotics for the Power Industry, pp. 1–6. IEEE (2010) 3. Li, T., Ma, S., Li, B., Wang, M., Wang, Y.: Axiomatic design method to design a screw drive in-pipe robot passing through varied curved pipes. Sci. China Technol. Sci. 59(2), 191–202 (2016) 4. Peng, J., Xu, W., Liu, T., Yuan, H., Liang, B.: End-effector pose and arm-shape synchronous planning methods of a hyper-redundant manipulator for spacecraft repairing. Mech. Mach. Theory 155, 104062 (2021) 5. Wang, Y., Yang, G., Zheng, T., Yang, K., Lau, D.: Force-closure workspace analysis for modular cable-driven manipulators with co-shared driving cables. In: 13th IEEE Conference on Industrial Electronics and Applications (ICIEA), pp. 1504–1509. IEEE (2018) 6. Sanjeevi, N., Vashista, V.: Stiffness modulation of a cable-driven serial-chain manipulator via cable routing alteration. J. Mech. Robot. 15(2), 021009 (2022) 7. Rezazadeh, S., Behzadipour, S.: Tensionability conditions of a multi-body system driven by cables. In: ASME International Mechanical Engineering Congress and Exposition, vol. 43033, pp. 1369–1375 (2007) 8. Ramadoss, V., Lau, D., Zlatanov, D., Zoppi, M.: Analysis of planar multi- link cable driven robots using internal routing scheme. In: International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, vol. 83990. American Society of Mechanical Engineers, V010T10A029 (2020) 9. Wang, Y., Yang, G., Zheng, T., Shen, W., Fang, Z., Zhang, C.: Tension reduction method for a modular cable-driven robotic arm with co-shared cables. Intell. Serv. Robot. 15(1), 27–38 (2022) 10. Sanjeevi, N., Vashista, V.: Effect of cable co-sharing on the workspace of a cabledriven serial chain manipulator. In: Proceedings of the Advances in Robotics 2019, pp. 1–6 (2019) 11. Rodriguez, S., Tang, X., Lien, J.-M., Amato, N.M.: An obstacle-based rapidlyexploring random tree. In: Proceedings 2006 IEEE International Conference on Robotics and Automation, ICRA 2006, pp. 895–900. IEEE (2006) 12. Guo, Y., Lau, D.: Heuristic-based design framework for the cable arrangement of cable-driven parallel robots. In: Gouttefarde, M., Bruckmann, T., Pott, A. (eds.) CableCon 2021. MMS, vol. 104, pp. 194–205. Springer, Cham (2021). https://doi. org/10.1007/978-3-030-75789-2 16

A Predictor-Corrector-Scheme for the Geometry Planning for In-Operation-Reconfiguration of Cable-Driven Parallel Robots Felix Trautwein(B) , Thomas Reichenbach, Andreas Pott, and Alexander Verl Institute for Control Engineering for Machine Tools and Manufacturing Units (ISW), University of Stuttgart, Stuttgart, Germany [email protected] http://www.isw.uni-stuttgart.de/ Abstract. This paper presents a predictor-corrector-scheme for determination of a geometry for a cable-driven parallel robot (short: cable robot) for In-Operation-Reconfiguration. The prediction step calculates a guess for the geometry, by determining a transformation of the robot based on a simple workspace abstraction. By using the prediction as initial values, the correction step determines the final set of reconfigurable parameters by constraint optimization, such that, the robot fulfils the requirements sufficiently. The approach considers a spatial cuboid as requirement and takes into the robot properties of cable forces, platform velocities and acting platform wrenches. Conclusively, the method is validated by a simulative scenario, where the numerical properties are evaluated and additionally, practical reconfiguration experiments (according to ISO 9283) to investigate the suitability of the general planning approach. The results show on the one hand that the suitability to rapidly determine a suitable robot’s geometry and on the other hand the feasibility for real reconfiguration planning.

Keywords: Cable-Driven Parallel Robot Robots · Design of Cable Robots

1

· Reconfiguration of Cable

Introduction

Cable-Driven Parallel Robots (short: cable robots) offer outstanding properties in terms of large workspaces, high payloads or the reconfigurability of the structure. These properties make cable robots suitable for large-scale applications from various areas like research on motion perception [1], field phenotyping [2] or the additive manufacturing of concrete building elements. Regarding the last application, existing approaches are Bosscher et al. [3] or in the context of the EU project HINDCON with hybrid manufacturing [4]. The industrial relevance can be shown by initiatives like Cobod [5] from Peri or Karlos from Putzmeister [6]. However, the mentioned systems Cobod and Karlos exhibit disadvantages like a c The Author(s), under exclusive license to Springer Nature Switzerland AG 2023  S. Caro et al. (Eds.): CableCon 2023, MMS 132, pp. 261–272, 2023. https://doi.org/10.1007/978-3-031-32322-5_21

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massive mechanical setup that is hard to upscale or to reconfigure. Consequently, cable robots could represent a scalable, reconfigurable and flexible solution for the application of on-site concrete printing. In particular, the ability to reconfigure is of interest here, since it could be used to divide a large required workspace into smaller segments, which allows for smaller and cheaper dimensioning of the components. Existing work regarding dynamic or real-time reconfiguration are Reichert et al. [7], Zhou et al. [8] or recently Xiong et al. [9]. Further studies for reconfiguration planning with discrete and continuous parameters are presented by [10,11] and Nguyen et al. [12], respectively. It is to note, that this paper enhances and generalizes the results from Trautwein et al. [13]. However, there is a lack of a combination of a representative formulation of process requirements and numerical efficient geometry planning, which needs no cumbersome preparations or training of the model. Hence, the contribution of the paper is a reconfiguration planning approach, which allows for determination of a cable robot’s geometry by approximation of an initial guess (prediction) based on a rough workspace measure, and subsequently application of constrained optimization to minimize the difference between the requirement and robot’s properties (correction). The paper is structured as follows. At first, the modelling of the examined system is introduced by the kinematic and static definitions, the modelling of reconfigurable parameters and the robot’s properties. Afterwards, the predictor-corrector-scheme for the geometry planning is presented, followed by a simulative and experimental validation (for set up see, Fig. 2).

2 2.1

Modelling Kinematic Foundations

The fundamental kinematic definitions are shown in Fig. 1. Hereby, the vector loop starts with the position vector r ∈ R3 to the coordinate frame of the platform KP and then to the i-th attachment point Bi on the platform by bi . The i-th

Fig. 1. Vector loop for kinematics of a general spatial cable robot

Fig. 2. Experimental set up of the cable robot and laser tracker

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cable is denoted by li with the unit vector ui = ||llii||2 , pointing towards the base attachment point Ai . Finally, the vector loop is closed with the vector ai , which describes the distance between the global coordinate frame K0 and the respective attachment point Ai . It is to note, that the modelling was done according to the standard model, and formulates the cables as ideal unilateral constraints and the cable attachments as point shaped. 2.2

Reconfigurable Parameters and Homogeneous Transformations

The problem of reconfiguration planning is related to general design of cable robots. However, a major difference is that for reconfiguration the system already exists, and the reconfiguration operations have to be compliant with the hardware setup. This means that e.g. the attachment points on the frame can not be arbitrarily placed but rather have to be put where the existing structure allows for. Consequently, these hardware restrictions are considered by a parametrization of the design variables. In this paper, we make two general assumptions: 1. The reconfigurable parameters are assumed to be of continuous nature. 2. Each reconfiguration operation is implemented by a homogeneous transformation (see Denavit-Hartenberg-Convention [14]). On the one hand, this ensures enough degrees of freedom to apply this method to theoretical scenarios where no restrictions are present, and on the other hand, to implement parameter restrictions by excluding the respective coordinates from the transformation process by an index vector (see [13]). The considered transformations are defined as       0t R0 diag(sx , sy , sz ) 0 , TR = and TS = . (1) TT = 01 0 1 0 1 Hereby, TT ∈ R4×4 describes a translation by t ∈ R3 and TR ∈ R4×4 a rotation by the orthogonal rotation matrix R ∈ R3×3 . Besides those, the matrix TS ∈ R4×4 describes a scaling, that scales the respective coordinates along the main coordinate axes by the factors sx , sy , sz ∈ R. By chaining TT and TR as T = TT TR one can define a rigid body motion as     Rt RT −RT t T = with T −1 = . (2) 0 1 0 1 Additionally, the motion expressed by T can be reverted by its inverse transformation T −1 . Besides this interpretation, the matrices can also be interpreted as transformations between two coordinate systems. In the following, we denote such a transformation from frame KB to frame KA as A T B that transforms a point as pA = A T B pB . In the following chapter, the relevant properties for the process requirements are introduced. 2.3

Properties of the Cable Robot

In general, the criteria of cable-force-, wrench-set- and velocity-set-feasibility are considered. Since all three criteria evaluate to a boolean true/false decision,

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we apply the line-search concept from Pott et al. [15] to capture the respective spatial area by its hull. Additionally, we require that at each position the criteria has to be fulfilled for a set of orientations R0 . It is to note, that all calculations were carried out by the implementations from WireX [16] on a Intel Core i77500U CPU with 2.7 GHz and Python3.8 64bit In the following sections, the respective criteria are introduced. Cable-Force-Condition. By this criterion we express the cable-forcefeasibility in terms of a workspace set as WCF = {y | y = (r, R) ∀ R ∈ R0 ∧ fi ∈ [fmin , fmax ]} .

(3)

Hereby, R0 denotes the set of required orientations and fmin and fmax the cable force limits, respectively. The spatial domain, where the condition is fulfilled is expressed by WCF . Wrench-Set-Condition. For taking into account a priori unknown disturbances, it is necessary to consider an entire set of acting forces and moments on the platform. Therefore, we apply the approach from Bouchard et al. [17], which yield to the workspace WWS = {y | y = (r, R) ∀ R ∈ R0 ∧ TWS ⊂ AWS } .

(4)

Hereby, the workspace area is denoted by WWS , the Task-Wrench-Set TWS and the Available-Wrench-Set AWS . Velocity-Set-Condition. Similar to the section above, we require that the platform is able to generate a set of prescribed motions in certain directions. By application of the approach from Gagliardini et al. [18], this leads us to the respective workspace as WVS = {y | y = (r, R) ∀ R ∈ R0 ∧ TVS ⊂ AVS } .

(5)

Hereby, WVS denotes the workspace itself and TVS as the Task-Velocity-Set as well as the Available-Velocity-Set AVS . Superposition of the Conditions. In order to express one common workspace we define the intersection as WKin = WCF ∩ WWS ∩ WVS .

(6)

The workspace set is called the kinetic workspace and denoted with WKin . In order, to ensure compatibility to the line-search method, we define the boolean criterion as WKin = { y | y = (r, R) ∀ R ∈ R0 ∧ fi ∈ [fmin , fmax ] ∧ TWS ⊂ AWS ∧ TVS ⊂ ACVS } .

(7)

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Based on the introduced description of the properties, the predictor-correctorscheme is presented in the next section.

3

Predictor-Corrector-Scheme

In this section, we introduce a predictor-corrector-scheme in order to solve the task of reconfiguration planning for cable robots. As starting point, we examine the initial situation from Fig. 3. The global origin is defined by the coordinate frame K0 . Relatively to that, the coordinate systems of the robot KR and the requirement KREQ are introduced. For both coordinate systems, it is allowed that they are translated and rotated around K0 . So, the relation between the frames are expressed by the homogeneous transformation 0 T R from KR → K0 and the transformation 0 T REQ from KREQ → K0 . The application of both introduced transformations is explained in the following Sect. 3.2 Hereby, it is to note, that the transformation matrices are applied to the position vectors of the cable attachment points. In the following, the determination of the transformations according to the predictor-corrector-scheme is introduced.

Fig. 3. Initial cable robot with its workspace as green mesh and the required workspace area from the respective application as blue cuboid

3.1

General Scheme

The principal idea of the predictor-corrector-scheme is inspired by numerical integration techniques. A common solution strategy is the use of an explicit method to calculate a qualified guess, in order to subsequently compute the solution with an implicit method (see Heun’s method [19]). Based on our findings, we want to adapt and apply this general procedure to our problem of geometry planning, which results in the following two steps: 1. Prediction: Based on the difference between the actual state and the requirement, three transformations (translation, rotation, scaling) are determined and applied to the robot’s geometry in order to find a good initial guess.

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2. Correction: Based on constraints and quality measures for the coverage of the requirement, the remaining difference between the robot’s workspace and the requirement is eliminated by determining a suitable scaling transformation. By using the prediction step before employing the numerical solver, we want to ensure reliable initial conditions as well as fast and stable convergence to a suitable solution. Both steps are explained in the following two sections. 3.2

Prediction

As shown in Fig. 3, the first step is to calculate the difference between the center of the requirement and the center of the workspace. To do so, the workspace hull is evaluated according to the criteria from Sect. 2.3 and the center is computed as k vk with k = 1, . . . , nv . (8) rWS = 1 nv Hereby vk denotes the k-th hull vertex and the index runs until the number of vertices nv = 22(ni +1) + 2. The symbol ni denotes the iteration depth of the triangulation that defines the discretization resolution. It is assumed, that the orientation of the coordinate frame in the workspace center is aligned with robot frame which yield to       RR rR R WS 1 rWS RREQ rREQ 0 R 0 REQ T = = = , T and T . (9) 0 1 0 1 0 1 Based on the transformations from Eq. (9), the vector loop from the origin via the coordinate frames and back reads as 0

T R R T WS WS T REQ (0 T REQ )−1 = 1.

(10)

Which yields to the relative transformation between workspace and requirement WS

T REQ = (0 T R R T WS )−1 0 T REQ .

(11)

The prediction for the displacement WS T REQ is given by Eq. (11). Reflecting Fig. 3, the next step is the determination of the predictive scaling factors. The core idea is to calculate the largest cuboid that fits into the workspace hull. Hereby, the search directions are defined as the lines between the center of the requirement and its eight vertices. Along these lines, the shortest distance is chosen and a box with equal side lengths is defined. Based on these side lengths, the inner box is defined by lI , wI and hI , which yield to the scaling factors ⎡ ⎤ sx 0 0 0 ⎢ 0 sy 0 0⎥ lREQ wREQ hREQ ⎥ TS = ⎢ and sz = . (12) ⎣ 0 0 sz 0⎦ with sx = lI , sy = wI hI 0 0 0 1 Finally, the three homogeneous transformations for the prediction are available and the attachment points are transformed as, ai,p = T S WS T REQ ai

with

i = 1, . . . , m.

(13)

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Correction

As aforementioned, the center of the workspace and requirement match. Despite the predictive scaling step, the workspace covers the cuboid not entirely, and we apply the iterative correction based on constraint optimization to eliminate the difference. At first, all search directions are determined according to, uv,i =

vi − rREQ . ||vi − rREQ ||2

(14)

Hereby, rREQ is the cuboid’s center and vi the i-th vertex. Along these directions, the intersection point with the workspace hull is determined and the scalar difference with the diagonal length of the cuboid is calculated as cv,i = λi − λREQ,i

with

i = 1, . . . , 8.

(15)

The distance from rREQ to the workspace hull along uv,i is denoted by λi . The combination of all scalars into one vector is the coverage vector, denoted by cv ∈ R8 . Consequently, thescalar lengths of the requirement are calculated 2 2 + wREQ + h2REQ . Based on the introduced straightforward as λREQ,i = 12 lREQ coverage measure, we define the constraint optimization problem as, min TS

subject to

cv 2 cv,i > 0.

(16)

So, the given optimization problem Eq. (16) minimizes the norm of the coverage vector cv by adjusting the scaling T S applied to the proximal anchor points Ai (c.f. Eq. (13)) in order to ensure the smallest possible robot and concurrently, the constraints on the coverage components ensure that the entire requirement box is covered by the workspace. In the following section, the simulative and practical evaluation of the method is presented.

4

Validation Studies

The evaluations in this section investigate the numerical behaviour on the one hand and the practical relevance on the other hand. By practical relevance is meant, that the determined geometry actually functions on a real robot and, thus, justifies the general approach. 4.1

Simulation Studies

For this simulation study, the initial situation from Fig. 3 is applied. Hereby, the robot’s frame system KR is translated by rR = [1 m, 0, 0]T and remains in neutral orientation. The requirement KREQ is moved by rREQ = [−1 m, 0, 0.5 m]T , rotated by γ = 45◦ around the z-axis and the cuboid has the dimensions lREQ = 0.5 m, wREQ = 0.5 m and hREQ = 0.3 m. The cable force limits are fmin = 25 N

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and fmax = 500 N, the required Task-Wrench-Set is defined symmetrically around the origin as TWS = [10 N, 10 N, 100 N, 0, 0, 0]T and the respective TaskVelocity-Set as TVS = [0.05 m s−1 , 0.05 m s−1 , 0.05 m s−1 , 0, 0, 0]T . The limits of the task velocity set were defined according to 50 mm s−1 , a common feed rate for 3D printing processes for the size of this machine. Thus, all three criteria are expressed via the common workspace hull and are thus taken into account in the calculation process. For the cable robot, we assume that the system has no reconfiguration option in vertical z-direction and allow continuous parameters in x- and y-direction and the attachment points on the platform remain unchanged. By using the introduced parameters, the result of the prediction step is shown in Fig. 4. Hereby, the dimensions of the found inner rectangle are lI = 0.52 m, wI = 0.52 m and hI = 0.31 m, which results in the respective scaling factors as sx = 0.97, sy = 0.97 and sz = 1.0. The prediction process took approx. 5 s. Then, the predicted geometry is used as initial guess for the numerical solver, whereby we used the COBYLA solver (COBYLA: Constrained Optimization BY Linear Approximation) from SciPy [20]. One of the reasons, we chose the COBYLA solver is that this routine doesn’t rely on the gradient of the objective function and, thus, we don’t have to take care or ensure this property. The results of the geometry correction are depicted in Fig. 5. For this scenario, the solver converges in 40 steps and needs approx. 20 s. An interesting effect is, although the workspace is already relatively close to the solution, the constraints enforce the system to be stretched very long in the x-direction of the local frame KR . Hence, for the design of the optimization problem it could be better to drop the constraints on the coverage measure and rather than just minimize the coverage measure and tolerate, that there can be small areas in the corner of the requirement, which are not covered by the theoretical workspace.

Fig. 4. Results of the geometry prediction, which show the translated and rotated robot

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Fig. 5. Results of the geometry correction

4.2

Experimental Studies

The examined cable robot is a small-sized robot at ISW (see Fig. 2) with a regular frame volume of roughly 1 m3 and m = 8 cables. All the eight proximal anchor points Ai can be moved manually along the vertical profile struts and the distal anchor points Bi remain constant. The experiment was conducted to investigate the robot behaves after a reconfiguration and, thus, to evaluate if a manual In-Operation-Reconfiguration can be performed economically and without exhaustive recalibration. In order to make a comparison from before to after, we consider the absolute positioning precision (AP) and the repeatability precision (RP) as performance-describing quantities. Both measures are determined according to the ISO 9283 [21]. In the following list, the procedure for the experimental precision evaluation is given: 1. Definition of a requirement cube and derive measurement positions according to the ISO 9283. For the measurement positions pi , see Fig. 6. 2. Derivation of the reference values for the AP and RP at the cable robot according to the measurement. Hereby, the geometry of the reference system was calibrated by a laser tracker. 3. Determination of a new geometry for the robot using the introduced predictorcorrector-scheme. 4. Reconfiguration of the system according to the determined geometry and repeated measurement of the AP and RP. For the simulative reconfiguration planning, it is assumed, that the systems K0 and KR coincide at the origin. The requirement is defined as a cuboid at rREQ = [0, 0, 0.4 m]T with lREQ = 0.2 m, wREQ = 0.2 m and hREQ = 0.2 m. For this experiment, the procedure with three distinct properties (see Sect. 2.3) is simplified in order to enable a representative validation. The admissible cable forces are determined with consideration of the installed drive train as, fmin = 5 N and fmax = 70 N. The external load is just the own gravitational force of the platform as fg = 8 N and the platform is fixed to zero orientation. Based on these assumptions, the achieved results from the planning process are illustrated in the right picture of Fig. 6. As expected, the process shifts the robot, such that

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Fig. 6. Initial and final configuration with the requirement as blue box as well as the measurement points placed at the center and vertices of the box

Fig. 7. Absolute and repeatability precision in reference and reconfigured configuration

its workspace center matches the requirement and shrinks its height until the upper vertices of the blue box touch the workspace hull. The evaluation needs approx. 10 ms for the prediction step and 20 iterations for the correction, which is approx. 350 ms computation time. The reason for the significantly smaller computation time lies mainly in the drop of the consideration of the wrench and velocity set calculation. Consequently, the next step is to adjust the vertical positions of the attachment points according to the determined positions. To do so, the robot was moved to rREQ and one upper cables was loosened, and the attachment points moved to the new location. When fixed at the new position, the geometry of the control algorithms was updated during runtime and the cables were fastened with a defined cable force distribution to ensure the zero orientation of the platform. This procedure was done until the configuration is changed from the reference configuration Cref to the reconfigured configuration Creconf . However, the precision results (not shown here) for the intermediate configurations were not good, since, the measurement points were outside the corresponding workspace. The position accuracy was measured with a laser tracker (FARO Vantage E6 ) as a reference measurement system. The results for the six measurement poses are illustrated in Fig. 7. The AP over all six poses in Cref is 6.24 mm and in reconfigured configuration Creconf 11.75 mm. The RP in Cref is 0.13 mm and for Creconf 0.17 mm.

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Conclusion and Outlook

In this paper, a predictor-corrector-scheme for the determination of a geometry for a cable robot for In-Operation-Reconfiguration is presented. First, the foundations of the applied modelling and the reconfigurable parameters in terms of homogeneous transformations are introduced. Second, the predictor-correctorscheme for the geometry planning according to a cable-force-, wrench-set- and velocity-set-criterion is presented. Third, the feasibility for simulative and experimental reconfiguration planning is investigated. The next step is to investigate a “buffer-zone“, which ensures a distance between the workspace and requirement to avoid that the robot approaches the borders of the workspace in order to avoid the degeneration of the robot’s properties. Acknowledgements. This work was partially supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy - EXC 2120/1 - 390831618 and the project grant 317440765.

References 1. Miermeister, P., et al.: The cablerobot simulator large scale motion platform based on cable robot technology. In: 2016 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS 2016), pp. 3024–3029. IEEE (2016) 2. Kirchgessner, N., et al.: The ETH field phenotyping platform FIP: a cablesuspended multi-sensor system. Funct. Plant Biol.: FPB 44(1), 154–168 (2016) 3. Bosscher, P.M., Williams, R.L., II., Bryson, L.S., Castro-Lacouture, D.: Cablesuspended robotic contour crafting system. Autom. Constr. 17(1), 45–55 (2007) 4. Mu˜ niz, M.M., et al.: Concrete hybrid manufacturing: a machine architecture. Procedia CIRP 97, 51–58 (2021) 5. COBOD. World leader in 3D construction printing | COBOD international (2022). https://cobod.com/ 6. Putzmeister GmbH: Karlos (2022). https://www.putzmeister.com/web/europeanunion/news-article-detail/-/asset publisher/karlos-efficient-and-economicalconcrete-walls-from-the-3d-printer-1?redirect=/ 7. Reichert, C., Glogowski, P., Bruckmann, T.: Dynamische Rekonfiguration eines seilbasierten Manipulators zur Verbesserung der mechanischen Steifigkeit. In: Bertram, T., Corves, B., Janschek, K. (eds.) Fachtagung Mechatronik 2015: Dortmund, Aachen, 12–13 March 2015, pp. 91–96. Inst. f¨ ur Getriebetechnik und Maschinendynamik (2015) 8. Zhou, X., Jun, S.-K., Krovi, V.: Tension distribution shaping via reconfigurable attachment in planar mobile cable robots. Robotica 32(02), 245–256 (2014) 9. Xiong, H., et al.: Real-time reconfiguration planning for the dynamic control of reconfigurable cable-driven parallel robots. J. Mech. Robot. 14(6) (2022) 10. Gagliardini, L., Caro, S., Gouttefarde, M., Girin, A.: Discrete reconfiguration planning for cable-driven parallel robots. Mech. Mach. Theory 100, 313–337 (2016) 11. Skopin, M., Long, P., Padir, T.: Design of a docking system for cable-driven parallel robot to allow workspace reconfiguration in cluttered environments. In: Gouttefarde, M., Bruckmann, T., Pott, A. (eds.) CableCon 2021. MMS, vol. 104, pp. 158–169. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-75789-2 13

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12. Nguyen, D.Q., Gouttefarde, M., Company, O., Pierrot, F.: On the analysis of large-dimension reconfigurable suspended cable-driven parallel robots. In: 2014 IEEE/RAS International Conference on Robotics and Automation (ICRA 2014), pp. 5728–5735. IEEE (2014) 13. Trautwein, F., Reichenbach, T., Pott, A., Verl, A.: Workspace planning for inoperation-reconfiguration of cable-driven parallel robots. In: Gouttefarde, M., Bruckmann, T., Pott, A. (eds.) CableCon 2021. MMS, vol. 104, pp. 182–193. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-75789-2 15 14. Denavit, J., Hartenberg, R.S.: A kinematic notation for lower pair mechanisms based on matrices. J. Appl. Mech. 22, 215–221 (1955) 15. Pott, A.: Efficient computation of the workspace boundary, its properties and derivatives for cable-driven parallel robots. In: Zeghloul, S., Romdhane, L., Laribi, M.A. (eds.) Computational Kinematics. MMS, vol. 50, pp. 190–197. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-60867-9 22 16. Pott, A.: WireX: an open source initiative scientific software for analysis and design of cable-driven parallel robots (2019) 17. Bouchard, S., Moore, B., Gosselin, C.M.: On the ability of a cable-driven robot to generate a prescribed set of wrenches. J. Mech. Robot. 2(1), 1–10 (2010) 18. Gagliardini, L., Caro, S., Gouttefarde, M.: Dimensioning of cable-driven parallel robot actuators, gearboxes and winches according to the twist feasible workspace. In: 2015 IEEE International Conference on Automation Science and Engineering (CASE 2015), pp. 99–105 (2015) 19. Epperson, J.F.: An Introduction to Numerical Methods and Analysis, 2nd edn. Wiley, Hoboken (2013) 20. Jones, E., Oliphant, T., Peterson, P., et al.: SciPy: open source scientific tools for Python (2001) 21. International Organization for Standardization. Manipulating industrial robots: Performance criteria and related test methods (1998). ISO 9283:1998

Variable Radius Drum Design for Cable-Driven Parallel Robots Based on Maximum Load Profile Jonas Bieber(B) , David Bernstein , and Michael Beitelschmidt Chair of Dynamics and Mechanism Design, Technische Universit¨ at Dresden, Dresden, Germany [email protected]

Abstract. In the winches of cable-driven parallel robots (CDPRs), the transmission ratio between motor torque and cable force is determined by the drum radius. 3D printing allows the design of variable radius drums (VRDs), where the local drum radius can be individually adjusted for each cable length. Within a desired workspace, the maximum load profile is determined. Based on this, a winding path is calculated so that a specific maximum torque is reached, for each cable length. A drum shape is created such that a groove guides the cable on the winding path. For a purely translational 2T demonstration robot, the design steps are performed and a corresponding VRD is manufactured. This is compared with a conventional constant radius drum (CRD). Keywords: Cable Driven Parallel Robot Drum · VRD

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Motivation

Cable-driven parallel robots (CDPRs) are based on force transmission via cables. In order to apply these forces, typical actuation systems are motor-driven winches in which cables are coiled on a drum. The gearless or geared motor generates a rotational torque at the drum that is converted into a translational cable force by the drum. The transmission ratio is determined by the drum radius. In common state of the art CDPRs, the drum is a cylinder-shaped body, often with a controlled single-layer coiling. In this typical case, the coiling radius is constant and independent of the coiled cable length. The cylinder shape is also a result of classic manufacturing processes since cylindrical drums are easy to manufacture using conventional production methods. However, modern methods such as additive manufacturing overcome many limits in drum design at low manufacturing costs. This allows the design of variable radius drums (VRD) as applied in this paper. In mechanism design, variable radius transmission elements similar to VRDs are well studied. They are for example used to implement adjustable nonlinear c The Author(s), under exclusive license to Springer Nature Switzerland AG 2023  S. Caro et al. (Eds.): CableCon 2023, MMS 132, pp. 273–282, 2023. https://doi.org/10.1007/978-3-031-32322-5_22

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springs [4,5,10], which are often applied in robot arms [2,6,13], in particular as part of legged robots [7,11,12]. In cable-driven mechanisms, VRDs have been used to realize specific constraints between two cables coiled on a single shaft [3,8,11]. The design there is mainly based on kinematic considerations. In this paper, however, the drum design is based on the CDPR’s inherent load profile. The non-constant radius and thus the variable transmission ratio between torque and cable force takes into account the static cable force varying over the workspace. In the following, a method for generating a cable length-dependent load profile and synthesizing an optimized drum geometry is presented. It is demonstrated and validated using a purely translational 2T CDPR with 2 cables as simple demonstration example. Finally, the advantages and issues of the VRD design are discussed.

2

Maximum Load Profile

The wrench on the platform from external loads and platform dynamics is balanced by the cable forces. Depending on the platform’s pose, the m cable forces fi are acting in varying directions ui with 1 ≤ i ≤ m. Thus, the load on the CDPR’s drums due to the cable force is depending on the platform pose and wrench. Assuming a static case with only gravitational force fg in vertical direction g 0 , the cable forces can be determined by solving the translational part of the strucure equation ⎡ ⎤ f1 ⎢f ⎥  u1 u2 ... ⎣ 2 ⎦ = fg · g 0 . (1) .. . When solving the equation for varying positions, two effects can be observed for suspended CDPRs for each of the single cable forces, denoted f = fi below: – Vertical: The higher the platform is, the greater the cable force due to increasing horizontal cable force components. – Horizontal: The closer the platform is to one cable exit point in the horizontal direction, the greater the cable force due to its increased contribution to gravity compensation. Figure 1 shows the effects for the 2T demonstration robot for the static case with only gravitational force and cable forces acting on the platform. The platform mass is 2 kg leading to a gravitational force of fg = 19.6 N. The desired square workspace (gray box) is defined with 50 cm edge length and is located 25 cm below and horizontally centered between the cable exit points. The distance between the cable exit points is 88.5 cm. In the right part of Fig. 1, the cable force of the left cable is visualized within the desired workspace box. The right cable has the mirrored cable force pattern due to the symmetrical robot setup. Due to the above effects, the high cable forces occur in the upper corner close to the cable exit point while low cable forces occur in the lower corner on the opposite side.

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cable exit point

f platform fg

Fig. 1. Left: Setup of the 2T demonstration robot; Right: Cable force f as function of the platform position in the square desired workspace

In Fig. 2, the cable force is shown along the cable length. The validation points are sampled along a cartesian grid in the desired workspace. The gray lines represent the cable force along the desired workspace borders. The dark gray line in particular depicts the maximum cable force fmax () for the respective cable length. In the following, fmax () calculated for discrete cable lengths  will be referred to as the maximum load profile and is the basis for the further design of the drum. Although the maximum load does not occur along a continuous path, it always occurs on one of the desired workspace borders, highlighted in dark gray in the right part of Fig. 1. The maximum cable force decreases with increasing cable lengths, despite a short increase of the maximum cable force for small cable lengths due to a locally dominant horizontal effect.

f in N

20 15 10

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 in m Fig. 2. Cable force f of the left cable over cable length  (color scheme according to Fig. 1)

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Winding Path

For the drum design, the drum radius r is determined so that the maximum motor torque τmax results at least in the maximum cable force fmax (): τmax ≥ fmax () · r() .

(2)

In order to obtain a small drum width, to reduce the cable bending fatigue, to achieve the highest possible cable speeds, and to exploit the available motor torque, the radius is set as large as possible in the following design. For the 2T demonstration robot, a maximum motor torque of τmax = 0.3 Nm is assumed. The maximum cable force of f = 20.4N which occurs at the upper desired workspace border at  = 0.43 m results in a radius of r = 14.7 mm. For a constant radius drum (CRD), r would be the radius for the entire drum, although the maximum cable force is only related to a single cable length. With a VRD, in contrast, the radius can be set individually for each cable length. Based on the load profile fmax (), generated in the previous section, a variable radius can be set as τmax . (3) r() = fmax () With a pitch p per cable revolution, the profile of the radius r() results in a three-dimensional winding path which can best be described in cylindrical coordinates. The drum rotates around the zd -axis with the angle γ. Since the load profile fmax () is numerically calculated for cable length steps Δ, discrete path segments can be considered (see Fig. 3). zd

yd γ() 

Δzd Δγ Δ Δr r() z()

xd

Fig. 3. Segments of cable winding path

The zd -coordinate results from the drum rotation angle γ and the constant pitch p by γ() ·p (4) zd () = 2π The length of a path segment Δ is calculated approximately from 

2

Δγp 2 Δ = (Δγr)2 + Δzd + Δr2 = (Δγr)2 + + Δr2 (5) 2π

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with the approximation error depending on the discretization resolution. From this, the segment angle Δγ results in  Δ2 − Δr2 Δγ = (6) p 2 . r2 + ( 2π ) The cable length dependent drum rotation angle γ() is then the sum over the segment angles Δγ:   Δγ . (7) γ() = 0

The cartesian coordinates of the path result from the cylinder coordinates as xd () = r() · cos(γ())

(8)

yd () = r() · sin(γ()) .

(9)

and The assumption for the calculation of the winding path is that the cable leaves the drum in its roll-off point perpendicular to the line from the roll-off point to the drum center. In addition, the winding path, projected into the rotational drum plane, must always be strictly convex. Figure 4 shows the winding path calculated accordingly for the 2T demonstration robot. A pitch of p = 2 mm is used. Here, the maximum load profile was calculated for 100 000 equidistant cable length steps of Δ = 7.1 µm resulting in 26 ≤ Δγ ≤ 99 . The resulting discretization error in cable length is 4.2 µm in total and can be further reduced with a smaller step size. Due to the drum radius being inversely proportional to the cable force (see Eq. (3)), a strongly increasing drum radius can be seen for low cable forces (at high cable lengths).

4

Drum Shape

The drum design is based on the previously calculated winding path x(), y() and z(). Assuming a circular cable cross section, the local cable center is on the desired radius r() accordingly. For this purpose, the cable must be routed in a guiding groove which ensures a controlled single-layer coiling. A triangular profile with a notch angle α is chosen as a simple groove geometry, for manufacturing reasons. Figure 5 schematically illustrates a drum profile with multiple cable windings. There is a radial offset between the grooves of the single windings. For the 2T demonstration robot, a notch angle of α = 90◦ is chosen for a cable diameter of d = 1 mm. If the cable shape is assumed to be non-deformable in its cross section, the radius rcenter and the radius rup in Fig. 5 results in rcenter = r −

d = r − 0.71 mm 2 sin α2

(10)

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Fig. 4. Cable winding path with highlighted points at γ = 2πn with n ∈ N rup r rcenter

d p

α

zd

xd

Fig. 5. Schematic view of the drum profile

and rup = rcenter +

p = r + 1.71 mm . 2 tan α2

(11)

Regarding the assumed circular cable cross section, [9] shows that the cable is pulled deeper into the groove at increasing cable forces and changes the shape of its cross section. However, this effect has no noticeable influence on the relatively short cable lengths at the 2T demonstration robot. From the pitch p and the radii rup and rcenter , three groove points result for each discrete point of the winding path, which form the vertices of a meshed drum shape.

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Validation

The calculated VRD is compared to a CRD with constant radius r = 14.7 mm. Figure 6 shows the calculated motor torques at the cartesian desired workspace grid of Fig. 1 for gravity compensation only. For the CRD, the maximum motor torque τmax = 0.3 Nm is only reached at a single cable length. In contrast for the VRD, τmax is achieved at each cable length.

Fig. 6. Calculated torque for CRD and VRD for points in cartesian desired workspace grid (color scheme according to cable forces in Fig. 1)

For validation purpose, both the VRD and a reference CRD are manufactured additively (FDM). The VRD is based on the meshed drum shape (mentioned above), extended by mounting structures. Figure 7 shows the drums printed accordingly on a Prusa MK3S with PETG as material. The figure also shows the experimental setup corresponding to the geometry described in Sect. 2. Pulleys are used as cable exit points located in the upper corners of a frame made of aluminum profiles. The pulleys have a radius of 17 mm. The roll-off point of the cable at the pulley is assumed to be constant. The distance between pulleys and the respective drum centers is 135 cm. A 2 kg weight disc is used as a platform. Brushless DC motors driven by ODrive motor controllers are mounted at the lower corners of the frame. The drums are attached to the motor shafts without gearbox. In the test, the VRD is used for the left cable and the CRD for the right cable. Due to the symmetrical workspace, both of the cables have the same maximum load profile as shown in Fig. 1. Position holding is tested at the outer contour of the desired workspace. Figure 8 shows the measured motor currents for holding discrete platform positions. The motor currents are linearly proportional to the motor torques, superimposed by cogging and frictional effects. It can be seen that the motor current qualitatively follows the calculated torque (Fig. 6), which is plotted in a different scale. Thus, the scaled motor currents qualitatively follow the calculated torque (Fig. 6).

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Fig. 7. Left: Experimental setup with platform (black disc) in upper left corner of the desired workspace; Right: manufactured drums

Fig. 8. Measured motor currents at static workspace border positions for CRD and VRD. Left scale: Measured motor currents imax for maximum load profile and irem for remaining profile. Right scale: Calculated torque for maximum load profile (dark grey) and remaining profile (light gray). (Color figure online)

6

Discussion and Outlook

The experimental validation shows that the use of VRDs based on the maximum load profile can be realized. However, some points have been identified which must be taken into account:

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– The drums are customized for a specific load condition on a specific setup and therefore can hardly take changing conditions into account. – For CRDs, cable length errors only lead to platform position errors. For VRDs, this also results in errors in the ratio between motor torque and cable force, especially in the area of large radius changes. – A variable drum radius might increase the heterogeneity of manipulability over the workspace. An increase in the drum radius also results in a reduced cable length resolution. Both may affect control and accuracy. – Using CRDs, a reference position based calibration is feasible without mechanical adjustments. Using VRDs, the cable length must additionally be precisely aligned with a drum angle. – 3D-printed drums made of plastic reach their limits when it comes to increased strength requirements. For VRDs from stronger material, the manufacturing effort will increase. – Common cable guiding mechanisms with linear motion mechanically coupled to the rotation of the drum can not be used. However, the use of VRDs is still useful for some applications. In particular, maximum load requirements largely varying with the cable length, make the use of VRDs worth considering. This applies e.g. for suspended CDPRs with heavy-weighted platform. Concerning the cable guidance, the use of VRDs is most suitable for simple cable-driven robots as described in [1], where only one fixed deflection pulley is used. Here, the drum width is limited by the maximum fleet angle of the cable leaving the drum groove and savings in drum width are beneficial. For the 2T demonstration robot, the drum width is reduced from 15.3 mm for the CRD to 11.7 mm for the VRD by about 25%. This allows the VRD to be placed closer to the pulley while maintaining the same fleet angle. VRDs are also beneficial for motors where a significant cogging effect occurs. As shown in [1], the effect of cogging on low cable forces can be reduced if a larger motor torque is required. The advantage of VRDs is even stronger for applications in which the difference between maximum and minimum cable force is greater. This applies e.g. for the 2T demonstration robot with an increased desired workspace. However, if the maximum cable force decreases too rapidly along the cable length, the required convexity of the winding path can prevent exactly meeting the maximum force profile. For systems with larger dimensions, also the influence of cable elasticity and cable cross section change inside the groove should be further investigated. Here, circular groove geometries instead of triangular groove geometries might increase accuracy. For safety reasons, dead turns should be added at the end of the drum in field applications. Systems with more cables and degrees of freedom are to be investigated. Also experiments on the deflection angle limits of the proposed single-layer coiling and the influence of the groove geometry are still outstanding. In addition, the influence of the mentioned assumption concerning the roll-off point of the cable at the VRD should be investigated in the future for improved accuracy.

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References 1. Bieber, J., Bernstein, D., Schuster, M., Wauer, K., Beitelschmidt, M.: Motor current based force control of simple cable-driven parallel robots. In: Gouttefarde, M., Bruckmann, T., Pott, A. (eds.) CableCon 2021. MMS, vol. 104, pp. 271–283. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-75789-2 22 2. Endo, G., Yamada, H., Yajima, A., Ogata, M., Hirose, S.: A passive weight compensation mechanism with a non-circular pulley and a spring. In: 2010 IEEE International Conference on Robotics and Automation, pp. 3843–3848, May 2010 3. Fedorov, D., Birglen, L.: Differential noncircular pulleys for cable robots and static balancing. J. Mech. Robot. 10(6) (2018) 4. Hirose, S., Ikuta, K., Sato, K.: Development of a shape memory alloy actuator. Improvement of output performance by the introduction of a σ-mechanism. Adv. Robot. 3(2), 89–108 (1988) 5. Kilic, M., Yazicioglu, Y., Kurtulus, D.F.: Synthesis of a torsional spring mechanism with mechanically adjustable stiffness using wrapping cams. Mech. Mach. Theory 57, 27–39 (2012) 6. Kim, B., Deshpande, A.D.: Design of nonlinear rotational stiffness using a noncircular pulley-spring mechanism. J. Mech. Robot. 6(4) (2014) 7. Kljuno, E., Zhu, J.J., Williams, R.L., Reilly, S.M.: A biomimetic elastic cable driven quadruped robot: the RoboCat. In: ASME 2011 International Mechanical Engineering Congress and Exposition, pp. 759–769. American Society of Mechanical Engineers Digital Collection, August 2012 8. Scalera, L., Gallina, P., Seriani, S., Gasparetto, A.: Cable-Based Robotic Crane (CBRC): design and implementation of overhead traveling cranes based on variable radius drums. IEEE Trans. Robot. 34(2), 474–485 (2018) 9. Schmidt, V., Mall, A., Pott, A.: Investigating the effect of cable force on winch winding accuracy for cable-driven parallel robots. In: Flores, P., Viadero, F. (eds.) New Trends in Mechanism and Machine Science. MMS, vol. 24, pp. 315–323. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-09411-3 34 10. Schmit, N., Okada, M.: Synthesis of a non-circular cable spool to realize a nonlinear rotational spring. In: 2011 IEEE/RSJ International Conference on Intelligent Robots and Systems, pp. 762–767, September 2011 11. Seriani, S., Gallina, P.: Variable radius drum mechanisms. J. Mech. Robot. 8(2) (2015) 12. Shirafuji, S., Ikemoto, S., Hosoda, K.: Designing noncircular pulleys to realize target motion between two joints. IEEE/ASME Trans. Mechatron. 22, 1 (2016) 13. Ulrich, N., Kumar, V.: Passive mechanical gravity compensation for robot manipulators. In: 1991 IEEE International Conference on Robotics and Automation Proceedings, vol. 2, pp. 1536–1541, April 1991

Reconfiguration and Performance Evaluation of TBot Cable-Driven Parallel Robot Jinhao Duan1,2 , Hanqing Liu1,2 , Zhaokun Zhang3 , Zhufeng Shao1,2(B) , Xiangjun Meng4 , and Jingang Lv4 1 State Key Laboratory of Tribology and Institute of Manufacturing Engineering, Department of

Mechanical Engineering, Tsinghua University, Beijing 100084, China [email protected] 2 Beijing Key Lab of Precision/Ultra-Precision Manufacturing Equipment and Control, Department of Mechanical Engineering, Tsinghua University, Beijing 100084, China 3 Pengcheng Laboratory, Shenzhen 518055, China 4 Dalian Shipbuilding Heavy Industry Group, Dalian 116083, China

Abstract. Facing the demand for the flexible arrangement of robots on the production lines and adjustment of workspace, reconfiguration research is carried out based on the high-speed cable-driven parallel robot (CDPR), named TBot. The anti-disturbance performance index and analysis method for CDPR with common driven parallel cables are proposed on the basis of the convex hull. The performance of two typical TBots are compared and analyzed accordingly, considering the symmetrical layout. The influence of the vertex angle of base and the location of the upper universal joint on the performance of TBot is illustrated, and a reconfiguration design principle for TBot is concluded to further promote its industrial application. Besides, the proposed method can be extended to the optimization of other CDPRs with common driven parallel cables. Keywords: Cable-driven parallel robot · Reconfiguration · Performance evaluation

1 Introduction Cable-driven parallel robots (CDPRs) inherit the advantages of parallel mechanism and cable-driven technology [1, 2], which have significant advantages in large workspace, lightweight, low cost, and low energy consumption. CDPRs have extensive potential in applications in long-span [3–6] and high-speed [7–9] scenarios. In addition, CDPRs have also been used in a large number of scenes such as 3D printer [10], pneumatic experimental system [11], rehabilitation device [12, 13], and motion simulator [14]. To address the industry demand for picking, handling, and palletizing, our research team designed a high-speed CDPR named TBot [15, 16], which realizes high-speed 3-DoF (Degree of Freedom) translation. As shown in Fig. 1a and Fig. 1b, TBot utilizes three groups of common driven parallel cables to drive the end effector. A passive rod is installed between the base and © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Caro et al. (Eds.): CableCon 2023, MMS 132, pp. 283–294, 2023. https://doi.org/10.1007/978-3-031-32322-5_23

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the end effector to keep cables in tension. TBot has simple structure, low inertia, and high dynamics, and has been applied to industrial production, as illustrated in Fig. 1c. In addition, actuation units and the passive rod of TBot adopt a modular design, facilitating their reconfiguration properties. In previous research, we have conducted the geometric optimization [17], kinematics calibration [18, 19], and trajectory planning [20] for the TBot with the base of equilateral triangle. Prototypes of TBot, named TBot600 and TBot800, with regular workspace diameters of 600 and 800 mm respectively, have been evaluated to execute the Adept Motion [21] at a rate of 240 cycle/min while carrying a load of 0.1 kg. The terminal acceleration can reach up to 15 g during the motion, and the repeated positioning accuracy is 0.05 mm, showing excellent performance advantages and application prospects.

(a)

(b)

(c)

Fig. 1. (a) Structure of the TBot (b) the passive rod and (c) typical application of TBot.

TBot needs frequent acceleration and deceleration to carry out pick-and-place tasks, and at the same time resists forces caused by the shift of center of gravity and the terminal loads. Therefore, the acceleration performance and the ability to resist external forces are the keys to evaluate the performance of TBot. Bouchard et al. [22] proposed an anti-disturbance performance analysis based the geometric method. Available Wrench Set (AWS) is proposed to analyze the allowance external force. This method represents the force output capability as a convex hull in the force space through the Minkowski sum. The method is widely used to analyze the workspace [23, 24], force output [25], acceleration performance [26] and et al. The convex hull method has also been adopted in the design of hybrid driven CDPRs [27], trajectory planning of redundant CDPRs [28] and et al. In the researches, the radius of the inscribed sphere of the convex hull is usually used as an index to evaluate the performance of the CDPRs. However, these studies are mainly aimed at CDPRs whose all cables are driven, and mainly focus on the performance to resist force rather than moment. During the automation transformation of the production line, problems related to robot size interference and layout are commonly encountered [29]. Benefiting from the modularization and easy reconfiguration, the width and volume of TBot robot can be effectively reduced by altering the base shape to isosceles triangle distribution. This reconfiguration results in changes in performance, and the position of the passive rod needs to be adjusted accordingly. To guide and promote the application of TBot, this paper presents a reconfiguration study analyzing the resultant performance changes.

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In this paper, evaluation indexes for anti-disturbance of the CDPRs is proposed based on the convex hull analysis. The layout optimization of the passive rod and the performance analysis of TBot are demonstrated based on the proposed index and workspace size. The paper is arranged as follows: In Sect. 2, the structure and modeling of TBot are briefly introduced. In Sect. 3, an anti-disturbance performance analysis based on the convex hull is proposed. In Sect. 4, the performance of two typical TBots are compared. In Sect. 5, the reconfiguration principles and performance trends are discussed. Section 6 summarize the full text.

2 System Description The structure of TBot is shown in Fig. 1a, which consists of the base, actuation units, parallel cables, passive rod, and end effector. Six cables are divided into three groups of parallel cables, which are led out from the winches of actuation units and connected to the end effector after passing through pulleys. Each group of parallel cables is controlled by an actuation unit to achieve synchronous motion. The passive rod consists of a spring and a rigid rod. The lower end of the rod is connected to the end effector using the lower universal joint, and the upper end is connected to the base through a composite joint composed of the upper universal joint and a prismatic joint.

(a)

(b)

Fig. 2. (a) Isometric view and (b) top view of the simplified model of TBot.

The kinematic model of TBot is shown in Fig. 2, and the main parameters is listed in Table 1. In kinematic and dynamic analysis, each group of parallel cables can be simplified to an equivalent cable. Ai and Bi represent the equivalent cable-exit points of the i-th group of parallel cables on the base and the equivalent anchor points on the end effector. Aij and Bij represent the actual cable-exit points and anchor points of the two cables in the i-th group of parallel cables (i = 1, 2, 3, j = 1, 2). P is the geometric center of the end effector and the rotation center of the lower universal joint. Pso is the center of the upper universal joint. ps is the passive rod vector pointing from Pso to P. O is the center of the distribution circle of equivalent cable-exit points on the base. The global

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coordinate system O − XYZ is established at point O, and the X axis is along OA1 . The end effector coordinate system P − xyz is established at point P. The unit vector of each group of cables is ei , and the position vector of each cable anchor point in P − xyz is bij . The position vector of the end effector is p, and its unit vector is ep . The stiffness coefficient of the spring is k, the original length is l0 , the actual length is ls , and its unit direction vector is es . Vectors in this paper are column vectors. Table 1. Main parameter symbols of TBot. symbol

physical meaning

R

Radius of the base

r

Radius of end effector

le

Parallel Cable Spacing

ϕ

The vertex angle  A1 of equivalent base

dso

Distance between upper universal joint Pso and point O along X-axis

m

Mass of end effector

k

Spring stiffness

l0

Original length of spring

With the cable direction vector, the Jacobian matrix J of the model with all six cables can be written as: T  e1 e2 e2 e3 e3 e1 J= (1) b11 × e1 b12 × e1 b21 × e2 b22 × e2 b31 × e3 b32 × e3 The gravity on the end effector is mg, the auxiliary force provided by the spring is k(l0 − ls )es , , other external forces (such as inertial force, external load, etc.) is Fe , and the resultant external force is Fsum . Assumed that there is no external moment acting on the end effector. The tension of each equivalent cable can be obtained as:   (mg + k(l0 − ls )es + Fe ) T = J −T (2) = J −T W e 0  T where T = T11 T12 T21 T22 T31 T32 corresponds to the tension of each cable. And T  we define T e = Te1 Te2 Te3 as the tension of each equivalent cable. The force of the i-th equivalent cable is T ei = Tei ei

(3)

In the reconfiguration analysis, the equivalent cable-exit points are symmetrically distributed relative to the OX axis, and the size of  A1 is adjustable. The shape of the base changes with ϕ (the size of the vertex angle of the isosceles triangle A1 A2 A3 ). The position of upper universal joint Pso is revised along the X-axis, and the distance away from point O is dso .

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3 Performance Evaluation Indexes The minimum and maximum cable tensions of the equivalent cable is [fmin , fmax ]. The set of the i-th cable tension is {T ei |T ei = Tei ei , Ti ∈ [fmin , fmax ]}. A single force vector can be expressed as a single point in the three-dimensional force space. Therefore, all the elements in the set correspond to a line segment along the direction of the cable in the force space. And the points on the line segment represents possible forces that the i-th equivalent cable can exert to the end effector. As shown in Fig. 3a, the tension sets T ei is expressed in the three-dimensional force space. By calculating the Minkowski sum of the reverse force sets −T ei , a parallelepiped convex hull is obtained. All points in this convex hull represent the external forces the three groups of cables can counteract. The sum of the external forces on the end effector can be expressed as a point Pf in the force space. If the point is inside the convex hull, the cable tension requirements are met. The distance from Pf to the surface of the convex hull in each direction corresponds to the limit of the external force that the end effector can withstand in the direction. The radius rf of the inscribed sphere of the convex hull with Pf as the center represents the minimum value of the external force limit on the end effector in arbitrary direction. rf is defined as Force Margin (FM). The vector erf from the center of the sphere Pf to the tangent point Pf ,edge represents the weakest direction of the end effector against external forces.

(a)

(b)

Fig. 3. Schematic diagram of (a) force margin and (b) moment margin.

As shown in Fig. 3b, the capacity to resist external moments can be analyzed similarly. When external moment is applied on the end, the cable tensions of the two cables in each group of parallel cables will be automatically distributed to generate the resistance moment. When the maximum and minimum cable force constraints of a single cable is [fs min , fs max ], the set of moment vectors that can be generated is {M i |M i ∈ [λbi1 × ei − α i , λbi2 × ei − α i ]}, in which λ = min(fs max , Tei − fs min ). α i is the current generated moment of the i-th set of parallel cables, which can be calculated according to the distribution of tension, and α i = (Ti1 − Ti2 )(bi1 × ei − bi2 × ei )/2. All the elements in the set correspond to a line segment in the moment space, and the points on the line segment represents all possible moments that the i-th parallel cables can output to the end effector.

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As shown in Fig. 3b, the moment sets M i is expressed in the three-dimensional moment space. By calculating the Minkowski sum of the reverse monent sets −M i , a parallelepiped convex hull is obtained. The sum of the external moments on the end effector can be expressed as a point Pm in the moment space. The radius rm of the inscribed sphere of the convex hull with Pm as the center represents the minimum value of the external moment limit on the end in arbitrary direction. rm is defined as Moment Margin (MM). The vector erm from the center of the sphere Pm to the tangent point Pm,edge represents the weakest direction of the end effector against external moments.

4 Performance Analysis of Typical TBots The performance of TBots with two typical layouts are compared, whose vertex angle ϕ are 30° and 60° respectively. Except the vertex angle and the position of upper universal joint, other parameters are the same, as shown in Table 2. It should be noted that the end effector is chosen as an equilateral triangle with the same size regardless of the number of vertex angles of the base in this paper. And the angles between A11 A12 , A21 A22 , and A31 A32 on the base are 60°, which is beneficial for resisting moment. Table 2. Size parameter of TBot. parameter

value

parameter

value

R

0.555 m

r

0.034 m

le

0.118 m

ϕ

60°/30°

ls

1m

m

1kg

k

400 N/m

As is shown in Eq. (4), we integrate the FM and MM in the workspace, and define the global average force margin FMavg and global average moment margin MMavg . FMavg and MMavg reflect the overall performance of the configuration to resist external force and external moment. Furthermore, we select the maximum value of FM and MM in the workspace to express the optimal performance of TBot in the workspace.   FMavg = FMdV dV  (4)  dV MMavg = MMdV For the TBot with central symmetry layout (ϕ = 60°), the workspace is constrained by the telescopic range of the spring and swing range of upper universal joint [8]. Specifically, the constraints used in the analysis of workspace are set as follows: (1) The spring compression is constrained by [0, 50%]; ; (2) The maximum angle between passive rod and the Z-axis should be lower than 60°; (3) The cable force constraints of each group of equivalent cables is [10, 150] N; (4) The tension constraints of a single cable is [0, 150]

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N. When no cable is slack, the maximum tension on a single cable should be lower than the equivalent tension, and the minimum tension is taken as 0. The influence of the position of upper universal joint Pso of the TBot is analyzed with the vertex angle ϕ = 30°. The influences on the reachable workspace volume, FMavg , MMavg , FMmax , and MMmax are discussed, as shown in Fig. 4. The analysis area is a 1.2 m × 0.8 m rectangle centered on O. The dotted line in the figure represents the edge of the base, and dyed area represents the effective positions of Pso . The black solid point on the OX axis is the circumcenter of the base. The blue hollow point is the barycenter of the base. The red “X” point is the incenter of the base. And the red solid point is the optimal position of Pso , which maximizes the workspace. It can be found that TBot has better performance when Pso is on the X-axis under the same X-coordinate value rather than the layouts that the Y-coordinate value is not zero. Therefore, when reconstructing TBot with a symmetrical layout, the position of Pso needs to be selected on the axis of symmetry. The optimal position of upper universal joint is between the barycenter and incenter of the base, and is near to the barycenter. A maximum workspace of 0.65 m3 can be realized. When upper universal joint is arranged near this position, FMavg , MMavg , FMmax and MMmax of are also high, which shows good anti-disturbance performance. In the z = −0.65 m section of the workspace, the FM and MM values of TBot with 30° and 60° vertex angle are compared. As shown in Fig. 5, it can be found that the workspace of TBot is a part of the hollow spherical crown centered on upper universal joint. Compared with 60° vertex angle TBot, the workspace of 30° vertex angle TBot translate along the negative direction of the X-axis due to the position of Pso . The volume change of workspace is small. And the workspace of 30° vertex angle TBot is slightly flattened in the Y-axis direction and stretched in the X-axis direction due to the shape change of the base. The TBot with 60° vertex angle has a workspace volume of 0.71 m3 , FMavg up to 18.07 N, and MMavg up to 1.59 Nm. At the center of the workspace, the TBot can resist an external force of 45.20 N in any direction, and an external moment of 3.62 Nm in arbitrary direction. The TBot with 30° vertex angle has a workspace volume of 0.67 m3 , FMavg up to 15.27 N, MMavg up to 1.43 Nm. At the center of the workspace, the TBot can resist an external force of 30.76 N in any direction, and an external moment of 3.04 Nm in any direction. The TBot with 30° vertex angle has a lower anti-disturbance ability except in the center of the workspace. By reducing the vertex angle to 30°, the size of the base can be effectively reduced. The shape of workspace will be slightly stretched, and the performance to against external force and moment will be slightly reduced with acceptable affection on carrying out high-speed sorting operations.

5 Performance Trends During Reconfiguration Based on above analysis of two typical TBots, this section further discusses the influence of the reconfiguration on the performance of TBot. Workspace is regarded as the main optimization target when determining the Pso . The performance of TBot is illustrated when the vertex angle changes in the range of 5–85° and Pso is at the optimal position.

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Fig. 4. Performance iInfluence of upper universal joint position of central branch chain.

When the vertex angle changes, the position of Pso with the largest workspace changes as shown in Fig. 6. The blue line represents the position of the circumcenter of the base. The black line indicates the optimal position of Pso . The red line is the position of the incenter. And the magenta line is the position of barycenter. It can be found that when the position of upper universal joint is between the barycenter and the incenter and very close to the barycenter, a large workspace can be obtained. In order to clarify the influence of the vertex angle on the performance of TBot robot, the workspace is analyzed, and the result is shown in Fig. 7. Define space utilization rate as W /SL, where W is the workspace volume of TBot, S is the area of the base, and L is the length of the passive rod. It can be found that the workspace volume is the largest when the vertex angle is 60°, and the volume of the workspace decreases rapidly when the vertex angle is less than 20°. It can be found in Fig. 7b that the space utilization rate is high in all vertex angle. Although the workspace volume decreases rapidly in the

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Fig. 5. Performance Comparison of Two Typical TBots with ϕ = 30° and 60°.

Fig. 6. Optimal position of upper universal joint with different base vertex angles.

range of 5–20°, the space utilization rate is still high because the space occupied by the robot frame is decreasing. Figure 7c shows the base and workspace of TBot under several typical vertex angles. When the vertex angle becomes small, the workspace becomes slender. The space occupied by the robot frame is small, and the workspace is still suitable for the point-to-point motion required for pick-and-place tasks. The average and maximum values of FM and MM are analyzed and the result is shown in Fig. 8. It can be found that FM changes greatly and MM changes little when the vertex angle changes, and TBot with 60° vertex angle has best anti-disturbance performance. Consider that TBot has good usability when FMavg and MMavg is greater than 60% that of TBot with 60° vertex angle, which is 10 N and 0.9 Nm. Thus, when the vertex angle is selected at 20–85°, TBot robot has adequate workspace and good anti-disturbance performance.

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Fig. 7. TBots workspace volume under different base vertex angles.

Fig. 8. Anti-disturbance performance under different vertex angles.

Generally speaking, when TBot is reconfiguration by adjusting the vertex angle and the position of upper universal joint, the vertex angle should be selected between 20° and 85°, and the position of upper universal joint should be arranged near the barycenter of the end effector, so as to obtain a large workspace and good anti-disturbance performance.

6 Conclusion Facing the flexible layout requirements of the production site, this paper discusses the reconfiguration scheme and performance changes of TBot. Based on the convex hull, a method to analyze the performance of resisting external forces and moments for CDPR with common driven parallel cables is proposed. The analysis process is simple and the physical meaning is clear. This method is used to compare the performance of two typical TBots with the vertex angle ϕ of 30° and 60°, and clarify the influence on the performance caused by the change of base shape and the position of upper universal joint.

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Furthermore, the performance trends of TBot with symmetrical layout are analyzed, which shows that the vertex angle should be selected in the range of 20°–85°, and the position of upper universal joint should be located near the barycenter, to effectively adapt to different shapes of workspaces and different production line layout requirements. In the future research, more flexible reconfiguration schemes will be analyzed, and the mechanical structure of TBot will be optimized to improve the reconfiguration convenience. Acknowledgement. This work was supported by the National Natural Science Foundation of China (grant number U19A20101, 52105025) and the National High-tech Ship Research Project of China (grant number MC-202003-Z01). The authors would like to thank the editor and reviewers for their pertinent comments and suggestions.

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15. Zhang, Z., Shao, Z., Wang, L.: Optimization and implementation of a high-speed 3-DOFs translational cable-driven parallel robot. Mech. Mach. Theory 145, 103693 (2020) 16. Zhang, Z., Shao, Z., Wang, L., Shih, A.J.: Optimal design of a high-speed pick-and-place cable-driven parallel robot. In: Gosselin, C., Cardou, P., Bruckmann, T., Pott, A. (eds.) CableDriven Parallel Robots, pp. 340–352. Springer International Publishing, Cham (2018). https:// doi.org/10.1007/978-3-319-61431-1_29 17. Zhang, Z., Shao, Z., Peng, F., et al.: Workspace analysis and optimal design of a translational cable-driven parallel robot with passive springs. J. Mech. Robot.-Trans. ASME 12(5), 051005 (2020) 18. Xie, G., Zhang, Z., Shao, Z., et al.: Research on the orientation error of the translational cable-driven parallel robots. J. Mech. Robot.-Trans. ASME 14(3), 031003 (2022) 19. Zhang, Z., Xie, G., Shao, Z., et al.: Kinematic calibration of cable-driven parallel robots considering the pulley kinematics. Mech. Mach. Theory 169, 104648 (2022) 20. Duan, J., Shao, Z., Zhang, Z., et al.: Performance simulation and energetic analysis of TBot high-speed cable-driven parallel robot. J. Mech. Robot.-Trans. ASME 14(2), 024504 (2022) 21. http://www.adept.com/products/robots/parallel/quattro-s650h/general 22. Bouchard, S., Gosselin, C., Moore, B.: On the ability of a cable-driven robot to generate a prescribed set of wrenches. J. Mech. Robot/-Trans. ASME 2(1), 011010 (2010) 23. Rasheed, T., Philip, L., Stephane, C.: Wrench-feasible workspace of mobile cable-driven parallel robots. J. Mechan. Robot.-Tran. ASME 12(3), 031009 (2020) 24. Hussein, H., Joao-Cavalcanti, S., Jean-Baptiste, I., et al.: Smallest maximum cable tension determination for cable-driven parallel robots. IEEE Trans. Rob. 37(4), 1186–1205 (2021) 25. Erskine, J., Abdelhamid, C., Stephane, C.: Wrench analysis of cable-suspended parallel robots actuated by quadrotor unmanned aerial vehicles. J. Mech. Robot.-Trans. ASME 11(2), 020909 (2019) 26. Eden, J., Darwin, L., Ying, T., et al.: Available acceleration set for the study of motion capabilities for cable-driven robots. Mech. Mach. Theory 336, 105320 (2016) 27. Sun, Y., YaoXin, G., Chen, S., et al.: Wrench-feasible workspace-based design of hybrid thruster and cable driven parallel robots. Mech. Mach. Theory 172, 104758 (2022) 28. Sun, G., Zhen, L., Haibo, G., et al.: Direct method for tension feasible region calculation in multi-redundant cable-driven parallel robots using computational geometry. Mech. Mach. Theory 158, 104225 (2021) 29. Droeder, K., Hoffmeister, H.-W., Tounsi, T.: Flexible and space-saving machine concept for micro production. In: 7th CIRP Conference on High Performance Cutting (HPC), pp. 181– 184. Elsevier, Netherlands (2016)

Development Methodology of Cable-Driven Parallel Robots Intended for Functional Rehabilitation Ferdaws Ennaiem(B) , Juan Sandoval, and Med Amine Laribi Department of GMSC, Pprime Institute CNRS, ENSMA. University of Poitiers, UPR 3346 Poitiers, France [email protected]

Abstract. This paper proposes a methodology for the development of cabledriven parallel robots (CDPRs) for functional rehabilitation purposes, starting from the requirements identification to the experimental validation of the designed prototype. A study of the task to be assisted by the robot is first presented, followed by the formulation of an optimization problem leading to the selection of the optimal robot structure. Later, once the prototype is designed, its control design constitutes the final step of the development. This methodology has been experimentally validated on two types of CDPRs, namely, a fully constrained planar robot and an under-constrained spatial robot. This approach can be extended to incorporate other fields of application other than functional rehabilitation. Keywords: Methodology · Cable-Driven Parallel Robots (CDPRs) · Experimental Validation · Prototype · Optimization Problem · Control design · Planar Robot · Spatial Robot

1 Introduction Cable-driven Parallel robots (CDPRs) are devices where the end effector is linked to the fixed base only via actuated cables [1]. The end-effector positions are monitored by adjusting either the length of the cables or the positions of the exit or anchor points [2]. Their particular architecture grants them a lot of competitive advantages over other types of robots, such as the large translational workspace, the safety due to the lightness and the low inertia of its moving parts, the transportability, and the adaptability with different morphologies [3]. Its development, either its optimal architecture identification or its control has been widely discussed in the literature [4]. Its optimal structure selection depends basically on the target application, the choice of the design vector parameters, the criteria to be optimized, and the constraints to be respected [5–8]. As for its control, an open-loop approach is generally adopted to drive this type of robot. With prior knowledge of the end-effector desired trajectory, the angular positions of the actuators are calculated based on the inverse kinematic model. However, the use of cables instead of rigid links introduces major challenges in the control, compared © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Caro et al. (Eds.): CableCon 2023, MMS 132, pp. 295–307, 2023. https://doi.org/10.1007/978-3-031-32322-5_24

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to traditional parallel robots with rigid links [9]. In addition, it is difficult to control the position and orientation of the end-effector precisely due to its low stiffness [2]. A closed-loop control can then be suggested to address this inaccuracy problem. External sensors, such as the angular position sensors [10] or force sensors [11], can be integrated in the control loop for real-time measurement of the end-effector poses, allowing thus a continuous position control servoing. This paper proposes a complete methodology to develop a CDPR, with the robot’s desired task as input and the final optimal prototype as output. The methodology is enriched with a dedicated control scheme based on an exteroceptive sensor and then validated by designing two types of CDPRs. This paper is structured as follows: Sect. 2 details the adopted methodology to follow for the CDPR development. Two cases of study are presented in Sect. 3, where the proposed approach is validated on two types of CDPRs to verify its efficiency. Section 4 concludes the paper.

2 CDPR Design Methodology 2.1 Task Workspace Analysis Task-based movements, where the patient is guided to achieve a specific activity of daily living show encouraging outcomes over conventional training that focuses on moving passively the impaired joint inside its functional range of motion [12]. For this reason, the CDPR to be designed will have the mission of assisting the affected member along some prescribed exercises, chosen with the help of therapists. Thus, the first step for the robot development is to study these movements, which will subsequently allow to identify the robot task workspace and thus, estimate its size and its architecture. A motion capture system can be used to record the prescribed exercises, performed by healthy subjects. It consists of using infrared cameras to track the positions of passive reflective markers attached to the member, whose movement is to be investigated. Local frames can be constructed thanks to these markers, allowing the computation of the positions and the orientations of the member in question while performing the prescribed exercises [13]. Once the robot task workspace is identified, the next step is to select the optimal robot structure. 2.2 Optimization Problem Formulation Several studies have been presented in the literature addressing the optimal development of CDPRs where the design parameters, the criteria, and the constraints are diverse. The design parameters are the variables describing the robot architecture such as the exit points coordinates, the anchor points coordinates, or the size of the fixed platform [5]. Their values have to be found to satisfy a set of criteria. This latter may have several formulations, for instance, to minimize the cables’ tension, or the robot size, or to maximize the dexterity or the elastic stiffness [3]. As for the constraints of the problem, they are the restrictions to be respected by the optimal architecture to ensure the robot’s well performance. For example, they can be formulated to prevent collisions between the robot’s moving parts or to maintain positive tension in cables and inside a bounding interval [14].

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2.3 Prototype Control Generally, an open-loop control schema is implemented to monitor CDPRs, where the input is the actuators’ angular position. However, this approach of control is not accurate for several factors such as the flexible nature of cables, the uncertainties of the robot model, and the difficulty to set manually the starting pose of the end-effector. Since this latter’s resulting displacement is hard to be estimated due to the complexity of solving the direct kinematic model of parallel robots [15], an interoceptive correction of the position error is not obvious to implement. To cope with this problem, a closed-loop control is proposed [16], where an exteroceptive sensor is integrated into the feedback loop as illustrated in Fig. 1. This sensor is an OptiTrack camera used to track continuously the end-effector poses which, compared to the desired trajectory, allows to compute in real-time the additional angular position needed to correct the eventual deviations as illustrated in Fig. 2.

Fig. 1. Accurate control schema using an exteroceptive sensor proposed for CDPR.

Where χdi and χri represent the ith desired and real configurations of the endeffector, respectively. Li is the cables’ length vector. θi , θci , and θT are the actuators’ present position, the angular position needed to correct the eventual deviations, and the total actuators’ angular position, respectively. rp is the winding radius. The proposed design methodology was validated experimentally by designing two types of CDPRs, a fully constrained planar robot, and an under-constrained spatial robot, whose developments are detailed in the next section.

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Fig. 2. Proposed position control schema.

3 Case of Study 3.1 Planar Cable-Driven Parallel Robot The prescribed movement to be assisted by the planar CDPR consists of drawing with the patient’s hand an “8” shaped curve on a horizontal plane as described in Fig. 3. To collect the data needed to analyze this task, a motion capture system was set up, composed of 4 infrared cameras and 5 passive reflective markers attached to the patient’s hand. The recorded information, namely the orientations of the hand and the positions of its center during the exercise execution, allow identifying the robot task workspace. This latter is represented by Fig. 4, and defines the range of motion that the robot end-effector must achieve to reproduce the prescribed movement.

Fig. 3. Prescribed exercise to assist by the planar CDPR.

A planar CDPR with 4 cables and 3 degrees of freedom (DoF), represented by Fig. 5, is considered in this section. The cables’ lengths are controlled by 4 actuators mounted on the robot fixed platform. The robot architecture can be described using the placement of the 4 actuators and the size of the end-effector, which defines the design vector Ip given by Eq. (1).   Ip = xi , yi , c , i = 1..4 (1)

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Fig. 4. (a) Translational and (b) rotational task workspace of the planar CDPR.

An optimization problem has been formulated to identify the optimal design vector Ip∗ allowing to satisfy a set of criteria while respecting some constraints. The chosen problem criteria consist of minimizing the cables’ tension distribution and maximizing the elastic stiffness and the dexterity of the robot [17]. Regarding the constraints, they consist of avoiding slack or over-tensioned cables, as well as collisions between the effector and the cables.

Fig. 5. Representation of the geometric parameters describing the planar CDPR structure.

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The optimization problem resolution was carried out using the Particle Swarm Optimization algorithm (PSO) [18]. The obtained optimal design vector Ip∗ , given by Eq. (2), was satisfied to design the robot prototype illustrated in Fig. 6. Ip∗ = [−375, −385, 375, −260, 375, 180, −375, 385, 100] [mm]

(2)

Fig. 6. Prototype of the planar CDPR.

The next step is to control the robot prototype to perform a specific trajectory. As detailed in Sect. 2.3, an open-loop control may lead to an inaccurate performance of a CDPR. To solve this issue, the planar CDPR was controlled in a closed loop with the integration of an exteroceptive visual sensor in the feedback loop. The results of the experimental validation, illustrated in Fig. 7, show a significant improvement in the robot accuracy after the applied correction unlike in the open-loop control where a non-negligible offset between the real and the desired trajectories is observed. The mean absolute error was computed in Table 1 to assess the performance of the proposed control method, where Pdi , ωdi , Pir , and ωir refer to the ith desired and real positions and rotation angle, respectively (Table 1). Table 1. Mean absolute error computation. Error

Equation

Value before correction

Value after correction

Percentage of reduction

Mean absolute error

n 1  |P − P | ir di n i=1

10.6 mm

0.5 mm

95.3%

n 1  |ω − ω | ir di n i=1

1.64°

0.08°

95.1%

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(a)

(b) Fig. 7. Visual servoing results, comparison between the desired and the real (a) trajectories, and (b) orientations of the end-effector controlled in open and closed-loop.

3.2 Spatial Cable-Driven Parallel Robot The development methodology of a CDPR proposed in Sect. 2 was also validated on a spatial robot. This latter has for mission to assist the patient’s hand along three prescribed activities of daily living, chosen in collaboration with practitioners from Poitiers University Hospital. These exercises, illustrated in Fig. 8, consist of starting from an initial pose, touching with the dominant patient’s upper limb either the mouth, the head, or the shoulder, and returning to the starting pose. Five healthy volunteers were asked to perform five cycles of each movement, where two of them are left-handed. To characterize the gestures of each participant, a motion capture system was set up. It is composed of 8 infrared cameras and 19 passive reflective markers attached mainly to the volunteer’s dominant upper limp. A comparative study intra et inter subjects, using the ANOVA test [19], was then conducted with the aim of identifying a single reference trajectory to consider as the robot task workspace [13]. The results of this test proved

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Fig. 8. Prescribed exercises to assist by the spatial CDPR.

the difficulty to identify one common trajectory among all participants. To get over this issue, a general translational task workspace was delimited covering all the recorded trajectories (see Fig. 9(a)). Regarding the hand orientations, the largest computed range of motion of each rotation angle was considered as the rotational task workspace (see Fig. 9(b)). A CDPR with 7 cables and 6 degrees of freedom (DoF) is considered. Its structure can be described using the coordinates of the cable exit points, expressed in the global frame (X, Y, Z) and the coordinates of the anchor points, expressed in the local frame (XL , YL , ZL ) attached to the end-effector as shown in Fig. 10. These parameters define the design vector Is given by Eq. (3).   Is = xu , yu , zu , a, b, c, d , xl , yl , zl , Rpl , θi , i = 1..7

(3)

An optimization problem has been formulated for a spatial CDPR. The criterion chosen for this robot is to minimize its size in order to facilitate its manipulation by both the therapist and the patient. Regarding the problem constraints, they are based mainly on ensuring the patient’s safety and the minimum energy consumption. In fact, the collisions between the cables and the end-effector, between the cables and the patient’s body and between the cables themselves are considered within the problem constraints. Over tensioned and slack cables are also handled [14]. The solution of this optimization problem, performed using the PSO algorithm, ∗ expressed in Table 2. The obtained structure measures gives the optimal design vector Is1

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(a)

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(b)

Fig. 9. (a) Translational and (b) rotational task workspace of the spatial CDPR.

Fig. 10. Representation of the geometric parameters describing the spatial CDPR structure.

5 m×4 m×3 m, , justified by the large prescribed rotational task workspace (see Fig. 9(b)) and by the safety constraints. The huge obtained sizeable architecture is unfeasible to use for rehabilitation applications (Table 2). To cope with the size problem, the solution of a reconfigurable CDPR is then proposed, where the motors are fixed on actuated sliders. Their position changes whenever a pose of the end-effector cannot be reached. Two nested optimization problems are then configured [14]. The first searches for the optimal robot architecture and the second determines the optimal linear displacement of each actuator. The new obtained robot

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Table 2. Optimal design vector parameters (distances are expressed in meter and angles in degree). ∗ Is1 ∗ Is1

xu

yu

zu

a

b

c

d

xl

yl

−1.36

2.48

3.05

0.25

4

0.25

4

3.55

−0.15

zl

Rpl

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θ2

θ3

θ4

θ5

θ6

θ7

0.02

0.28

310

139

0

162

256

17

193

structure measures 2.5 m × 2.35 m × 3 m evaluated as 70.6% of size reduction compared ∗ is given in Table 3. to the first formulation. The new optimal design vector Is2 Table 3. Optimal design vector parameters (distances are expressed in meter and angles in degree). ∗ Is1 ∗ Is1

xu

yu

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c

d

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yl

1.5

1.12

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2.35

0.25

3

0.25

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-1

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θ2

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288

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Even though a significant reduction of the robot structure can be noticed, the obtained architecture is still large for the target application. In this context, the solution of a hybrid robot is proposed and adopted for the prototype design. The 6-Dof are decoupled, where the 3 translational degrees are supported by 4 cables and the remain 3 rotational degrees are controlled by an actuated orthosis defined by 3 motors (see Figs. 11(b) and 12(b)). The new design vector gathers the positions of the 4 actuators and 2 anchor points (the other 2 are deduced as given by Eq. (5)), as illustrated in Fig. 10. Similar to the previous two formulations, the PSO algorithm was used to solve the optimization problem. The optimal design vector is expressed in Eq. (4).   I∗s3 = x1 , y1 , z1 , a, b, c, y4 , z4 , θ3 , θ4 (4) = [0.41, 0.82, 1.43, 1.2, 1.2, 0.54, 0.82, 0.02, 128, 45], [m,◦ ] θ1 = θ3 − 120◦ θ2 = θ3 + 120◦

(5)

The resulting structure measures 1.2 m × 1.2 m × 1.4 m evaluated as 96.64% of reduction compared to the first solution. The prototype of the hybrid robot, represented in Fig. 12, is then designed. The last step is to control the robot prototype. The closed-loop control method was applied to the hybrid robot, where the movement (1) is programmed (the movement “hand-mouth” illustrated in Fig. 8). The implementation results are illustrated in Fig. 13. The examination of these curves shows an improvement in the robot accuracy since its

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Fig. 11. Representation of the design vector parameters, (a) on the robot structure, (b) on the actuated orthosis.

(a)

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Fig. 12. (a) Prototypes of the spatial CDPR, and (b) the orthosis.

behavior after the correction is close to the desired trajectory contrary to the open loop control where a significant gap between the real and the desired trajectory is noticed. The mean absolute error was also computed, in Table 4, to quantify the position error reduction, where Pdi , and Pir , refer to the ith desired and real positions of the end-effector.

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Fig. 13. Visual servoing results, comparison between the desired and the real (a) trajectories, and (b) orientations of the end-effector controlled in open and closed-loop for the movement 1. Table 4. Mean absolute error computation. Error

Equation

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n 1  |P − P | ir di n i=1

Value before correction

Value after correction

Percentage of reduction

43.87

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95.1%

4 Conclusions This paper discussed the development methodology from the design to the control of cable-driven parallel robots by providing an overview of the process to be followed. The proposed methodology has been validated on a planar fully constrained and a spatial under-constrained CDPRs. For each case of study, the robot mission was identified and analyzed to delimit its task workspace. Afterwards, the optimal robot architecture was identified in response to an optimization problem, where the design parameters, the criteria and the constraints were described. Once the optimal structure is designed, the last step was to control robot prototype. As the open-loop control technique is unable to provide the desired accuracy, a closed-loop control was proposed with the integration of an exteroceptive visual sensor in the feedback loop. This method showed significant improvement in the robot precision.

References 1. Jin, X., et al.: Upper limb rehabilitation using a planar cable-driven parallel robot with various rehabilitation strategies. In: Pott, A., Bruckmann, T. (eds.) Cable-Driven Parallel

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Robots. MMS, vol. 32, pp. 307–321. Springer, Cham (2015). https://doi.org/10.1007/978-3319-09489-2_22 Qian, S., Zi, B., Shang, W.-W., Xu, Q.-S.: A review on cable-driven parallel robots. Chin. J. Mech. Eng. 31(1), 1–11 (2018) Xiong, H., Diao, X.: A review of cable-driven rehabilitation devices. Disabi. Rehabil. Assistive Technol. 15(8), 885–897 (2020) Pott, A., Bruckmann, T. (eds.): Cable-driven parallel robots. MMS, vol. 32. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-09489-2 Li, Y., Xu, Q.: GA-based multi-objective optimal design of a planar 3-DOF cable-driven parallel manipulator : 2006 IEEE International Conference on Robotics and Biomimetics, ROBIO 2006. In: 2006 IEEE International Conference on Robotics and Biomimetics, ROBIO 2006, pp. 1360–1365 (2006) Fattah, A., Agrawal, S.K.: On the design of cable-suspended planar parallel robots. J. Mech. Des. 127(5), 1021–1028 (2004) Hussein, H., Santos, J.C., Gouttefarde, M.: Geometric optimization of a large scale CDPR operating on a building facade. In: 2018 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pp. 5117–5124 (2018) Laribi, M.A., Carbone, G., Zeghloul, S.: On the optimal design of cable driven parallel robot with a prescribed workspace for upper limb rehabilitation tasks. J. Bionic Eng. 16(3), 503–513 (2019) Hwang, S.W., Bak, J.-H., Yoon, J., Park, J.H., Park, J.-O.: Trajectory generation to suppress oscillations in under-constrained cable-driven parallel robots. J. Mech. Sci. Technol. 30(12), 5689–5697 (2016). https://doi.org/10.1007/s12206-016-1139-9 Fortin-Côté, A., Cardou, P., Campeau-Lecours, A.: Improving cable driven parallel robot accuracy through angular position sensors. In: 2016 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pp. 4350–4355 (2016) Picard, E., Caro, S., Claveau, F., Plestan, F.: Pulleys and force sensors influence on payload estimation of cable-driven parallel robots. In: 2018 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS 2018) (2018) Khan, S.B., Rahman, M.H., Haque, M.O., Rahman, E.: Effectiveness of task oriented physiotherapy along with conventional physiotherapy for patients with stroke. Int. J. Neurol. Phys. Ther. 5(2), 37–41 (2019) Ennaiem, F., et al.: Cable-driven parallel robot workspace identification and optimal design based on the upper limb functional rehabilitation. J. Bionic Eng. 19(2), 390–402 (2022) Ennaiem, F., et al.: A hybrid cable-driven parallel robot as a solution to the limited rotational workspace issue. Robotica 41, 850–868 (2022) Cavalcanti Santos, J., Gouttefarde, M.: A Real-Time Capable Forward Kinematics Algorithm for Cable-Driven Parallel Robots Considering Pulley Kinematics. In: Lenarˇciˇc, J., Siciliano, B. (eds.) ARK 2020. SPAR, vol. 15, pp. 199–208. Springer, Cham (2021). https://doi.org/10. 1007/978-3-030-50975-0_25 Ennaiem, F., et al.: Cable-driven parallel robot accuracy improving using visual servoing. In: Müller, A., Brandstötter, M. (eds.) Advances in Service and Industrial Robotics: RAAD 2022, pp. 97–105. Springer International Publishing, Cham (2022). https://doi.org/10.1007/ 978-3-031-04870-8_12 Ennaiem, F., et al.: Task-based design approach: development of a planar cable-driven parallel robot for upper limb rehabilitation. Appl. Sci. 11(12), 5635 (2021) Eberhart, R., Kennedy, J.: A new optimizer using particle swarm theory. In: MHS’95. Proceedings of the Sixth International Symposium on Micro Machine and Human Science, pp. 39–43 (1995) Kim, T.K.: Understanding one-way ANOVA using conceptual figures. Korean J. Anesthesiol. 70(1), 22 (2017)

Transmission Systems to Extend the Workspace of Planar Cable-Driven Parallel Robots Foroogh Behroozi1 , Philippe Cardou1 , and St´ephane Caro2(B) 1

2

Department of Mechanical Engineering, Robotic Laboratory, Laval University, Qu´ebec, QC G1V 0A6, Canada [email protected], [email protected] ´ Nantes Universit´e, Ecole Centrale Nantes, CNRS, LS2N, UMR 6004, 1, rue de la Noe, 44321 Nantes, France [email protected]

Abstract. In cable-driven parallel robots (CDPRs), each cable is typically driven by one actuator, and one is led to think that the number of actuators necessarily depends on the number of cables. In this paper, we consider that the number of cables depends on the required workspace while the number of actuators depends on the number of degrees of freedom of the robot. When designing a CDPR, one typically starts from the shape of the desired workspace to determine the required number of cables. Thence, he or she adds as many actuators, assuming that each of them drives one cable. Here, the number of actuators is supposed to be equal to one more than the number of degrees of freedom of the CDPR irrespective of the number of cables. A transmission system is designed for the actuators to drive several cables, first theoretically in the form of a transmission matrix, and then mechanically in the form of the corresponding cable-pulley routing. Two examples are proposed with two and three degrees of freedom respectively; both aim at covering a rectangular workspace. The wrench-closure-workspaces of the resulting robots compare favorably to existing CDPRs with more actuators. Keywords: Transmission System · Wrench-Closure-Workspace · Wrench-Feasible-Workspace · Transmission matrix · Schematic design CDPR

1

·

Introduction

One of the most commonly adopted architectures for planar Cable-Driven Parallel Robots (CDPRs) aims to cover a rectangular workspace and typically consists of four cables and four actuators (see Fig. 1(a)). As a result, CDPR with four cable and a point-mass end-effector have a degree of redundancy of two. As cables can only exert tension and not compression, an n-DoF CDPR should generally include at least n + 1 actuated cables to ensure that the end-effector c The Author(s), under exclusive license to Springer Nature Switzerland AG 2023  S. Caro et al. (Eds.): CableCon 2023, MMS 132, pp. 308–320, 2023. https://doi.org/10.1007/978-3-031-32322-5_25

M1

M2 (a)

M1

M3

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M3 M2

M4

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(b)

Fig. 1. Planar CDPRs with four cables and a point-mass end-effector: (a) having four actuators; (b) using a transmission system to drive four cables with three actuators

is in a wrench-closure configuration. In the 2-DoF planar CDPR with a pointmass end-effector under consideration, this means that the minimum number of actuators is three, so it has one more than necessary. The additional actuator(s) can result in a significant cost for the robot, and therefore, we seek to remove one actuator while preserving the rectangular workspace. Our strategy is to keep the same four cables while searching for a transmission system that would allow them to be driven by just three actuators, as shown in Fig. 1(b). A planar 3-DoF CDPR driven by no more than four actuators is also synthesized and studied in this paper. The use of pulleys and additional cables to reduce the number of actuators in CDPRs has been previously explored in the literature [5–8]. The concept of using cable differentials in the design of spatial and planar CDPRs is presented in [6,7]. Those papers focused on comparing the results of using the same actuator and increasing the number of cables and differentials to determine the coverage of the workspace [6]. In [7], the authors used differentials to actuate several cables of a planar CDPR by an actuator to minimize the number of actuators. Their results showed that using differentials leads to larger workspaces and improved kinetostatic performance with the same number of actuators. They covered the triangular shape of the WCW using three actuated cables. Cable-pulley transmission systems offer a unique combination of zerobacklash motion, high stiffness, low stiction, and low friction, making them desirable in force and torque control applications [11]. In [2], a mechanism comprising two non-circular pulleys and a constant-length cable was introduced. The authors proposed a differential cable routing method in order to enhance the capabilities of the non-circular pulley mechanism. In [5], Kevac and Filipovic provided an overview of various construction modeling approaches in cable-suspended parallel robots (CSPR). They presented six types of CSPR systems for camera carriers that can cover the workspace using only three actuators instead of the typical four, although they did not specify the percentage of the desired workspace area that can be covered by their proposed concepts. In [13], an original design for a 3-DoF CDPR that utilizes six cables and three actuators are proposed. The authors employed a parallelogram configuration to simultaneously wind two cables by an actuator, allowing for the fixation of the moving-platform (MP) orientation while performing the translational movement. In [1], the design, modeling, and prototyping of a planar CDPR with infinite rotation capabilities, free from parasitic tilt, and without the need for an additional actuator is presented. The 2-DoF motions of the MP and the internal

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degrees of freedom of the embedded mechanism are controlled by a total of three actuators, which are fixed to the frame. The proposed crane robot comprises of a mobile platform (MP) equipped with an embedded mechanism and a transmission module. The MP is connected to the frame via four cables, three of which act in parallel, forming a double parallelogram structure. The 2-DoF motions of the moving platform and the internal degrees of freedom of the embedded mechanism are controlled by a total of three actuators, which are fixed to the frame. In [9], two novel architectures for planar spring-loaded CDPRs that do not require actuation redundancy are proposed. The authors combine springs with a cable-loop system to eliminate the need for actuator redundancy, allowing for N actuators to control N -DoF motion. The proposed method ensures that the cables and springs are kept in tension within a rectangular workspace, but preloading of the springs is required to cover the entire workspace. The authors suggest that appropriate adjustments to the portion of stiffness and preload can increase the workspace. In [8], a configuration for planar CDPRs that utilizes parallelogram links instead of conventional links is presented. The use of parallelogram links ensures that the cables remain in tension during the robot’s movements, thereby enhancing its dexterity and elastic stiffness, as well as the magnitude of its stiffness indices. The authors manipulate cable redundancy to maintain the robot’s structure while utilizing only three actuators in the design of the planar CDPRs. In this research, a general formulation for parallelogram links is provided to support the suitable design of planar CDPRs. Although the proposed structure is applied to both fully constrained actuated and redundantly actuated configurations, it is unable to cover the entire rectangular workspace. In this paper, two novel cable robot architectures are proposed that aim to reduce the number of actuators while preserving a large wrench-closureworkspace (WCW). This architecture is distinct from traditional cable robot architectures, which typically feature a one-to-one correspondence between cables and actuators. In these proposed architectures, a single cable may be linked to multiple actuators (or conversely), resulting in a reduction in the number of actuators required. In general, if p actuators drive an n-DoF CDPR that does not rely on gravity or any other external force to keep the cables taut, then p ≥ n + 1 is a necessary condition for the moving-platform to be in equilibrium. We resort to a transmission matrix to effectively describe the relationship between actuators and driving cables. The organization of this paper is as follows. The mathematical foundation of the kinetostatic analysis of n-DoF m-cable p-actuator CDPRs is presented in Sect. 2. In Sect. 3, the transmission matrices, WCW, Wrench-FeasibleWorkspace (WFW), and architecture of a 2-DoF four-cable three-actuator CDPR are discussed. The transmission matrix for a 3-DoF six-cable fouractuator CDPR is presented in Sect. 4, along with an illustration of its WCW in various orientations.

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Kinetostatic Analysis

In this section, we present the kinetostatic analysis of the proposed n-DoF mcable p-actuator planar CDPR. The equations of equilibrium of the end-effector can be written as Eq. (1) as reported in [3]. W(p)t + we = 0n ,

t > 0m ,

(1)

where W(p) ∈ Rn×m is the wrench matrix of the cable robot when its endeffector is located at p, we is the external wrench and t is tension vector. The WCW is the set of poses of the platform where any wrench that can be generated at the platform by tightening the cables [4]. Roberts et al. [10] showed that a given pose lies inside the WCW (i.e., has wrench closure) if and only if one can find a vector t⊥ > 0m in the nullspace of W, where > indicates a componentwise strict inequality. In our case, however, not all the tensions found in the nullspace of W can be generated from p < m actuators. Let us define matrix T ∈ Rm×p , which represents the linear transmission between actuators and winches. The matrix T maps actuator torques τ into cable tensions t ∈ Rm×1 , namely, t = Tτ .

(2)

To determine whether there exists a solution (T, τ ) to equations (2) and for external wrench we = 0, we must identify a positive vector t that lies simultaneously within the nullspace of the wrench matrix W(p) and in the range1 of matrix T. To accomplish this, we recast equations (2) into matrix form, which yields:      0 W 0n×p t = n (3) Im×m −T 0m τ where Im×m is the m × m identity matrix and 0n×p is the n × p zero matrix. The solution of Eq. (3) amounts to computing the nullspace of the (m + n) × (m + p) matrix and verifying whether t > 0m . As p = n + 1, the number of columns is one more than the number of rows and this matrix has a rank of m + n unless one of W and T is rank-deficient. To compute this nullspace symbolically, let us modify it by adding two variable vectors z1 ∈ Rm and z2 ∈ Rp , as the first row of the matrix of Eq. (3). This yields the square matrix ⎡ T ⎤ zT2 z1 A = ⎣Wn×m 0n×p ⎦ (4) Im×m −Tm×p To calculate the cable tensions, we can take the partial derivative of the determinant of Eq. (4). If we let adj(A) and det(A) be the adjoint matrix and determinant of A respectively, then, by definition, 1

The range (also called column space or image) of a m × n matrix T is the span (set of all possible linear combinations) of its column vectors.

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Aadj(A) = det(A)1(m+n+1)×(m+n+1) .

(5)

By knowing that t⊥ is the first column of adj(A), the WCW of the proposed CDPR is the set of poses where: t⊥ =

∂ det(A) > 0m , ∂z1

(6)

Since t⊥ can be computed in closed form, leaving the pose parameters as variables and setting the entries of t⊥ equal to zero one at the time yields the potential boundaries of the WCW. To our knowledge, this method of tracing the WCW for CDPRs where T = Im×m has not been reported before.

3

A 2-DoF Four-Cable Three-Actuator CDPR

Consider the case of the 2-DoF CDPR shown in Fig. 1b, where a particle endeffector is to cover a workspace that is as close as possible to the encompassing rectangle. 3.1

Determination of the Transmission System T of a 2-DoF CDPR

The CDPR workspace depends on the value of the transmission matrix T. This value was obtained through trial and errors, while following a few guidelines. First, we should pick T so as to maximise the area of the corresponding WCW. Second, T should be as sparse as possible in order to simplify the mechanical implementation of the transmission system. Third, we hypothesized that the transmission should abide at least partly by the same symmetry found in the CDPR geometry for the WCW area to be maximised. Fourth, we hypothesized that restricting ourselves to entries of T that were either −1, 0 or 1 would still allow us to maximise the WCW area. Indeed, the scale of a row of T has no effect on the WCW, as long as its sign remains the same. The same goes for the scale of a column of T, which can always be adjusted by changing the input torque of a actuator. This is the rationale behind this hypothesis. Optimal T matrices were found, which maximize the WCW of the CDPR, and each of them yields the same WCW: ⎡ ⎡ ⎤ ⎤ 1 1 0 1 1 0 ⎢1 0 1 ⎥ ⎢1 0 1 ⎥ ⎢ ⎥ ⎥ (7) T=⎢ ⎣1 −1 0 ⎦ and T = ⎣0 −1 0 ⎦ 1 0 −1 0 0 −1 The value of the element Ti,j of T is Ti,j = 1 when the ith cable is connected to the j th actuator directly and is Ti,j = −1 when the ith cable is connected to the j th actuator in reverse. Ti,j = 0 when the ith cable is not connected to the j th actuator. The second actuator needs to connect to cables one and three, while the third actuator connects to cables two and four. When the second actuator

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Fig. 2. (a) WCW of the proposed 2-DoF four-cable three-actuator CDPR, (b) WCW of the planar CDPR proposed in [8]

winds the first cable, it has to unwind the third cable at the same rate. Also, the relation between cable numbers two and four is the same, but with the third actuator. Having defined T, we turn our attention to the computation of its associated WCW. This is done by following the method detailed in Sect. 2, which results in the WCW shown in Fig. 2(a). As shown, in this figure the WCW of this robot covers the whole rectangle formed by the fixed cable attachment. In Fig. 2(b), the WCW of a similar CDPR proposed in [8] is shown, which could not cover some parts of the rectangular shape workspace. Using Maple2018, we obtained symbolic expressions of the workspace through the four inequalities t⊥ = ∂z∂ 1 det(A) > 04 . The workspace boundaries in the Cartesian plane take the form t⊥ (x, y) = 04 are non-polynomial in x and y. The WFW is the set of platform poses where all the wrenches of a given set, can be balanced with tension forces in the cables such that the cable tensions are within prescribed limits [4]. To determine the area where all cable tensions are within a specified range, the WFW of the manipulator is determined. We trace the WFW with the values of drum radius R1 = R2 = R3 = 25 mm, where the tension limits are set to 0.5N and 20N (Fig. 3). We see that the WFW is approximately rectangular, but its size shrinks as the interval of external forces widens. Having confirmed that the proposed CDPR architecture allows wrench closure over the full rectangle formed by its fixed attachment points with three actuators, we set out to propose a mechanical realization of this theoretical design. 3.2

Embodiment of the Transmission System in a 2-DoF Planar CDPR

For the implementation of the proposed transmission matrix, we decide to use cable-pulley transmission systems to distribute the power of three actuators among four cables. Two mechanical implementations of the transmission matrices are schematised in Fig. 4. The schematic corresponding to the transmission

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Fig. 3. WFW of the proposed CDPR (Case 1) considering different external forces: (a) WFW of proposed CDPR (Case 1) with square external force 0.5N with the origin center; (b) WFW of proposed CDPR (Case 1) with square external force 2.5N with the origin center; (c) WFW of proposed CDPR (Case 1) with square external force 5N with the origin center.

matrix given in Eq. (7) is presented in Fig. 4(a). As shown in Fig. 4(a), we use two moving three-level pulleys, which directly control the cables in the workspace. The first actuator helps to drive all moving pulleys, while the second and third actuators are used to drive two cables at the same time. Another schematic that represents a possible embodiment of the transmission matrix given in Eq. (7) is proposed in Fig. 4(b). As shown in Fig. 4(b), we use two moving two-level pulleys to keep cables one and four in tension in the workspace. The first actuator controls the force applied by these two-level pulleys. To ensure that the cable routings of Fig. 4 correspond to the transmission matrices of Eq. (7), a static analysis of the schematic in Fig. 4(a) is done, which yields the relationship presented in Eq. (8) between the actuator torques and cable tentions.

τ1

M1

P2

t4 t2

τ1 /R1 t3

P1

t1 t3 τ2

t2 t3

t4

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τ3 M3

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t4

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τ3 M3

t2

t1

(a)

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τ2 M2 t 3

t2

t1

(b)

Fig. 4. Proposed schematic of three actuators 2-DoF four cables CDPR: (a) Case 1; (b) Case 2

1

23

(a)

M1

315

M4 M3 M2

4

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Transmission System

M1 M2 M3 M4

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(b)

Fig. 5. Using transmission system in 3-DoF six-cable four-actuator CDPRs

⎡ T =

1 2R1 ⎢ 1 ⎢ 2R 1 ⎣ 1 2R1 1 2R1

1 2R2

0

1 − 2R 2

0

0

⎤

⎥ ⎥ 0 ⎦

1 2R3

(8)

1 − 2R 3

In this matrix, Ri is the radius of the drum corresponding to the ith actuator. If all the drums have the same radius, then R1 = R2 = R3 = R, and we have T = 2RT .

4

A 3-DoF Six-Cable Four-Actuator CDPR

The most common task of a 3-DoF planar CDPR is probably to move its platform over a rectangular area. The standard 3-DoF planar CDPR consists of four cables driven independently by as many actuators, attached at the vertices of the said rectangle and at the other ends to the vertices of a rectangular moving-platform. Such an arrangement was proposed in [3,12], and is shown in Fig. 5a. One important problem with this architecture is that its translational workspace shrinks dramatically as the platform deviates from its reference orientation shown in Fig. 5(a). To resolve this, we propose to use a transmission system in 3-DoF planar six-cable CDPR with four actuators as shown in Fig. 5(b). This architecture uses two more cables than that found in [3,12], but, importantly, it has same number of actuators. The problem becomes that of finding an appropriate transmission system that would allow large rotations of the moving platform. 4.1

Determination of the Transmission System for 3-DoF CDPR Through T Matrix

The WCW of the CDPR is directly related to the choice of its T matrix. A trial and error method was applied, and guidelines outlined in Sect. 3.1 were followed again for this 3-DoF CDPR. This led to the following T matrices:

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Fig. 6. WCW of the 3-DoF six-cable four-actuator CDPR for nine MP orientations: (a) 0o ; (b) 45o ; (c) 90o ; (d) 135o ; (e) 180o ; (f) 225o ; (g) 270o ; (h) 315o ; (i) 360o .

⎤ 1 0 0 1 ⎢ 0 1 0 1⎥ ⎥ ⎢ ⎢ 0 0 1 1⎥ ⎥ T=⎢ ⎢−1 0 0 1⎥ , ⎥ ⎢ ⎣ 0 0 −1 1⎦ 0 −1 0 1 ⎡

⎤ 1 0 0 0 ⎢ 0 1 0 0⎥ ⎥ ⎢ ⎢ 0 0 1 0⎥ ⎥ and T = ⎢ ⎢−1 0 0 1⎥ . ⎥ ⎢ ⎣ 0 0 −1 1⎦ 0 −1 0 1 ⎡

(9)

Considering the first transmission matrix proposed in Eq. (9), one actuator should drive all cables for the purpose of keeping them in tension. The first actuator should wind the first cable while unwinding the fourth cable at the

Transmission Systems to Extend the Workspace of Planar CDPRs τ4

M4

P3

t5 t3

317

τ3 M3

τ4 /R4 t6

P2 t2 t6 M2

τ2

t5

t4

P1 t1 t4 M1

τ1

t1

t2

t3

Fig. 7. Proposed schematic of four actuators 3-DoF six-cable CDPR

same rate thus forming a cable loop. The same goes for actuators 2 and 3 with cable pairs (2, 6) and (3, 5), respectively. In the second matrix of Eq. (9), the number of connections between actuators and cables is reduced by taking advantage of the opposition between cables. Indeed, actuator 4 keeps in tension only cables 4–6, but since these cables form loops with cables 1–3, all the cables are tensioned by this actuator. With both transmission matrices defined in Eq. (9), the constant-orientation WCWs were computed for nine different angles of the moving platform. The resulting WCWs are traced in Fig. 6. The results are the same for both transmission matrices. As shown, although these transmission matrices do not allow to cover the complete rectangle formed by the workspace defined by the fixed attachment points, the area covers a wide range of angles. 4.2

Embodiment of the Transmission System of the 3-DoF Planar CDPR

In order to implement the proposed transmission matrix for the 3-DoF fouractuator six-cable planar CDPR, we adopt an approach similar to that adopted for the 2-DoF planar CDPR in Subsect. 3.2. A cable-pulley transmission system is used to distribute the power of the four actuators among the six cables. The schematic for the implementation of the transmission matrix outlined in Eq. (9) is presented in Fig. 7. One actuator is used to control the tension in the other cables, while the remaining actuators drive two cables in different directions. A system comprising two suspended three-level pulleys is used to directly control the cables within the workspace. The statics analysis of the mechanism can be performed by calculating the sums moments on the actuator drums and the sums of forces on the moving pulleys similar to what was outlined in Subsect. 3.2.

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Fig. 8. The WCW of the classic version of 3-DoF four-cable four-actuator CDPR in different end-effector angles (a) 0o ; (b) 45o ; (c) 90o ; (d) 135o ; (e) 180o ; (f) 225o ; (g) 270o ; (h) 315o ; (i) 360o .

4.3

Comparison Results

In Fig. 8, the WCW of the classic 3-DoF CDPR with four actuated cables is traced in eight different orientations. In this version, we note a fairly good coverage of the workspace except for moving-platform orientations of 90◦ and 270◦ , where there is no workspace. On the other hand, the WCW is shown in Fig. 6 for the proposed 3-DoF six-cable four-actuator, we observe slightly smaller workspaces in most orientations. We note also that there is a WCW in all tested orientations for the proposed architecture. This implies that the proposed six-

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cable CDPR may theoretically perform full rotations when the end-effector is in the central region of its workspace while the classic 3-DoF planar CDPR would be limited to 180◦ continuous rotations.

5

Conclusions

In this research, we proposed two planar CDPRs with fewer actuators than cables by using appropriate transmission systems. The presented architectures hold advantages over other classic CDPRs that are available in the literature. It shows that using the transmission system in a 2-DoF CDPR allows it to cover a rectangular WCW with one less actuator than the classic four-actuator version. Also, using a transmission system in a 3-DoF CDPR with six cables and four actuators allows for a fairly large WCW that extends over 360◦ range of possible rotations of the moving-platform. Other researchers had proposed CDPRs with fewer actuators than cables in the past [7–9]. What sets this work apart from theirs is the design method that was followed. Instead of starting with the selection of mechanical components for the transmission system, we focused first on the choice of the transmission matrix mapping the actuator torques onto the cable tensions. Only once a suitable transmission matrix had been found did we try to implement it with machine components—cables and pulleys were used, but gears, belts, and other standard mechanical components could have been used as well. Transmission matrices for two and three DoF are presented and two CDPR schematics for 2-DoF and a schematic for 3-DoF planar CDPRs are proposed to cover a rectangular WCW. For specific three different external force sets, the WFW was found for the 2-DoF CDPR. The next step of this research consists in automating the choice of the transmission matrix to expand the WCW while simplifying its matrix as sparse as possible to keep the mechanical implementation.

References ´ 1. Etienne, L., Cardou, P., M´etillon, M., Caro, S.: Design of a planar cable-driven parallel crane without parasitic tilt. J. Mech. Robot. 14(4), 041006 (2022) 2. Fedorov, D., Birglen, L.: Differential noncircular pulleys for cable robots and static balancing. J. Mech. Robot. 10(6), 061001 (2018) 3. Gouttefarde, M., Gosselin, C.M.: On the properties and the determination of the wrench-closure workspace of planar parallel cable-driven mechanisms. In: International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, vol. 46954, pp. 337–346 (2004) 4. Gouttefarde, M., Gosselin, C.M.: Analysis of the wrench-closure workspace of planar parallel cable-driven mechanisms. IEEE Trans. Rob. 22(3), 434–445 (2006) 5. Kevac, L.B., Filipovic, M.M.: Development of cable-suspended parallel robot, CPR system, and its sub-systems. Zbornik Radova (2022) 6. Khakpour, H., Birglen, L.: Workspace augmentation of spatial 3-DoF cable parallel robots using differential actuation. In: 2014 IEEE/RSJ International Conference on Intelligent Robots and Systems, pp. 3880–3885. IEEE (2014)

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7. Khakpour, H., Birglen, L., Tahan, S.A.: Synthesis of differentially driven planar cable parallel manipulators. IEEE Trans. Rob. 30(3), 619–630 (2014) 8. Khodadadi, N., Hosseini, M.I., Khalilpour, S., Taghirad, H., Cardou, P.: Kinematic analysis of planar cable-driven robots with parallelogram links. In: 2021 CCToMM Mechanisms, Machines, and Mechatronics (M3) Symposium (2021) 9. Liu, H., Gosselin, C., Lalibert´e, T.: Conceptual design and static analysis of novel planar spring-loaded cable-loop-driven parallel mechanisms. J. Mech. Robot. (2012) 10. Roberts, R.G., Graham, T., Lippitt, T.: On the inverse kinematics, statics, and fault tolerance of cable-suspended robots. J. Robot. Syst. 15(10), 581–597 (1998) 11. Schempf, H.: Comparative design, modeling, and control analysis of robotic transmissions. Ph.D. thesis, Massachusetts Institute of Technology (1990) 12. Stump, E., Kumar, V.: Workspace delineation of cable-actuated parallel manipulators. In: International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, vol. 46954, pp. 1303–1310 (2004) 13. Vu, D.-S., Barnett, E., Zaccarin, A.-M., Gosselin, C.: On the design of a threeDOF cable-suspended parallel robot based on a parallelogram arrangement of the cables. In: Gosselin, C., Cardou, P., Bruckmann, T., Pott, A. (eds.) Cable-Driven Parallel Robots. MMS, vol. 53, pp. 319–330. Springer, Cham (2018). https://doi. org/10.1007/978-3-319-61431-1 27

Toward the Creation of a Hybrid 4-UPS CDPR Paul W. Sanford1(B) , Juan Antonio Carretero1 , Andrew C. Mathis1 , and St´ephane Caro2 1

University of New Brunswick, Fredericton, NB, Canada {psanford,juan.carretero,andrew.mathis}@unb.ca 2 ´ Nantes Universit´e, Ecole Centrale Nantes, CNRS, LS2N, UMR 6004, 1, rue de la No¨e, 44321 Nantes, France [email protected] https://www.unb.ca/fredericton/engineering/depts/mechanical/

Abstract. A hybrid cable-driven parallel robot (CDPR) is being developed that expands the capabilities of conventional CDPRs by replacing one of the cables with a linear actuator that can be spooled and extended. This actuator uses three curved leaf springs connected using magnetic strips to provide compressive or tensile forces while being very compactly stored, and is kinematically similar to a cable that can push. When used in conjunction with a suspended CDPR, this actuator is being designed to be part of a novel pick-and-place robot that combines the benefits of cable robots (lightweight actuators) with those of a conventional manipulator (each of the actuators can be above the workspace to help avoid cable collisions). The developments of this novel actuator are outlined, showing how pointing motions were made possible through mechanical design. Keywords: cable-driven parallel manipulator

1

· telescoping actuator

Introduction

In the field of Cable-Driven Parallel Robots (CDPRs) two of the greatest challenges toward practical implementation come from cable interference, and developing manipulator configurations that are capable of completing tasks in a cluttered industrial environment. This is particularly apparent in spatial applications where tasks must be completed over a flat workspace (such as a conveyor belt) due to the requirement for cable tension to be maintained in order to retain control. This means that CDPRs with an antagonistic actuator configuration are the only ones capable of providing a downward force greater than the weight of the end-effector, while cable-suspended parallel robots (CSPRs), CDPRs with each of their cable anchor points above the workspace, are limited in the downward forces that they can apply [16]. The manipulator discussed in this paper attempts Supported by NSERC Discovery Grant funding. c The Author(s), under exclusive license to Springer Nature Switzerland AG 2023  S. Caro et al. (Eds.): CableCon 2023, MMS 132, pp. 321–331, 2023. https://doi.org/10.1007/978-3-031-32322-5_26

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to solve some of these problems by combining a CSPR with a novel extensible linear actuator that can sustain forces both in tension and in compression. The linear actuator used in this manipulator is being designed specifically for this application. The goal of the design is to create an actuator that retains most of the benefits of cable robots while also being able to provide a compressive force in order to have favourable actuator positions. It must be capable of being directed in pointing motions, similar to a universal joint. Hybrid CDPRs or h-CDPR refer to a CDPR where one or more of the cable actuators has been replaced by a rigid kinematic chain, commonly using a linear actuator in order to have one actuator capable of providing compressive forces [2, 5,9,26,27]. With universal and spherical joint connections to the frame and endeffector, respectively, the resultant manipulator is kinematically equivalent to a standard CDPM. While designing the extensible linear actuator there were several criteria that had to be met in order to be used as part of a h-CDPR. The first of which is that it must be lightweight to maintain low inertial properties and high efficiency. The second is that, like a cable, the material that is extruded should be coiled when not in use so that the linear actuator does not require excessive space above the workspace. The third requirement is that this actuator should be rigid and strong enough to supply a compressive force. In a CSPR, this will allow the cables to remain in tension and the manipulator to provide a downward force greater than the weight of the end-effector. This has an effect of an extended force closure workspace compared to a typical CSPM [7]. The goal of this actuator is to be used in a spatial pick-and-place parallel robot for lightweight applications similar to ABB’s Flexpicker (IRB 360) [1]. This will fit into a class of medium-sized CDPRs alongside these examples [14,15,20,25]. This also fits into a more niche group of linear actuators being created for CDPRs alongside work at Universit´e Laval [22]. Similarly, a high-speed CDPR with a similar architecture and goals has been constructed using a passive spring-loaded central rod for pick-and-place motions with a delta configuration [9,27]. In this work, for the designed linear actuator to be used in a spatial manipulator the rod must be capable of prismatic (linear) actuation as well as passive rotation for pointing motions, similar to the capabilities of a universal joint. This paper outlines the development of a novel leaf-spring actuator for prismatic actuation and pointing guidance developed at the University of New Brunswick. This actuator prototype uses standard measuring tapes as the leaf springs, connected using magnetic strips to form a strong, rigid rod to be used as part of a hybrid CDPR.

2 2.1

Project History Development of the Tape Rod

This project is not the first attempt that has been made to create a linear actuator that can be extruded from a roll. This work has been reviewed in [12,13,17],

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and can be divided into thin-walled tubular booms [12], tape springs (most commonly in tape measures) in [24], and more specifically as part of parallel manipulators (PMs) in [11,23]. Helical band actuators have been manufactured by Spiralift [10]. Other types of compact linear actuators include collapsible tubular masts [8,18,21], triangular rollable and collapsible booms [3,4], and closedsection triangular booms in works such as [6,19]. The linear actuator developed and described in this work most closely represents a closed-section triangular boom and uses several tape measures attached using magnetic strips to keep them together. Building from these developments in the field, some prior work at UNB was completed where sets of three tape measures were attached using various methods. The mechanical properties of several design options were tested using standardized methods, including bending and buckling. Further testing was performed on the top candidates using a custom-made dynamic testing jig shown in Fig. 2. This study resulted in a tape rod attached by magnetic strips along the outer edge that was capable of supporting a compressive load greater than 300 N at a rod length of 1 m [13]. This is significantly stronger than the tapes on their own as they readily buckle under compressive or bending loads. This design was the best among the tested methods using exclusively readily-available, offthe-shelf components. The rods were supported on both ends with a custom 3D-printed end block that matched the geometry of the attached tapes. Each rod was made of a fixed length for testing, and a cross-section of one of those rods is drawn in Fig. 1. The magnetic strips are glued to the tapes along the entire length, and each set of tapes is attached magnetically as shown.

Fig. 1. Cross section of three tape measures attached using magnetic strips.

Fig. 2. Dynamic testing jig of a non-actuated tape rod [13].

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Application in CDPRs

Conventional CDPRs require proximal anchor points (the points attached to the stationary frame) both above and below the workspace to be able to control the force in the vertical direction beyond what gravity will allow. With every proximal anchor point above the workspace (therefore above the distal anchor points on the mobile platform), it is a CSPR and has capabilities that are limited by both gravity and the minimum cable tension required to avoid slack. The goal of creating this actuator is to replace one of the cables in a CSPR to expand its capabilities. This can theoretically allow downward acceleration faster than gravitational acceleration and downward forces greater than the combined weight of the mobile platform and its hardware. The benefit of having every proximal anchor point above the workspace is to help avoid collisions between the cables and objects in the workspace, which could be applicable for a robot sitting over a flat surface such as a conveyor belt. 3.1

Planar Example

Although the manipulator under development is spatial, the expanded capabilities are most easily demonstrated with a planar CDPR. For this purpose, a 3-actuator planar CDPR with anchor points forming an equilateral triangle is considered. In order to qualitatively demonstrate the static equilibrium workspace (SEW) of this manipulator a few assumptions are required. First, the manipulator can be kinematically described as a 3-RPR parallel manipulator without any rotational limits for the revolute (R) joints, while the length of each actuated prismatic joints (P) can have any positive length. Each cable actuator has a minimum allowable tension force that would be the amount required to avoid having a slack cable. That is, it is a non-zero positive value. Also, it is assumed that there is a larger, positive tension value as a maximum actuator force that could correspond to the limits of safe operation based on the cable strength or the driving motor limits. A second, similar CDPR can be used for comparison that replaces one of the actuated cables with a linear actuator, making it a hybrid CDPR. This actuator has an actuation distance limit from 0 to some finite length, and a force limit that can be negative (a compressive force) or positive (a tension). The manipulators are shown with an approximation of their workspaces in Fig. 3. The SEW of the 3-cable actuator has three sides, each of which are curved-in toward the centre of the manipulator. The requirement for finite tension in the cables makes it so the mobile platform can not be in-line between two of the proximal anchor points. The workspace of the 2-cable plus 1-linear actuator mechanism has an expanded workspace where the side opposite the linear actuator attachment point extends past the convex hull of the actuator base positions. 3.2

Current Research (Spatial 4-UPS)

Extending into the spatial case, a similar effect on the workspace is expected, where the workspace can be extended based on the addition of a linear actuator

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Fig. 3. Qualitative SEW of two variants of a planar CDPR. Red circles represent cable actuators and the green squares represents a linear actuator. The shaded area is an approximation of their respective workspaces. (Color figure online)

for compressive forces. While the mass of the end-effector allows motions to take place underneath the actuators in a suspended configuration, the force capabilities are expanded by the addition of the extensible linear actuator making a spatial, hybrid CDPR. The goal of this manipulator was to have a workspace similar to that of ABB’s Flexpicker, which has a nearly cylindrical workspace with a diameter of 1.13 m and a height of approximately 0.3 m [1]. A simplified manipulator diagram is shown in Fig. 4. The reachable workspace of the designed CDPR is explained more in Sect. 4. To accomplish a similar workspace, the novel linear actuator is placed at the top centre of a cubic frame with a side length of 2 m. Three cable actuators are placed approximately 1 m from the central linear actuator, equally spaced around the outside of the top of the frame. A finalized CDPR with the extensible linear actuator could complete the same pick-and-place motions as the Flexpicker with lower costs and reduced energy consumption due to the mass of the cables being less than that of rigid actuators. In a spatial manipulator, the kinematic model treats each cable actuator as a UPS chain, with U representing a universal joint and S representing a spherical joint. The novel linear actuator should be capable of movements similar to a universal joint, or pointing motions. The kinematic complexity of this manipulator comes from the extensible linear actuator not being a true universal joint. The twisting and bending motions do not take place about coinciding axes, meaning that the point where the rod begins changes as a function of the location of the end-effector, as well as the length of the tape inside the guides between the spools and the beginning of the rod. Consideration will need to be made in the control algorithm in order to maintain reasonable accuracy by approximating the effect of this actuator.

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Fig. 4. Simplified model of the manipulator showing the IRB 360 workspace (cylindrical) and reachable workspace of the designed manipulator (wedge of a sphere).

4 4.1

Actuator Design Methodology

The portion of the actuator that needs to be guided is the driven tape, as it must withstand compressive forces while not having the added strength of being combined into the rigid rod of three tapes. In order to effectively guide the tape it is necessary to prevent unsupported bending and buckling, as that would cause the actuator to be unable to transmit force. Figure 5 shows a cross sectional view of the tape actuator. A tape enters the actuator on the right, is driven by two wheels that compress it, freely twists through one guide, bends in the next, then is combined with the two other tapes at outlet of the actuator. In order to allow the tapes to twist about their longitudinal axis the chosen method of support was to create a series of disks with a hole matching the profile of the tapes with the magnetic strips on the outer edges. The holes have a cross-section slightly larger than the tapes, allowing for some motion throughout their thickness, and the disks can slide relative to one another to allow the tape to experience a significant level of twist while still being supported against buckling throughout the entire length. During the design stage, a length of ≈14 cm was adequate to allow the tape to twist throughout the

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Fig. 5. Full actuator assembly cross section.

workspace within the ±90◦ range, which is the most that this actuator could move with its current design. To reduce friction during actuation and strain in the driven tape this is artificially decreased to ±45◦ . After printing a test disk with the chosen hole spacing it was found that there was adequate clearance and support for approximately 1 cm of length without issues. This discretized the twisting between 14 disks so that each one supports an equal angular displacement. Figure 5 shows an outer case that is bolted to the frame and the shape of the disks that are used to fill it. 4.2

Bending Guide

The bending guide proved to be a more complex problem as it was hung from the previous twisting guide frame, and the rotation of the tape required it to be bend while still being supported. In order to force the tape to bend in a controlled manner, the chosen method was to rotate it about a fixed radius of curvature by wrapping the tape around a guiding wheel. To avoid damaging the magnetic strips (or breaking the adhesion between the strips and the tape measures), surface contact on the face of the magnetic strips should be avoided as much as possible. This combination of constraints led to a design with a rolling wheel contacting the flat face and a meshing guide contacting the centre of the tape on the opposite side, as the magnetic strips are placed along the outer edges. This guide slots into the rotating part of the guide to support the tape at any feasible rotation angle, and is shown in Fig. 6.

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Fig. 6. Bending guide.

Fig. 7. Tape combining guide.

The current design is capable of guiding the tape through a bending rotation of ±23◦ , although this could be increased with a few minor modifications to the chosen joint limits. Figure 6 shows the bending guide, which is bolted to a flange at the end of the twisting guide. The tape passes through the inside of the guide along a wheel, and several rollers support the tape as it straightens after the bend. 4.3

Combining Tapes

After the driven tape passes through the two guides (twisting and bending), it must be brought into contact with the two non-driven tapes to form a rigid rod. The chosen attachment method of having magnetic strips along the outer edges of each tape and a triangular combined configuration necessitates accurate positioning of each of the tapes. Without accurate alignment the tapes may not end up with proper contact and the strength of the actuator (and therefore the amount of force that can be transferred) can be substantially reduced. The method used to combine the tapes was to have a “separating plate” that has three separated slots for the tapes, and a “rod brace” that smoothly guides each of the tapes together. This functions similarly to how the slider of a zipper separates and combines the teeth. The separating plate has three slots with the shape of the cross-section of the tapes with magnetic strips at the edges, and are angled toward the centre of the brace. The rod brace is shaped to support the three outer walls of the linear actuator and guide the final bend as the three tapes are combined. Figure 7 shows the end of the bending guide bolted to the separating and bracing pieces. The driven tape is passed directly into the guide, while the non-driven tapes are fed from outside into the separating plate. These

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design choices help keep the moment of inertia of the actuator lower than if each of the stored tapes were fixed to the moving rod. The actuation of the driven tape happens immediately before any of the guides, and each of the parts is fixed to the frame. The driven tape passes out of its case into a frame with wheels used for actuation. There are two wheels, each of which has a slot for rubber O-rings on the outer face to have a high friction surface to contact the painted metal tapes. One wheel is driven and is connected to a DC gearmotor using a keyed shaft. A second shaft supports another wheel that is free to rotate on bearings, but extension springs are used on either side to pull the two wheels together to increase the contact force and reduce the chance of slipping. Figure 5 shows the tape actuator, where the bottom wheel is driven by a motor, the top wheel is free to rotate, and the top of the brace is bolted to a plate on the frame. 4.4

Reachable Workspace

Given the limitations in the bending and twisting guides, the reachable workspace of the h-CDPR is the same as that of the designed linear actuator. This is shown in Fig. 4, and is compared directly with the workspace of ABB’s Flexpicker (IRB 360) in Fig. 8. A maximum linear actuator length of 1 m was used for this reachable workspace because that is the length that has been tested in [13]. The minimum length was chosen to be 0.5 m, giving the reachable workspace a spherical shape as the inner and outer surfaces. The current reachable workspace is shown as a wedge of the area common to two spheres with radii of 0.5 m and 1 m, with bending limits of 90◦ and twisting limits of 46◦ . Future validation may allow for a linear actuator maximum length to exceed 1 m and modifications of the bending guides to allow for greater rotations than the current ±23◦ currently used, which would expand the reachable workspace. The static equilibrium and wrench-feasible workspaces will be a subset of this, and based on the final joint limits and actuator capabilities of the constructed manipulator, as well as the desired task.

Fig. 8. Comparison of the workspace of the IRB 360 (cylinder) to the reachable workspace of the designed h-CDPR (wedge) Dimensions are in mm.

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Conclusion

The development of a hybrid CDPR shows promise as a method to complete pick-and-place tasks with lightweight and efficient actuators. The novel extensible linear actuator extends past what many prior actuators were capable of by combining multiple tapes into a rigid rod. If the spatial actuator performs similarly to the prior tests with fixed-length test samples, this manipulator should be more than capable of expanding the capabilities of conventional CSPRs in terms of force capabilities. With further development, this system could be used in facilities that would otherwise use a conventional parallel manipulator, such as ABB’s IRB 360, with improvements to the amount of overhead vertical space required. Theoretically, this manipulator can retain all the same benefits of a CDPR while providing more functionality, and these prospects will continue to be studied as this project develops.

References 1. ABB: Product specification: IRB 360. Website (2021). https://new.abb.com/ products/robotics/industrial-robots/irb-360 2. Alamdari, A., Krovi, V.: Modeling and control of a novel home-based cable-driven parallel platform robot: PACER, pp. 6330–6335. IEEE, September 2015. https:// doi.org/10.1109/iros.2015.7354281 3. Arya, M., Lee, N., Pellegrino, S.: Ultralight structures for space solar power satellites. In: 3rd AIAA Spacecraft Structures Conference (2016) 4. Banik, J.A., Murphey, T.W.: Performance validation of the triangular rollable and collapsible mast. In: Proceedings of the Small Satellite Conference, vol. 24, p. 8. AIAA/USU (2010). https://digitalcommons.usu.edu/smallsat/2010/all2010/10/ 5. Behzadipour, S., Khajepour, A.: Cable-based robot manipulators with translational degrees of freedom. In: Industrial Robotics: Theory, Modelling and Control. Pro Literatur Verlag, Germany/ARS, Austria, December 2006. https://doi.org/10. 5772/5035 6. Bobbio, S.M.: Spool-mounted coiled structural extending member. United States Patent No. 7,891,145 (2011) 7. Carretero, J.A., Gosselin, C.: Wrench capabilities of cable-driven parallel mechanisms using wrench polytopes. In: IFToMM Symposium on Mechanism Design for Robotics, September 2010 8. Chu, Z., Lei, Y.: Design theory and dynamic analysis of a deployable boom. Mech. Mach. Theory 71, 126–141 (2014). https://doi.org/10.1016/j.mechmachtheory. 2013.09.009 9. Duan, J., Shao, Z., Zhang, Z., Peng, F.: Performance simulation and energetic analysis of TBot high-speed cable-driven parallel robot. J. Mech. Robot. 14(2), 024504 (2021). https://doi.org/10.1115/1.4052322 10. Gala Systems: Spiralift Technology. Website (2023). https://www.galasystems. com/en/spiralift/ 11. Guinot, F., Bourgeois, S., Cochelin, B., Hochard, C., Blanchard, L.: Hybrid tapesprings for deployable hexapod. In: 50th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference. American Institute of Aeronautics and Astronautics, May 2009. https://doi.org/10.2514/6.2009-2611

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12. Jensen, F., Pellegrino, S.: Arm development review of existing technologies. techreport, Cambridge University Dept. of Engineering (2001) 13. Mathis, A.: A high packing ratio linear actuator for use in cable driven parallel manipulators. Master’s Thesis, University of New Brunswick (2018). https:// unbscholar.lib.unb.ca/islandora/object/unbscholar%3A9763 14. Mishra, U.A., Caro, S.: Forward kinematics for suspended under-actuated cabledriven parallel robots with elastic cables: a neural network approach. J. Mech. Robot. 14(4) (2022). https://doi.org/10.1115/1.4054407 15. Mottola, G., Gosselin, C., Carricato, M.: Dynamically feasible motions of a class of purely-translational cable-suspended parallel robots. Mech. Mach. Theory 132, 193–206 (2019). https://doi.org/10.1016/j.mechmachtheory.2018.10.017 16. Cable-Driven Parallel Robots. STAR, vol. 120. Springer, Cham (2018). https:// doi.org/10.1007/978-3-319-76138-1 17. Rauchenbach, H.S.: Solar cell array design handbook. techreport, NASA, October 1976 18. Rehnmark, F., Pryor, M., Holmes, B., Schaechter, D., Pedreiro, N., Carrington, C.: Development of a deployable non-metallic boom for reconfigurable systems of small spacecraft. In: 48th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference (2007) 19. Saito, T.: Manipulator mechanism. United States Patent No. 20,100,242,659 (2010) 20. Sciarra, G., Rasheed, T., Mattioni, V., Cardou, P., Caro, S.: Design and kinetostatic modeling of a cable-driven Sch¨ onflies-motion generator. In: the ASME 2022 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference IDETC/CIE 2022. St. Louis, Missouri, United States, August 2022. https://hal.archives-ouvertes.fr/hal-03758220 21. Seifart, K., G¨ ohler, W., Schmidt, T., John, R., Langlois, S.: Deployable structure for flexible solar generators. In: In Proceedings of the European Conference on Spacecraft Structures, Materials & Mechanical Testing (2005) 22. Servant, C.: Conception d’un actionneur prismatique r´etractable. Master’s Thesis, Universit´e Laval (2021). https://robot.gmc.ulaval.ca/fileadmin/documents/ Memoires/clemence servant.pdf 23. Soetebier, S., Raatz, A., Maris, C., Krefft, M., Hesselbach, J.: Spread-band: a smart machine element enables compact parallel robot design. In: 28th Biennial Mechanisms and Robotics Conference, Parts A and B. ASME-DETC, vol. 2, January 2004. https://doi.org/10.1115/detc2004-57069 24. Stanev, S., Heinz, R.: Linear drive mechanism. United States Patent No. 20,120,160,042 (2012) 25. Yang, J., Su, H., Li, Z., Ao, D., Song, R.: Adaptive control with a fuzzy tuner for cable-based rehabilitation robot. Int. J. Control Autom. Syst. 14(3), 865–875 (2016). https://doi.org/10.1007/s12555-015-0049-4 26. Yuasa, K., Arai, T., Mae, Y., Inoue, K., Miyawaki, K., Koyachi, N.: Hybrid drive parallel arm for heavy material handling. In: Proceedings of the 1999 IEEE/RSJ International Conference on Intelligent Robots and Systems. Human and Environment Friendly Robots with High Intelligence and Emotional Quotients (Cat. No.99CH36289), vol. 2, pp. 1234–1240 (1999). https://doi.org/10.1109/IROS.1999. 812848 27. Zhang, Z., Shao, Z., Wang, L., Shih, A.J.: Optimal design of a high-speed pickand-place cable-driven parallel robot. In: Gosselin, C., Cardou, P., Bruckmann, T., Pott, A. (eds.) Cable-Driven Parallel Robots. MMS, vol. 53, pp. 340–352. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-61431-1 29

Effect of Antagonistic Cable Actuation on the Stiffness of Symmetric Four-Bar Mechanisms Vimalesh Muralidharan(B) , Christine Chevallereau, Philippe Wenger, and Nicolas J. S. Testard Laboratoire des Sciences du Num´erique de Nantes (LS2N), CNRS, 44321 Nantes, France {Vimalesh.Muralidharan,Christine.Chevallereau,Philippe.Wenger, Nicolas.Testard}@ls2n.fr

Abstract. In biological systems, the joints are actuated antagonistically by muscles that can be moved coherently to achieve the desired displacement and co-activated with appropriate forces to increase the joint stiffness. Taking inspiration from this, there is an interest to develop bioinspired robots that are suitable for both low-stiffness and high-stiffness tasks. Mechanisms actuated by antagonist cables can be a reasonable approximation of biological joints. A study on the anti-parallelogram mechanism showed that the antagonistic forces (>0) have a positive influence on its stiffness, similar to the biological joints. In this work, more general symmetric four-bar mechanisms with crossed/regular limbs, larger/smaller top and base bars are investigated for this property. Totally, six different types of mechanisms were identified and the limits of movement were determined in each case. Inside these limits, it was found through numerical simulations that the cable forces have a positive (resp. negative) influence on the stiffness of the mechanism when its limbs are crossed (resp. regular). This shows that the symmetric fourbar mechanisms with crossed limbs are suitable for building bio-inspired joints/robots, while their counterparts cannot serve this purpose. Among these, the anti-parallelogram mechanism offers the largest orientation range of ] − π, π[ for the top bar w.r.t. its base and is thus the best choice.

Keywords: four-bar mechanism cable-driven · stiffness

1

· antagonistic actuation ·

Introduction

There has always been an interest in developing robotic arms that are fast, accurate, repeatable, and energy efficient, for industrial applications. But, in the recent past, research on robotic arms with more sophisticated capabilities such as stiffness modulation, deployability, safe interaction with environment, have been c The Author(s), under exclusive license to Springer Nature Switzerland AG 2023  S. Caro et al. (Eds.): CableCon 2023, MMS 132, pp. 332–343, 2023. https://doi.org/10.1007/978-3-031-32322-5_27

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gaining prominence [1,2]. An important source of inspiration for developing such robots stems from the nature/biological systems, e.g., human arm in [3], giraffe’s neck in [4], elephant’s trunk in [5]. One of the key differences between the conventional robots and biological systems lies in their joints. While most of the robots are made up of revolute or prismatic joints, the biological systems hardly contain any of them. Instead, their joints are composed of complex surfaces in contact with one another. Some works have been dedicated exclusively to the study of kinematics such joints, e.g., human knee in [6], vertebrae of a bird’s neck in [7]. A review on the animal joints and their approximation with linkage mechanisms can be found in [8]. Another interesting feature of biological joints is their actuation. Unlike conventional robots with linear or rotary actuators, they are actuated antagonistically by muscles. Normally, one set of muscle(s) contract while their antagonistic counterparts relax and vice versa, to achieve the desired joint movement. But, under special circumstances, both sets of muscles contract simultaneously to increase the stiffness of the joint. This phenomenon is referred to as co-activation of muscles in biological systems [9]. This interesting actuation scheme has been adopted in variable stiffness actuators, where a pulley is used as the joint and two cables with in-series non-linear springs are used for antagonistic actuation [10]. The two cables are pulled simultaneously as in muscle co-activation, to enhance the stiffness of this joint. However, it must be emphasized that this is possible for the pulley joint only in the presence of non-linear springs [10]. There are other joints where stiffness modulation can be achieved without in-series springs, by only varying the cable forces, see e.g., the tensegrity-inspired joints presented in [11]. It was found that with the increase in antagonistic cable forces, the revolute joint experiences a drop in stiffness, which was a counterintuitive result [11]. The same behavior was also reported for a 2R joint with offsets that represents one circle pure rolling over another [12]. In contrast, for an anti-parallelogram joint, that is equivalent to one ellipse pure rolling over another, the antagonistic actuation has a positive influence on the joint stiffness [11], just as in the biological joints. Thus, drawing inspiration from the anti-parallelogram mechanism, the goal of this work is to find all the four-bar mechanisms with symmetric limbs, that exhibit an increase in stiffness with antagonistic actuation by cables. The remaining paper is organized as follows: the description of mechanism and cable arrangement is presented in Sect. 2. The kinematic and static models are discussed in Sects. 3 and 4, respectively. The effect of antagonistic forces on the stiffness of various symmetric four-bar mechanisms are studied in Sect. 5. Finally, the conclusions are presented in Sect. 6.

2

Description of the Symmetric Four-Bar Mechanism

The schematics of four-bar mechanisms with symmetric limbs of length l, and a top bar of length b is shown are Fig. 1. The two pivots fixed to the ground

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(a) b0 < 0

(b) b0 > 0

Fig. 1. Schematic diagram of four-bar mechanisms with symmetric limbs that are crossed when b0 < 0 (left) and regular when b0 > 0 (right). The two actuating cables are shown in dashed lines.

are set at locations B1 (0, 0) and B2 (b0 , 0), where b0 is a parameter that can be varied to produce different four-bar mechanisms. Notably, b0 < 0 produces mechanisms with crossed limbs, while b0 > 0 produces mechanisms with regular (non-crossed) limbs as illustrated in Figs. 1a and 1b, respectively. The special cases of anti-parallelogram and parallelogram mechanisms are obtained when b0 = −b and b0 = b, respectively. However, the case b0 = 0 degenerates the four-bar to a revolute joint and will not be considered in  For all the  this work. |b−b0 | be satisfied mechanisms, it is necessary that the geometric condition l > 2 for its assembly. This mechanism is actuated antagonistically with two cables C1 , C2 , connected between the pivots (P1 , B2 ) and (P2 , B1 ), respectively, as indicated by dashed lines in Fig. 1. The force imparted by the cable Ci is given by Fi ≥ 0 and its varying length in the mechanism is denoted by li , for i = 1, 2. The cables are assumed to be massless and inelastic in this study. The orientation of the top bar w.r.t. the base is denoted by α, while those of the two limbs w.r.t. the base are given by φ, ψ, respectively (see Fig. 1). The coordinate α is used to measure the range of movement of the mechanism. The upper bound for α, denoted by αmax , can be found by rotating the top bar from α = 0 in the counterclockwise direction until any of the three pivots (B1 , B2 , P1 , P2 ) become collinear. Physically, at this configuration, the wrench imposed by one of the cables vanishes, and the static balance of the mechanism cannot be maintained. Thus, the wrench-feasible range of movement for this mechanism is given by α ∈] − αmax , αmax [, owing to the symmetry in architecture and actuation scheme about α = 0.

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However, the above representation is not valid for the parallelogram mechanism (b0 = b), since α remains zero at all the configurations. This case will be treated separately in Sect. 5.1. But, for all other cases, further study will be conducted inside α ∈] − αmax , αmax [.

3

Kinematic Model of the Mechanism

The loop-closure equation for the four-bar mechanism can be written as follows (see Fig. 1): −−−→ −−−→ −−−→ −−−→ − → B1 P1 + P1 P2 − B2 P2 − B1 B2 = 0 This can be expanded into:           cos ψ cos α cos φ b 0 l +b −l − 0 = sin ψ sin α sin φ 0 0

(1)

(2)

Since the above equations are homogeneous in terms of the length parameter, it can be normalized by setting b = 1, without any loss of generality. Considering α as the known input, it is possible to find the remaining angles (φ, ψ) as a function of α using the above equations (see e.g., [13], pp. 411-412). There are two possible solutions (φ, ψ)1 and (φ, ψ)2 , as presented below: ⎧ ⎧ α−b0 α+b0 cos φ = μ sin α+cos cos φ = − μ sin α−cos ⎪ ⎪ 2l 2l ⎪ ⎪ ⎪ ⎪ ⎨sin φ = sin α+μ(b0 −cos α) ⎨sin φ = − μ(b0 −cos α)−sin α 2l 2l (φ, ψ)1 := (φ, ψ) := 2 μ sin α−cos α+b μ sin α+cos α−b0 0 ⎪ ⎪ cos ψ = cos ψ = − ⎪ ⎪ 2l 2l ⎪ ⎪ ⎩ ⎩ 0 −cos α) sin ψ = μ(b0 −cos2lα)−sin α sin ψ = − sin α+μ(b 2l (3)

2 2 4l −b0 −1+2b0 cos α . For a given α ∈] − αmax , αmax [, one of the above where μ = b20 +1−2b0 cos α solutions corresponds to the top bar P1 P2 being above the base B1 B2 , while the other corresponds to top bar being below the base. In this study, only the former solution is of interest. Note that the specified joint limits α ∈] − αmax , αmax [ preclude the case where one end of the top bar is above, while the other one is below. By setting α = 0 in Eq. (3), it can be deduced from the resulting expressions that the desired solution branch is given by (φ, ψ)2 when (b0 < 1) and by (φ, ψ)1 when (b0 > 1). Revoking the normalization w.r.t. b, the above conditions translate into (b0 < b) and (b0 > b) in the two cases, respectively. From Fig. 1, the cable lengths (in all cases) can be written as follows:  Length of cable C1 =⇒ l1 := P1 B2 = l2 + b20 − 2lb0 cos ψ (4) Length of cable C2 =⇒ l2 := P2 B1 = l2 + b20 + 2lb0 cos φ The lengths l1 , l2 can be obtained as functions of α by substituting for cos ψ and cos φ from Eq. (3), appropriately.

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Static Model of the Mechanism

The static model of four-bar mechanism in Fig. 1 can be developed starting from its potential energy: U = Ug + F1 l1 + F2 l2

(5)

where Ug represents the contribution of gravity and springs (if any), Fi li with i = 1, 2 represents the work potential of the actuating cables. Treating α as the generalized coordinate in this study, differentiating U w.r.t. α and setting it to 0, yields the static equilibrium equation: dUg dl1 dl2 + F1 + F2 =0 dα dα dα

(6)

Further differentiation w.r.t. α yields the stiffness (K) of the mechanism: K :=

d2 Ug d2 l1 d2 l2 + F + F 1 2 dα2 dα2 dα2

(7)

Since the stiffness must be evaluated only when the equilibrium equation is satisfied, one can solve for F2 from Eq. (6) and substitute into Eq. (7), to obtain:

where γ1 =



d2 l 1 dα2

K = γ1 F1 + other terms (8)  2   d l2 1 /dα + −dl dl2 /dα dα2 . Similarly, it is also possible to solve for F2

from Eq. (6) and substitute  in Eq. (7) to obtain the coefficient of F2 in K as −dl2 /dα d2 l1 d2 l2 γ2 = dα2 + dl1 /dα dα2 . The effect of actuation forces on stiffness can be studied based on the terms γ1 and γ2 . If γ1 > 0 (resp. γ2 > 0), it implies that F1 (resp. F2 ) has a positive influence on the stiffness. Similarly, if they are negative then forces have a negative influence on the stiffness. Due to symmetry in the architecture and cable connections, γ1 and γ2 are mutually symmetric about α = 0. Mechanisms with positive γ1 , γ2 , are quite interesting because even when they become unstable due to external factors such as an increase in payload, they can be stabilized by simply increasing the actuation forces. This key property makes them ideal candidates for mimicking muscle actuated joints in biological systems, e.g., elbow joint of a human arm, where its increased stability can be felt by simultaneous contraction of the associated muscles. In the subsequent sections, the nature of γ1 , γ2 is studied for various four-bar mechanisms.

5

Effect of Actuation Forces on Stiffness

The effects of antagonistic forces on the stiffness of parallelogram and antiparallelogram mechanisms are studied in Sect. 5.1, while the effects on general symmetric four-bar mechanisms are studied in Sect. 5.2.

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Fig. 2. Schematic of the parallelogram mechanism (b0 = b).

5.1

Parallelogram and Anti-parallelogram Mechanisms

The parallelogram mechanism shown in Fig. 2 is obtained by setting b0 = b. Unlike other four-bar mechanisms, α remains 0 at all configurations for this mechanism. Hence, the orientation (θ) of the line joining mid points of the top and base bars w.r.t. the vertical, is used as the independent coordinate. The range of movement is limited by θ ∈ − π2 , π2 , due to the flat-singularities. From Fig. 2, it is apparent that φ = ψ = π2 + θ. Thus, from Eq. (4), the cable lengths are given by: (9) l1 = l2 + b2 + 2lb sin θ l2 = l2 + b2 − 2lb sin θ Following the same process described in Sect. 4, with θ in place of α, one obtains γ1 , γ2 as: ⎧ 2bλ2 (λ2 +1) cos2 (θ) ⎨γ1 = − (λ2 +1−2λ sin θ)(λ2 +1+2λ sin θ)3/2 (10) 2bλ2 (λ2 +1) cos2 (θ) ⎩γ = − 2 3/2 2 2 (λ +1−2λ sin θ) (λ +1+2λ sin θ) where λ = (l/b). It is apparent that all the factors in the numerators of γ1 and γ2 are positive. The two factors in the denominators are also positive since they are bounded inside [(λ − 1)2 , (λ + 1)2 ] for all real θ. Thus, it is clear that γ1 , γ2 < 0, due to the leading negative sign. This shows that antagonistic forces have a negative impact on the stiffness of the parallelogram mechanism. This result is consistent with the experimental data presented in [14], which shows that the cable tensions were reduced to increase the stiffness of this mechanism. Contrary to the parallelogram mechanism, it has been proven analytically in [11] that the antagonistic forces have a positive impact on the stiffness of anti-parallelogram mechanism. As a numerical illustration consider a parallelogram (b0 = b) and an antiparallelogram (b0 = −b) mechanism with b = 1 m and l = 2 m each. For the sake of simplicity, the bar masses are neglected and no springs are added to these mechanisms. In order to perform a fair comparison, the anti-parallelogram mechanism will also be described by the coordinate θ, as defined above for the parallel-

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  Fig. 3. Stiffness of the mechanisms when θ ∈ − π2 , π2 for different actuation forces.

ogram mechanism. The associated expressions for cable lengths and mechanism stiffness can be found in [11]. One of the ways to study the change in stiffness with increasing antagonistic forces is to firstly specify a minimum value for the actuation forces, say Fmin . At a given configuration θ, one could compute the balancing forces (F1 , F2 ) from Eq. (6) (neglecting Ug ) such that one of them is equal to Fmin while the other is greater than or equal to Fmin . These forces can be substituted in Eq. (7) to find the respective value of stiffness. This process has been carried for different values of Fmin : 0 N, 75 N and 150 N. The corresponding values of stiffness are plotted for the parallelogram and anti-parallelogram mechanisms in Fig. 3a and 3b, respectively. The equilibrium forces are also represented at certain configurations. It is apparent that an increase in Fmin causes a decrease (resp. increase) in stiffness for the parallelogram (resp. anti-parallelogram) mechanism, for all values of θ. This is a consequence of the negative (resp. positive) force coefficients γ1 , γ2 for the parallelogram (resp. anti-parallelogram) mechanism. This shows that the anti-parallelogram mechanism can serve as a bio-inspired joint while the parallelogram mechanism cannot. 5.2

General Symmetric Four-Bar Mechanisms

Unlike the parallelogram and anti-parallelogram mechanisms, it is very difficult to conduct analytical studies on γ1 , γ2 for the general mechanisms (b0 = ±b) due to the emergence of nested square roots in expressions of l1 , l2 (see Eqs. (3),(4)). Hence, the nature of γ1 , γ2 will be studied through numerical examples for these mechanisms. Firstly, six different cases I, . . . , VI have been identified based on the value of b0 , as shown in Tables 1 and 2. In each case, the limiting configurations at (±αmax ), plot of γ1 , γ2 inside α ∈] − αmax , αmax [ for one candidate design, and the limiting value of γ1 , γ2 inside the feasible design space, are presented in the successive columns of these tables. The following observations are made from them:

Condition/ Schematic Bounds on α ∈] − αmax , αmax [

3

0.0

0.1

0.2

0.3

0.4

3

0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.1

0.5

0

1

2

3

4

3

2

2

2

1

1

1

0

0

0

1

1

1

2

2

2

3

3

3

Plot of − γ1 , − γ2 for one design with b = 1 m, l = 2 m

1.0

0.5

0.0

0.5

1.0

0

5

10

15

20

(γmin /b) in design space 2l > |b − b0 | & l, |b0 | ∈ [0, 20b]

Table 1. Effect of antagonistic forces on the stiffness of four-bar mechanisms with (b0 < 0) (crossed limbs). Effect of Antagonistic Actuation on the Stiffness of Symmetric Four-Bar 339

Condition/ Schematic Bounds on α ∈] − αmax , αmax [

8

6

4

2

0

0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2

2.0

1.5

1.0

0.5

0.0

3

1.5

3

2

1.0

2

0

1

0

0.5 0.0

1

1

0.5

1

2

1.0

2

3

1.5

3

Plot of − γ1 , − γ2 for one design with b = 1 m, l = 2 m

1.0

0.5

0.0

0.5

1.0

0

5

10

15

20

(γmin /b) in design space 2l > |b − b0 | & l, |b0 | ∈ [0, 20b]

Table 2. Effect of antagonistic forces on the stiffness of four-bar mechanisms with (b0 > 0) (regular limbs).

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– The maximum orientation of top bar αmax varies in ]0, π[ for the mechanisms in cases (I, II, III), while it is limited to [0, π2 [ in cases (IV, V, VI). Thus, mechanisms with crossed limbs must be preferred for applications requiring large α. – From the plots of γ1 , γ2 for one candidate design, it is observed that they remain non-negative (resp. non-positive) for cases (I, II, III) (resp. (IV, V, VI)). The values of γ1 , γ2 tend to ±∞ near the boundary, in dl2 1 cases (I, III, IV, VI) due to the vanishing of dl dα or dα , present in the denominator of the respective expressions (see below Eq. (8)). However, γ1 , γ2 remain bounded and tend to 0 in cases (II, V) due to the vanishing of both first derivatives of l1 , l2 at the boundary. – In order to verify if γ1 , γ2 remain positive (resp. negative) for other designs in cases (I, II, III) (resp. (IV, V, VI)), their minimum γmin (resp. maximum γmax ) inside the range of movement is tested. Since the expressions of γ1 , γ2 are homogeneous w.r.t. the derivatives of cable lengths, one of the length variables (b = 0) can be factored out as in Eq. (10). This reduces the design space to just two variables ( bl , bb0 ). Firstly, a feasible design space satisfying 0| the assembly condition l > |b−b and bounded by 0 < bl , |bb0 | ≤ 20 is con2 γmin γmax structed. The values of b and b are computed for the feasible designs numerically to obtain the plots in last columns of the two tables. From these, it is clear, that γmin ≥ 0 for cases (I, II, III) and γmax ≤ 0 for cases (IV, V, VI). This illustrates that the antagonistic forces have a positive (resp. negative) influence on the stiffness of mechanisms with crossed (resp. regular) limbs. – Among the four-bar mechanisms with a positive correlation between forces and stiffness, the anti-parallelogram (case II) has the largest range of movement α ∈] − π, π[ and is to be preferred in general. However, the mechanisms in cases (I, III) might also be interesting for applications where large orientation range may not be essential, e.g., joints in the hyper redundant robots inspired from elephant’s trunk [5]. In cases (I, III) the value of γmin is large for designs close to the limiting assembly condition 2l = |b − b0 |, which indicates that there is a compromise between the range of movement and γmin in these designs.

6

Conclusion

A class of four-bar mechanisms with symmetric limbs, actuated antagonistically with two cables imposing forces F1 , F2 > 0, has been considered in this work. The effect of an actuation force F1 (resp. F2 ) on the stiffness a mechanism has been studied through its coefficient γ1 (resp. γ2 ) in the expression of stiffness after eliminating the other force F2 (resp. F1 ) using the equilibrium equation. Due to the symmetry in architecture and arrangement of cables, the force coefficients are also mutually symmetric about the configuration where the top and base bars are parallel. When γ1 , γ2 are positive, the stiffness increases with the increase in cable forces, similar to the muscle actuation of a biological joint. It was found through numerical simulations that γ1 , γ2 > 0 occurs only when the two limbs are crossed, and not otherwise. Among such mechanisms, the

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anti-parallelogram mechanism offers the largest orientation range of ] − π, π[ for the top bar w.r.t. its base, and is thus best suited for building bio-inspired robot manipulators. The other mechanisms with crossed limbs might also be of interest for applications where large range of movement is not a necessity, as in redundant and hyper-redundant systems. The formulation of stiffness showed that the effect of actuation forces depends only on the varying cable lengths in the mechanism and its first, second order derivatives. Hence, it would be interesting to search for more general conditions on the instantaneous properties of actuation force lines and the instant center of rotation, to have a positive correlation between actuation and stiffness. For the four-bar mechanisms, it is well known that the movement of its coupler w.r.t. the base can be represented by the pure rolling of one curve (moving centrode) over another (fixed centrode) [15]. A study of the nature of these curves and the cable force lines in each case might provide a more intuitive understanding of the co-activation phenomenon, and aid in the development of more such mechanisms (with one or more degrees of freedom) for bio-inspired robots.

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13. McCarthy, J.M., Soh, G.S.: Geometric Design of Linkages, 2nd edn. SpringerVerlag, New York (2010). https://doi.org/10.1007/978-1-4419-7892-9 14. Boehler, Q., Abdelaziz, S., Vedrines, M., Poignet, P., Renaud, P.: Towards the control of tensegrity mechanisms for variable stiffness applications: a case study. In: Wenger, P., Flores, P. (eds.) New Trends in Mechanism and Machine Science. MMS, vol. 43, pp. 163–171. Springer, Cham (2017). https://doi.org/10.1007/9783-319-44156-6 17 15. Norton, R.L.: Design of Machinery: An Introduction to the Synthesis and Analysis of Mechanisms and Machines, 6th edn. McGraw-Hill, New York (2019)

Tensegrity-Inspired Joint Can Protect from Impacts by Isolating Jonas Walter1(B) , Lukas Rothfischer1 , Richard Stierstorfer1 , org Franke1 , and Sebastian Reitelsh¨ ofer1 Takeru Nemoto1,2 , J¨ 1

Institute for Factory Automation and Production Systems, Friedrich-Alexander-Universit¨ at Erlangen-N¨ urnberg, 91058 Erlangen, Germany [email protected] 2 Siemens Technology, 81739 Munich, Germany

Abstract. Cable-driven parallel robots are known for their highly dynamic motions. At the same time, tensegrity robots are considered to be robust due to their intrinsic compliance. In a previous work, it was shown that a cable-driven joint with a tensegrity-inspired design allows fast rotations. The question remains whether such a joint is not only fast, but could also protect its base link from impacts due to its special design. In this paper, it is shown that a tensegrity-inspired joint can protect from impacts by isolating. A hardware prototype of such a joint was loaded with a modal hammer. It was found that the joint isolates impact forces up to 83%, demonstrating its ability to effectively absorb and dissipate energy from external impacts. Putting the results in a more general context, such a joint, however, can be used not only (a) passively to protect a base link, but also (b) actively for high-speed motions of a follower link by pulling a lightweight mass with parallel attached and actuated cables. It is anticipated that this study could be a starting point for the creation of novel robotic kinematics, composed of conventional and tensegrity kinematics, to build fast and robust robots.

Keywords: Tensegrity

1

· Cable-Driven Joint · Impact Protection

Introduction

Both cable-driven parallel robots (CDPR) and tensegrity robots have promising properties, which is why they have received attention [12,16]. The conventional CDPR, in which a mobile platform is moved by pulling on cables attached in parallel, are known for high-speed motions in a relatively large workspace [3,4,7]. Concurrent with the exploration of CDPR, robots based on the design paradigm tensegrity (= tension + integrity) have been presented [12]. Such tensegrity robots are considered robust since the energy of external impacts is distributed internally, preventing damage [10,13]. The conventional tensegrity robots are mainly used for locomotion [5,9,11,14]. However, the robustness of tensegrity structures also seems promising for designing novel robotic joints – especially in c The Author(s), under exclusive license to Springer Nature Switzerland AG 2023  S. Caro et al. (Eds.): CableCon 2023, MMS 132, pp. 344–354, 2023. https://doi.org/10.1007/978-3-031-32322-5_28

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Fig. 1. Schematic of a tensegrity-inspired joint and two acting forces. The joint comprises a base, a body, and cables. The body and the base are not in contact with each other but are connected by the cables. The cables are attached in parallel to the joint body in a tensegrity design so that the joint body is compression-loaded while the cables are tension-loaded. This paper investigates whether such a joint isolates impacts. Therefore, an impact force Fin was applied to the joint and simultaneously measured. This force was compared to the force measured at the joint base Fout .

combination with the technology of CDPR. Following this idea, a cable-driven joint with a tensegrity-inspired design was presented and shown to be fast [8,15]. It remains unknown whether such a tensegrity-inspired joint [8] is not only fast, but could also protect its base from impacts, which would be an interesting property. A protection of the base could be used, for example, to increase the mechanical resilience of robots. Particularly robots with rigid, vulnerable structural elements (e.g., gears) are at risk from impacts, since even small deformations can lead to failure. For this reason, several robotic mechanisms were presented that can handle impacts to a certain level, e.g., a spring-based transmission [1], a musculoskeletal design [2], and a cable-driven human-like manipulator [6]. The tensegrity-inspired joint considered here is potentially an interesting alternative approach to provide impact protection, as it could not only passively protect the base link, but also allows fast motions of the follower link [8]. In other words, such a joint design could solve two problems at the same time. In this paper, new data are reported that address the question whether a tensegrity-inspired joint could be used to protect its base link from impacts, while the joint design allows highly dynamic motions. In experiments, a hard-

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Fig. 2. Impacting a hardware prototype of such a tensegrity-inspired joint with a modal hammer. The impact was vertically applied to the joint body. The joint base was connected to the ground.

ware prototype of a tensegrity-inspired joint was loaded with impact forces. By analyzing the isolation behavior, it is discussed whether such a joint could protect its base link from impacts. Furthermore, it is discussed why a cable-driven joint with a tensegrity-inspired design could advance the robotics.

2

Results

It is investigated whether a tensegrity-inspired joint could protect its base from impacts. A hardware prototype of such a joint was loaded with impact forces. The applied forces on the joint body and the resulting forces in the joint base were measured (Fig. 1). It is determined whether the joint isolated the incoming forces from the base. An isolation would have the effect that the maximum force at the joint base is less than the maximum force at the joint body. From that, it is discussed whether such a joint can provide an impact protection.

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Table 1. Calculated isolation I and fitted damping coefficient δ. max(Fin ) in N

max(Fout ) in N

I

δ in rad/s

4.15 4.79 5.32 5.84 6.11 6.23 6.43 6.51 6.52 6.57 6.96 10.13 11.31 12.50 15.46

1.15 1.28 1.35 1.35 1.53 1.50 1.36 1.72 1.77 1.56 1.72 1.99 2.25 2.31 2.63

0.28 0.27 0.25 0.23 0.25 0.24 0.21 0.26 0.27 0.24 0.25 0.20 0.20 0.18 0.17

35.60 32.22 33.20 33.32 46.32 44.49 34.42 42.86 49.70 34.71 38.94 35.21 67.01 48.51 58.18

Fout = 0.25 · Fin

3

R = 0.97

max(Fout ) in N

Fout = 0.5 · Fin Fout = Fin

2

Fout = 0.1288 · Fin + 0.7177 1 Measurement Trend Line Helper Line 0

0

2

4

6

8 10 max(Fin ) in N

12

14

16

18

Fig. 3. The maximum force at the joint base plotted over the maximum force at the joint body. Theoretically, the measurement points should lie on a straight line, whereby the slope depends on the isolation. The lower the slope, the higher the isolation. For orientation, helper lines with slopes of 1.0, 0.5 and 0.25 were drawn. It is evident that the measurement points lie approximately on a straight line with a slope of 0.1288 and a y-axis intercept of 0.7177. The correlation coefficient R is 0.97.

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The joint comprised two rigid elements and six cables. The two rigid elements were a joint body and a joint base, to which a follower link and base link could be attached, respectively. The body and the base were not in contact with each other, but were connected by the cables (e.g., Fig. 1). All six cables were attached in parallel to the joint body in a tensegrity design so that the joint body was compression-loaded while the cables were tension-loaded. The cables can be categorized into two types: actuated cables and non-actuated cables. By pulling on three actuated cables like with a conventional CDPR, this joint could have been driven [8]. In this study, the actuators were kept stationary, since the goal of this study was to detect whether there is any protection at all. A distinctive feature in contrast to conventional CDPR is that three non-actuated cables were used to support and translationally constrain the joint body. The tensegrity-specific cable arrangement gives the joint an intrinsic compliance and could allow protecting the base from impacts. In each experiment, an impact force was applied by hitting the joint body vertically with a modal hammer (Fig. 2). Forces were measured at the joint body and at the joint base. The force at the joint body Fin was measured using a piezoelectric force sensor installed in the modal hammer with a sampling rate of 125 kHz. The force at the joint base Fout was measured using a strain gauge based load cell with a sampling rate of 1 kHz. In total, 15 experiments were conducted, each with one hammer strike. Based on the force data obtained from this protocol, it was analyzed whether the joint isolated the impacts. The isolation factor for each experiment was calculated by max(Fout ) . (1) I= max(Fin ) Since any isolation is generally caused by springs and dampers, the function of a damped oscillation y(t) = yˆ0 e−δt sin(ωt) (2) was fitted to the 15 measurements of Fout , where yˆ0 is the initial amplitude, δ is the damping coefficient, and ω is the angular frequency. This procedure revealed the data presented in Table 1. One finding was obtained. It was found that the joint isolates impacts up to 83%. Isolation values of 0.17 to 0.28 are observed (Table 1). Accordingly, the maximum forces measured at the base max(Fout ) are lower than the corresponding maximum force measured at the joint body max(Fin ) (Fig. 3). There should be a linear relation between these two forces, which means that the measurement points should lie on a straight line in a scatter plot, with the slope depending on the isolation. In fact, the measurement points can be approximated by a straight line with a slope of 0.13 and a y-axis intercept of 0.72 (Fig. 3). Such a y-axis intercept is unexpected, but may be due to missing points in the range of 0 N to 4 N for max(Fin ) or a nonlinearity at the edge. The general linear relation between max(Fin ) and max(Fout ) is further confirmed by a correlation coefficient R of 0.97. The isolation can be explained by a spring-damper behavior of the joint. The measured force values in the base correspond to a damped oscillation, as

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3

2 Fin

10

Force in N

Force in N

1

5 y ˆ0 e−δt sin(ωt)

0 −1

0

Fout

−2 Fout −3

−5 0

2

4 Time in s

0.5

6 −2

·10

1 Time in s

1.5

2 −2

·10

Fig. 4. Plot of all 15 measured forces at the joint base and at the joint body over the time (left) with a corresponding zoom (right). As the joint was stimulated impulsively with a modal hammer, corresponding impulsive signals can be seen for Fin . Due to the isolation, no impulse can be seen in Fout , but an oscillation with a reduced amplitude compared to the maximum of Fin . The signals of Fout resembles a damped oscillation. In fact, the 15 measurement signals can be approximated by the Eq. (2) of a damped oscillation, where yˆ0 = 2 N, δ = 38 rad/s, and ω = 635 rad/s. Such a tensegrity joint behaves passively like a spring-damper system, which is why a protection of the base from impacts is possible.

in a classical spring-damper system (Fig. 4). In fact, the measured data can be approximated with the function values of a damped oscillation according to (2), where yˆ0 = 2 N, δ = 38 rad/s, and ω = 635 rad/s. The spring behavior of the joint can be explained by the usage of cables and their tensegrity-specific arrangement. The cables themselves behave like springs. Furthermore, they change the direction and extend the path of the impacting force. The damping behavior can be explained by friction losses in the moving parts.

3

Discussion

The contribution of this paper is new data showing that a tensegrity-inspired joint isolates impacts. It was found that a hardware prototype of such a joint

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isolates impact forces up to 83%, demonstrating its ability to effectively absorb and dissipate energy from external impacts. There is evidence that such a joint behaves like a spring-damper system. The results indicate that a tensegrityinspired joint can passively protect its base from impacts. Such a tensegrity-inspired joint, however, can be used not only (a) passively to protect a base link, but also (b) actively for high-speed motions of a follower link. Similar to a conventional CDPR, this joint design allows highly dynamically motions by pulling the lightweight joint body with parallel attached cables. This has already been demonstrated for this specific design [8]. With the results presented here and earlier, it is concluded that such a joint presents a strategy to build robust and fast robots. However, there are still open points. The isolation of the joint could be examined over a wider range of impact forces than that applied here (4 N to 15 N). By this, more data points could be obtained to determine a trend line (Fig. 3) and to identify a possible nonlinearity. A wider range of impact forces would also allow verifying whether the isolation determined here is also valid for impact forces higher than 15 N that could damage robots with vulnerable, rigid structural elements (e.g., gears). Furthermore, it would also be interesting to investigate how different cable tensions affect the isolation. Additionally, determining the mechanical stiffness and achievable accuracy of such a joint would be of interest. Finally, it is anticipated that this study could be a starting point for the creation of novel robotic kinematics composed of conventional and tensegrity kinematics. The joint could be attached to any kinematics, for example to a conventional articulated robot (Fig. 5). This would still allow fast movement of the end effector, but at the same time the robot itself would be protected to some extent from unexpected impacts (e.g., collisions during high-speed manipulation in unstructured environments with occlusions).

4

Materials and Methods

The materials and methods used in this study are described here. First, the fabrication and setting of the tensegrity-inspired joint are explained. Second, it is described how the joint was impacted with a modal hammer. Third, it is shown how the forces at the joint body and the joint base were measured. 4.1

Fabrication and Setting of the Joint

The joint was fabricated according to the design presented in [8] and described in Sect. 2. The joint mainly comprised the base, the body, three actuated cables, and three non-actuated cables. However, additional components (e.g., pulleys, components to fix the cables) were needed for the experimental setup. There were also actuators (Dynamixel MX-28, Robotis, USA) attached to the joint base. Since these were kept stationary in the experiments, they are not further addressed here. In the following, the fabrication of the joint is explained. The base comprised an aluminum plate to which components for fixing the non-actuated cables and guiding the actuated cables were attached. The

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Fig. 5. Rendering of a novel robot kinematics composed of conventional and tensegrity kinematics. The tensegrity-inspired joint could be attached to any kinematics. For example, the joint could be used to protect the serial chain of an articulated robot from impacts. At the follower side of the joint, any gripper or tool could be mounted. By pulling with parallel attached cables on the lightweight joint body, the end effector can be rotated highly dynamically [8].

aluminum plate had a radius of 80 mm and a thickness of 6 mm. The nonactuated cables were fixed using additively manufactured cable holders. These holders were manufactured from polylactic acid (PLA) using the fused deposition modeling (FDM) additive manufacturing process with an infill of 80% and a downward thickening cross-section to minimize bending. The holders were joined to the aluminum plate via screws. To guide the actuated cables, a pulley system was used (Fig. 6). The pulleys were made of aluminum and were guided by needle roller bearings (K5X8X8-TV). The base was fixed to the ground via a linear guiding system (LHFRD8). In total, the base weighed 1018 g. The joint body was a cylindrical component whose height and radius are 75 mm and 20 mm, respectively. The waist of the joint body was widened to allow a smooth guidance and fixation of the non-actuated cables in the bottom. The actuated cables were connected to the top of the platform. The joint body was also manufactured from PLA via the FDM process with an infill of 100%. A steel plate was attached to the top of the joint body to properly receive impact forces. This plate had a radius of 23 mm and a thickness of 4 mm. The body

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Fig. 6. Section views of the hardware prototype. (A) Overall view of the joint. Three cables were attached to the joint body for bearing and three cables for possible drive. (B) Detail view of the routing of a non-actuated cable used for constraining the platform. One end of the cable was fixed to the holder by a screw. The other end was guided into the joint body via a rounded edge and also fixed there with a screw. (C) Detail view of the force transmission to the load cell installed in the joint base. The entire weight of the joint is transferred to the load cell via a punch made of aluminum. The load cell was mounted on a linear guiding system.

carried an inertial measurement unit (XSENS MTi-300, XSENS, Netherlands). This sensor was only used for the check of the joint orientation to be upright before conducting the experiments. In total, the joint body weighed 252 g. All cables were made from ultra-high-molecular-weight polyethylene, branded as Dyneema. The Young’s modulus of this material is 110 GPa, according to the

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manufacturer. Each cable had a diameter of 1 mm. The non-actuated cables had a length of 229.3 mm and were prestressed by spacers at the cable holders. The actuated cables were 226.0 mm long before each reached a cable winch. Each actuated cable was fixed with a trucker’s hitch knot to a winch. The winches had a radius of 15 mm and were each connected to a powerless actuator. For a certain pre-tension, each actuated cable was wound up with 0.5 turns. 4.2

Impacting the Joint While Measuring the Impact Force

The joint was impacted with a modal hammer (9722A2000, Kistler, Swiss). The stainless steel hammer head weighed 100 g. A steel tip was attached to the hammer head. To measure the impact force, an piezoelectric quartz sensor integrated in the hammer was used. This sensor has a voltage output according to the integrated electronics piezoelectric (IEPE) standard. The measurement range of the force sensor is 0 N to 2000 N. The hammer was connected to an oscilloscope (MSOX2024A, Keysight, USA). The oscilloscope provided the electrical voltage of the force sensor with a frequency of 125 kHz. Finally, the voltage was converted into the impact force using the characteristic value of the sensor of 2.448 mV/N. In order to enable precise evaluation of the measurement data, we have decided to limit the load to 15 N. The reason for this was that double impacts occur at higher loads. These are second impulses caused by the rebound effect of the modal hammer. Double impacts are undesirable when using a modal hammer. To avoid these, the load was limited. 4.3

Measuring the Forces at the Joint Base

The force at the joint base was measured with a load cell (KM10, ME-Meßsysteme, Germany) connected to an amplifier (GSV-3USBx2, MEMeßsysteme, Germany). The load cell is based on strain gauges and is pressure loaded. The nominal force is 1000 N. The sensor has an accuracy class of 1%. The dimensions of the load cell are 9.8 mm in diameter and 4 mm in height. The force is applied via a cap that has a diameter of 2.4 mm. The load cell was connected to a digital amplifier. The amplifier has an input sensitivity of 2 mV/V and a resolution of 16 bits. The measurement signals were received at a computer via an USB interface. On the computer, a software (GSV-multi, ME-Meßsysteme, Germany) was installed, which converted the voltage signals into forces and enabled measurement recordings. Before conducting the experiments, the load cell was calibrated using a two-point calibration with a reference weight. Acknowledgements. This work was supported by the German Federal Ministry of Education and Research (033KI216).

References 1. Albu-Schaffer, A., et al.: Soft robotics. IEEE Robot. Autom. Mag. 15(3), 20–30 (2008). https://doi.org/10.1109/mra.2008.927979

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2. Imran, A., Yi, B.J.: A closed-form analytical modeling of internal impulses with application to dynamic machining task: biologically inspired dual-arm robotic approach. IEEE Robot. Autom. Lett. 3(1), 442–449 (2018). https://doi.org/10.1109/ lra.2017.2760907 3. Kawamura, S., Choe, W., Tanaka, S., Pandian, S.: Development of an ultrahigh speed robot FALCON using wire drive system. In: Proceedings of 1995 IEEE International Conference on Robotics and Automation. IEEE (1995). https://doi.org/ 10.1109/robot.1995.525288 4. Kawamura, S., Kino, H., Won, C.: High-speed manipulation by using parallel wire-driven robots. Robotica 18(1), 13–21 (2000). https://doi.org/10.1017/ s0263574799002477 5. Kim, K., Moon, D., Bin, J.Y., Agogino, A.M.: Design of a spherical tensegrity robot for dynamic locomotion. In: 2017 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS). IEEE, September 2017. https://doi.org/ 10.1109/iros.2017.8202192 6. Kim, Y.J.: Anthropomorphic low-inertia high-stiffness manipulator for high-speed safe interaction. IEEE Trans. Robot. 33(6), 1358–1374 (2017). https://doi.org/10. 1109/tro.2017.2732354 7. Lamaury, J., Gouttefarde, M.: Control of a large redundantly actuated cablesuspended parallel robot. In: 2013 IEEE International Conference on Robotics and Automation. IEEE, May 2013. https://doi.org/10.1109/icra.2013.6631240 8. Nemoto, T., Walter, J., Bachmann, C., Gerlich, M., Reitelshofer, S., Franke, J.: Highly dynamic 2-DOF cable-driven robotic wrist based on a novel topology. IEEE Robot. Autom. Lett. 7(2), 5727–5734 (2022). https://doi.org/10.1109/lra.2022. 3159816 9. Paul, C., Valero-Cuevas, F., Lipson, H.: Design and control of tensegrity robots for locomotion. IEEE Trans. Robot. 22(5), 944–957 (2006). https://doi.org/10.1109/ tro.2006.878980 10. Rieffel, J., Mouret, J.B.: Adaptive and resilient soft tensegrity robots. Soft Robot. 5(3), 318–329 (2018). https://doi.org/10.1089/soro.2017.0066 11. Sabelhaus, A.P., et al.: System design and locomotion of SUPERball, an untethered tensegrity robot. In: 2015 IEEE International Conference on Robotics and Automation (ICRA). IEEE, May 2015. https://doi.org/10.1109/icra.2015.7139590 12. Shah, D.S., et al.: Tensegrity robotics. Soft Robot. 9(4), 639–656 (2022). https:// doi.org/10.1089/soro.2020.0170 13. Skelton, R., Adhikari, R., Pinaud, J.P., Chan, W., Helton, J.: An introduction to the mechanics of tensegrity structures. In: Proceedings of the 40th IEEE Conference on Decision and Control (Cat. No.01CH37228). IEEE (2002). https://doi.org/ 10.1109/cdc.2001.980861 14. Vespignani, M., Friesen, J.M., SunSpiral, V., Bruce, J.: Design of SUPERball v2, a compliant tensegrity robot for absorbing large impacts. In: 2018 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS). IEEE, October 2018. https://doi.org/10.1109/iros.2018.8594374 15. Walter, J., Nemoto, T., Bachmann, C., Gerlich, M., Reitelsh¨ ofer, S., Franke, J.: Seilrobotergelenk f¨ ur automatisiertes h¨ ammern/cable-driven robotic joint for automated hammering. wt Werkstattstechnik online 112(09), 625–628 (2022). https:// doi.org/10.37544/1436-4980-2022-09-97 16. Zarebidoki, M., Dhupia, J.S., Xu, W.: A review of cable-driven parallel robots: typical configurations, analysis techniques, and control methods. IEEE Robot. Autom. Mag. 29(3), 89–106 (2022). https://doi.org/10.1109/mra.2021.3138387

Calibration and Accuracy

Dynamic Parameter Identification for Cable-Driven Parallel Robots Tahir Rasheed1 , Loic Michel1 , Stéphane Caro1(B) , Jean-Pierre Barbot1,2 , and Yannick Aoustin1 1

Nantes Université, École Centrale de Nantes, CNRS, LS2N, UMR 6004, Nantes, France {Tahir.Rasheed,Yannick.Aoustin}@univ-nantes.fr, [email protected], [email protected] 2 QUARTZ EA 7393, ENSEA, Cergy-Pontoise, France [email protected] https://www.ls2n.fr, https://www.ensea.fr/fr/laboratoire-de-recherche-quartz-51

Abstract. This paper presents an initial work of identifying the dynamic parameters for cable-driven parallel robot (CDPR), CRAFT with a rigorous protocol. An orbital trajectory of the platform is designed in order to get a pure translation movement of the platform. This trajectory evolves in a plane and allows the identification of four dynamic parameters, the mass of the platform and the first three moments along the three main axes. The identification results obtained respectively from the data from the experimental measurements of the moving platform, and from the experimental measurements treated with the application of a semi-implicit homogeneous differentiator, show that in spite of the complexity of CRAFT an identification of all its essential dynamic parameters is possible. Moreover, in the long term, an online identification of CRAFT during handling tasks is envisaged. Keywords: Cable-Driven Parallel Robots · Inverse Dynamic Model Dynamic Parameter Identification · Least-squares estimator · Semi-Implicit Euler Homogeneous Differentiator · Experiments

1

·

Introduction

The objective is to initiate a scientific approach in order to identify the essential dynamic parameters of a cable-driven parallel robot (CDPR). A CDPR consists of a moving-platform (M P ) that is connected to a rigid frame by means of cables and actuators, the latter being generally mounted on the ground. These robots are very attractive for handling tasks [16] because of their low inertia, a higher payload to weight ratio and a large workspace compared to conventional manipulator robots with articulated rigid limbs. Their possible application fields can Supported by the EquipEx+ TIRREX project, grant ANR-21-ESRE-0015 https:// www.ls2n.fr/le-projet-tirrex-est-laureat-de-lami-equipex/. c The Author(s), under exclusive license to Springer Nature Switzerland AG 2023  S. Caro et al. (Eds.): CableCon 2023, MMS 132, pp. 357–368, 2023. https://doi.org/10.1007/978-3-031-32322-5_29

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be industrial, or dedicated to search-and-rescue operations. To deal with various restrictions on cable tensions, cable elasticity, collisions and obstacle avoidance, over-actuation of the M P is actually a challenging scientific problem [5,7]. The control of a CDPR is complex because, among other things, whatever its movement, the tension of its cables must always be positive. There are still open problems in the control of CDPRs. One key example of an open problem is that control design requires a consistent dynamic model with a relatively good knowledge of the dynamic parameters of the robot such as the terms of inertia, mass, and even friction at the level of the actuators, which manage the winding of the cables. Several contributions about identification of a robot cable exist. Kraus et al. [10] present an identification method for the complete actuator of a cable robot. A second-order system is established with a dead time as an analogous model. An inverse dynamic method (IDM) is used to identify the model parameters of a cable-driven finger joint for surgical robot [18]. But To our best knowledge the identification parameters of the handling platform of a cable robot during a movement in space is not currently investigated. A CDPR, named CRAFT and located at LS2N, Centrale Nantes campus, is equipped with eight actuators and a M P . Each motor has an encoder sensor measuring the angular velocity of its output shaft allowing to evaluate the performances of the differentiation solutions. The M P has six degrees of freedom. This M P is thus over-actuated [14]. The identification of the parameters of a robot is essential to evaluate its behavior in simulation or to synthesize a control based model. The objective of the identification is not to find physical parameters of a mechanism such as a robot, whose value is the most exact possible but to provide a modeling tool that is consistent in order to make simulation, control or other. The identification methodology is usually based on an inverse dynamic model that is linear in the dynamic parameters to be identified [2,8], or [1]. Any matrix inversion, which is generally a generator of numerical problems, is thus avoided. However the identification of dynamic parameters is not easy when the dynamic model is complex. Among the many challenges is the search for exciting trajectories that identify the largest number of dynamic parameters and overcome noise problems [9]. However the identification of dynamic parameters is not easy when the dynamic model is complex. The dynamic behavior of the CRAFT that moves in 3D space with rotation and translation combinations effectively requires a complex modeling with a lot of inertial parameters. In addition, one or more cables may slacken unexpectedly. As a consequence to initiate an identification work, only the model of the platform maneuvered by the cables is considered. Moreover, the defined trajectories generate only translation movements of the platform, i.e. without any rotational movement of the platform with respect to itself, in a horizontal plane. The model to describe the dynamic behavior of CRAFT becomes simpler with only four dynamic parameters to identify. To manage the noise problems semi-implicit Euler homogeneous differentiators are proposed (see e.g. [11] for a comparison of the techniques) and [13]. Off-line identification was conducted in order to compare, respectively from the planned trajectories of the

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platform, the measured filtered signals and their usual Euler discretization, and measured signals processed with an original semi-implicit homogeneous differentiators without extra filter. The main advantage of this differentiator is to offer the perspective of a real-time identification of dynamic parameters ofCRAFT. The contribution of this work is based on three parts: Elaboration of a protocol in order to work with a dynamic model for translational movements of the platform and to have only four parameters to identify and experimental definition of this movement; Identification of the four parameters starting respectively from the reference trajectory, the tension measurement of the cable tension, the position measurement and the application of the usual Euler discretization to obtain the speed and acceleration signals. Furthermore these speed and acceleration signals are compared with those obtained an original semi-implicit Euler homogeneous differentiator. The long-term idea of this work is to develop a complete framework that brings together numerical identification and derivation tools to identify the dynamic parameters of the on-line CRAFT as it performs handling tasks. The remaining of the paper is outlined as follows. Section 3 is devoted to the presentation of CRAFT namely, its geometric, kinematic, dynamic model and its adaptation to the presented strategy. The identification methodology, which is based on the least-squares (LS) estimator is presented in Sect. 4, with the solving way. A semi-implicit homogeneous differentiator, which is considered in this study to get the velocities and the accelerations from the measured position, is presented Sect. 5. The experimental results are presented in Sect. 6. Conclusions and future work are offered in Sect. 7.

2

Robot CRAFT

The cable-driven parallel robot prototype, named CRAFT is located at LS2N, France. The base frame of CRAFT is 4 m long, 3.5 m wide, and 2.7 m high, see Fig. 1). The three-DoF translational and the three-DoF rotational motions of its suspended M P are controlled with eight cables being respectively wound around eight actuated reels fixed to the ground. The M P is 0.28 m long, 0.28 m wide, and 0.2 m high, its overall mass being equal to 5 kg. For each of the eight electrical motors an encoder sensor measures the angular variable of its shaft. The eight motors are equipped with a gearbox reducer of ratio n = 8. The measured value is divided by n in order to obtain the angular position of the output shaft of the gearbox reducer. The robot CRAFT has no tachometer. The main hardware of the prototype consists of a PC (equipped with c MATLAB and  c ControlDesk software), eight  c PARKER SME60 motors  c dSPACE DS1007-based real-time controller and eight and TPD−M drivers, a  custom made winches. Each cable can exert a tension up to 150 N to the M P . The maximum velocity of each cable is equal to 5.9 m/s. The cable tensions are measured using eight FUTEK FSH04097 sensors, one for each cable, attached to cable anchor points. Their signal is amplified using eight FSH03863 voltage

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Fig. 1. (left) CRAFT prototype located at LS2N, Nantes, France and (right) CRAFT Geometric Parametrization.

amplifiers and sent to the robot controller by a coaxial cable. CRAFT is equipped with a motion capture, OptiTrack that the sampling period is equal to 0.01 s.

3

Robot Modeling

Figure 1 depicts the main geometric parameters of CRAFT and the ith cable where i ∈ {1, . . . , m}, m being the number of cables attached to the n Degreeof-freedom (DoF) M P (for CRAFT, m = 8 and n = 6). The dynamic equilibrium wrench equation can be expressed as: ¨ − Cp˙ + wg = 0m Wτ − Ip p with W being the robot wrench matrix expressed as:   ... ui ... um u1 W= b1 × u1 . . . bi × ui . . . bm × um

(1)

(2)

where ui is the unit vector of the ith cable vector pointing from cable anchor points Bi to exit points Ai , expressed in the reference frame Fb . Vector bi pointing from point P to point Bi and is expressed in the platform frame Fp . From Eq. (1), τ = [τ1 , . . . , τi , . . . , τm ] is the cable tension vector, Ip is the inertia tensor, C is the Coriolis matrix and wg is the gravity wrench. p˙ = [t˙ , ω] and p ¨ = [¨ t, α] are the vectors of the M P velocity and acceleration, where, t = [t¨x , t¨y , t¨z ] are the vectors of the M P linear velocity t˙ = [t˙x , t˙y , t˙z ] and ¨ and acceleration, while ω = [ωx , ωy , ωz ]T and α = [αx , αy , αz ] are the vectors of the M P angular velocity and angular acceleration, respectively. The gravity wrench is:   I3 wg = m b ˆ g (3) Rp SG m is the M P mass, I3 is the 3 × 3 identity matrix, b Rp is the rotation matrix ˆ G is the skew-symmetric matrix of the M P Fp frame w.r.t the base frame Fb , S

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associated to sG = [xG , yG , zG ] which represents the coordinate vector of the center of mass of the M P expressed in Fp as: ⎤ ⎡ 0 −zG yG ˆ G = ⎣ zG 0 −xG ⎦ (4) S −yG xG 0 ˆ G represents the first momentum of the M P defined with respect to base mb Rp S frame Fb . g = [0, 0, −g] is the gravity acceleration vector where g = 9.81 m s−2 . The matrix Ip in Eq. (1) can be expressed as:   ˆG m I3 −m b Rp S (5) Ip = ˆG m b Rp S Ip with Ip being the M P inertia tensor expressed in Fp . The term C, which represents the centrifugal and Coriolis effects in Eq. (1) is expressed as:   ˆG ˆS 03 −m ω C= (6) ˆ Ip 03 ω Using Eqs. (3), (5), and (6) in Eq. (1), gives:         ˆG ¨ ˆ G t˙ I3 m I3 −m b Rp S t ˆS 03 −m ω + −m b ˆ = Wτ (7) ˆG α ω ˆ Ip 03 ω Rp SG m b Rp S Ip As a first step of the identification, in this article, we will only consider platform translations, i.e., the platform does not rotate in relation to itself. Hence, the angular velocities and accelerations of the MP in Eq. (7) vanish and is reduced to:      mI3 m I3 ¨ − (8) ˆG t ˆ G g = Wτ m b Rp S mb R p S As the platform orientation is always constant, we can consider b Rp = I3 . Equation (8) can be expressed as in the form as: Ax = b

(9)

where, ⎡

t¨x 0 0 ⎢ t¨y 0 0 ⎢ ⎢t¨z + g 0 0 A=⎢ ⎢ 0 0 t¨z + g ⎢ ⎣ 0 −t¨z − g 0 0 t¨y −t¨x

⎤ 0 ⎡ ⎤ 0 ⎥ m ⎥ ⎢ ⎥ 0 ⎥ ⎥ , x = ⎢mxG ⎥ , b = Wτ ⎣ myG ⎦ −t¨y ⎥ ⎥ mzG t¨x ⎦ 0

(10)

In Eq. (10), we need the linear acceleration of the M P ¨ t and the cable tensions τ . As the robot is already equipped with the cable tension senors so we can directly acquire τ .

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Identification Methodology

The identification methodology is based on the used of the inverse dynamic model (ID) Eq. (9), [1–3]. This model is linear as function of the four parameters m, mxG , myG , and mzG . An off-line identification of these parameters is then considered, given measured or estimated off-line data for the linear acceleration of the M P ¨ t and the cable tensions τ , collected while the robot CRAFT is tracking a planned trajectory. The model Eq. (9) is sampled at a sufficient number of time samples ti , for i = 1, · · · , ne , with (6ne )  4, in order to get an over-determined linear system of (6ne ) equations [1]: YID (τ ) = WID x + ρ where

⎛

⎞ b1 (6 × 1) ⎜ ⎟ .. YID (τ ) = ⎝ ⎠; . bne (6 × 1)

(11) ⎛

WID

⎞ A1 (6 × 4) ⎜ ⎟ .. =⎝ ⎠, . Ane (6 × 4)

(12)

ρ is the (6ne × 1) vector of errors between the data of the measurement of the torques in YID (6ne ×1) and the data WID (6ne ×4) predicted by the model. These errors are due to noise measurement and modelling error. The identification problem consists in finding x the norm squared of the error ρ: ρ2 = YID − WID x2

(13)

The LS estimator models a process by fitting the parameter vector x according to the minimisation of Eq. (13). Then model (14) becomes  + ρ YID (τ ) = WID x

(14)

The estimated parameter x  is equal to: +  = WID YID x

(15)

+ + is the pseudo-inverse of WID . Since 6ne > 4 then WID = where WID  −1   (WID WID ) WID . x is the unique LS solution of Eq. (15). The standard deviation σx is estimated assuming that WID is a deterministic matrix and ρ is a zero-mean additive independent Gaussian noise, with a covariance matrix Cρρ

Cρρ = E(ρρ ) = σρ2 I

(16)

E is the expectation operator and I(6ne × 6ne ) is the identity matrix. By replacing YID in Eq. (15) with its definition Eq. (14) we obtain: +  = x + WID x ρ

(17)

Applying the expectation operator at both sides of Eq. (17) ρ being a zero-mean additive independent Gaussian noise we get: E( x) = E(x)

(18)

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Thus the estimated values are unbiased. An unbiased estimation of the standard deviation σρ is: 2 YID − WID x (19) σ ρ2 = (6ne − 4) WID and being respectively a non-stochastic matrix and a stochastic matrix, the covariance matrix of the LS estimation error is then given by:      ) = (WID Cxx = E(x − x WID )−1 WID σρ2 I WID (WID WID )−1 = σ ρ2 (WID WID )−1

(20) σ x2 i = Cxx (i, i) is the ith diagonal coefficient of Cxx . The relative standard deviation % σxi for each identified parameters is given by: % σxi =

σ xi | xi |

(21)

The calculation of Eq. (14) and the condition number of WID can be obtained using the singular value decomposition (SVD) of WID .

5

Semi-implicit Homogeneous Differentiator

The purpose is to estimate the velocity of the angular variable exclusively from the measured position of the output shaft for each of the eight motors. To estimate the velocity let us introduce the continuous-time state: ⎧ ⎨ x˙ 1 = x2 (22) Σ : x˙ 2 = p(t) ⎩ y = x1 where x1 and x2 are respectively the angular variable and its velocity; y is the measure of x1 with additional noise η. Here p(t) is a bounded perturbation, which is unknown and assumed to be a constant parameter or a slowly variable. This implies that for a sufficient small sampling-time h > 0, p ≡ p+ , with the notation for a discretized variable: •(t = (k + 1)h) = •+ and •(t = kh) = •. Homogeneity approach is interesting because if a local stability is obtained due to the dilatation, this framework allows extending this local property to global settings, [17]. This differentiator can be written such as [4,15], ⎧ ⎨ z˙1 = z2 + λ1 μ 1 α z˙2 = λ2 μ2  1 2α−1 (23) ⎩ yˆ = z1 where α ∈ ]0.5 1[ has to be fixed [6], 1 = y − z1 , and the notation •α = | • |α sgn(•) is adopted along the paper. The degree of homogeneity of the differentiator (23) d is equal to α − 1 with respect to dilatation Λr with r = (r1 = 1, r2 = 1) [15]. Moreover, λi > 0, i = 1, 2 are the linear part gains, which are considered and allow to have the eigenvalues of the differentiation

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error 1 sufficiently stables, while the coefficient μ is chosen sufficiently large to cancel the effect of the unknown perturbation p(t). The discretization of the differentiator (23) is based on the so-called implicit projection that acts as corrective terms and aim to “generalize” the sign function in sliding-based differentiator in order to reduce the chattering and preserve stability properties for high time steps). Two projectors N1 and N2 are used respectively to design the correction terms with the differentiator SIHD-2. In [12], it has been highlighted that the SIHD-2 algorithm offers better performances than with only one projector since the projectors N1 and N2 are respectively dedicated to the estimation of z1 and z2 . Considering the signal to differentiate + y, the error 1 = y − z1+ , and the definition of the notation  + 1  ≡ | 1 |sign( 1 ), As a result, the semi-implicit homogeneous Euler discretization based on two projectors (SIHD-2) reads as:    + z1 = z1 + h z2+ + λ1 μ| 1 |αd N1 (24) z2+ = z2 + E1+ hλ2 μ2 | 1 |2 αd −1 N2 . The projector N1 , and N2 are defined: ⎧  1 1−αd ⎨ , E1+ = 1

1 ∈ SD → N1 = N1 ( 1 ) := λ1 μh ⎩

1 ∈ / SD → N1 = sign( 1 ), E1+ = 0,

(25)

1

with the domain of attraction SD = { 1 / | 1 | ≤ (λ1 μh) 1−α }. ⎧  1 2(1−αd ) ⎨

1 ∈ SD → N2 = N2 ( 1 ) := λ2 h2 μ2 ⎩ 

1 ∈ / SD → N2 = sign( 1 ), 1

(26) 1

αd

where SD = { 1 ∈ SD/ | 1 | ≤ (λ1 μ2 h2 ) 2(1−αd ) ≡ | 2 | ≤ (λ1 μ2 ) 2(1−αd ) h 1−αd }. The filtering properties of the differentiator avoid using an extra filter before the differentiation, and the cascade offers the possibility of adjusting the homogeneous parameter αd with some flexibility. The measured angular positions are noisy such as y becomes ym = x1 + η. The λi , i = 1, 2 parameters are chosen such as the linear part is stable. The value of homogeneous exponent αd is chosen to allow better filtering properties of the estimated differentiation. The μ parameter is also chosen by numerical test trial and error in order to determine the best possible action of both projectors N1 and N2 . The numerical values of these five parameters are tuned as follows: λ1 = 210, λ2 = 210, λ3 = 525, λ4 = 525 αd = 0.95, μ = 1.

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(27)

Experiments

The platform performs four orbital movements. A video of these movements is available at1 . A sampling period of 10 ms is used to measure the position vector t 1

https://youtu.be/I-IOcAGha3o.

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Fig. 2. a) Cable tensions τ as function of time, b) Orientation of the mobile platform.

and the cable tension τ with the FUTEK FSH04097 sensors. The eight measured cable tensions are plotted Fig. 2a). The Euler angles of the platform during its motion are shown in Fig. 2b). This motion of the platform is indeed translational as it is predicted by the trajectory planning. The identification of the parameters t. For m, mxG , myG , and mzG needs the linear acceleration vector of the M P ¨ the best knowledge of this acceleration vector a comparison is made from the measured platform position t between the estimation of the velocity vector t˙ and the acceleration vector ¨ t respectively thanks to the proposed SIHD-2 algorithm and the estimation of these vectors thanks the usual Euler discretization method. With the usual Euler discretization method the numerical tests shows that the estimation of t˙ and ¨ t requires filtering the position vector t of the platform with a cutoff frequency 4 Hz. The filter applies a zero-phase forward and reverse digital infinite impulse response (IIR) filtering. Figure 3 shows the estimated velocity and acceleration vectors in 3D by using the SIHD-2 algorithm and the usual Euler discretization method. The curves are similar between the two algorithms. Table 1 gathers the four identified parameters. The standard deviations are not computed based on the standard deviation from the actual parameter values that are unknown. The numerical values obtained by the two methods for each identified parameter are very close to each other. However, having to filter the position measurement of the platform is a strong constraint in the perspective of doing real time identification of the dynamic parameters of CRAFT. That is why the SIDH -2 algorithm, which is iterative, is a relevant solution in the perspective of real time identification.

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Fig. 3. t˙ and t¨ along the orbital trajectory of the platform: velocity (a) and acceleration (b) with SIDH-2; velocity (c) and acceleration (d) with Euler discretization. Table 1. Numerical values of the identified parameters Identified parameters

7

differentiation SIHD-2 i σ xi x

% σxi

m (kG)

9.88

mxG (kGm)

−0.064 0.0027 4.21

myG (kGm)

0.019

mzG (kGm)

−1.059 0.070

Euler discretization i σ xi x

0.0027 0.027 9.88

0.0027 0.027

−0.064 0.0027 4.16

0.0027 13.86 0.019 6.67

% σxi

0.0027 13.65

−1.210 0.078

6.41

Discussion and Future Work

This research paper presents a novel methodology for identifying the dynamic parameters of Cable-Driven Parallel Robots (CDPRs). Initially, the focus is on identifying four parameters, specifically the platform mass and the location of its center of mass, using an orbital trajectory that has been specifically designed to achieve pure platform translation movement in all the three xyz−plane. The trajectory, which occurs within a single plane, allows for the identification of

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these four dynamic parameters. The values determined by the identification of the inertial parameters are consistent with the actual platform mass (measured) and the estimated center of mass location from the computer aided Design. The results have also been compared with other types of trajectories, such as pointto-point linear trajectories. The reason for choosing a circular trajectory is the availability of continuous platform accelerations which facilitates the use of the proposed methodology. The results obtained from experimental measurements of the platform position, as well as measurements treated with the application of a homogeneous semi-implicit differentiator, indicates that despite the complexity of the CRAFT, these results allow for the real-time identification of all the essential dynamic parameters. In future work, the proposed approach will be used to identify the remaining six dynamic terms associated with the inertia tensor, specifically three terms associated with the platform moment of inertia and other three are products of inertia. To manage the future identification tasks, an IMU sensor will be used in order to provide acceleration measurements that are synchronized with those of the cable tensions. Additionally, we will also aim to identify the friction that acts throughout the actuators of CRAFT.

References 1. Ardiani, F.: Contribution to the parametric identification of dynamic models: application to collaborative robotics. Ph.D. thesis, University of Toulouse, Institut Supérieur de l’Aéronautique et de l’Espace, Toulouse (2023) 2. Gautier, M., Khalil, W.: On the identification of the inertial parameters of robots. In: Proceedings of the 27th IEEE Conference on Decision and Control, vol. 3, pp. 2264–2269 (1988) 3. Gautier, M., Venture, G.: Identification of standard dynamic parameters of robots with positive definite inertia matrix. In: 2013 IEEE/RSJ International Conference on Intelligent Robots and Systems, pp. 5815–5820 (2013). https://doi.org/10.1109/ IROS.2013.6697198 4. Ghanes, M., Barbot, J.P., Fridman, L., Levant, A., Boisliveau, R.: A new varyinggain-exponent-based differentiator/observer: an efficient balance between linear and sliding-mode algorithms. IEEE Trans. Autom. Control 65(12), 5407–5414 (2020) 5. Gouttefarde, M., Collard, J.F., Riehl, N., Baradat, C.: Geometry selection of a redundantly actuated cable-suspended parallel robot. IEEE Trans. Robot. 31(2), 501–510 (2015) 6. Hong, Y., Huang, J., Xu, Y.: On an output feedback finite-time stabilization problem. IEEE Trans. Autom. Control 46(2), 305–309 (2001). https://doi.org/10.1109/ 9.905699 7. Hussein, H., Santos, J.C., Izard, J.B., Gouttefarde, M.: Smallest maximum cable tension determination for cable-driven parallel robots. IEEE Trans. Robot. 37(4), 1186–1205 (2021) 8. Janot, A., Vandanjon, P.O., Gautier, M.: A generic instrumental variable approach for industrial robot identification. IEEE Trans. Control Syst. Technol. 22(1), 132– 145 (2014)

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9. Khalil, W., Dombre, E.: Modeling, Identification and Control of Robots. Butterworth Heinemann (2002) 10. Kraus, W., Schmidt, V., Rajendra, P., Pott, A.: System identification and cable force control for a cable-driven parallel robot with industrial servo drives. In: 2014 IEEE International Conference on Robotics and Automation (ICRA), pp. 5921– 5926 (2014). https://doi.org/10.1109/ICRA.2014.6907731 11. Michel, L., Selvarajan, S., Ghanes, M., Plestan, F., Aoustin, Y., Barbot, J.P.: An experimental investigation of discretized homogeneous differentiators: pneumatic actuator case. IEEE J. Emerg. Sel. Top. Ind. Electron. 2(3), 227–236 (2021) 12. Michel, L., et al.: Experimental validation of two semi-implicit homogeneous discretized differentiators on the CRAFT cable-driven parallel robot. In: CFM2022, Congrès Français de Mécanique. Nantes, France (2022). https://hal.archivesouvertes.fr/hal-03751623 13. Michel, L., Ghanes, M., Plestan, F., Aoustin, Y., Barbot, J.P.: Semi-implicit homogeneous Euler differentiator for a second-order system: validation on real data. In: IEEE Control and Decision Conference CDC, Austin, Texas, USA (2021) 14. Mishra, U.A., Métillon, M., Caro, S.: Kinematic stability based AFG-RRT* path planning for cable-driven parallel robots. In: IEEE International Conference on Robotics and Automation ICRA, Xi’an, China (2021) 15. Perruquetti, W., Floquet, T., Moulay, E.: Finite-time observers: application to secure communication. IEEE Trans. Autom. Control 53(1), 356–360 (2008) 16. Picard, E., Plestan, F., Tahoumi, E., Claveau, F., Caro, S.: Control strategies for a cable-driven parallel robot with varying payload information. Mechatronics 79, 102648 (2021) 17. Rosier, L.: Homogeneous Lyapunov function for homogeneous continuous vector field. Syst. Control Lett. 19(6), 467–473 (1992) 18. Yu, L., Wang, W., Wang, Z., Wang, L.: Dynamical model and experimental identification of a cable-driven finger joint for surgical robot. In: 2017 IEEE International Conference on Mechatronics and Automation (ICMA), pp. 458–463 (2017). https://doi.org/10.1109/ICMA.2017.8015860

Forward Kinematics and Online Self-calibration of Cable-Driven Parallel Robots with Covariance-Based Data Quality Assessment Ryan J. Caverly(B) , Keegan Bunker, Samir Patel, and Vinh L. Nguyen Department of Aerospace Engineering and Mechanics, University of Minnesota, Minneapolis, MN 55455, USA {rcaverly,bunke029,samir004,nguy2876}@umn.edu

Abstract. This paper presents an algorithm for the forward kinematics and online self-calibration of cable-driven parallel robots. Covariancebased metrics known as the position dilution of precision (PDOP) and orientation dilution of precision (ODOP) are introduced as a means to quantify the quality of data collected with regards to self-calibration. These metrics enable systematic pruning of the data used for self-calibration and an assessment of when sufficiently rich data has been collected to perform self-calibration. The proposed algorithm is demonstrated through inversekinematics- and dynamics-based numerical simulations. Keywords: Cable-driven parallel robots · Pose estimation kinematics · Calibration · Least-squares optimization

1

· Forward

Introduction

Estimating the end-effector pose of a cable-driven parallel robot (CDPR) with the sensing information available is a critical aspect of CDPR operation. Many CDPRs only have access to indirect cable length measurements via incremental winch encoders, which rely on an initial cable length calibration procedure. Iterative forward kinematics algorithms are often employed to solve for the endeffector pose given indirect cable length measurements with the assumptions that initial cable length calibration has been performed and the effect of biases in the initial cable lengths are small [8,9]. Offline calibration algorithms can be implemented that collect cable length measurements while placing the CDPR end-effector at specific known poses and determining biases in the cable length measurements [12,13]. Although these methods can solve the issue of calibration, they require a portion of CDPR operating time to be dedicated specifically for calibration before it can be used for regular operation. Online selfcalibration algorithms that combine forward kinematics and calibration into a This material is based upon work supported by the National Science Foundation Graduate Research Fellowship under Grant No. 2237827. This work was also supported in part by the University of Minnesota Office of Undergraduate Research. c The Author(s), under exclusive license to Springer Nature Switzerland AG 2023  S. Caro et al. (Eds.): CableCon 2023, MMS 132, pp. 369–380, 2023. https://doi.org/10.1007/978-3-031-32322-5_30

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single operation have been proposed for over-constrained CDPRs [5] and suspended CDPRs [4] with initial cable length biases, as well as for CDPRs with initial cable length biases and pulley kinematics [11,14]. As noted in [5], the number of data points used in this self-calibration process and the richness of motion/geometry present in these data points has a major impact on the accuracy of the calibration. A systematic approach to optimally select the data points used as part of self-calibration would allow for accurate calibration without requiring inordinate amounts of data collection and computation time. The forward kinematics and self-calibration algorithm proposed in this paper attempts to address this need by introducing data quality assessment metrics inspired by GPS navigation [7], which allows for systematic data selection and assessment of when sufficient data has been collected for self-calibration. The novel metrics are defined as the position dilution of precision (PDOP) and the orientation dilution of precision (ODOP), which are computed based on the covariance of the self-calibration pose estimation error. The PDOP and ODOP are used within our proposed algorithm to 1) systematically select the data points used for self-calibration in a manner that maintains accuracy with minimal number of data points (and thus minimal computational burden) and 2) determine when sufficiently rich data has been collected and self-calibration computations can be performed reliably. The resulting novel forward kinematics and self-calibration algorithm features an inner loop that solves forward kinematics with the current calibration values and an outer loop that performs self-calibration. The proposed algorithm is presented in a general manner in Sect. 2 to accommodate CDPRs with different geometries and degrees of freedom (DOFs), a variety of sensors (e.g., cable orientation sensors [3] or direct cable length sensors [6]), and/or generalized loop-closure equations (e.g., geometrico-static equations [4]). The equations involved in applying this algorithm to an over-constrained 6-DOF CDPR with inelastic cables are presented in Sect. 2.2. The remainder of this section introduces important notation and a mathematical description of the forward kinematics and self-calibration problem considered in this work. 1.1

Notation

In this paper, the identity and zero matrices are 1 and 0, respectively. A reference i 1, − i 2, − i 3. frame, Fi , is defined by three dextral, orthonormal basis vectors, − → → → The subscript (·)k denotes a variable evaluated at time step k, the superscript (·)i denotes a variable concerning the CDPR’s ith cable, the superscript (·)(j) denotes a variable evaluated at the j th step within an iterative algorithm, and ˆ denotes an estimate. The continuous random variable v ∼ N (¯ (·) v, Qv ) satisfies v)(v−¯ v)T ], where a normal distribution with mean ¯v and covariance Qv = E[(v−¯ T Qv = Qv ≥ 0. 1.2

System Description and Problem Definition

Consider an n-DOF CDPR with m cables, an example of which is shown in Fig. 1. The reference frame Fo is an inertial frame with point o at its origin. The position

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Fig. 1. An example of an over-constrained 6 DOF CDPR.

of the end-effector’s center of mass relative to o is − r and its components resolved → 3 th in Fo are given by r ∈ R . The position of the i cable’s anchoring point (e.g., routing pulley or winch drum) relative to o is − a i and its components resolved → i th in Fo are given by a . The position of the i cable’s attachment point on the end-effector relative to its anchoring point is −  i and its components resolved in → Fo are given by i . In the case of a CDPR with a rigid-body end-effector, Fp denotes an end-effector-fixed frame, the position of the ith cable’s attachment point on the end-effector relative to the end-effector center of mass is − b i , and → its components resolved in Fp are given by bi . In general, the pose of the   endeffector is described by ρ ∈ Rn . In the case of a 6-DOF CDPR, ρT = rT θ T , where θ ∈ R3 represents a parameterization of the rotation matrix Rpo (θ) or direction cosine matrix Cpo (θ) that describes the orientation of Fp relative to Fo (e.g., Euler angles). It is assumed that a relationship between the CDPR’s measurements and its pose is known to satisfy the generalized model f(ρ, y, v, β) = 0,

(1)

where ρ ∈ Rn contains the pose parameters, y ∈ Rp contains linearly independent CDPR measurements, v ∈ Rp is zero-mean Gaussian white noise that satisfies v ∼ N (0, Qv ) with Qv ∈ Rp×p , and β ∈ Rq contains constant bias terms in the measurements. The dimension of the generalized model (i.e., the number of equations in (1)) varies depending on the modeling assumptions used, and the sensor information available. As an example, in the case of a typical forward kinematics formulation, (1) represents the loop closure equations, y ∈ Rm contains the cable length measurements, and β ∈ Rm represents the biases in the cable length measurements, which may be due to initial calibration errors or unknown initial lengths. More generally, (1) can represent geometricostatic equations with y containing cable length and cable tension measurements (e.g., [4]) or it can represent loop-closure equations augmented with additional measurements (e.g., the inclusion of cable orientation sensors [3] or direct cable length measurement sensors [6]). The overarching goal of this paper is develop an

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online self-calibration method that accurately estimates both ρ and β without requiring large amounts of data or excessive computation. The formulation of the proposed method presented in Sect. 2 is kept general to account for all possible cases of (1), with only the assumptions that all measurements contained in y are available at each time step and the function f(ρ, y, v, β) is differentiable with respect to y, v, and β. It is also assumed that n ≤ p < n + q, which implies that the biases β is not observable from the measurements at a given time step (i.e., it is not possible to simultaneously estimate ρ and β given a single time step of data).

2

Online Forward Kinematics and Self-calibration Formulation

The proposed forward kinematics and self-calibration formulation is first presented in a general form in Sect. 2.1, followed by the case of an over-constrained 6-DOF CDPR with initial cable length calibration biases in Sect. 2.2. 2.1

General Formulation

The proposed forward kinematics and online self-calibration algorithm features an inner-loop and outer-loop structure. The inner loop estimates the end-effector ˆ pose ρˆk at each time step using the current best knowledge of the bias term β. The algorithm’s outer loop makes use of measurements across N time steps to estimate the bias and uses least-squares estimation theory and newly defined PDOP and ODOP metrics to automatically determine when this bias estimate is accurate enough for implementation in the inner loop. Both the inner and outer loops of the algorithm involve linearizing (1) about ¯ which results in ¯ v = E[v] = 0, and β = β, ρ = ρ, ¯ ¯ + Jb (ρ, ¯ ¯ + Jρ (ρ, ¯ y, β)δρ ¯ y, 0, β)v ¯ y, β)δβ, ¯ y, 0, β) + Jv (ρ, (2) f(ρ, y, v, β) ≈ f(ρ,    ∂f  ∂f  ∂f  where Jρ (ρ, y, β) = ∂ρ , Jv (ρ, y, 0, β) = ∂v , Jb (ρ, y, β) = ∂β .   ρ,y,0,β ρ,y,0,β

ρ,y,0,β

At each time step k, the inner loop of the algorithm assumes a fixed estimate of βˆ is available and solves the forward kinematics as a weighted nonlinear leastsquares optimization problem ˆ −1 (ρˆk , yk , β)f( ˆ ρˆk , yk , 0, β), ˆ min fT (ρˆk , yk , 0, β)H ρˆk

(3)

ˆ is the covariance of the noise associated with f(ρˆk , yk , 0, β) ˆ where H(ρˆk , yk , β) T ˆ ˆ ˆ and is calculated as H(ρˆk , yk , β) = Jv (ρˆk , yk , β)Qv Jv (ρˆk , yk , β). The solution to (3) represents the best least unbiased estimate of ρˆk [1, p. 84]. The weighted (0) nonlinear least-squares problem is solved iteratively by initializing ρˆk and solving for ρˆk through the iterative Levenberg-Marquardt algorithm given by (j+1)

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(4)

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where η > 0 provides numerical damping to the iterative calculations and the iteration is terminated when a specified stopping criterion is satisfied. The covariance of the pose estimation error after j iterations at the k th time step is [1, p. 84]  −1 (j) (j) (j) (j) ˆ −1 (ρˆ(j) , yk , β)J ˆ ρ (ρˆ(j) , yk , β) ˆ ˆk , yk , β)H Pk = E[(ρk − ρˆk )(ρk − ρˆk )T ] = JT . ρ (ρ k k

(5) After accumulating N time steps of measurements, the time indices of the outer loop estimator data set are initialized as k1 = 1, k2 = 2, . . . , kN = N . The covariance of the estimation error in the bias and the end-effector poses across these N time steps is computed as  −1 ¯ −1 (ˆ ¯ = E[(x − ˆx)(x − ˆx)T ] = ¯JT x, y)H x, y)¯ Jρβ (ˆ x, y) , P ρβ (ˆ  T  T T   T T T where xT = β T ρT k1 ρk2 · · · ρkN , y = yk1 yk2 · · · ykN , ¯ y) = diag{H(ρk1 , yk1 , β), H(ρk2 , yk2 , β), . . . , H(ρkN , ykN , β)}, H(x, ⎡ ⎤ 1 Jρ (ρk1 , yk1 , β) 0 ··· 0 ⎢1 ⎥ 0 0 Jρ (ρk2 , yk2 , β) · · · ⎥ ¯Jρβ (xk , y) = ⎢ ⎢ .. ⎥. .. .. . . . . ⎣. ⎦ . . . . 1

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The metrics used to assess the quality of the data in these N time steps of measurements are denoted as the position dilution of precision (PDOP) and the orientation dilution of precision (ODOP), which are defined as   N 1      ¯ n(i−1)+9:n(i−1)+8+n ≈ E rk − ˆrk 2 , PDOP =  (6) trace P r 2 N i=1   N 2   1     ¯ n(i−1)+9+n :ni+9 ≈ E  ODOP =  trace P θk − θˆk  , r N i=1 2

(7)

¯ i:j denotes a submatrix of P ¯ given by its i through j rows and columns, where P and nr ≤ n is the number of CDPR end-effector translational DOFs (e.g., nr = 3 in the case of a 6-DOF CDPR with three translational DOFs). The terms PDOP and ODOP are inspired by similar terms used within the GPS community to describe the richness of data available when computing a navigation solution based on the geometry and number of visible satellites [7]. The values of PDOP and ODOP provide a way to quantify the quality or richness of the data collected with regards to self-calibration. Larger values of PDOP and ODOP indicate that either not enough data points are being used for this computation (i.e., N is too small) or the data points do not feature rich enough motion. The PDOP and ODOP are used for two purposes within the proposed self-calibration method. Their first use is that they provide a means to

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keep the amount of data points involved in the outer-loop computation low, with minimal sacrifice to accuracy. Specifically, after N time steps of measurements are accumulated, the PDOP and ODOP values can be used to determine whether the current time step of data provides additional richness to the existing N time steps of data. In the proposed scheme, the PDOP and ODOP are calculated with the existing N time steps of data and then again, with the current time of data replacing a randomly-selected time step of data within the existing N time steps. If the PDOP and ODOP are both smaller with the new time step of data, then the data from the randomly-selected time step of data is removed from the outer-loop computation and the current time step of data is added, else the current time step of data is ignored in the outer loop. The time step indices of data stored by the outer loop are denoted by the indices k1 , . . . , kN . The second use of the PDOP and ODOP is their ability to assess whether the outer-loop selfcalibration has rich enough data to be able to trust the bias estimates it computes for use within the inner-loop forward kinematics calculations. Threshold values of PDOP and ODOP can be set to prevent the computation of the biases until specified accuracies in end-effector position and orientation are achievable. This prevents inaccurate bias estimates from being computed and implemented within the inner loop, which could lead to even worse forward kinematics pose estimates than without the use of bias estimates. This choice of thresholds is discussed further in the numerical example of Sect. 3. If it is determined that PDOP and ODOP are both lower than their threshold values, then ˆx is solved for through the weighted nonlinear least-squares optimization problem ¯ −1 (ˆx, y)¯f(ˆx, y), min ¯fT (ˆx, y)H ˆ x

  (j) T where ¯fT (x, y) = fT (ρk(j) . This is solved itera, y , 0, β) · · · f (ρ , y , 0, β) k k 1 N kN 1 tively using the Levenberg-Marquardt update law  −1 ¯f(ˆ ¯ −1 (ˆx(j) , y)¯Jρβ (ˆx(j) , y) + η1 ˆx(j+1) = ˆx(j) − ¯JT x(j) , y)H x(j) , y), ρβ (ˆ

(8)

where ˆx(0) is initialized with the previous estimates of the bias and pose parameters. The bias estimate is then extracted from ˆx(j+1) following the termination ˆk , which is then used from the current time step of this iteration and set as b onwards within the inner loop, until a new outer-loop computation is performed. A depiction of the overall forward kinematics and self-calibration algorithm is provided in Fig. 2. Although the proposed self-calibration method is described as an online strategy where the CDPR end-effector is following a dynamic trajectory, it is worth noting that all of the computations described can be performed using offline static trajectories where the CDPR end-effector is moved to select poses in the workspace.

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Fig. 2. Flow chart of the proposed forward kinematics and self-calibration algorithm.

2.2

Application to the Forward Kinematics of an Over-Constrained 6-DOF CDPR with Inelastic Cables

Consider a 6-DOF over-constrained CDPR with m > n cables, where a relative encoder on each winch is used to obtain cable length measurements relative to an initial length. Assuming that the cables are inelastic and pulley kinematics can be ignored, the position vector of the ith cable’s attachment point on the i i end-effector relative to its attachment is given by i (ρ) = r + CT po (θ)b − a . Each cable length measurement is modeled to satisfy   y i = i (ρ)2 + v i + β i , (9) 2

where y i is the ith cable’s length measurement, v i ∈ N (0, σ i ) is the noise associated with this measurement, and β i is the initial length calibration error bias associated with the ith cable. Defining the measurement Eq. (9) for all m cables and rewriting the relation in the form of (1) results in 0 = f(ρ, y, v, β) = y − g(ρ) − v − β,

(10)

where y ∈ Rm contains all the CDPR cable length measurements, gT (ρ) =  [1 (ρ)2 · · · m (ρ)2 ], vT = [v 1 · · · v m ] contains zero-mean Gaussian cable 2 2 length measurement noise with v ∈ N (0, Qv ) and Qv = diag{σ 1 , . . . , σ m }, T 1 m while β = [β · · · β ] contains the calibration error bias in the cable length measurements. For this specific application, the Jacobians included in the generic lineariza∂g is the Jacobian tion of (2) are specified as Jβ = Jv = −1, while Jρ (ρ) = − ∂ρ that maps pose rates to cable length rates and its transpose maps cable forces to the end-effector wrench. Analytic expressions for Jρ (ρ) with different parameterizations of θ are found in [8].

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Fig. 3. (a) Depiction of the 6-DOF over-constrained CDPR used in the numerical simulations of Sect. 3. (b) Plots of the simulated desired end-effector pose trajectory versus time, where r = [rx ry rz ]T and θ = [θ1 θ2 θ3 ]T .

3

Numerical Results

Numerical simulations of a 6-DOF CDPR with 8 cables and a geometry similar to that of IPAnema 2 described in [10, p. 319] are performed in this section. The dimensions of the CDPR, as shown in Fig. 3(a), are 1.43 m x 0.76 m x 0.93 m, with exact dimensions provided in [8]. Two sets of simulations are performed: inversekinematics-based simulations with rigid cables and dynamics-based simulations with linear elastic cables. In both cases, the desired pose trajectories of the endeffector is presented in Fig. 3(b), where a 3-2-1 Euler-angle sequence is used to parameterize the orientation of the end-effector. All implementations of the proposed algorithm assume loop-closure equations of the form in (10). The damping and stopping criteria tolerances are chosen as (j+1) (j) (j+1) (j) − ρˆk ||2 < 10−9 , and ||ˆxk −ˆ xk ||2 < 10−12 . In all cases, η = 0.001, ||ρˆk the initial pose estimate used in the Levenberg-Marquardt algorithm is the pose estimate from the previous time step. The threshold values of PDOP and ODOP for the outer loop are chosen as 10 mm and 5 deg, respectively, although a two other sets of threshold values are also compared in Sect. 3.1. The number of time steps to include in the outer-loop computations is typically chosen as N = 30, although values of N = 10 and N = 100 are also tested in Sect. 3.1. 3.1

Inverse-Kinematics-Based Simulations

Inverse-kinematics-based simulations are performed with a time step of T = 0.005 s to produce cable lengths for the pose trajectory in Fig. 3(b). Zeromean Gaussian measurement noise with standard deviation σi = 0.5 mm, i = 1, . . . , 8 and arbitrarily-chosen constant initial cable length biases β =  T 5 30 −10 −30 −10 5 −30 10 mm are added to the cable lengths before proceeding with forward kinematics. The proposed algorithm is first implemented with the standard PDOP and ODOP thresholds, with the number of outer-loop time steps varied between

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N = 10, N = 30, and N = 100. The pose estimation error versus time with ±3σ bounds computed from the covariance matrix are presented in Fig. 4, where it is shown that similar levels of pose estimation accuracy are achieved. The main difference when varying the value of N is that it affects the amount of time it takes to meet the threshold values of PDOP and ODOP, as well as the time elapsed before performing the first outer-loop bias estimate computation. This emphasizes one of the main benefits of the proposed self-calibration algorithm, which is the reduction in computation time (e.g., the size of the matrices inverted can be reduced by a factor of 10 when using N = 10 instead of N = 100), with minimal effect on the estimation accuracy.

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To emphasize the role that the threshold values of PDOP and ODOP play in the proposed algorithm, simulations are performed with N = 30 and three different threshold values: 50 mm and 30 deg, followed by 10 mm and 5 deg, and finally 5 mm and 3 deg. Plots of PDOP and ODOP, as well as ||r − ˆr||2 and ˆ 2 versus time are presented in Fig. 5. The results demonstrate the impor||θ − θ|| tance of collecting rich enough data (i.e., setting low enough PDOP and ODOP thresholds) before performing an outer-loop bias computation and implementing it within the inner-loop forward kinematics, otherwise large transient errors are possible (as seen in Figs. 5(a) and 5(d)). Figure 5 also demonstrates that the PDOP and ODOP values give accurate quantifications of the steady-state position and orientation estimation errors, respectively. 3.2

Dynamics-Based Simulations with Linear Elastic Cables

To demonstrate robustness of the proposed forward kinematics and selfcalibration algorithm, dynamics-based simulations are performed that make use of the linear elastic cable model in [2] with an axial cable stiffness of EA = 10 kN and a viscous friction coefficient of Ci = 1 × 10−5 N · m · s. The end-effector payload has a mass of 1 kg and its second moment of mass resolved in Fp is Jp = 10−2 diag{2.81, 1.13, 2.81} kg · m2 .

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The proposed algorithm is implemented with no a priori knowledge of the vibrations in the cables. To account for this, the algorithm assumes that the zero-mean Gaussian white noise in the cable length measurements is artificially increased to a standard deviation of 1.5 mm. To be clear, the actual zero-mean Gaussian white noise added to the cable lengths measurements remains the same as in Sect. 3.1, however the algorithm assumes additional noise is present to indirectly account for cable vibrations. Results of the pose estimation errors versus time are included in Fig. 6, which shows a marked improvement in the pose estimation error after the outer-loop self-calibration begins at around 6 s. The ±3σ bounds computed from the covariance matrix are no longer tightly capturing the error, but this is to be expected as the elastic cables result in unmodeled dynamics that are not captured by the loop-closure equations or measurement noise model. These results demonstrate that the proposed algorithm is still able to perform well under these challenging circumstances.

4

Conclusion

A novel forward kinematics and self-calibration algorithm was presented that makes use of newly-defined PDOP and ODOP metrics to evaluate the richness of data used in the calibration process. The PDOP and ODOP metrics are computed based on the covariance of the pose and bias estimation error, which enables to a quantifiable manner to achieve accurate self-calibration without requiring large amounts of data. Future work on this topic will validate the proposed algorithm experimentally on a CDPR that has been built at the University of Minnesota, but does not yet have a working motion tracking system to obtain ground-truth pose data.

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References 1. Crassidis, J.L., Junkins, J.L.: Optimal Estimation of Dynamic Systems. CRC Press, Boca Raton (2004) 2. Fang, S., Franitza, D., Torlo, M., Bekes, F., Hiller, M.: Motion control of a tendonbased parallel manipulator using optimal tension distribution. IEEE-ASME Trans. Mech. 9(3), 561–568 (2004) 3. Garant, X., Campeau-Lecours, A., Cardou, P., Gosselin, C.: Improving the forward kinematics of cable-driven parallel robots through cable angle sensors. In: Gosselin, C., Cardou, P., Bruckmann, T., Pott, A. (eds.) Cable-Driven Parallel Robots. MMS, vol. 53, pp. 167–179. Springer, Cham (2018). https://doi.org/10. 1007/978-3-319-61431-1 15 4. Id´ a, E., Merlet, J.-P., Carricato, M.: Automatic self-calibration of suspended underactuated cable-driven parallel robot using incremental measurements. In: Pott, A., Bruckmann, T. (eds.) CableCon 2019. MMS, vol. 74, pp. 333–344. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-20751-9 28 5. Lau, D.: Initial length and pose calibration for cable-driven parallel robots with relative length feedback. In: Gosselin, C., Cardou, P., Bruckmann, T., Pott, A. (eds.) Cable-Driven Parallel Robots. MMS, vol. 53, pp. 140–151. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-61431-1 13 6. Martin, C., Fabritius, M., Stoll, J.T., Pott, A.: Accuracy improvement for CDPRs based on direct cable length measurement sensors. In: Gouttefarde, M., Bruckmann, T., Pott, A. (eds.) CableCon 2021. MMS, vol. 104, pp. 348–359. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-75789-2 28 7. Misra, P., Enge, P.: Global Positioning System: Signals, Measurements, and Performance, 2nd edn. Ganga-Jamuna Press, Lincoln (2011) 8. Nguyen, V.L., Caverly, R.J.: CDPR forward kinematics with error covariance bounds for unconstrained end-effector attitude parameterizations. In: Gouttefarde, M., Bruckmann, T., Pott, A. (eds.) CableCon 2021. MMS, vol. 104, pp. 37–49. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-75789-2 4 9. Pott, A.: An algorithm for real-time forward kinematics for cable-driven parallel robots. In: Lenarˇciˇc, J., Staniˇsi´c, M.M. (eds.) Advances in Robot Kinematics: Motion in Man and Machine, pp. 529–538. Springer, Dordrecht (2010). https:// doi.org/10.1007/978-90-481-9262-5 57 10. Pott, A.: Cable-Driven Parallel Robots. Springer, Cham (2018) 11. Wang, B., Caro, S.: Exit point, initial length and pose self-calibration method for cable-driven parallel robots. In: Zeghloul, S., Laribi, M.A., Arsicault, M. (eds.) MEDER 2021. MMS, vol. 103, pp. 90–101. Springer, Cham (2021). https://doi. org/10.1007/978-3-030-75271-2 10 12. Yuan, H., Zhang, Y., Xu, W.: On the automatic calibration of redundantly actuated cable-driven parallel robots. In: CableCon 2019. MMS, vol. 74, pp. 357–366. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-20751-9 30 13. Zhang, B., Zhou, F., Shang, W., Cong, S.: Auto-calibration and online-adjustment of the kinematic uncertainties for redundantly actuated cable-driven parallel robots. In: IEEE 4th International Conference on Advanced Robotics and Mechatronics (ICARM), pp. 280–285 (2019) 14. Zhang, Z., Xie, G., Shao, Z., Gosselin, C.: Kinematic calibration of cable-driven parallel robots considering the pulley kinematics. Mech. Mach. Theory 169, 104648 (2022)

Elasto-Static Model and Accuracy Analysis of a Large Deployable Cable-Driven Parallel Robot Zane Zak¸e1(B) , Nicol` o Pedemonte1 , Boris Moriniere2 , Adolfo Suarez Roos1 , and St´ephane Caro3 1

3

IRT Jules Verne, 1 Mail des 20 000 Lieues, 44340 Bouguenais, France [email protected] 2 Airbus Operations, rue de l’Aviation, 44340 Bouguenais, France ´ Nantes Universit´e, Ecole Centrale Nantes, CNRS, LS2N, UMR 6004, 1, rue de la Noe, 44321 Nantes, France [email protected]

Abstract. Cable-driven parallel robots (CDPRs) have the potential of being the go-to rapidly deployable and reconfigurable robots. This is because cables are used instead of rigid links and thus the overall robot architecture can consist of only four masts, eight motors and very long cables actuating the moving-platform. This paper introduces a large deployable CDPR called ROCASPECT. Its accuracy and repeatability have been evaluated according to ISO 9283:1998. Moreover, the effect of modeling errors and mast compliance on the CDPR accuracy and repeatability is studied. Despite modeling issues, accuracy of 5 cm and repeatability of 1 cm were obtained for the deployable CDPR of size 23.3 m × 19.0 m × 4.0 m. Keywords: CDPR

1

· deployable · accuracy · repeatability · elasticity

Introduction

Cable-driven parallel robots (CDPRs) are a type of parallel robots with cables instead of rigid links. Due to this, CDPRs can have a very large workspace (WS) and a light deployable and reconfigurable structure. However, with the increased size, a decreased accuracy and repeatability can usually be observed. Among the existing large CDPRs we can mention: (a) IPAnema 8 m × 6 m × 5 m [1]; (b) CoGiRo 16 m × 11 m × 6 m [2]; (c) CDPR in the art installation Prince’s Tears 20.8 m × 7.3 m × 5.1 m [3]. To our knowledge only IPAnema has been assessed using ISO 9283:1998 [4] so far. In [1] the pose repeatability of IPAnema was below 0.75 mm for all tested velocities. Interestingly, the authors observed that the best values could Supported by IRT Jules Verne (French Institute in Research and Technology in Advanced Manufacturing Technologies for Composite, Metallic and Hybrid Structures). c The Author(s), under exclusive license to Springer Nature Switzerland AG 2023  S. Caro et al. (Eds.): CableCon 2023, MMS 132, pp. 381–393, 2023. https://doi.org/10.1007/978-3-031-32322-5_31

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be achieved with the highest velocity of their CDPR. The authors report path repeatability of 0.5 mm. In [5] the pose accuracy is reported to be 37.86 mm. In both papers ground truth measurements were performed using a Leica Absolute Laser Tracker AT901-MR (Leica Laser-Tracker) with a certified absolute accuracy of less than 10 µm. For CoGiRo the reported mean accuracy is 50 mm and mean repeatability is 3 mm [6]. However, ISO 9283 relies on the worst accuracy and repeatability measurement instead of the mean. No comparable accuracy or repeatability has been reported for the CDPR used in the art installation. In this paper, a large deployable CDPR named ROCASPECT, shown in Fig. 1a, is presented. The structure consists of four masts, each being an assembly of multiple pieces, as shown in Fig. 1b, thus it can be stored in a small space when not in use. The moving-platform (MP) is pulled by 8 cables. In the configuration shown in Fig. 1a, the CDPR size is 23.3 m × 19.0 m × 4.0 m with a footprint of 443 m2 . The MP size is 1.0 m × 1.0 m × 0.5 m and its mass is 35 kg. The coordinates of cable exit points Aij measured by the Leica Laser-Tracker can be found in Table 1. For security, the masts are fixed to the ground to avoid them tipping over. The wrench-feasible WS [7] corresponds approximately to the light green shape shown in Fig. 2 and has a footprint of 332.5 m2 . During the experiments it was observed that the masts are subject to torsion and bending due to their light structure. In this paper we address the modeling of the CDPR with bending masts as well as the analysis of the effect of mast deformation on the robot accuracy and repeatability.

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Design and Modeling

As discussed in [3], the height of a CDPR can become a limiting factor as the ground footprint is increased. To counteract the large cable tensions pulling on Ai2 Ai1

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Fig. 2. Schematic of ROCASPECT: (i) footprint shown in dark green; (ii) WS shown in light green with its diagonal plane in gray; (iii) three sets of poses for pose accuracy and repeatability analysis shown in violet, pink and gold; (iv) two paths for path accuracy and repeatability analysis shown in brown and cyan (Color figure online) Table 1. Coordinates of ROCASPECT cable exit points Nominal Aij in Fb , m A11

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[20.094; 0.038; 3.852]

0.0567

the masts, ROCASPECT masts are designed with supporting ropes that can be seen in Fig. 1b and Fig. 2. These are certified inelastic Dyneema ropes. However, these ropes have spliced loops on each end and as the splicing tightened during the experiments they elongated by approximately 15 mm or 0.39% of their nominal length of 3.8 m. While the elongation is small and normally would be negligible, it was not possible to tighten the supporting ropes and as a result the masts were bent slightly towards the center of the WS and not as stiff as expected. After two weeks of experiments the cable exit points were remeasured with the MP positioned at [10.0; 8.56; 1.0]. The new measurements, denoted as Arij # » are shown in Table 1 along with the Euclidean distance ||Aij Arij ||2 . Furthermore, a certain compliance of masts and as a consequence a displacement of pulleys depending on the MP pose was observed. As an example, the Cartesian coordinates of A21 and A22 and their distance to Arij depending on the MP pose are shown in Table 2.

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MP position, m [4.5; 4.5; 1.0] [4.5; 12.5; 1.0]

A21 , m [−0.725; 16.529; 3.555]

δa21c , m A22 , m

δa22c , m

0.0204

0.0322

[0.030; 17.090; 3.859]

[−0.699; 16.524; 3.559]

0.0103

[0.063; 17.093; 3.853]

0.0131

[15.5; 12.5; 1.0] [−0.689; 16.554; 3.567]

0.0286

[0.072; 17.129; 3.847]

0.0289

[15.5; 4.5; 1.0]

0.0167

[0.040; 17.119; 3.854]

0.0232



2.1

[−0.711; 16.547; 3.561]





Kinematic Model

A CDPR is defined by cable exit points Aij and anchor points Bij , where i denotes the ith mast with i = 1, . . . , k and k = 4; j denotes the jth cable exiting from ith mast with j = 1, . . . , p and p = 2. The ijth cable vector is expressed as: lij = b aij − b tp − b Rp p bij

(1)

where b aij is the Cartesian coordinates vector of Aij expressed in frame Fb ; bij is the Cartesian coordinates vector of Bij expressed in frame Fp ; b tp and b Rp are the position vector and rotation matrix of the MP expressed in Fb . The static equilibrium of the MP is given by: p

Wτ + wg = 0

(2)

where τ is the cable tension vector, wg is the MP gravity wrench, and W is the wrench matrix of the CDPR, defined as [7]:   b b u ... ukp (3) W = b p 11 b Rp b11 × u11 . . . b Rp p bkp × b ukp l

where b uij is the unit vector of lij , namely b uij = ||lijij||2 . Note that ROCASPECT has large pulleys, thus pulley kinematics must be taken into account, but not detailed here due to the limited space. Please refer to [8–11] for the expression of the unit vector b uij . 2.2

Mast Model

The elongation of the supporting ropes leads to a displacement of cable exit points from Aij to A∗ij , because the mast beam tilts forward at its lower assembly point. Moreover, as the beam and rope assembly (mast) is no longer stiff, applying forces to CDPR cables leads to a small displacement around A∗ij . The latter can be modeled by three perpendicular and intersecting elastic joints at the base of the mast, as shown in Fig. 1b. Although two pulleys are on the same mast, the resulting displacement of each pulley is different and is described as: δaij = Cij wij

(4)

where Cij is the Compliance matrix expressed as:  Cij = Jij K−1 θi Jij

(5)

Elasto-Static Model and Accuracy of a Large Deployable CDPR

Jij is the mast Jacobian matrix expressed for each pulley as follows:   i j k #∗ » #∗ » Jij = # ∗ » Aij Oi × j Aij Oi × k Aij Oi × i

385

(6)

with i, j and k being the unit vectors along xb , yb , and zb axes, respectively. The matrix Kθi is the (3 × 3) diagonal joint stiffness matrix of ith mast. wij is the resultant wrench exerted on the mast expressed as:     p f fi = p b ∗z=1 b iz∗ (7) wij = mij z=1 ( aiz − aij ) × fiz with fij = −τij (vij + uij ), b a∗ij is the Cartesian coordinates vector of A∗ij # » expressed in Fb , and vij is the unit vector of A∗ij Oi . Cable tensions are either measured or estimated, for example by using a tension distribution algorithm [12]. Finally, the actual cable exit point coordinates are thus: ∗ A# ij = Aij + δaT ij

(8)

where δaT ij is the translational part or the last three components of δaij . Note that the cable exit point coordinates Arij shown in Table 1 are A# ij for the MP b ◦ ◦ ◦ pose tp =[10.0 m; 8.56 m; 1.0 m; 0 ; 0 ; 0 ]. Mast compliance can be used in the control scheme as shown in Fig. 3. To find the actual MP pose, given the new cable exit point coordinates, one would need to solve the direct geometric problem defined by (1), which is a complex task. Here, the desired MP pose is used as the first guess and the actual MP pose is found via a least squares algorithm. Planar Case. Let us begin with a planar CDPR with two cables and a point # » # » b b mass MP, as shown in Fig. 4a. Equation (1) becomes li = OAi − OPc = b ai −b tp.  Similarly, wg = [0, −mg] and the wrench matrix (3) becomes W = u1 u2 . Here, each mast has one degree of freedom (DoF) about Ei . A rotation about Ei leads to a displacement of cable exit points from Ai to A∗i , as shown in Fig. 4b. If the cable lengths li remain the same, then the MP is no longer at Pc , but instead at P∗ , thus producing a pose error. ∗ For compliant masts, shown in Fig. 4c, cable exit points are at A# i = Ai + δai , compliance computed by (4). Of where δai is a small displacement due to mast   of the course, in this case mast Jacobian is Ji = hi Evi , where hi isthe length  . Similarly, ith mast, vi is a unit vector pointing from Ei to A∗i and E = 01 −1 0 wij is simply wij = fi = −τi (vi + ui ) using the tensions τ obtained from (2). Aij Desired MP pose

aT ij Aij

IGM

lij

Mast Compliance Internal position control

qij

CDPR

Fig. 3. Control scheme taking into account mast compliance

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Fig. 4. Planar CDPR (a) schematic; (b) the desired MP pose does not match with the actual one due to a modeling error; (c) forces used to find further mast displacement due to their compliance; (d) initialization of the CDPR at a given pose in the presence of modeling errors leads to a difference between actual cable lengths li# and the ones computed by the controller li

(a)

(b)

(c)

(d)

Fig. 5. Trajectory execution (a) MP pose computed from cable lengths; (b), (c) and (d) initializing the robot at different poses with modeling errors

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If the actual MP pose P# corresponds to Pc , as shown in Fig. 4d, then the actual cable lengths li# cannot be equal to cable lengths li computed by control. Figure 5 illustrates the behavior of a simulated CDPR with compliant masts that is controlled by a controller using the simple CDPR model. Here, the desired trajectory is shown in red and it is executed clockwise from the initial MP pose b tp = [3.0; 0.6]. Mast height is set to hi = 4 m. Masts are rotated by 1.5◦ , which leads to a distance of approximately 10 cm between Ai and A∗i . Then, to find δai and A# i , we used m = 35 kg and kθi = 710000 [Nm]. First, if the MP pose is computed from cable lengths, the MP is always significantly lower than desired, as shown in Figs. 4b and 5a. Here, the cyan path corresponds to simulated CDPR with cable exit points at A∗i , while the blue path corresponds to a simulated CDPR with cable exit points at A# i. However, usually the initial MP pose is measured by an external system to initialize the controller. Thus the real cable lengths li# are shorter than li , which are computed by the controller at the point of initialization, as shown in Fig. 4d. Now the behavior of the robot depends on where the initialization is done, as can be seen in Figs. 5b to 5d. This is because the error ei between li and li# depends on and is fixed at the initialization pose. Then, at each path point, the cables are always shorter by ei than the length li computed by the control. In Fig. 5b, the initialization is in the center of the WS. Most of the path, the MP is above the desired pose and the closer it is to the masts, the larger the difference. While the two curves get very close between X = 8 m and X = 12 m, none of the path points match. In Fig. 5c the initialization is done on the path at b tp = [13; 1.2] and the rest of the blue path is above the desired one. Finally, in Fig. 5d the initialization is done at b tp = [23; 0.6] , which leads to the largest deviation compared to Figs. 5b and 5c. Thus, it appears that to minimize the modeling errors, the CDPR should be initialized in the center of the WS or in the center of the path, if it does not coincide with the center of the WS.

3 3.1

Accuracy and Repeatability ISO Standard 9283:1998

The ISO 9283:1998 [4] describes a collection of experiments to evaluate robot precision. Pose accuracy AP and repeatability RP , as well as path accuracy AT and repeatability RT were assessed. Note that here the notation from the ISO standard is being used. For this, the test plane shown in gray in Fig. 2 was defined. The poses to visit for the evaluation of AP and RP must be placed at the center and at (0.1 ± 0.02)d from each corner of the test plane, where d is the WS diagonal, d = 26.238 m. This corresponds to the gold spheres in Fig. 2. However, as it is often shown that CDPRs are more precise at the center of the WS, two additional sets of poses were defined and are shown in pink and violet in Fig. 2. Regarding the choice of the path for AT and RT , we were constrained by our case study, where the main goal was to compare a path in the middle of the WS (shown in brown) and close to WS border (shown in cyan). Furthermore, each path was executed with initialization at both the brown and cyan spheres.

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3.2

Pose Accuracy and Repeatability

Each set of five poses was visited 30 times and with several MP velocities defined as a percentage v% of the maximum velocity of 0.5 m/s. Moreover, the violet set was repeated with the remeasured cable exit points Arij , shown in Table 1. However, mast compliance was not taken into account during these tests. Position accuracy and repeatability results can be seen in Fig. 6. First, let us compare the experiments with the nominal CDPR model shown in the first six columns of each bar graph. It can be seen that indeed, the closer the MP to the WS boundaries, the worse the translational accuracy AP , while the components along each global axis do not always follow this trend. On the other hand, the best repeatability is for the largest (gold) test set, while the worst one is for the medium (pink) set. Regarding the rotational accuracy (APR ) and repeatability (RPR ), in general they both become worse with increased distance from the WS center. However, all the values are very close to one another and the APR always remains below 2.5◦ and RPR below 0.6◦ . It appears that higher velocity leads to better results, as described in [1]. Indeed, both AP and RP are better with v% = 100% for the violet and pink test sets compared to v% = 10%. For the large test set only RP is better. Comparing the results obtained with the nominal and the remeasured CDPR model, the latter gives a better result, except for APz with v% = 100%. Furthermore, the results for the remeasured model are almost the same no matter the velocity with only two exceptions - the aforementioned APz with v% = 100%, and RPz with v% = 10%. Surprisingly, the APR and RPR are worse with the new model, even though they are still very good, not surpassing 1.5◦ and 0.55◦ , resp. Finally, the WS covered in the violet test set is slightly larger than the WS of IPAnema [1,5]. With the remeasured model we obtain almost the same accuracy, with the exception of the aforementioned APz with v% = 100%. 3

accuracy, degrees

accuracy, mm

150

100

50

0

2

1

0 VN10 VN100 PN10 PN100 GN10 GN100 VR10 VR40 VR100

VN10 VN100 PN10 PN100 GN10 GN100 VR10 VR40 VR100

(b) repeatability, degrees

repeatability, mm

(a) 20 15 10 5 0 VN10 VN100 PN10 PN100 GN10 GN100 VR10 VR40 VR100

(c)

0.6

0.4

0.2

0 VN10 VN100 PN10 PN100 GN10 GN100 VR10 VR40 VR100

(d)

Fig. 6. Pose accuracy AP and repeatability RP . Notation: first letters V, P, G refer to violet, pink and gold sets of poses shown in Fig. 2, resp.; second letters N and R refer to the use of nominal or remeasured CDPR models shown in Table 1; numbers 10, 40, 100 refer to velocity as percentage of max MP velocity of 0.5 m/s

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3.3

389

Path Accuracy and Repeatability

As explained in Sect. 3.1 and shown in Fig. 2, two paths parallel to the global ZY plane were selected: one in the center and one on the side of the WS. Moreover, two initialization poses were used for each path: in the center and on the side of the WS. The starting position is [10.0; 8.56; 1.2] or [1.0; 8.56; 1.2], corresponding to the brown and cyan spheres shown in Fig. 2, resp. The results are shown in (Table 3) and Fig. 7 on the left, where the desired path is shown in red and the executed one is shown in blue. Furthermore, the evolution of the translational accuracy AT along the path can also be seen in Fig. 7 on the right with the notation nominal. Here, the translational accuracy is the difference between the desired path and the measured one. It is clear that the best behavior is when both the path and the initialization are in the center of the WS, shown in Figs. 7a and 7b. Accuracy AT is 2.5 cm on average, and the worst peak of 4.6 cm is due to ATz . Note that the shape of the executed trajectory is very close to the simulated one in Fig. 5b. Then, if the initialization is done on the side of the WS (Fig. 7c), the average AT is 5.2 cm, while the peak reaches 7.7 cm, thus roughly doubling when compared to Figs. 7a and 7b. The main reason behind this increase is a considerable increase of ATx , that remains at about 4 cm throughout the path. Thus, it is important to initialize the robot on the trajectory to have the best accuracy. Next, in Fig. 7e both the initialization and the path are on the side of the WS. The executed trajectory is even less precise and is very similar to the simulated one in Fig. 5c. As initialization is done on the path, the ATx is small. However, the large deviation along Z drives AT to peak at 12.8 cm. Finally, in Fig. 7g the initialization is now done in the center, while the path is on the side of the WS. This leads to the worst accuracy out of the four scenarios, AT peaking at 20.6 cm due ATz . Thus, in both scenarios shown in Figs. 7c and 7g initialization far from the path leads to worse accuracy. 3.4

Accuracy and Repeatability with the Compliant Mast Model

In this section the model proposed in Sect. 2.2 is verified by estimating the behavior of ROCASPECT given the desired paths shown in red in Fig. 7. It is assumed # ∗ ∗ that the final cable exit points A# ij are computed as Aij = Aij + δaij , where Aij corresponds to cable exit point coordinates after the elongation of supporting ropes by 1.5 cm and δaij is the additional cable exit point displacement due to mast compliance. To compute δaij , the stiffness matrix of each mast is set so that the diagonal components are k11 = 200000 [Nm], k22 = 500000 [Nm], k33 = 300000 [Nm]. These stiffness coefficients were obtained by simulating the robot with the MP poses given in Table 2 and tuning the coefficients to get similar cable exit point coordinates. Cable tensions τ are also needed, however the real cable tensions were not measured. Accordingly, the cable tensions were estimated by using the tension distribution algorithm described in [13,14]. The resulting paths are shown in cyan in Fig. 7 on the left. The translational accuracy AT is plotted in Fig. 7 on the right with the notation Compliant.

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(a)

(b)

(c)

(d)

(e)

(f )

(g)

(h)

Fig. 7. Path and accuracy: (a) and (b) path and initialization in the middle of the WS; (c) and (d) path in the middle of the WS, but initialization on the side; (e) and (f ) path and initialization on the side of the WS; and (g) and (h) trajectory on the side of the WS, but initialization in the middle

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Table 3. MP translational errors with the nominal and the compliant models Path

Initialization N ATmax , mm N ATavg , mm CATmax , mm CATavg , mm

Center Center

43.1

24.8

28.0

13.6

Center Side

77.1

52.0

46.3

24.0

Side

Side

127.6

67.8

69.2

34.2

Side

Center

206.3

154.0

111.4

95.7

Here AT is the difference between the estimated and the measured paths. Right away it can be seen in Fig. 7a that the estimated and measured paths are almost the same. Compliant ATx and ATy almost coincide with the nominal ones in Fig. 7b, while ATz is considerably lower. Indeed, now AT peaks at 2.8 cm due to ATy . Thus, while the compliant model is indeed considerably closer to the real robot, there are some sources of errors, affecting ATy that are not modeled. In Fig. 7c, the difference between the cyan and blue paths is a bit larger, but they are still very similar. As can be seen in Fig. 7d, compliant ATx is almost 0, thus the error of the nominal ATx comes mainly from the difference between the control model Aij and the compliant model A# ij . Compliant ATz is also smaller than the nominal one, consequently compliant AT averages at 2.4 cm and peaks at 4.6 cm, which is almost twice better than the nominal AT . In Figs. 7e and 7f, thanks to the dramatic decrease of compliant ATz to less than 3 cm, the accuracy AT is now twice better than the nominal one. Finally, the difference between the estimated and the measured paths is large in Fig. 7g. Indeed, the compliant ATz remains at about 10 cm, as shown in Fig. 7h, however even that is significantly better than the nominal ATz that averages at 15.4 cm and peaks at 20.6 cm. This is also the only case where a large difference between nominal and compliant ATx can be seen. It seems that initializing the MP in the center and then working on the side of the WS accentuates the modeling issues.

4

Conclusions

In this paper the accuracy and repeatability of a very large deployable CDPR was measured according to the ISO 9283:1988 standard is presented. A simple compliant mast model is then proposed to explain the observed behavior. Overall, the results are comparable to the state of the art, especially with the remeasured model: about 5 cm for accuracy and less than 1 cm for repeatability, which is 0.19% and 0.04% of the WS diagonal, resp. However, to obtain this model, it is important to first have a working-in period for a CDPR to ensure that all cables (supporting and actuated) stretch and tighten in their knots, splices or winding systems. CDPR accuracy depends on the distance of the MP to its WS center, as could be seen in both sets of experiments. Moreover, as shown in the path experiments, the initialization pose matters as well. Indeed, a badly chosen initialization pose can accentuate the differences between the model and the actual robot.

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CDPR accuracy depends also on the robot model used within the controller. Indeed, if the model is not correct or too simple, the behavior of the robot cannot be precise. Using a compliant mast model the obtained behavior is very similar to the measured one given the same control output. Note that our estimation was obtained without the real tension measurements, without a mast stiffness identification and assuming that all masts have the same stiffness coefficients. Thus, the resulting accuracy can be significantly improved by using the proposed model in the controller. As a consequence, future work includes mast stiffness identification and implementation of a control scheme that takes into account cable tensions and mast compliance.

References 1. Pott, A., M¨ utherich, H., Kraus, W., Schmidt, V., Miermeister, P., Verl, A.: IPAnema: a family of cable-driven parallel robots for industrial applications. In: Bruckmann, T., Pott, A. (eds.) Cable-Driven Parallel Robots (CableCon), pp. 119– 134. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-31988-4 8 2. Dallej, T., Gouttefarde, M., Andreff, N., Dahmouche, R., Martinet, P.: Visionbased modeling and control of large-dimension cable-driven parallel robots. In: Intelligent Robots and Systems (IROS), Vilamoura, Algarve, Portugal, pp. 1581– 1586. IEEE (2012) 3. Merlet, J.-P., Papegay, Y., Gasc, A.-V.: The Prince’s tears, a large cable-driven parallel robot for an artistic exhibition. In: International Conference on Robotics and Automation (ICRA), Paris, France, pp. 10378–10383. IEEE (2020) 4. ISO 9283:1998. Manipulating industrial robots - Performance criteria and related test methods 5. Schmidt, V., Pott, A.: Implementing extended kinematics of a cable-driven parallel robot in real-time. In: Bruckmann, T., Pott, A. (eds.) Cable-Driven Parallel Robots (CableCon), pp. 287–298. Springer, Heidelberg (2013). https://doi.org/10.1007/ 978-3-642-31988-4 18 6. Auffray, V.: Vers la manipulation pr´ecise de grandes pi`eces dans de tr`es grands espaces de travail. https://www.cnrs.fr/mi/IMG/pdf/cable tecnalia cnrs.pdf 7. Pott, A.: Cable-Driven Parallel Robots: Theory and Application, vol. 120, pp. 52– 56. Springer, Cham (2018) 8. Pott, A.: Influence of pulley kinematics on cable-driven parallel robots. In: Lenarcic, J., Husty, M. (eds.) Latest Advances in Robot Kinematics, pp. 197–204. Springer, Dordrecht (2012). https://doi.org/10.1007/978-94-007-4620-6 25 9. Picard, E., Caro, S., Claveau, F., Plestan, F.: Pulleys and force sensors influence on payload estimation of cable-driven parallel robots. In: Intelligent Robots and Systems (IROS), Madrid, Spain, pp. 1429–1436. IEEE (2018) 10. Paty, T., Binaud, N., Caro, S., Segonds, S.: Cable-driven parallel robot modelling considering pulley kinematics and cable elasticity. Mech. Mach. Theory 159, Article 104263 (2021) 11. Zake, Z.: Design and stability analysis of visual servoing on cable-driven parallel ´ robots for accuracy improvement. Ph.D. thesis, Ecole Centrale de Nantes (2021) 12. Picard, E., Caro, S., Plestan, F., Claveau, F.: Stiffness oriented tension distribution algorithm for cable-driven parallel robots. In: Lenarˇciˇc, J., Siciliano, B. (eds.) ARK 2020. SPAR, vol. 15, pp. 209–217. Springer, Cham (2021). https://doi.org/10.1007/ 978-3-030-50975-0 26

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13. Mikelsons, L., Bruckmann, T., Hiller, M., Schramm, D.: A real-time capable force calculation algorithm for redundant tendon-based parallel manipulators. In: International Conference on Robotics and Automation (ICRA), Pasadena, CA, USA, pp. 3869–3874. IEEE (2008) 14. Rasheed, T., Long, P., Marquez-Gamez, D., Caro, S.: Tension distribution algorithm for planar mobile cable-driven parallel robots. In: Cable-Driven Parallel Robots (CableCon), Qu´ebec, Canada, pp. 268–279 (2017)

Application

A Warehousing Robot: From Concept to Reality Amir Khajepour1(B) , Sergio Torres Mendez2 , Mitchell Rushton1 , Hamed Jamshidianfar1 , Ronghuai Qi3 , Alireza Pazooki1 , Laaleh Durali1 , and Amir Soltani1 1 University of Waterloo, Waterloo, ON, Canada

[email protected] 2 Instituto Tecnológico de Puebla, Puebla, Mexico 3 Georgia Institute of Technology, Georgia, USA

Abstract. In this paper, we review how Cable Driven Parallel Robots (CDPR) can be used for automated warehousing, specifically for Automated Storage and Retrieval Systems (ASRS), and how a concept design in the past decade has turned into a strong contender to the existing ASRS technologies. The developments in addressing low rigidity, in-plane and out-of-plane vibrations, unwanted degrees of freedom, and workspace are discussed in detail. The prototypes and experimental results are discussed and solutions to address the shortcomings of each prototype are reviewed. The outcomes of the studies have resulted in a new Cable-Driven ASRS (CD-ASRS) that is being built and is expected to be ready for industrial trial soon. Keywords: Cable Driven Parallel Robots (CDPR) · Kinematic Cable Constraints · Automated Warehousing · Automated Storage and Retrieval Systems (ASRS) · Cable Robots Vibration Control · Cable Topology

1 Introduction In recent years, e-commerce explosion has fueled the growth of automated warehousing solutions. The global warehousing and storage market will be over $700 billion in 2023 and it is expected to reach just under a trillion-dollar industry by 2027 [1]. Similarly, the global market for automated material handling equipment was estimated to be $39.3 billion in 2020 and is projected to reach $64 Billion by 2027 [2]. Such growth of automated warehousing applications provides a unique opportunity for different robotic ideas to be developed at industrial scales. The majority of warehouses use racking/shelving systems to store thousands of different items for distribution or fulfillment of e-commerce orders. Automated Storage and Retrieval Systems (ASRS) are used to automatically store or retrieve any box, tote, or pallet in and out of the warehouse racking/shelving system. The global market for ASRS was over $7.2 billion in 2022 and is projected to reach $11.5 billion by 2027, growing at a rate of 8.0% over 2022–2027. Cable driven parallel robots (CDPR) can be a strong competitor in ASRS industry as they possess several unique attributes including minimum moving mass, design independence to racking dimensions, and on-ground actuation systems. These unique features © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Caro et al. (Eds.): CableCon 2023, MMS 132, pp. 397–406, 2023. https://doi.org/10.1007/978-3-031-32322-5_32

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result in higher operational speeds, lower energy consumption, simplicity, reduced capital costs, very large workspaces, and low maintenance. CDPR however, have weaknesses compared to rigid structure ASRS including rigidity (especially in unwanted degrees of freedom), actuator redundancy, cable uni-directional force limitations, and control complexity. In the past 15 years, in addition to some of our works in cable driven warehousing robots [3–21], there have been many other works [22–31] studying the application of CDPRs to automated warehousing. An excellent review of this topic can be found in [31]. Among these activities, few prototypes have been built and studied including the FASTKIT mobile storage robot [23, 30] shown in Fig. 1a, and the rack feeder system based on the Stewart-Gough platform (SGP) [24–26] shown in Fig. 1b. With all the benefits of cable driven warehousing robots, there is still no commercial cable warehousing robot in the market. This can be attributed to the issues and weaknesses associated with such robots mentioned above and the high requirements of ASRS including large workspace, extreme reliability, speed, load capacity, repeatability, and cost. In this paper, we review our solutions to address the issues and weaknesses of cable warehousing robots and discuss the implementation and results of these solutions to a laboratory and full-scale ASRS. We further present how the outcomes of these studies have resulted in a new Cable-Driven ASRS (CD-ASRS) that is being built through ReelIn Robotics Inc., www.reelinrobotics.com, to bring cable-driven warehousing robots to reality.

Fig. 1. a) FASTKIT mobile storage robot [30], b) Rack Feeder System robot [24, 25].

2 CD-ASRS Prototypes In a warehousing ASRS, a rigid platform is connected to on-ground winches via flexible cables that by controlling the length and/or tension of the cables, the platform’s motion is controlled. On the platform, one or more loaders are used to store and retrieve boxes, totes, or pallets. As there are only two degrees of freedom (vertical and horizontal) needed for the platform to reach any location in the racks, the topology of the cables becomes critical in suppressing the extra unwanted four degrees of freedom of the robot platform. The unwanted degrees of freedom are the three rotational and out-of-plane motions of the platform.

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In [3], the concept of kinematic constraints in cable robots was introduced and later in [7, 8, 21] this concept was extended to warehousing robots to maximize the robot’s stiffness in the four unwanted degrees of freedom. Figure 2a, shows the cable topology used in our first 3 m by 1.5 m prototype, shown in Fig. 2b. As seen in Fig. 2a, 12 cables and 4 motors are used to drive the platform: actuators 1 and 4 synchronously drive 4 cables each, actuators 2 and 3, synchronously drive 2 cables each. The topology of the cables adds kinematic cable constraints to suppress the rotational and out-ofplane motions of the platform. For example, the 4 cables on each of the upper actuators form a parallelogram in the plane of motion. The two parallelograms are the kinematic constraints to suppress the rotation of the platform in the plane of motion. Similarly, the other 3 unwanted degrees of freedom are significantly suppressed passively through the arrangement of the cables.

Fig. 2. a) Cable topology, b) Laboratory CD-ASRS prototype.

In a constrained CDPR with n degrees of freedom, at least n + 1 actuators are needed to actively control the robot. In the design shown in Fig. 2 with n = 2 degrees of freedom, there are four actuators. The two redundant actuators are used to increase the robot workspace and its stiffness, especially in the unwanted degrees of freedom. In [7, 8], the workspace analysis of the robot shown in Fig. 2 is discussed and it is shown how the maximum allowed cable tension and cable topology can change the robot workspace. A detailed analysis of real-time stiffness optimization by using different norms, and their computational costs are discussed in detail in [10, 20]. In-plane and out-of-plane vibrations are major concerns for CDPR applications to large warehousing ASRS robots. Many studies have been conducted to suppress the inplane and out-plane vibrations. In [9, 12, 14], in-plane vibration of CD-ASRS has been studied thoroughly and different control systems have been developed successfully. In Fig. 3, the vertical vibration of the platform of the prototype shown in Fig. 2b using different controllers are shown [12]. In the CD-ASRS prototype in Fig. 2b, only two degrees of freedom (vertical and horizontal) are controllable and the amplitude and decay rate of any disturbances in the rotational and out-of-plane motions of the platform will be dependent on the platform stiffness and damping in these four degrees of freedom. In large warehousing ASRS with unbalanced payloads and high accelerations, an active system might be needed to improve the performance and efficiency of CD-ASRS systems. In [13, 16–18], we have developed a Multi-Axis Reaction System (MARS) using two co-axial unbalanced masses (pendulums) controlled by two servo motors to actively control disturbances

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Fig. 3. In-plane vertical vibration control of CD-ASRS shown in Fig. 2b [12]

in the uncontrolled axes. Figure 4a shows the design of MARS and Fig. 4b shows the CD-ASRS platform equipped with MARS.

Fig. 4. a) Multi-Axis Reaction System (MARS) and coordinates, b) MARS implementation.

Figure 5, shows experimental results of using MARS on the CD-ASRS prototype shown in Fig. 2b for the platform out-of-plane vibrations and rotations around x and y axes. The prototype shown in Fig. 2b has been tested extensively with different controllers for ASRS operations and the vibration control modules discussed above [9, 11, 14, 15]. One of the shortcomings we observed in the robot before building a full-scale CD-ASRS was the robot workspace and its nonhomogeneous payload capacity. Figure 6 shows the robot’s concave workspace for a given maximum tension in the cables. The workspace can be extended by increasing the cable tensions however, in practice, this is not possible due to motors torque limitations. In addition, the load capacity of the robot will be spatially dependent causing significant complexities in warehousing management. To address this drawback, the design was revised to have a linearly variable load balancing mechanism to improve the workspace as shown in Fig. 7. The passive cable loop connecting the top of the platform to the counterbalance mass and the high-density cable/chain provides a variable assistive vertical force to the platform. This force in the lower parts of the workspace is low and increases as the platform reaches the higher

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Fig. 5. a) Platform response for a step response in z, ψx, and θy [13].

Fig. 6. General workspace of the CD-ASRS shown in Fig. 2 [19]

sections of the workspace. Figure 8 shows the workspace of a CD-ASRS with a footprint of 53 m × 14 m and the cable topology of the prototype in Fig. 2 that is equipped with the load balancing mechanism shown in Fig. 7. As seen in the figure, there is a significant improvement in the robot workspace. The details of the proposed mechanism can be found in [19]. Using the revised design as explained above, a new full-scale CD-ASRS with a workspace of 25 m × 5 m with the platform width of 1m was developed and built in 2019. The footprint, workspace, and the overall design of the second CD-ASRS prototype are shown in Fig. 9. The load capacity of the robot was 150 kg for automated tote/box warehousing systems. Some photos of the robot are shown in Fig. 10. Since the purpose

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Fig. 7. CD-ASRS with Counterbalance Mechanism

Fig. 8. The workspace of the CD-ASRS with Counterbalance Mechanism

was the evaluation of the robot, we did not install any racks for the storing and retrieving operations.

Fig. 9. Footprint and Workspace of a Large-Scale CD-ASRS, Blue line: Robot Footprint, Red line: Robot Workspace

The experiments showed a maximum speed of 5 m/s and acceleration of 6 m/s2 where both were among the highest in available ASRS in the market. The out-of-plane displacement of the platform without any active system was limited to under 1cm which was very acceptable to such robot operations. Figure 11 shows a bowtie point-to-point robot operation where the platform follows points 1-2-3-4-1 with a 2 s dwell time at each endpoint. The figure on the top shows the desired (dashed line) and actual (solid

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line) paths of the robot. The figures in the bottom show displacement and velocity of the robot in horizontal and vertical axes versus time. As seen in the figures, the robot closely tracks the desired path and velocity with some small errors in the middle of the track.

Fig. 10. The new CD-ASRS prototype

The second prototype showed a huge potential of the robot in ASRS applications; however, before it could turn into a real product, we needed to address some shortcomings observed during installation, operation, and discussions with industry including the larger robot footprint than its workspace as seen in Fig. 9, extra towers, and the counterbalance mechanism that resulted in additional complexities, cost, and maintenance. With a new cable topology and use of the racking columns for running a passive trolly for guiding the platform vertical cables, we managed to address all the issues discussed above in a new CD-ASRS design shown in Fig. 12. The new cable topology provides a uniform load capacity over the whole workspace of the robot eliminating the need for a counterbalance mechanism. The elimination of the towers and uniformity of the load capacity resulted in an almost identical robot footprint and robot workspace. The robot in Fig. 12 is currently installed for testing and evaluation and it is expected to be ready in May 2023. The results and a video of the robot operation will be demonstrated in CableCon 2023.

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Fig. 11. Robot point-to-point operation for a bowtie motion

Fig. 12. New CD-ASRS

3 Conclusion In this paper, we reviewed the possibility and potential of CDPR for automated warehousing. Using kinematic cable constraints through specific cable topology, it was shown that the unwanted degrees of freedom could be suppressed significantly. In addition, the actuator redundancy was used to maximize the robot stiffness in real-time to reduce system disturbances. To extend the workspace, a counterbalance mechanism was designed

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and implemented in the second prototype. The results of the full-scale CD-ASRS clearly showed that CDPR could be a strong competitor to other ASRS technologies. By addressing the shortcomings observed in the second prototype, we believe the new CD-ASRS, currently being installed, will be ready for industrial trials.

References 1. https://www.thebusinessresearchcompany.com/report/warehousing-and-storage-global-mar ket-report. 18 Jan 2022 2. https://www.globenewswire.com/news-release/2022/10/11/2532046/0/en/Global-Automa ted-Storage-and-Retrieval-Systems-ASRS-Market-to-Reach-10-5-Billion-by-2027.html. 18 Jan 2022 3. Khajepour, A., Behzadipour, S., Dekker, R., Chan, E.: Light Weight Parallel Manipulators using Active/Passive Cables. US Patent #s: 7,172,385, 7,367,771, 7,367,772 (2007) 4. Hassan, M., Khajepour, A.: Optimization of actuator forces in cable-based parallel manipulators using convex analysis. IEEE Trans. Rob. 24(3), 736–740 (2008) 5. Hassan, M., Khajepour, A.: Analysis of a Large-Workspace Cable-Actuated Manipulator for Warehousing Applications. In: ASME 2009 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference: Volume 7: 33rd Mechanisms and Robotics Conference, Parts A and B: ASME, pp. 45–53 (2009) 6. Hassan, M., Khajepour, A.: Analysis of bounded cable tensions in cable-actuated parallel manipulators. IEEE Trans. Rob. 27(5), 891–900 (2011) 7. Torres Méndez, S.J., Khajepour, A.: Analysis of a high stiffness warehousing cable-based robot. In: ASME 2014 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference: Volume 5A: 38th Mechanisms and Robotics Conference: ASME, V05AT08A088 (2014) 8. Torres Méndez, S.J., Khajepour, A.: Design optimization of a warehousing cable-based robot. In: ASME 2014 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference: Volume 5A: 38th Mechanisms and Robotics Conference: ASME, V05AT08A091 (2014) 9. Jamshidifar, H., Fidan, B., Gungor, G., Khajepour, A.: Adaptive vibration control of a flexible cable driven parallel robot. IFAC 48(3), 1302–1307 (2015) 10. Jamshidifar, H., Khajepour, A., Fidan, B., Rushton, M.: Kinematically-constrained redundant cable-driven parallel robots: modeling, redundancy analysis, and stiffness optimization. IEEE/ASME Trans. Mechatron. 22(2), 921–930 (2016) 11. Rushton, M., Khajepour, A.: Transverse vibration control in planar cable-driven robotic manipulators. In: Gosselin, C., Cardou, P., Bruckmann, T., Pott, A. (eds.) Cable-driven parallel robots, pp. 243–253. Springer International Publishing, Cham (2018). https://doi.org/10. 1007/978-3-319-61431-1_21 12. Jamshidifar, H., Khosravani, S., Fidan, B., Khajepour, A.: Vibration decoupled modeling and robust control of redundant cable-driven parallel robots. IEEE/ASME Trans. Mechatron. 23(2), 690–701 (2018) 13. De Rijk, R., Rushton, M., Khajepour, A.: Out-of-plane vibration control of a planar cabledriven parallel robot. IEEE/ASME Trans. Mechatron. 23(4), 1684–1692 (2018) 14. Jamshidifar, H., Khajepour, A., Fidan, B., Rushton, M.: Vibration regulation of kinematically constrained cable-driven parallel robots with minimum number of actuators. IEEE/ASME Trans. Mechatron. 25(1), 21–31 (2019) 15. Qi, R., Rushton, M., Khajepour, A., Melek, W.W.: Decoupled modeling and model predictive control of a hybrid cable-driven robot (HCDR). Robot. Auton. Syst. 118, 1–12 (2019)

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16. Rushton, M., Jamshidifar, H., Khajepour, A.: Multiaxis reaction system (MARS) for vibration control of planar cable-driven parallel robots. IEEE Trans. Rob. 35(4), 1039–1046 (2019) 17. Khajepour, A., Rushton, M.M., Jamshidifar, H.: Multi-Axis Reaction System and Method for Vibration Control of Mechanical Systems. US Patent #: US 15/433,325 (2020) 18. Jamshidifar, H., Rushton, M., Khajepour, A.: A reaction-based stabilizer for nonmodel-based vibration control of cable-driven parallel robots. IEEE Trans. Rob. 37(2), 667–674 (2020) 19. Jamshidifar, H., Khajepour, A., Korayem, A.H.: Wrench feasibility and workspace expansion of planar cable-driven parallel robots by a novel passive counterbalancing mechanism. IEEE Trans. Rob. 37(3), 935–947 (2021). https://doi.org/10.1109/TRO.2020.3038697 20. Qi, R., Khajepour, A., Melek, W.W.: Redundancy resolution and disturbance rejection via torque optimization in hybrid cable-driven robots. IEEE Trans. Syst., Man Cybern. Syst. 52(7), 4069–4079 (2022). https://doi.org/10.1109/TSMC.2021.3091653 21. Khajepour, A., Mendez, S.T.: A Light-Weight Robot and Method for Material Handling/Warehousing and Operating the Same Using Active/Passive Cables. US Patent #: 14/613,450 (2021) 22. Rasheed, T., Long, P., Marquez-Gamez, D., Caro, S.: Tension distribution algorithm for planar mobile cable-driven parallel robots. In: Gosselin, C., Cardou, P., Bruckmann, T., Pott, A. (eds.) Cable-Driven Parallel Robots. MMS, vol. 53, pp. 268–279. Springer, Cham (2018). https:// doi.org/10.1007/978-3-319-61431-1_23 23. IRT Jules Verne, FASTKIT project sheet: Low cost and versatile cable- driven parallel robot solution for logistics. https://docs.wixstatic.com/ugd/2a96d6_549beb788698431aba7f1f6d 3a839753.pdf. Accessed 07 Mar 2018 24. Bruckmann, T., Lalo, W., Nguyen, K., Salah, B.: Development of a storage retrieval machine for high racks using a wire robot. In: ASME 2012 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference: Volume 4: 36th Mechanisms and Robotics Conference, Parts A and B, New York City (NY), p. 771. ASME, USA (2012) 25. Reichert, C., Bruckmann, T.: Optimization of the geometry of a cable-driven storage and retrieval system. In: (Chunhui) Yang, R., Takeda, Y., Zhang, C., Fang, G. (eds.) ISRM 2017. MMS, vol. 72, pp. 225–237. Springer, Cham (2019). https://doi.org/10.1007/978-3030-17677-8_18 26. Bruckmann, T., Lalo, W., Sturm, C.: Application examples of wire robots. In: Gattringer, H., Gerstmayr, J. (eds.) Multibody System Dynamics, Robotics and Control, pp. 291–310. Springer Vienna, Vienna (2013) 27. Alias, C., et al.: Adapting Warehouse Management Systems to the Requirements of the Evolving Era of Industry 4.0. In: ASME 2017 12th International Manufacturing Science and Engineering Conference collocated with the JSME/ASME 2017 6th International Conference on Materials and Processing: Volume 3: Manufacturing Equipment and Systems: ASME, V003T04A051 (2017) 28. Salah, B., Janeh, O., Bruckmann, T., Noche, B.: Improving the performance of a new storage and retrieval machine based on a parallel manipulator using FMEA analysis. IFAC-PapersOnLine 48(3), 1658–1663 (2015) 29. Salah, B., Janeh, O., Noche, B., Bruckmann, T., Darmoul, S.: Design and simulation based validation of the control architecture of a stacker crane based on an innovative wire-driven robot. Robot. Comput.-Integrated Manuf. 44, 117–128 (2017) 30. Girin, A., Dayez-Burgeon, P.: FASTKIT – Collaborative and mobile cable driven parallel robot for logistics: Echord++. http://echord.eu/fastkit/. 11 Apr 2018 31. Alias, C., Nikolaev, I., Correa Magallanes, E.G., Noche, B.: An overview of warehousing applications based on cable robot technology in logistics. In: 2018 IEEE International Conference on Service Operations and Logistics, and Informatics (SOLI), Singapore, pp. 232–239 (2018). https://doi.org/10.1109/SOLI.2018.8476760

IPAnema Silent: A CDPR for Spatial Hearing Experiments Christoph Martin1(B) , Marc Fabritius1 , Christian Lehnertz1 , Philipp Juraˇsi´c2 , Johannes T. Stoll1 , Marc O. Ernst3 , Werner Kraus1 , and Andreas Pott4 1

Robot and Assistive Systems Department, Fraunhofer Institute for Manufacturing Engineering and Automation IPA, Stuttgart, Germany [email protected] 2 Department of Physics, Technical University of Munich, Munich, Germany 3 Department of Applied Cognitive Psychology, University of Ulm, Ulm, Germany 4 Institute for Control Engineering of Machine Tools and Manufacturing Units (ISW), University of Stuttgart, Stuttgart, Germany

Abstract. Experiments within the field of applied cognitive psychology examine how humans perceive spatial audio signals emitted by dynamically moving sources. These experiments require a speaker moving silently in front of a test person in all six dimensions. Common manipulators such as serial robots or gantry systems cannot be used in this application as their drive trains (e.g. motors, gearboxes, movement mechanics) are too noisy. Cable-driven parallel robots (CDPRs) have the unique benefit that their motors can be placed far away from their end-effector, which can therefore be moved silently. This work presents the design and implementation of the IPAnema Silent, a silent CDPR with high dynamic capabilities and an optimized workspace for studying the spatial hearing of humans. Based on the requirements of cognitive psychology experiments, the design process for the IPAnema Silent’s layout and mechanical components is presented. Experimental evaluations of the IPAnema Silent’s noise, workspace, dynamic performance, and accuracy show that it satisfies its objectives and is suitable for the application of spatial hearing experiments. Keywords: cable-driven parallel robots · robot design optimization · spatial hearing experiments

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Introduction

Cable-driven parallel robots (CDPRs) consist of multiple cables which connect a moving platform to fixed winches. The winches change the cable lengths to perform coordinated movements of the platform. The main components of CDPRs (frame, platform, cables, and winches) can be designed in a modular fashion c The Author(s), under exclusive license to Springer Nature Switzerland AG 2023  S. Caro et al. (Eds.): CableCon 2023, MMS 132, pp. 407–418, 2023. https://doi.org/10.1007/978-3-031-32322-5_33

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Pulleys

Cables

Platform with speaker Frame

Location of the test person

Fig. 1. Picture of the IPAnema Silent

and hence easily adapted for each application. The use of lightweight cables can enable large workspaces and highly dynamic movements of the platform. An exemplary application with a large workspace and high dynamics is the spidercam, i.e. a stadium camera driven by a CDPR with four cables [11]. A CDPR for motion experiments with test persons is developed in [6]. Merlet [5] presents the CDPR family MARIONET for applications in rehabilitation, motion training, as haptic devices for virtual reality, or rescue operations for humans. Another unique benefit of CDPRs is that their motors can be positioned far away from their platform. This is beneficial in applications where the platform should operate in challenging environments, or where the motors would interfere with the application. In [1], a CDPR-based actuation of an underwater vehiclemanipulator system (UVMS) is proposed. While the UVMS (in this application the CDPR’s platform) can move underwater, its winches are positioned above the water’s surface. The department of applied cognitive psychology at the University of Ulm requires a robot for a similarly demanding application. To study the spatial hearing capabilities of humans, a robot has to move a speaker silently in front of a test person with various velocities and orientations. CDPRs offer a unique combination of properties, making them more suitable for this application compared to other types of parallel or serial robots. The CDPR designed and built in this work, named IPAnema Silent1 , can move its platform silently by placing its winches in an adjacent room, where their noise can be isolated. The design of the platform is optimized to maximize the size of the spherical-orientation

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https://s.fhg.de/cable-robot.

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workspace, which captures the unique requirements of spatial hearing experiments. A picture of the IPAnema Silent is shown in Fig. 1. The paper is structured as follows: In Sect. 2, the requirements and design of the IPAnema Silent are described. In Sect. 3, the experimental evaluation of the IPAnema Silent is presented. The paper closes with the conclusion in Sect. 4.

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Design

The geometry of the IPAnema Silent is described according to [9] using two coordinate systems: a fixed one K0 and a moving one KP , which is attached to the platform. The IPAnema Silent is redundantly-constrained, having a platform with six degrees-of-freedom and eight cables. For each cable i, the vector ai describes the position of its fixed proximal anchor point on the IPAnema Silent’s frame in K0 . The vector bi denotes the distal anchor point on the platform in KP . The platform’s pose (r, R) consists of its position r ∈ R3 and orientation R ∈ SO3 and determines the relationship between the coordinate systems K0 and KP . The controller of the IPAnema Silent is based on the TwinCAT 3 software from Beckhoff. 2.1

Requirements

The requirements of the spatial hearing experiments at the University of Ulm can be summarized as follows: – Silent movement of the platform with a wireless speaker. – The maximal installation space dimensions are: 3.3 m x 3.1 m x 3.15 m. – The platform with the speaker has to perform movements in the sphericalorientation workspace, where it is always oriented towards the head of a test person. This workspace is defined in Sect. 2.4. An exemplary trajectory within this workspace is visualized in Fig. 4. – The platform has to achieve a maximum velocity of 1.4 m/s within an acceleration distance of 0.5 m. For constant acceleration this corresponds to 1.96 m/s2 . – The workspace of the IPAnema Silent should be as large as possible. – Simple mechanical design. 2.2

Mechanical Design

The first requirement is met by placing the noisy drive trains (motors, winches) in an adjacent room (Winch-room). The CDPR’s cables are guided through small holes in the wall into the CDPR-room. As it is not possible to mount the CDPR’s pulleys directly to the walls, floor, or ceiling of the rooms, a frame made of aluminum profiles is constructed. All forces resulting from the cables are absorbed by the frame structure. To make full use of the installation space, the proximal anchor points ai are positioned in the corners of the frame. The distal anchor points bi are optimized to maximize the CDPR’s workspace as explained in the

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following subsections. The IPAnema Silent uses Dyneema D-Pro cables2 with a diameter of 4 mm and a breaking load of 13 kN. The winches are designed with a focus on simplicity, without cable-guiding spools or force measurement sensors. The cable forces are calculated via the motors’ current. To minimize inaccuracies in the calculation, the motors drive the drums directly without gearboxes. Based on the recommendations in [12], the effective drum radius of the winches is set to 29.4 mm, which results in a usable cable stroke of around 6.8 m for each winch. In the absence of cable-guiding spools, the cables are solely guided by the grooves in the drums. As previous experiments at Fraunhofer IPA have shown, this only works if the angle δ (see Fig. 2) between the cable in its maximum or minimum position on the drum is smaller than 6◦ . For the IPAnema Silent, this translates into a minimum distance of around 1.9 m between the drum and the first pulley. Figure 2 shows how the eight winches (green) are arranged in two vertical layers in the Winch-room.

Fig. 2. Arrangement of the winches

2.3

Platform Model

The platform is designed as a hollow cylinder with a radius of rp = 0.07 m and a length of lp = 0.1 m. This shape is selected to house the likewise cylindrical speaker and to minimize collisions between the cables and the platform in the spherical-orientation workspace, which is presented in the next section. Based on its CAD design, the nominal weight of the platform, including the speaker, is calculated as 1.52 kg. The platform is centered at the origin of KP , and its flat sides are orthogonal to the y-axis of KP . The distal anchor points bi are located on 2

https://www.liros.com/catalog/en/d-pro-4mm-p1854/, [Accessed: 29-Nov-2022].

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its surface. Their locations can be conveniently described using cylinder surface coordinates, consisting of two angles ϕ, θ. They are similar to spherical coordinates, but the piecewise-linear function boxsin (θ) replaces its trigonometric counterpart within the rectangular cross-section of the cylinder. Figure 3 illustrates how bi is expressed in cylinder surface coordinates using the boxsin (θ) function.

Fig. 3. Cylinder surface coordinates using the boxsin (θ) function

2.4

Spherical-Orientation Workspace

In spatial hearing experiments, the speaker should always be oriented towards the test person’s head in front of the robot, whose fixed location is denoted as T the point c = [0 m, −2 m, 0 m] in K0 (see Fig. 4). This defines the sphericalorientation workspace, where for each platform position r, its orientation R is chosen such that the speaker points towards c. The orientation of the platform can be calculated using the axis-angle notation R = R (e, η) with   r − c, ey  ey × (r − c) −1 and angle η = cos , (1) axis e = ey × (r − c) r − c where ey is the basis vector of K0 pointing along the y-axis. If r − c is parallel to ey , then R = I. To determine and optimize the volume of this workspace, three workspace criteria for CDPRs are combined: Wrench-Feasibility. The platform pose has to belong to the wrench-feasible workspace of the CDPR, which is defined in [4]. This criterion is evaluated using the force distribution algorithm from [8]. The cable force limits are set to fmin = 35 N and fmax = 500 N. The lower limit is defined to ensure that the cables are always kept under tension to avoid winding errors on the drums. The upper limit is chosen because from 500 N upwards there is only a small improvement in the workspace size. The platform’s wrench is determined by its weight, which is stated above. Cable-Platform-Collisions. There must not be collisions between the platform and the cables. For each platform pose, this criterion is evaluated using the method from [2] and the CAD design of the platform which is shown in Fig. 3.

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Cable-Cable-Collisions. There must not be collisions between cables of the CDPR. Unlike the previous criteria, it does not suffice to directly check for such collisions at individual positions in the spherical-orientation workspace. Instead, any continuous movement between two positions within the workspace has to be free of collisions. This is implied if there are no collisions at c for any relevant orientation and on the straight lines between c and all platform positions r in the workspace. On these line segments (r − c), the platform has a constant orientation R (e, η), as calculated in Eq. (1), and the methodology for detecting collisions between two cables from [7] can be applied. Let the differences between the anchor points of the cables i and j be denoted as aij = aj − ai , bij = bj − bi . There is a collision between the cables i and j at platform positions between c and r if and only if one of the following conditions is satisfied: (aij , bij ) = π, c + α (r − c) = aj − Rbi + aij β + Rbij γ

with α ∈ [0, 1] , β, γ > 0,

(2) (3)

c + α (r − c) = ai − Rbj − aij β − Rbij γ

with α ∈ [0, 1] , β, γ > 0.

(4)

These conditions can be translated into linear programming problems, which can be efficiently solved for all cable pairs (i, j) with i = j. 2.5

Optimization of the Distal Anchor Points

To design the CDPR according to the requirements from Sect. 2.1, the locations of the distal anchor points are optimized using cylinder surface coordinates to maximize the volume of the spherical-orientation workspace. The optimization considers an evenly spaced grid of points throughout the frame of the CDPR. The objective function is the number of grid points that are inside the spherical-orientation workspace. The Differential Evolution3 algorithm from the SciPy Python library is used with default settings to perform a global search for the optimal configuration of anchor points on the platform. To reduce the search space and speed up the optimization, it is assumed that the upper and lower anchor points are reflections of one another with respect to the horizontal xy-plane in KP . This symmetry reduces the number of variables for describing the distal anchor points to eight. The optimal anchor points determined by this optimization are listed in Table 1 in cartesian coordinates. The resulting workspace is shown in red in Fig. 4. It has a volume of 9.75 m3 , which corresponds to 42 % of the frame’s volume of 22.9 m3 . 2.6

Dynamic Performance

To meet the requirement of highly dynamic platform movements, appropriate motors have to be chosen. Therefore, a representative trajectory, displayed in Fig. 4, is evaluated regarding the maximal platform velocity, required motor 3

https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize. differential evolution.html.

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Table 1. Geometrical parameters of the IPAnema Silent ai [m]

Cable i 1 2 3 4 5 6 7 8

bi [m] T

[ 1.432, 1.441, 1.414] [ 1.453, −1.387, 1.413]T [−1.483, −1.362, 1.422]T [−1.505, 1.358, 1.430]T [ 1.427, 1.454, −1.395]T [ 1.453, −1.378, −1.398]T [−1.483, −1.354, −1.407]T [−1.510, 1.373, −1.400]T

[ 0.016, −0.013, 0.068]T [ 0.041, 0.020, 0.056]T [−0.040, 0.013, 0.057]T [−0.024, −0.009, 0.065]T [ 0.016, −0.013, −0.068]T [ 0.041, 0.020, −0.056]T [−0.040, 0.013, −0.057]T [−0.024, −0.009, −0.065]T

z c

K0

y

y

K0 c

x

x

Dynamic trajectory

Fig. 4. Optimized workspace of the IPAnema Silent

rotation speeds, and torques (including the inertia of the platform, drums, and motors). The trajectory is simulated using the open-source software WireX4 , developed at Fraunhofer IPA. The translational platform velocities (vx , vy , vz ), plotted in Fig. 5, show that the required maximal velocity of 1.4 m/s is reached. The motors’ torques over their rotation speeds are shown in Fig. 6. The maximum rotation speed of n = 400 rpm and the maximum torque of T = 14.64 Nm are used to choose the motors. As a result, the servomotor AM80625 from Beckhoff is chosen. For a voltage of 400 V, it has a nominal torque of 18.5 Nm and a rotation speed range of 0 − 1500 rpm. This higher torque serves as a safety margin to account for uncertainties and to enable future modifications of the IPAnema Silent.

4 5

https://gitlab.cc-asp.fraunhofer.de/wek/wirex.git. https://www.beckhoff.com/de-de/produkte/motion/rotatorische-servomotoren/ am8000-servomotoren/am8062.html, [Accessed: 05-Dec-2022].

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T[Nm]

v[m/s]

0.7 vx vy vz

−0.7 −1.4 0

2

4 t[s]

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Fig. 5. Simulated velocity along the spherical trajectory

3

10 5 0 −400 −200

0 n[rpm]

200

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Fig. 6. Simulated torque over rotation speed for all eight motors

Experimental Evaluation of the IPAnema Silent

The IPAnema Silent is built according to the design described in Sect. 2. Its noise, workspace, dynamic capabilities, and accuracy are evaluated in the following. 3.1

Noise Measurement

The noise of the IPAnema Silent is measured while executing the dynamic trajectory (Fig. 4) at various velocities between 0 m/s − 1.4 m/s. Therefore, a Sound Level Meter PCE-322A6 from PCE Instruments is used. In the CDPR-room it is positioned close to the location of the head of the test person c and in the Winch-room it is positioned in the center in front of the frame at a height of 1.05 m and with a distance of 0.55 m to the frame. The results of the mean sound pressure level (SPL) for both rooms are shown in Fig. 7. For the Winch-room the color blue is used and for the CDPR-room green. Circles show the measured values and dashed lines indicate the background noise level. Green triangles show the calculated extraction of the noise caused solely by the CDPR in the CDPR-room. Up to a velocity of 0.3 m/s, only the background noise of 33 dBA is audible. For higher velocities, the noise level rises up to 42 dBA, which is comparable with the humming of a fridge. The background noise level of 50 dBA in the Winch-room is largely due to the noise of the components in the control cabinet. The noise rises up to 56.7 dBA for 1.4 m/s. Due to the separate Winch-/CDPR-room design of the robot, the noise level of the CDPR-room is significantly reduced compared to a CDPR without such separation.

6

https://www.pce-instruments.com/english/measuring-instruments/test-meters/ sound-level-meter-noise-level-meter-pce-instruments-data-logging-sound-levelmeter-pce-322a-det 60903.htm, [Accessed: 19-Jan-2023].

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The accessible constant-orientation wrench-feasible workspace of the IPAnema Silent is evaluated in practice in the three main planes of K0 . For this evaluation, the IPAnema Silent is controlled by the force control method presented in [3]. The IPAnema Silent moves slowly with constant velocity (0.08¯ 3 m/s) and orientation R = I along 16 evenly distributed rays in each plane (visualized in Fig. 8 as dashed lines) until one of the cable forces exceeds the limits specified in Sect. 2.4. The results are displayed in Fig. 8. The workspace cross-sections accessible in practice are displayed in orange and the ones computed in simulation (using the advanced closed-form [8] method) are shown in blue. It can be seen that the shape is similar, but the workspace accessible in practice is slightly smaller. Two possible reasons are deviations between the control model and the physical robot as well as inaccuracies in the cable forces, which are measured via the motor currents and are used as inputs for the force controller. 3.3

Dynamic Performance

To evaluate the dynamic performance of the IPAnema Silent, the trajectory from Fig. 4 is executed in practice. The same plots used for the design in Sect. 2.6 with the measured data from the experimental evaluation are shown in Fig. 9 and Fig. 10. Figure 9 shows that the required velocity 1.4 m/s is reached. Figure 10 shows that the maximum motor torque is T = 11.06 Nm. This is 3.58 Nm less than in the simulation during the design process. Possible reasons are differences in the trajectory interpolators used in the simulation (WireX) and the robot controller (Beckhoff CNC). Furthermore, the simulated cable force distribution at each pose, calculated with the advanced closed-form algorithm, is one out of many possible solutions and might not match the cable force distributions measured in practice. Lastly, the CDPR’s torques are calculated based on motor currents, which can introduce inaccuracies.

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The absolute accuracy of the IPAnema Silent is evaluated using an OptiTrack system with six Flex137 cameras. The setup can achieve a position accuracy of < 0.2 mm and a rotational accuracy of < 0.057◦ . The controller of the IPAnema Silent uses the inverse pulley kinematic model [10] with the force control method from [3] to position the platform. The accuracy is measured at a grid of 64 positions that are equally distributed within a cube of 0.9 m side length, centered at the origin of K0 . The position error is calculated as the Euclidean norm of the difference between the measured position and the commanded position. The orientation error is the magnitude of the rotational deviation between the orientation in the home pose and the measured orientation. The measured absolute position errors have a maximum of 2.71 mm, their mean value is 1.29 mm and the standard deviation is 0.55 mm. For the orientation, the maximum absolute error is 3.42◦ , the mean error is 1.21◦ and the standard deviation is 0.66◦ . In Fig. 11, arrows show the direction and magnitude of the errors. The orientation errors are visualized as rotation vectors.

Fig. 11. Measured translational and rotational accuracy

4

Conclusion

This paper presents the design process and realization of the IPAnema Silent, a CDPR for spatial hearing experiments in the field of applied cognitive psychology. Based on the unique requirements of this application, the robot is designed using the software WireX. Evaluations show that the robot meets the defined requirements and is suitable to be used for spatial hearing experiments. First experimental studies with test persons have been conducted. 7

https://optitrack.com/cameras/flex-13/, [Accessed: 08-Dec-2022].

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References 1. El-Ghazaly, G., Gouttefarde, M., Creuze, V.: Hybrid cablethruster actuated underwater vehicle-manipulator systems: a study on force capabilities. In: 2015 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pp. 1672–1678. IEEE (2015) 2. Fabritius, M., Martin, C., Pott, A.: Calculation of the cable-platform collision-free total orientation workspace of cable-driven parallel robots. In: CableCon 2019. MMS, vol. 74, pp. 137–148. Springer, Cham (2019). https://doi.org/10.1007/9783-030-20751-9 12 3. Fabritius, M., et al.: A nullspace-based force correction method to improve the dynamic performance of cable-driven parallel robots. Mech. Mach. Theory 181, 105177 (2023) 4. Gouttefarde, M., Merlet, J.-P., Daney, D.: Wrench-feasible workspace of parallel cable-driven mechanisms. In: IEEE International Conference on Robotics and Automation, Roma, Italy, pp. 1492–1497 (2007) 5. Merlet, J.-P.: MARIONET, a family of modular wire-driven parallel robots. In: Lenarcic, J., Stanisic, M. (eds.) Advances in Robot Kinematics (ARK), pp. 53–61. Springer, Dordrecht (2010). https://doi.org/10.1007/978-90-481-9262-5 6 6. Miermeister, P., et al.: The CableRobot simulator large scale motion platform based on cable robot technology. In: 2016 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pp. 3024–3029. IEEE (2016) 7. Perreault, S., et al.: Geometric determination of the interference-free constantorientation workspace of parallel cable-driven mechanisms. J. Mech. Robot. 2(3) (2010) 8. Pott, A.: An improved force distribution algorithm for over-constrained cabledriven parallel robots. In: Thomas, F., P´erez Gracia, A. (eds.) Computational Kinematics. MMS, vol. 15, pp. 139–146. Springer, Dordrecht (2014). https://doi. org/10.1007/978-94-007-7214-4 16 9. Pott, A.: Cable-Driven Parallel Robots: Theory and Application. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-76138-1 10. Pott, A.: Influence of pulley kinematics on cable-driven parallel robots. In: Lenarcic, J., Husty, M. (eds.) Latest Advances in Robot Kinematics, pp. 197–204. Springer, Dordrecht (2012). https://doi.org/10.1007/978-94-007-4620-6 25 11. spidercam. spidercam. https://www.spidercam.tv/. Accessed 28 Nov 2022 12. Wehr, M.: Beitrag zur Untersuchung von hochfesten synthetischen Faserseilen unter hochdynamischer Beanspruchung. Ph.D. thesis. Stuttgart: Universit¨ at Stuttgart, Germany (2017). http://dx.doi.org/10.18419/opus-9190

Experimental Study on Thrustered Cable-Suspended Parallel Robot for Collaborative Task Kazuki Hayashi, Yusuke Sugahara(B) , and Yukio Takeda Tokyo Institute of Technology, I6-15, 2-12-1, Ookayama, Meguro, Tokyo 152-8552, Japan [email protected]

Abstract. This study presents a concept of a collaborative robot consisting of a cable-suspended parallel mechanism and a vertical downward thruster equipped on its end-effector to achieve low risk in the event of collision, and good positioning performance at high speed operation. A prototype of a 4-wires 2 degrees-of-freedom vertical planar translational cable-suspended parallel robot was developed, and the experiments to examine its effectiveness and issues were conducted. The findings include the effectiveness of thruster to reduce wire slack for stiffness improvement, negative effect of the reaction torque of the single-rotor thruster which could be solved by introducing the contra-rotating propellers, and several additional issues such as the non-negligible stiffness of the thruster’s power and signal cables and vibration caused by the uneven output of the thruster. Keywords: Cable-Driven Parallel Robot · Thruster · Cable-Suspended Parallel Robot · Collaborative Robot

1

Introduction

Recently, with the enactment of ISO 10218-1 and ISO 10218-2 [1,2], industrial robots that meet safety requirements based on risk assessment are permitted to be used without safety fences. These industrial robots are called collaborative robots and are being introduced in various processes of work because they can work in close proximity to humans and collaborate with them. However, since the operating speeds of collaborative robots are determined based on risk assessment, a major challenge is that they can only operate at low speeds. It is thought that high-speed operation would be possible if their safety can be guaranteed in the event of collision. To improve the safety of robots in the event of a collision, it is effective to reduce the weight of moving parts. If the moving parts are lightweight, the This work was partially supported by JKA and its promotional funds from KEIRIN RACE. c The Author(s), under exclusive license to Springer Nature Switzerland AG 2023  S. Caro et al. (Eds.): CableCon 2023, MMS 132, pp. 419–429, 2023. https://doi.org/10.1007/978-3-031-32322-5_34

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impact of contact with a human is small, and the risk of a serious accident in the event of a collision is low. Therefore in this study, we focus on cabledriven parallel robots (CDPR) [3,4]. This mechanism is capable of high-speed movement due to its extremely lightweight moving parts, and there are many research examples that take advantage of this characteristic. However, the usual CDPR requires many wires around the end-effector, which easily interfere with the operator, making it difficult to apply to collaborative robots, and there are few examples on production lines. On the other hand, the cable-suspended parallel robots (CSPR) [5,6] do not require wires to be placed under the end effector. However, since these use the gravity applied to the end-effector, if theend effector is lightweight, the stiffness is insufficient and the wires become slack and vibration occurs, so they are not good at high-speed positioning. As examples to improve stiffness by adding elements that generate internal force to CSPRs, there are studies of placing cylinders or springs between the frame and the end-effector [7–9]. However, since these methods increase the mass of the moving parts, high safety cannot be expected. In this study, it is considered to mount a thruster using propellers on the end effector and obtain a vertically downward force. If a lightweight thruster can be used, the end-effector will be lighter than as if the downwards force was solely caused by gravity. So that the resulting design of the robot has a low collision risk and good positioning accuracy during high-speed operation. Another advantage is that a larger workspace can be obtained compared to the use of cylinders or springs. There are several research examples on the use of thrusters in CDPR [10–16]. Among them, Lau et al. [16] formulated a method to increase wrench-feasible workspace by combining thruster and wire tension for general CDPR. Aiming at application as collaborative robots, on the CSPR that utilizes the vertical downward force from the thruster mounted on the end effector, this paper describes the conceptual design, the development of a vertical planar prototype, and the experimental examination of the effects and issues of thruster on the position and orientation of the end-effector.

2

Conceptual Design

In this study, the application is envisioned as collaborative robots whose primary task is pick-and-place of lightweight objects. By using a cable-supended parallel robot, the mass of the moving part, which is the sum of the mass of the endeffector and the handled object, is reduced, and the risk in the event of collision with the operator is reduced. As mentioned above, in the CSPR, when the end-effector is lightweight, the wire tension becomes small, and large-amplitude vibration generated by even a small external force and the wire slack reduce the positioning accuracy. Problems such as wires falling out of the pulleys also occur. In this concept, a thruster mounted on the end-effector generates a vertical downward force to increase the wire tension, to improve accuracy during positioning and stabilize operation at high speed.

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Z

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Fig. 1. The configuration example of the vertical planar thrustered cable-suspened parallel robot.

Figure 1 shows the configuration example of the vertical planar thrustered cable-suspened parallel robot. As a first step in this study, the authors propose the configuration of a planar prototype with a simple structure here, while the authors also plan to develop a spatial prototype as a next step. A thruster is attached to the end-effector so that a vertical downward thrust fT is applied on it. Gravity acting on the end-effector always acts vertically downward regardless of the end-effector’s orientation, whereas the thruster is fixed to the end-effector and the direction of its thrust depends on the end-effector’s orientation. Therefore, it is necessary to keep the orientation of the end-effector constant in order to make the thrust force work in the same way as gravity. Therefore, referring to the structure of the DELTA robot, the authors have decided to drive parallel and equal length wires with one winch. This mechanism is a planar two degreesof-freedom (DOF) mechanism in which four wires are driven by two winches. By driving the two winches, the end-effector translates in the Y -Z plane.

3 3.1

Prototype Hardware Design

Figure 2 shows the configuration and dimensions of the prototype developed in this study. The two winches and pulleys are attached on the frame. Four wires reeled out from two winches are connected to the end effector via pulleys. The power and signal cables of the thruster are connected from the top of the end

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600 mm

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Fig. 2. The prototype of the vertical planar thrustered cable-suspened parallel robot TTCS82.

effector to the frame. A 2 mm diameter Dyneema rope is used as the driving wire. Figure 3 shows the photograph of the end-effector equipped with the thruster. In this study, two end-effectors were built: one with a single propeller and the other with a contra-rotating propeller (CRP). The figure (a) shows the end-effector with a single propeller. A motor with a propeller attached as a thruster is fixed at the bottom, and an electronic speed controller (ESC) that controls the motor is fixed below it. The mass of the endeffector is 186 g, including the wiring connectors. Figure (b) shows an end-effector equipped with a CRP. The same motor/propeller pair used in the single-rotor type is arranged vertically opposite each other, and a propeller guard is provided around the propeller. The wire attachment positions are the same as those used in the single-rotor type. The mass of this end-effector is 293 g. The specifications of the thruster used here are shown in Table 1. A brushless motor and propeller for a multicopter are used and controlled by the ESC for brushless motors. As a reference value, the maximum thrust is 21 N according to the manufacturer’s test data. 3.2

Controller

The servo motors used here are the B3M-SC-1170-A, hobby-use products manufactured by Kondo Kagaku Co., Ltd. Since this motor operates in speed control

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T-motor F80 PRO KV1900

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Gemfan Hulkie 5055S Durable 3 Blade

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ZTW Beatles Series 60A

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6S Lipo Battery

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mode, a control system that combines angular feedback and angular velocity feed-forward was adopted. The commanded angular velocity θc is computed by the following equation: (1) θ˙c = Kp (θd − θ) + θ˙d where θd is the desired angle, θ is the current angle, θ˙d is the desired angular velocity, Kp is the proportional gain. The desired angle θd is computed from given desired trajectory by inverse kinematics.

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Experiments were conducted to move the end-effector while operating the thruster and to measure the position and orientation of the end-effector using the motion capture system VENUS3D (Nobby Tech. Ltd.). 4.1

Experiments Using Single-Rotor Thruster

First, using a single-rotor type end-effector, to investigate how the thruster affects the movement of the end-effector, the position and orientation were compared by varying the thrust. Stationary Experiment. The end-effector was stopped at the center of the workspace, the thruster was started at time t = 0 s, and the throttle was kept constant at 25% and 50% for 30 s. The measured position and orientation of the end-effector are shown in Fig. 4. Figure 4 (a) shows that the end-effector is displaced significantly in the negative direction of the X-axis. This displacement is larger when the throttle is larger. Vibrations also occur around a specific position on the negative side of the X-axis, and the amplitude of these vibrations is also larger with larger throttle. This is believed to be caused by the thruster tilting around the Y -axis due to an error in the center of gravity position of the end-effector. The figure (b) shows that at t = 0 s, immediately after the thruster is activated, the posture is tilted in a positive direction around the Y -axis. Since the mechanism cannot constrain the end-effector in the X direction, if the end-effector is tilted around the Y -axis, the thrust force has an X-axis component and causes a negative displacement in the X direction. The cause of vibration is that the thrust force is slightly unstable. In the figure (c), negative rotation of the end-effector around the Z-axis can be observed. This is thought to be caused by the reaction torque of the thruster. Path Following Experiment. Next, a desired trajectory to follow a rectangular path within the workspace moving at 100 mm/s was given, and experiments to follow it were performed. The throttle was kept constant at 0%, 25%, 50%,

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(a)

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and 100%, respectively. The measured positions of the end-effector are shown in Fig. 5. Figure 5 (a) shows that the path of the end-effector moves downward as the throttle increases. Also, this difference is large for throttles of 0% and 25%, but small for 25% to 100%. Since the elastic deformation of the wire due to external force should show a near linear behavior, it is considered that the slack in the wire is reduced by the thrust force from 0% to 25% of throttle, and the stiffness due to the wire is increased.

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Fig. 5. The position and orientation of the single-rotor type end-effector in the path following experiment. (a): front view, (b): isometric view.

The cause of the asymmetric error from the desired trajectory on the left and right may be the effect of the thruster cable, which has a non-negligible bending stiffness compared to the wire. From Fig. 5 (b), it can be seen that there is a large displacement in the Xaxis negative direction around (y, z) = (−200, 50), and in the X-axis positive direction around (y, z) = (200, 50). The reason for this is thought to be the reaction torque of the thruster, as same as seen in Fig. 4 (c). When the endeffector rotates in the negative direction around Z axis due to reaction torque, the wire connection points at the bottom of the end-effector move in the positive and negative directions along X axis and are consequently pulled up by the

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wires and lifted toward the pulley. Since this effect is larger for shorter wires, the connection point of the wire No. 1 is greatly displaced in the X-axis negative direction around Y = −200, which also displaces it in the Z-axis positive direction, resulting in a negative X component to the thrust force. Conversely, near Y = 200, the connection point of the wire No. 4 is greatly displaced in the X-axis positive and Z-axis positive directions, resulting in a positive X component of thrust. This thrust component is thought to be the reason why the end-effector generates displacement also in the X direction as it is displaced in the Y direction. 4.2

Experiments Using CRP Thruster

Next, using a CRP type end effector, same experiments as before to follow a rectangular path within the workspace moving at 100 mm/s were performed. Experiments were conducted with four throttle settings: 0% for both the upper and lower propellers, 30% for the upper propeller, 30% for the lower propeller, and 15% for both the upper and lower propellers. The results of the end-effector paths are shown in Fig. 6. Figure 6 (a) shows that, as in the single-rotor type, the path moves downward when the thruster is activated, which is thought to reduce wire slack. On the other hand, this figure shows no significant difference when comparing the three patterns: only the upper propeller is operated, only the lower propeller is operated, and both the upper and lower propellers are operated. However, figure (b) shows that when only one of the upper and lower propellers is operated, a large displacement occurs in the X direction, whereas the CRP hardly causes any displacement. It has been confirmed that the problem that occurred in the single-rotor type can be resolved by introducing CRP thruster.

5

Conclusions and Future Work

This study presents a concept of a CSPR that utilizes vertical downward force generated by thruster mounted on end-effector, aiming at application as a cooperative robot. A vertical planar 2-DOF prototype was developed, and its effectiveness and issues were examined through experiments. The main results obtained are as follows: – A concept of a collaborative robot consisting of a CSPR and a vertical downward thruster equipped on its end-effector, which is effective to achieve low risk in the event of collision, and good positioning performance at high speed operation, was proposed. – A prototype of a 4-wires 2-DOF vertical planar translational CSPR was developed. – The following findings were obtained from the experiments: • When the single-rotor thruster is used, the reaction torque of the propeller causes the end-effector to rotate around its axis, resulting in displacement in the out-of-plane direction.

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Fig. 6. The position and orientation of the CRP type end-effector in the path following experiment. (a): front view, (b): isometric view.

• By introducing CRP, problems caused by the reaction torque of the propeller do not occur. • There are issues such as the negative effects of the stiffness of the thruster’s power and signal cables on the motion of the end-effector, and vibration caused by the uneven output of the thruster. • The thruster can reduce wire slack. This is expected to improve the stiffness of the wire. In a future work, it is necessary to measure the wire tension and evaluate the actual force acting on the end-effector in the prototype, because if the parallel wires are not exactly the same length, the orientation of the end-effector may

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be affected. Modeling of thrustered CSPR and theoretical analysis of the relationship between wire tension, thrust, and displacement based on this model are also future important tasks. In addition, evaluation of the stiffness of the actual mechanism, positioning accuracy, performance at high speed operation, energy efficiency and safety of propellers are also future work. Finally, the authors will develop and evaluate a prototype of a spatial mechanism that extends the concept presented in this paper.

References 1. ISO 10218-1:2011, Robots and robotic devices – Safety requirements for industrial robots – Part 1: Robots. (2011) 2. ISO 10218-2:2011, Robots and robotic devices – Safety requirements for industrial robots– Part 2: Robot systems and integration. (2011) 3. Gosselin, C.: Cable-driven parallel mechanisms: state of the art and perspectives. Mech. Eng. Rev. 1(1), DSM0004 (2014) 4. Pott, A.: Cable-Driven Parallel Robots – Theory and Application. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-76138-1 5. Albus, J., Bostelman, R., Dagalakis, N.: The NIST SPIDER, a robot crane. J. Res. Natl. Inst. Stand. Technol. 97(3), 373–385 (1992) 6. Alp, A.B., Agrawal, S.K.: Cable suspended robots: design, planning and control. In Proceedings of the IEEE ICRA, pp. 4275–4280 (2002) 7. Arai, T., et al.: Development of hybrid drive parallel arm for heavy material handling. In: Proceedings of the 16th ISARC, pp. 263–268 (1999) 8. Behzadipour, S., Khajepour, A.: A new cable-based parallel robot with three degrees of freedom. Multibody Syst. Dyn. 13, 371–383 (2005) 9. Zhang, Z., et al.: Optimization and implementation of a high-speed 3-DOFs translational cable-driven parallel robot. Mech. Mach. Theory 145, 103693 (2020) 10. Ono, A., et al.: Rotor turning over mechanism for wall hovering robot. Trans. JSME Ser. C 75(755), 2013–2019 (2009). (in Japanese) 11. El-Ghazaly, G., et al.: Hybrid cable-thruster actuated underwater vehiclemanipulator systems: a study on force capabilities. In: IEEE/RSJ IROS (2015) 12. Yigit, A., et al.: Aerial manipulator suspended from a cable-driven parallel robot: preliminary experimental results. In: IEEE/RSJ IROS, pp. 9662–9668 (2021) 13. Schr¨ oder, S.: Under constrained cable-driven parallel robot for vertical green maintenance. In: Gouttefarde, M., Bruckmann, T., Pott, A. (eds.) CableCon 2021. MMS, vol. 104, pp. 389–400. Springer, Cham (2021). https://doi.org/10.1007/9783-030-75789-2 31 14. Shao, Z., et al.: Design and analysis of the cable-driven parallel robot for cleaning exterior wall of buildings. Int. J. Adv. Robot. Syst. 18(1) (2021) 15. Lee, K.U., et al.: Obstacle-overcoming and stabilization mechanism of a rope-riding mobile robot on a facade. IEEE Robot. Autom. Lett. 7(2), 1372–1378 (2022) 16. Sun, Y., et al.: Wrench-feasible workspace-based design of hybrid thruster and cable driven parallel robots. Mech. Mach. Theory 172, 104758 (2022)

The Robotic Seabed Cleaning Platform: An Underwater Cable-Driven Parallel Robot for Marine Litter Removal Marc Gouttefarde1(B) , Mariola Rodriguez2 , Cyril Barrelet1 , Pierre-Elie Herv´e2 , Vincent Creuze1 , Jose Gorrotxategi2 , Arkaitz Oyarzabal2 , David Culla2 , Damien Sall´e2 , Olivier Tempier1 , Nicola Ferrari3 , Marc Chaumont1 , and G´erard Subsol1 1

2

LIRMM, Univ Montpellier, CNRS, Montpellier, France [email protected] TECNALIA, Basque Research and Technology Alliance (BRTA), Gipuzkoa, Spain 3 Servizi Tecnizi Srl, Monza, Italy

Abstract. As a contribution to the development of new techniques to remove marine litter from the seabed of sees and oceans, the Robotic Seabed Cleaning Platform has been designed, built and experimented in the framework of the European Union project MAELSTROM. It essentially consists of a floating platform that supports the base elements of a 6 degree-of-freedom cable-driven parallel robot actuated by eight winches. The mobile platform of this robot can work underwater and is equipped with sensors to control its underwater motions and to detect & identify marine litter. To achieve efficient and selective litter removal, an aspiration system and a gripper are installed on the CDPR underwater mobile platform. Keywords: Marine litter robots

1

· underwater robots · cable-driven parallel

Introduction

It is estimated that about 70% of marine debris sinks to the seabed where it will fragment in micro and nano plastics. On overall, tens of million tons of waste and debris are estimated to lie on the seafloors of seas and oceans worldwide. Significant efforts to reduce sea and ocean pollution are thus of fundamental importance. Besides, developing new techniques for the removal of marine litter is needed. In this context, the European Union project MAELSTROM [1] aims at designing and integrating technologies to identify, remove, sort and transform all types of collected marine litter into valuable raw materials. In this project, two systems are considered for marine litter removal. The first one is developed by The Great Bubble Barrier and consists in catching plastic pollution in rivers before it reaches the ocean, using bubbles. The second one is presented in the c The Author(s), under exclusive license to Springer Nature Switzerland AG 2023  S. Caro et al. (Eds.): CableCon 2023, MMS 132, pp. 430–441, 2023. https://doi.org/10.1007/978-3-031-32322-5_35

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present paper: The Robotic Seabed Cleaning Platform. It mainly consists of a floating platform that supports the base elements of a 6-DOF Cable-Driven Parallel Robot (CDPR) actuated by eight winches. The mobile platform of this CDPR can work underwater and is equipped with sensors to control the underwater mobile platform motion and detect & identify marine litter to be removed. To achieve efficient and selective removal, an aspiration system to suck up smaller litter and a gripper to grasp and remove larger items, e.g. tires, are also installed on the CDPR underwater mobile platform. Existing solutions for removal of marine litter lying on the seabed can be highly harmful to the marine ecosystem, e.g., dredges and bottom trawl nets. Human divers can also remove litter from the seabed but this work can put them in danger and their actions are limited in time and litter weight. Another option for marine litter removal is to use robotic systems. For instance, there exist autonomous surface vehicles to remove floating pollution such as the low cost, zero greenhouse emissions WaterShark [4] and the multi-purpose Jellyfishbot [3]. Besides, to remove litter from the seabed, underwater robots [5] can be used. Both autonomous underwater vehicles and Remotely Operated Vehicles (ROV) allow to reach seabed at depth of several tens or hundreds of meters. For example, the SeaClear project [2] aims at deploying a system consisting of an unmanned surface vehicle, an unmanned aerial vehicle, a small ROV and a larger ROV. The unmanned surface vehicle is the mothership that deploys the aerial vehicle and the two underwater robots. The aerial vehicle can search litter from the air (in clear water) and helps the navigation of the surface vehicle while the small ROV scan the seabed at close range. The larger ROV removes litter with a gripper and suction device. The SeaClear system should thus be able to operate autonomously over large distances but can remove rather small lightweight objects from the seafloor. Moreover, ROV may take time to stabilize before being able to grasp a litter, thus impairing efficiency. Hence, one advantage of the Robotic Seabed Cleaning Platform introduced in this paper is its capability to efficiently and selectively remove relatively large and heavy objects, as well as smaller ones, since it is based on a CDPR which constitutes a (relatively) accurate and fast positioning system. Numerous previous papers on underwater cables exist in the literature, from their physical modeling, e.g. [6,7], to their use as ROV umbilical, e.g., [13,15], where the cables are usually used in low tension contrary to the case of CDPRs where tensed cables transmit motion and force. In fact, few previous works deal with marine and underwater CDPRs. In [9], the idea of a 6-DOF underwater mobile platform connected to surface ships by more than 6 cables is presented but there is no specific underwater modeling, design, control or experiments introduced in this paper. In [11,12], the workspace, dynamic performances and various cable configurations of a Gough-Stewart type floating cable-driven marine platform subjected to waves are studied. The winches are placed on the surface mobile platform and the cables anchored on the seabed (mooring cables). Computer simulations are reported but no prototype and experiments are presented. Moreover, the concept of underwater robot actuated in a hybrid fashion

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Fig. 1. The main components of the RSCP.

by thrusters and cables is introduced in [8] and further studied in numerical simulations in [14], but no experiment or prototype has been presented at the time of writing the present paper. Hence, the contribution of this paper is to introduce the Robotic Seabed Cleaning Platform (RSCP) design, control and experimental testing to remove marine litter lying on the seabed of Venice lagoon. To the best of the authors’ knowledge, it is the first time an underwater CDPR is designed and used. The paper is organized as follows. In Sect. 2, an overview of the RSCP is presented. Then, in Sects. 3 and 4, the RSCP design and control are described, respectively. Finally, Sect. 5 presents the results of the first RSCP cleaning campaign in Venice lagoon while Sect. 6 concludes the paper.

2

Presentation of the Robotic Seabed Cleaning Platform

The RSCP consists mainly of a CDPR installed and operated from a floating barge. The mobile platform of the CDPR can work underwater and is equipped with selective cleaning tools that allow the removal of legacy small (micro-plastics >5 mm) and large items on the seabed, as well as floating plastics in the water column, in a beneficial way for the marine ecosystem. The main components of the RSCP are presented in Fig. 1 where the mobile platform is shown outside of the water. In the CAD view of Fig. 2, the mobile platform is shown working underwater. The 6-DOF mobile platform of the RSCP is driven by eight cables in a suspended configuration similar to the one of the CoGiRo CDPR [10]. Four base frame posts are secured to the floating platform (floating barge), the latter being made of several modules (pontoons) assembled together. Each post has two winches and pulleys that route the cable from the winch to the top of the post where a swiveling output pulley directs the cable toward the mobile platform. One of the pulley located near the winch at the bottom of the post is equipped with a load pin and thereby allows the measurement of the cable tension. The mobile platform can move down in the water below the floating barge which

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Fig. 2. A CAD view of the RSCP with the CDPR mobile platform working underwater.

has a rectangular hole (inner pool) in its middle. An aspiration system to suck up smaller litter in the water column and seabed below the hole in the floating barge, as well as a gripper to remove larger items, are installed on the CDPR underwater mobile platform. A control room hosts the electronic and control cabinets, the control PCs and the human operator(s) of the RSCP. A picture of the RSCP during first tests on the ground in Tecnalia’s facilities in Spain is shown in Fig. 3. The mobile platform is the beige structure in the middle of the picture. After its setting up in Tecnalia’s facilities, the RSCP was disassembled and moved to Venice in Italy for its first underwater cleaning campaign (cf Sect. 5), as shown in Fig. 4.

3

Design of the Robotic Seabed Cleaning Platform

The RSCP components have been designed to be used in a harsh outdoor maritime environment. The maximum depth of operation of the underwater mobile platform is directly correlated to the size of the floating barge. Besides, the rigidity of the barge depends on the size of its inner pool. A compromise has been found in the design of the floating barge to reach the required depth of operation (between 15 to 20 m) while maximizing its rigidity. The CAD of the floating barge final design is shown in Fig. 5. The floating barge, designed for both still waters and sea operations, is composed of several pontoons. Two metallic posts are fixed on two steel bases located on the same pontoon to avoid relative displacements between the swiveling output pulleys of the CDPR. The two other posts are also mounted on another single pontoon. During the experiments reported in Sect. 5, the relative displacement between two opposite posts mounted on different pontoons has been estimated

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Fig. 3. The RSCP during tests in Tecnalia’s facilities.

by means of measurements with a total station. When middle size waves are shaking the floating barge, the relative movement is around +/- 5 mm, which was deemed to be an acceptable value. The floating barge is maintained in position, depending on the depth of the cleaning, by anchor feet or wire anchors to the seafloor. The floating barge is displaced from a cleaning spot to another one by a tugboat. As shown in Fig. 6, each cable goes out of the winch and is guided by pulleys to a movable carriage. This carriage has a vertical translation to move the upper swiveling pulleys from a the lower position to an upper position, the latter being 2.5 m higher than the former. On the one hand, when the carriage is at the top of the post (parking position), this reconfiguration capability allows the CDPR mobile platform to be out of the water for assembly and maintenance operations and for moving the floating barge from spot to spot. On the other hand, when the carriage is at the bottom of the post (working position), it allows having the robot inside the water for marine litter removal operations with increased operational workspace. The marine litter removal operations are also possible in parking position, but the CDPR workspace size is reduced. The design of the supporting structure of the winches and pulleys is suitable for any kind of pontoons and floating barge and is independent of the height of the pontoon. The location of the posts on the floating barge has been defined by optimizing the workspace of the CDPR with methods well known in the state of the art. The calibration of the positions of the swiveling pulleys (points Ai ) is made with the help of a total station that measures reflective targets closed to the pulleys as shown in Fig. 7. For practical reasons, it is not possible to measure directly the positions of the cable-platform connection points Bi as they are virtual points at the center of rotation of universal joints. Hence, as shown in

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Fig. 4. The RSCP in Venice lagoon.

Fig. 5. CAD of the final design of the floating barge.

Fig. 8, a 3D printed plastic part installed around point Bi is used to measure 2 reflective targets. The position of point Bi is at the middle of the measured targets. The plastic part is removed from point Bi once it has been measured and installed to the next point Bi . Besides, three reflective targets are fixed on the mobile platform to measure its pose. The positions of these targets are measured just before or after the position measurements of the points Bi . This platform pose measurement is required to calibrate the initial pose of the platform. The underwater mobile platform, shown in Fig. 9, is a classic steel welded structure, protected by a marine resistant paint. The cable attachment points (points Bi ) are made with stainless steel universal joints. All bearings at points Ai and Bi are Igus plastic bearings to avoid corrosion. The cables are nonrotating steel wires of diameter 12 mm. They are generously greased with a bio sea water compatible grease, to prevent corrosion.

4

Sensors and Control System

Various sensors are located on the floating floating barge including a pressure sensor to compensate for the atmospheric pressure and two RTK GPS placed at two

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Fig. 6. Final CAD design of the metallic posts (left subfigure) and the swiveling output pulleys (right subfigure).

different locations on the floating barge to estimate the position and orientation (around the vertical axis) of the floating barge based on position measurements from a global navigation satellite system. It thus allows to position the RSCP on top of a previously generated bathymetry map of the seabed environment, allowing to identify “hotspots”, to position the barge inner pool above one of these hotspots and to make the CDPR mobile platform dive to reach it. This capability turned out to be a key enabling feature in the highly turbid water of Venice lagoon. Moreover, the CDPR has eight force sensors placed at fixed (non swiveling) routing pulleys to measure the cable tensions as well as encoders in the winch motors. The sensors located inside the underwater platform shown in Fig. 9 comprise five IP cameras which enable the human operator located in the control room on the floating barge to see the surrounding of the underwater platform (provided that water turbidity is not too high). Several other sensors, listed below, are also integrated into the so-called “smart camera” system, shown in Fig. 10, which is fixed to the mobile platform on one of its edge. The smart camera mainly consists of the following components. A camera used for visual servoing: The camera enables a marine litter to be seen by the operator who can click on the litter in the camera image so that the CDPR mobile platform approaches the litter automatically. One depth (pressure) sensor to measure the depth of the mobile platform (distance with respect to the sea surface). One IMU used mainly to measure the orientation of the underwater mobile platform. A Doppler Velocity Log (DVL) which is a hydro-acoustic sensor integrating four acoustic beams to measure the distance of the mobile platform to the seabed and its velocity with respect to the seabed. The DVL estimates velocity relative to the sea bottom by sending acoustic waves from four angled transducers and then measure the frequency shift (Doppler’s effect) from the received echo. By combining the measurements of all four transducers and the time between each acoustic pulse, it is possible to estimate the speed and direction of movement.

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Fig. 7. Calibration: Determination of the positions of the points Ai from the target measurements.

Fig. 8. Calibration: 3D plastic parts used to measure the positions of points Bi .

Since the DVL also indicates the altitude with respect to the seabed (from the range measurements achieved by the four beams), it can be used to build a local map of the seabed slope (sea ground surface) such as illustrated in Fig. 12. To this end, the pressure sensors on the floating barge and on the underwater mobile platform are also used to determine the depth of the mobile platform with respect to the floating barge and the altitude of the floating barge with respect to the seabed. Subsequently, the two GPS on the floating barge can be used to locate the local map in a global earth coordinate system. This allows either estimating previously located underwater litter positions and/or to build a map of the (removed) litters. The control system of the CDPR of the RSCP used in the experiments reported in Sect. 5 consists of the CDPR teleoperation control and the perception system for underwater vision. Regarding the CDPR teleoperation control, the pilot (human operator) uses estimated underwater mobile platform pose as well as the cameras located on the mobile platform to drive the underwater mobile platform using a joystick. The joystick commands issued by the pilot are interpreted as desired operational velocities, i.e., linear and angular velocities

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Fig. 9. Close-up view of the underwater mobile platform including 3 reference points on the platform (reflective targets).

Fig. 10. The waterproof underwater capsule of the smart camera system (left) and the smart camera system description with reference frames fixed to it (right).

of the underwater mobile platform. These velocities are integrated to provide desired positions and orientations of the mobile platform that are converted into desired winch motor positions by means of the inverse kinematics of the CDPR. These desired joint motor positions are sent on the fly to the drive setpoint controllers which are in charge of the feedback position control of the motors. The Human-Machine Interface (HMI) of the CDPR is shown in Fig. 11. It allows the human operator (pilot) to monitor various sensor values and to choose between different control modes. The human operator can see the videos from the IP cameras placed on-board the underwater mobile platform in a dedicated screen. Regarding the perception system for underwater vision, the camera in the smart camera system is used as the underwater perception system. It notably enables visual servoing: Once a marine litter is seen by the human operator (at relatively closed range from the camera because of water turbidity), the latter can click on the litter in the camera image and the CDPR mobile platform approaches the litter automatically. Figure 12 shows the HMI of the underwater perception system. On the left, the depth map built with the DVL measurement

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Fig. 11. HMI of the CDPR: 1. Winch axis information area; 2. Secondary commands for individual or combined motion of the cables, motion command to calibration position, and enabling post vertical carriage linear movements using the joystick; 3. Mobile platform position and orientation information area; 4. Manual commands area including initialization (calculation of the Cartesian pose computed using the winch motor positions by means of forward kinematics), manual move with joystick enabling & speed override setting, and robot status; 5. Automatic command area including pointto-point motion command (actual to destination pose) and CNC trajectory program execution; 6. Cable tension monitoring area: Force sensor values in Kg.

is shown. In the middle, the human operator can see the image from the camera of the smart camera system (here shown in turbid water with very little visibility range). The operator can click on a point in this image, where a possible litter to be removed is located, so that the mobile platform moves toward this point. On the right of the HMI, the bathymetry map is shown. It is centered on the RSCP inner pool (the orange rectangle). The red rectangle is the safe working zone where no collision between the RSCP and the CDPR cables can happen. In the HMI bottom part, various sensor values and other information are displayed.

5

Experiments in Venice

The RSCP was tested in September 2022 in Venice lagoon in Italy where it performed a cleaning campaign. Despite highly turbid water, the cleaning campaign was a success since many marine litters of several different types were removed from the seabed, as shown in Fig. 13. To deal with the very low visibility due to the turbidity of the lagoon water, real-time image enhancement was used to improve the perception of the marine litter with the cameras on-board the CDPR underwater mobile platform.1

1

As shown in the following videos of the RSCP experiments in Venice: https://youtu. be/1EVQm-0yyRY and https://youtu.be/16k3-Bp4FCI.

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Fig. 12. HMI of the underwater perception system.

Fig. 13. The RSCP removing a tire from the water (left) and various litters collected by the RSCP during the cleaning campaign in Venice lagoon.

6

Conclusion

This paper presented the Robotic Seabed Cleaning Platform. It mainly consists of a floating platform that supports the base elements of a 6-DOF CDPR actuated by eight winches. The mobile platform of this CDPR can work underwater and is equipped with sensors to control the underwater mobile platform motion and detect & identify marine litter to be removed. To achieve efficient and selective removal, an aspiration system to suck up smaller litter and a gripper to grasp and remove larger items, e.g. tires, are also installed on the CDPR underwater mobile platform. To the best of the authors’ knowledge, it is the first time an underwater CDPR is designed and used. The Robotic Seabed Cleaning Platform was successfully experimented in Venice lagoon where many marine litters of several different types were removed from the seabed. Acknowledgments. This work was supported by the European Union’s H2020 Program (H2020-FNR-2020) under the grant agreement No. 101000832 (MAELSTROM project).

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References 1. European union’s H2020 project MAELSTROM (2023). https://www.maelstromh2020.eu 2. European union’s H2020 project SeaClear (2023). https://seaclear-project.eu/ 3. Jellyfishbot: Surface robot for the cleaning and the depollution of water surfaces (2023). https://www.jellyfishbot.io/ 4. Wasteshark: Autonomous surface vessel designed by RanMarine (2023). https:// www.ranmarine.io/products/wasteshark-3/ 5. Antonelli, G.: Underwater Robots: Motion and Force Control of VehicleManipulator Systems. In: Springer Tracts in Advanced Robotics. Second edition, Springer, Berlin, Heidelberg (2006). https://doi.org/10.1007/11540199 6. Boyer, F., Nayer, G., Leroyer, A., Visonneau, M.: Geometrically exact Kirchhoff beam theory: application to cable dynamics. ASME J. Comput. Nonlinear Dyn. 6 (2011) 7. Buckham, B., Nahon, M.: Formulation and validation of a lumped mass model for low-tension ROV tethers. Int. J. Offshore Polar Eng. 11(4), 282–289 (2001) 8. El Ghazaly, G., Gouttefarde, M., Creuze, V.: Hybrid cable-thruster actuated underwater vehicle manipulator systems: a study on force capabilities. In: Proceedings of the IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS). Hamburg, Germany (2015) 9. Ghaffar, A., Hassan, M.: Study on cable based parallel manipulator systems for subsea applications. In: Proceedings of the 3rd International Conference on Mechanical Engineering and Mechatronics. Prague, Czech Republic (2014) 10. Gouttefarde, M., Collard, J.F., Riehl, N., Baradat, C.: Geometry selection of a redundantly actuated cable-suspended parallel robot. IEEE Trans. Robot. 31(2), 501–510 (2015) 11. Horoub, M., Hassan, M., Hawwa, M.: Workspace analysis of a Gough-Stewart type cable marine platform subjected to harmonic water waves. Mech. Mach. Theory 120, 314–325 (2018) 12. Horoub, M., Hawwa, M.: Influence of cables layout on the dynamic workspace of a six-dof parallel marine manipulator. Mech. Mach. Theory 129, 191–201 (2018) 13. Laranjeira, M., Dune, C., Hugel, V.: Catenary-based visual servoing for tether shape control between underwater vehicles. Ocean Eng. 200 (2020) 14. Sacchi, N., Simetti, E., Antonelli, G., Indiveri, G., Creuze, V., Gouttefarde, M.: Analysis of hybrid cable-thruster actuated ROV in heavy lifting interventions. In: Proceedings of the IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS). Kyoto, Japan (2022) 15. Viel, C.: Self-management of the umbilical of a ROV for underwater exploration. Ocean Eng. 248 (2022)

A Cable-Based Haptic Interface with a Reconfigurable Structure Bastien Poitrimol(B) and Hiroshi Igarashi Tokyo Denki University, Adachi-ku, Tokyo, Japan [email protected], [email protected]

Abstract. Cable robots have been used as haptic interfaces for several decades now, the most noticeable examples being the SPIDAR and its many iterations throughout the years. However, these robots still have major drawbacks, particularly their high number of cables due to a inherently low workspace to installation-space ratio. Using a variable-structure cable robot (VSCR) could prevent some collisions that occur between the cables and the user’s body. More specifically, some applications requiring multimodal feedback could benefit from the flexibility that a reduced number of cables offers. Thus, this paper presents a novel SPIDAR-like VSCR and its sensorless force control method based on motor current. The purpose of this work is to clarify the gains that a variable structure can provide for haptic interaction. In this regard, experimental results regarding the workspace of the device and its force feedback are presented. Another experiment was also conducted to study the changes in user performance when using both the variable and usual cable configuration. Results showed that feedback accuracy remains and that there are apparently no drawbacks about using hybrid configurations. Keywords: Variable-structure · Haptic interface parallel robots · Motor current · Control

1

· Cable-driven

Introduction

The rise of technologies related to virtual reality (VR) has created a need for new ways to interact with virtual worlds. Head mounted displays (HMD) [1] have been the most popular type of device used to immerse the user. However, human’s complex haptic perception requires more than simple visual feedback to reach complete immersion [2]. In order to allow for more complex interactions, cable-driven parallel robots (CDPRs) were used as haptic interfaces as soon as the early nineties [3]. Indeed, thanks to their high reconfigurability, large workspace, low intrusiveness and low inertia, CDPRs present interesting features when used as force feedback interfaces. Since the development of the first prototype of SPIDAR [3,4], CDPRs, and haptic interface in general, have been adapted to a wide range of applications, with even more configurations to fit each task [5]. Recent publications suggest a c The Author(s), under exclusive license to Springer Nature Switzerland AG 2023  S. Caro et al. (Eds.): CableCon 2023, MMS 132, pp. 442–454, 2023. https://doi.org/10.1007/978-3-031-32322-5_36

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trend towards more immersion (or performance for medical applications) through multimodal haptic feedback [6–8] and innovative displays (co-localization with half-tainted mirrors) [9,10]. Cable-based haptic device indeed lacked diversity of feedback and the virtual environment is usually displayed using a simple screen [5,11]. Nonetheless, some problems remain. The high number of cables necessary to provide force feedback induces a poor workspace to installation-space ratio as well as interferences between the cables and less freedom of movement for the user [12]. Moreover, most interfaces do not allow rotations which keeps the control scheme and end-effector design as simple as possible but limits the possibilities of the interfaces. Progresses have been made, especially using task-specific endeffectors [13,14] or configurations, but solutions like reconfigurable CDPRs are still mostly left untouched [15]. Thus, in this paper we present a hybrid SPIDAR-like haptic interface with a variable structure. The aim of this VSCR, is to reduce the number of cables while increasing the workspace of the robot. Besides providing more freedom of movement for the user, this could provide solutions to integrate other devices for multimodal haptic display (on the effector for example). In Sect. 2, we describe the specifications of the system and its kinematic model. Section 3 provides a detailed explanation of the control method used to provide haptic feedback. Then, Sect. 4 presents the user-centered experiment and results obtained regarding haptic rendering and workspace. We conclude this paper by discussing future works and remaining challenges.

Fig. 1. Overview of the reconfigurable haptic device. It can be used from any direction thanks to the HMD that displays the virtual environment. The cables are represented in green and in yellow is the workspace where force feedback can occur.

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System Description

The haptic device is composed of three main parts that we will detail in this section. Then, we discuss the limitations and design choices and finish by showing their impact on the kinematic model of the VSCR. 2.1

Specifications and Hardware

The layout of the entire system is as depicted in Fig. 1. It was made to be a transportable interface, its total dimensions are about 80 cm in length, 55 cm in width and 160 cm in height. Six motors are used to provide feedback through the cables, four are attached to the frame (cables on top) and two are fixed to a frame that is being moved dynamically by the linear actuator to follow the endeffector. Thus, the theoretical workspace (in yellow in Fig. 1.) that can be used for manipulating the end-effector is delimited by the length of the actuator and the position of the motors. The dimensions of the workspace are: 77 cm in length, 49 cm in width and 53 cm in height. The hardware specifications are detailed in Table 1. The hardware choices were largely inspired by other existing devices [9]. The end-effector, as well as the pulleys and motor mounts are made of 3D printed resin. This CDPR being first and foremost a haptic interface, the SLP15 and its linear shaft motor was chosen for its high accuracy, high speed and absence of friction. Meanwhile, direct drive motors like Maxon’s DCX32 provide sufficient torque (low torque is used for safety considerations) and precision for haptic applications. Furthermore, such motors have already been used [9,10] and a low cogging torque fits the sensorless (i.e. no loadcells or torque sensors) control method [15] described in Sect. 3. Table 1. Detailed hardware specifications. Parts

Specifications

Motors

Maxon DCX 32 48V (32 mm diameter, no gearhead)

Encoders

Maxon ENX16 EASY 1024 (1024 pulses per round)

Linear actuator

Nippon Pulse Motor SLP-15

DC motor drivers

Maxon Motor ESCON Module 50/5

Linear actuator driver Panasonic Servo driver MADHT1105L01 Micro controller

ARM mbed NUCLEO-F767ZI

HMD

Oculus Quest 2

The raw data from the motors encoders is sent to the micro controllers and then to a C++ program running on a Linux OS server at a 2kHz frequency with a UDP communication protocol. The C++ program realizes all the computation related to the control of the robot (kinematics) but also, the force feedback. Indeed, here the simulation is displayed using an HMD while the simulation is

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run remotely on a monitoring PC. Only the position of the robot and the handle are sent from the control server through UDP to the monitoring PC, which then sends back the updated information via Wi-Fi to the HMD. Limitations: Note that this device is direct adaptation of the SPIDAR and its many variations, hence the fact that the kinematic model considers the endeffector as a point. Even though some recent works had some more complex handles and models for dedicated applications, a simple model should be sufficient since our aim is to provide a proof of concept. Regardless, the inherent lack of accuracy of the SPIDAR [16] has not been pointed out as a problem when providing haptic feedback. Similar considerations oriented the choice of using a HMD: while colocalization between the simulation and the real world is hard to achieve [10], the perspective provided by the HMD will allow more complex interactions than the usual translational movements used in many studies. Anyhow, the real impact of the display method should be investigated in future works.

Fig. 2. Kinematic modeling of the hybrid SPIDAR

2.2

Kinematic Model

The system needs to track the position of the user’s hand, which implies that the cables lengths are given and the pose of the effector is sought. In Fig. 2, the model used to calculate the forward kinematics of the robot is depicted. The half-length, half-width and half-height are respectively noted l, w and h. The end-effector of coordinates E = [xe , ye , ze ]T has its first position initialized at the center of the device where E = [0, 0, 0]T . Note that the position of the linear actuator yl will be almost equal to ye during operation. The lengths of the cables

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Li are calculated from the encoder’s data, and each cable’s magnitude Mi E can be written as (1) Mi E = L2i for i = 1, ..., 6 where the detachment points are M1 = [w, −l, h]T ; M2 = [−w, −l, h]T ; M3 = [−w, l, h]T M4 = [w, l, h]T ; M5 = [w, yl , −h]T ; M6 = [w, yl , −h]T

(2)

Since the effector is considered as a point, the Euclidian norm Eq. (1) yields ⎧ 2 L1 = (xe − w)2 + (ye + l)2 + (ze − h)2 ⎪ ⎪ ⎪ ⎪ L2 = (xe + w)2 + (ye + l)2 + (ze − h)2 ⎪ ⎪ ⎨ 22 L3 = (xe + w)2 + (ye − l)2 + (ze − h)2 (3) ⎪ L24 = (xe − w)2 + (ye − l)2 + (ze − h)2 ⎪ ⎪ ⎪ ⎪ L25 = (xe − w)2 + (ye − yl )2 + (ze + h)2 ⎪ ⎩ 2 L6 = (xe − w)2 + (ye + yl )2 + (ze + h)2 Solving the above system of equations knowing all the cables length lead to obtaining E = [xe , ye , ze ]T in the form: xe =

1 (L2 − L21 ) 4w 2

1 ye = (L22 − L23 ) 4l  ze = h ± L23 − (xe + w)2 − (ye − l)2

(4) (5) (6)

It can be noted from Eqs. (4), (5) and (6) that the configuration of the robot allows to use only the upper cables to infer the position E; doing so will prevent the small difference between yl and ye to add inaccuracy in the measurements (M5 and M6 are actualized in real time with a slight delay). To put it simply, the bottom part of the robot composed of the linear actuator is enslaved to the upper part. Lastly, we define u ˆ the estimation of the force applied by the user on the handle, and the unit cable vectors ui : ui =

EMi EMi 

(7)

They will be used in the next part for the control algorithm.

3

Control

This section focuses on the force control algorithm of the interface. An overview of the controller is shown in Fig. 3. The controller is composed of four main parts: an observer part to estimate the force applied on the effector, a PID to control the linear actuator, a virtual model to simulate the behavior of the

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Fig. 3. Block diagram of the control scheme used to provide force feedback.

virtual object, and a PD controller that outputs the force feedback of the virtual object through the end-effector. Note that during the entire manipulation, when a cable is not used for feedback it is maintained in tension. This minimum force of around 0.7 N used to straighten the cables is called τof f set ; the total force on the end-effector is maintained under 10N force safety considerations. Also as explained earlier the end-effector is initialized in [0, 0, 0]T . Linear Actuator PID: Using the forward kinematics described in Sect. 2, the angle θm of each motor is used to calculate the position of the handle. This position is then designed as the target position for the linear actuator which acts as a slave system. The aim is to keep the end-effector close to the center of the workspace where static equilibrium is achieved during manipulation. Thus, the user will feel less resistance due to cable tension when moving the effector without having to use methods to dynamically distribute the cable tension. Disturbance Observer and Force Estimation: The disturbance observer (DOB) and reaction torque observer (RTOB) were first presented in [17] by Ohnishi et al. Each motor is represented by an ideal model, here we chose a pure inertia I after experimental identification. The DOB suppresses the internal disturbance and parasitic noise of each motor using their speed and current input (LPF stands for low-pass filter on the diagram). Once the disturbance is suppressed, the RTOB estimates any variation of speed as an input force from the user. An estimated torque τˆi (for i = 1, ..., 6) is obtained for each motor, then the following equation gives u ˆ: ⎡ ⎤ ⎡ ⎤ ⎡τˆ ⎤ 1 Fx u1x u2x ... u6x .. ⎥ r−1 ⎣Fy ⎦ = ⎣u1y u2y ... u6y ⎦ ⎢ (8) ⎣ . ⎦ pulley Fz u1z u2z ... u6z τˆ6       T u ˆ

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Where J T is the transpose of the Jacobian matrix, τˆi the cable force distribution and rpulley =0.0065 m the radius of the pulley mounted on the motor shaft. Virtual Model: Efforts produced by the interaction between the operator and the virtual object are usually deduced using the god-object method [18]. In our work, such method is unnecessary because the object has a simple shape (sphere: see Fig. 4). We chose to use two virtual models: a mass-damper system model to ˆ object and a spring-mass-damper estimate the movement of the sphere called X model to simulate the texture of the object. The movement is represented as: ¨ ˆ object = J −1 (ˆ X u − Bv x˙ ef f ector ) v

(9)

With Jv = Jv I [3x3] the term of mass and Bv = Bv I [3x3] the term of damper ˆ object is where I [3x3] is the 3 by 3 identity matrix. By successive integration, X ˆ is obtained, and the actual movement of the object considering the texture X similarly found as: ¨ ˆ = J −1 (ˆ X u − Bo x˙ ef f ector − Ko (xsurf ace − xef f ector )) o

(10)

Where the mass, damper and spring terms are respectively defined as Jo = Jo I [3x3] , Bo = Bo I [3x3] and Ko = Ko I [3x3] . Force Feedback: After obtaining the estimated pose of the end-effector, the difference between the estimation and the actual position of the handle is sent back to a PD controller. Finally, using the Moore-Penrose pseudo-inverse the torque variation required for the force feedback is distributed to the motors (see Fig. 3).

Virtual sphere

Fig. 4. Interaction between the operator and the virtual sphere. xsurf ace is simply calculated using the equation of a sphere in three dimensions.

4

Experiments and Results

Three experiments were undergone to evaluate the performance of the reconfigurable SPIDAR. The experimental process and the results will be presented in this section.

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4.1

Workspace

The results presented in this section refer to Fig. 5. In the case of a SPIDAR-like cable robot, force feedback can be displayed as long as the end-effector stays in the space delimited by the anchor points: the yellow space shown in Fig. 1. Thus, it is not pertinent to evaluate wrench-feasible workspace as it is usually done for CDPRs. Moreover, it is easily understood that the workspace is nearly doubled (from 0.1 m3 to 0.2 m3 ) when comparing the hybrid configuration and the fixed configuration (i.e. with the actuator locked in yl = 0). Instead, we propose to study the resistance felt by the user when manipulating the endeffector. The manipulation resistance is defined as the Euclidian norm of the vector [Fx , Fy , Fz ]T from Eq. (8) when only τof f set is applied to the handle. For the simulation, τof f set =10 mNm or around twice the minimal tension used for the next experiment. Then, the resistance is calculated for every points inside the workspace with a 0.5 cm step using Mathematica. The same computation is done using an actual set of data obtained by manipulating freely the end-effector inside the workspace. The results show a wider area of low resistance with the hybrid configuration which could give some interesting prospects since most SPIDARs tend to use only the center of the workspace, where the tension is low and the feedback accurate, during operation [16]. It could be argued that cable tension distribution algorithm could yield similar or better results but they would impact the computation speed by complexifying the control algorithm.

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4.2

Haptic Feedback Evaluation

As a proof of concept, a simple task where the user touches a fixed virtual sphere about 5 cm in diameter was undergone. The graphs in Fig. 6 shows the sphere, the trajectory of the effector (black trajectory and colored points) and the force feedback modelized with black arrows rescaled to fit inside the graph. These arrows start at the contact point between the effector and the sphere and their direction is given by u ˆ. As expected the forces are consistent with the simulated sphere with no noticeable errors in direction. Note that another experiment with an object being pushed along a straight line was designed; but due to the movement of the sphere, the arrows were superimposed to the object and endeffector trajectories resulting in unreadable graphs. For similar considerations, the graphs of Fig. 6 were made using data acquired with 20 Hz frequency. Future works will include more complex tasks and thorough survey-based evaluation of the force feedback as done in [10].

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User-Based Evaluation

Lastly, for the purpose of studying the influence of the hybrid configuration on free manipulation, a picking task was designed. Methodology: Users gather twelve targets (cubes shown in Fig. 7) in a predetermined order. Every time the effector touches a target, the next target changes color to become yellow, thus, indicating the order in which to pick the targets. This imposes a trajectory the users have to follow for each trial (see right of Fig. 7). To compare the hybrid and fixed configuration of the VSCR in a fair way, all the targets are situated inside the workspace of the fixed configuration, hence the triangular repartition. Each user operates the system eight times at least; the only direction given is to try to keep the same speed of manipulation for each trial. At first, the task is undergone twice in order to get used to the interface and each configuration (once for fixed configuration and once for the hybrid). Right after, the operator does a first set of four trials (designed as ‘untrained’ in Fig. 8) alternating between each configuration. Then, the user trains as much as needed to remember the order of the task picking and aims for a completion time of around 10 s. Finally, once again the operator does a second set of four trials (designed as ‘trained’ in Fig. 8). At this point, the data was completely acquired and the experiment ends. Regarding the participants, they are nine naive volunteers (all male) in their twenties both left-handed and right-handed. They do not report any physical of visual impairment that could affect their performance. Results: We computed paired sample t-tests using the free software JASP [19]; the results are presented in Fig. 8. Two parameters were compared for both trained and untrained states: the time taken by the participants to complete the task and the total travel distance of the end-effector. The two graphs on the left in Fig. 8 tend to indicate that there is no significant difference in terms of performance when using either configurations for untrained users (p = 0.120 for the distance and p = 0.404 for the completion time). The same observation can be made for trained users regarding the travel distance (with p = 0.157). However, the travel time comparison has a p-value closer to 0.05 (p = 0.085). From the above results, it can be concluded that there is no merits nor demerits about using a hybrid SPIDAR when doing free exploration of the virtual environment with the effector.

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Fig. 8. Results of the experiments comparing the time[s] and distance[m].

5

Conclusion

In this paper, we presented a SPIDAR haptic interface with a reconfigurable configuration that follows the movement of the operator through the effector. The kinematic model of the robot and a sensorless control method based on motor current were described. An analysis was conducted to show that the hybrid configuration has a larger workspace, resulting in a lower manipulation resistance for the user. As a proof of concept, two experiments were conducted to evaluate the accuracy of the feedback and investigate the effect of the linear actuator during free manipulation of the effector. It was concluded that there are apparently no drawbacks about using hybrid configurations. Future work will focus on three main aspects: the collect of participants’ impressions on the feedback by conducting a survey; the evaluation of the impact of the type of display used to represent the virtual environment; and lastly the integration of other types of feedback (e.g. vibrations, temperature...). Other new configurations using even less cables will also be investigated.

References 1. Desai, P.R., et al.: A review paper on oculus rift-a virtual reality headset. arXiv (2014). https://doi.org/10.48550/arXiv.1408.1173 2. Gepshtein, S., et al.: The combination of vision and touch depends on spatial proximity. J. Vis. 5, 1013–23 (2005). https://doi.org/10.1167/5.11.7

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3. Hirata, Y., Sato, M.: 3-dimensional interface device for virtual work space. In: Proceedings of the IEEE/RSJ International Conference on Intelligent Robots and Systems, pp. 889–896 (1992). https://doi.org/10.1109/IROS.1992.594498 4. Ishii, M., et al.: A virtual work space for both hands manipulation with coherency between kinesthetic and visual sensation. In: Proceedings of the Fourth International Symposium on Measurement and Control in Robotics (1994). https://ntrs. nasa.gov/citations/19950009569 5. Sato, M.: SPIDAR and virtual reality. In: Proceedings of the 5th World Automation Congress, pp. 17–23 (2002). https://doi.org/10.1109/WAC.2002.1049515 6. Fan, L., et al.: Development of an integrated haptic sensor system for multimodal human-computer interaction using ultrasonic array and cable robot. IEEE Sens. J. 22(5), 4634–4643 (2022). https://doi.org/10.1109/JSEN.2022.3144888 7. Gallo, S., et al.: Towards multimodal haptics for teleoperation: design of a tactile thermal display. In: 12th IEEE International Workshop on Advanced Motion Control (AMC), pp. 1–5 (2012). https://doi.org/10.1109/AMC.2012.6197145 8. Wang, D., Ohnishi, K., Xu, W.: Multimodal haptic display for virtual reality: a survey. IEEE Trans. Ind. Electron. 67(1), 610–623 (2020). https://doi.org/10. 1109/TIE.2019.2920602 9. Saint-Aubert, J., Regnier, S., Haliyo, S.: Cable driven haptic interface for colocalized desktop VR. In: 2018 IEEE Haptics Symposium (HAPTICS), pp. 351–356 (2018). https://doi.org/10.1109/HAPTICS.2018.8357200 10. Saint-Aubert, J., et al.: Pre-calibrated visuo-haptic co-location improves execution in virtual environments. IEEE Trans. Haptics 13(3), 588–599 (2020). https://doi. org/10.1109/TOH.2019.2957801 11. Zanotto, D., et al.: Sophia-3: a semiadaptive cable-driven rehabilitation device with a tilting working plane. IEEE Trans. Rob. 30(4), 974–979 (2014). https://doi.org/ 10.1109/TRO.2014.2301532 12. Akahane, K., Hyun, J., Kumazawa, I., Sato, M.: Two-handed multi-finger stringbased haptic interface SPIDAR-8. In: Galiana, I., Ferre, M. (eds.) Multi-Finger Haptic Interaction. Springer, London (2013). https://doi.org/10.1007/978-1-44715204-0 6 13. Chibani, D., Achour, N., Daoudi, A.: SPIDAR-welder a haptic interface for virtual welding training. In: 1st International Conference on Communications, Control Systems and Signal Processing (CCSSP), pp. 293–297 (2020). https://doi.org/10. 1109/CCSSP49278.2020.9151477 14. Faure, C., et al.: Adding haptic feedback to virtual environments with a cabledriven robot improves upper limb spatio-temporal parameters during a manual handling task. IEEE Trans. Neural Syst. Rehab. Eng. 28(10), 2246–2254 (2020). https://doi.org/10.1109/TNSRE.2020.3021200 15. Rushton, M., Khajepour, A.: Variable-structure cable-driven parallel robots. In: Gouttefarde, M., Bruckmann, T., Pott, A. (eds.) CableCon 2021. MMS, vol. 104, pp. 206–214. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-757892 17 16. Frad, M., et al.: SPIDAR calibration based on regression methods. In: Proceedings of the 11th IEEE International Conference on Networking, Sensing and Control, pp. 679–684 (2014). https://doi.org/10.1109/ICNSC.2014.6819707 17. Murakami, T., Yu, F., Ohnishi, K.: Torque sensorless control in multidegreeof-freedom manipulator. IEEE Trans. Industr. Electron. 40(2), 259–265 (1993). https://doi.org/10.1109/41.222648

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18. Ortega, M., et al.: A six degree-of-freedom god-object method for haptic display of rigid bodies with surface properties. IEEE Trans. Vis. Comput. Graph. 13(3), 458–469 (2007). https://doi.org/10.1109/TVCG.2007.1028 19. JASP Homepage. https://jasp-stats.org/. Accessed 9 Mar 2023

Author Index

A Aertbeliën, Erwin Albayrak, Özlem Ament, Christoph Ankaralı, M. Mert Aoustin, Yannick

184 197 173 197 357

B Barbot, Jean-Pierre 357 Barrelet, Cyril 430 Behroozi, Foroogh 308 Beitelschmidt, Michael 273 Bernstein, David 273 Bieber, Jonas 273 Boumann, Roland 209 Bouzgarrou, Chedli 69 Bruckmann, Tobias 209 Bruyninckx, Herman 184 Bunker, Keegan 369 C Cardou, Philippe 308 Caro, Stéphane 55, 221, 308, 321, 357, 381 Carretero, Juan Antonio 321 Carricato, Marco 3, 149 Caverly, Ryan J. 369 Chaumette, François 221 Chaumont, Marc 430 Chevallereau, Christine 121, 234, 332 Chevrel, Philippe 55 Claveau, Fabien 55 Coevoet, Eulalie 55 Creuze, Vincent 430 Culla, David 430 D Decré, Wilm 184 Deroo, Boris 184 Duan, Jinhao 283 Durali, Laaleh 397

Duriez, Christian 55 Durmaz, Atakan 197 E Ennaiem, Ferdaws 295 Ernst, Marc O. 407 F Fabritius, Marc 407 Ferrari, Nicola 430 Franke, Jörg 344 G Gabaldo, Sara 3 Gan, Dongming 40 Gorrotxategi, Jose 430 Gouttefarde, Marc 134, 149, 430 Guo, Yaoxin 249 H Hamann, Marcus 173 Hayashi, Kazuki 419 Hervé, Pierre-Elie 430 Höpfner, Valentin 173 I Idà, Edoardo 3, 149 Igarashi, Hiroshi 442 Izard, Jean-Baptiste 109, 134 J Jamshidianfar, Hamed 397 Jeziorek, Christoph 209 Jin, Lei 40 Juraši´c, Philipp 407 K Khajepour, Amir 397 Khaw, Ing Tien 97 Kraus, Werner 407

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Caro et al. (Eds.): CableCon 2023, MMS 132, pp. 455–456, 2023. https://doi.org/10.1007/978-3-031-32322-5

456

Author Index

L Laribi, Med Amine 295 Lau, Darwin 249 Lehnertz, Christian 407 Liu, Hanqing 283 Lv, Jingang 283 M Martin, Christoph 407 Mathis, Andrew C. 321 Mattioni, Valentina 149 Mendez, Sergio Torres 397 Meng, Xiangjun 283 Merlet, Jean-Pierre 30 Michel, Loic 357 Moriniere, Boris 381 Moussa, Karim 55 Muralidharan, Vimalesh 332 N Nakka, Sanjeevi 16 Nemoto, Takeru 344 Nguyen, Vinh L. 369 Nomanfar, Pegah 161 Notash, Leila 161 O Oyarzabal, Arkaitz 430 P Patel, Samir 369 Pazooki, Alireza 397 Pedemonte, Nicolò 221, 381 Poitrimol, Bastien 442 Pott, Andreas 261, 407 Pott, Peter P. 97 Q Qi, Ronghuai

397

R Rasheed, Tahir 357 Reichenbach, Thomas 261 Reitelshöfer, Sebastian 344 Rodriguez, Mariola 430

Rothfischer, Lukas 344 Rousseau, Thomas 221 Rushton, Mitchell 397 S Sallé, Damien 430 Sandoval, Juan 295 Sanford, Paul W. 321 Schäfer, Max B. 97 Shao, Zhufeng 283 Soltani, Amir 397 Stierstorfer, Richard 344 Stoll, Johannes T. 407 Suarez Roos, Adolfo 381 Subsol, Gérard 430 Sugahara, Yusuke 419 T Taha, Tarek 40 Takeda, Yukio 419 Tempier, Olivier 430 Testard, Nicolas J. S. 121, 234, 332 Thieffry, Maxime 55 Trautwein, Felix 261 U Ünal, Perin 197 V Vashista, Vineet 16 Verl, Alexander 261 W Walter, Jonas 344 Weiland, Sophie 97 Wenger, Philippe 121, 234, 332 Worbs, Lukas 97 X Xiong, Hao 82 Xu, Yuchen 82 Z Za¸ke, Zane 381 Zhang, Zhaokun 283