Gravity Compensation in Robotics 3030957497, 9783030957490

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Mechanisms and Machine Science

Vigen Arakelian   Editor

Gravity Compensation in Robotics

Mechanisms and Machine Science Volume 115

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, 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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Vigen Arakelian Editor

Gravity Compensation in Robotics

Editor Vigen Arakelian LS2N Institut National des Sciences Appliquées Rennes, France

ISSN 2211-0984 ISSN 2211-0992 (electronic) Mechanisms and Machine Science ISBN 978-3-030-95749-0 ISBN 978-3-030-95750-6 (eBook) https://doi.org/10.1007/978-3-030-95750-6 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 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

The actuator power required to resist joint torque caused by the weight of robot links can be a significant problem. Gravity compensation is a well-known technique in robot design to achieve equilibrium throughout the range of motion and as a result to reduce the loads on the actuator. Thus, gravity compensation is beneficial, by which a robotic system can be operated with relatively small actuators generating less torque. Therefore, it is desirable and commonly implemented in many situations. Nature of the forces that must compensate gravity and its emplacement in the robotic systems may be diverse: elastic spring forces, counterweights, pneumatic or hydraulic cylinders, electromagnetic forces, etc. The compensation systems can be mounted on the links of the initial robotic structures or on the auxiliary linkage connected with them. This work presents new research results in the field of gravity compensation in robotic systems. It includes the research results obtained in France, Australia, Russia, Korea, Belgium, Armenia and Italia. Various problems were considered: gravity compensation of planar articulated robotic manipulators; the stiffness modeling of manipulators with gravity compensators; the multi-degree-of-freedom counterbalancing; the design of actuators with partial gravity compensation; a cable-driven robotic suit with gravity compensation for load carriage; various compensation systems for medical cobots and assistive devices; gravity balancing of parallel robots. The book includes both theoretical and experimental research results. The editor thanks the authors who have contributed with various and interesting research results on several issues of gravity compensation. He hopes that the present book will be useful to the readers and it will expand knowledge in the field of robot design. Rennes, France December 2021

Vigen Arakelian

v

Contents

A Modularization Approach for Gravity Compensation of Planar Articulated Robotic Manipulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vu Linh Nguyen and Chin-Hsing Kuo

1

Stiffness Modeling for Gravity Compensators . . . . . . . . . . . . . . . . . . . . . . . . Alexandr Klimchik and Anatol Pashkevich

27

Multi-DOF Counterbalancing and Applications to Robots . . . . . . . . . . . . . Jae-Bok Song, Hwi-Su Kim, and Won-Bum Lee

73

Series Parallel Elastic Actuator: Variable Recruitment of Parallel Springs for Partial Gravity Compensation . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 Furnémont Raphaël, Glenn Mathijssen, Tom Verstraten, Bram Vanderborght, and Dirk Lefeber Design, Optimization and Control of a Cable-Driven Robotic Suit for Load Carriage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 Yang Zhang and Vigen Arakelian Tool Compensation for a Medical Cobot-Assistant . . . . . . . . . . . . . . . . . . . . 147 Juan Sandoval and Med Amine Laribi Design of Statically Balanced Assistive Devices . . . . . . . . . . . . . . . . . . . . . . . 165 S. D. Ghazaryan, M. G. Harutyunyan, Yu. L. Sargsyan, and V. Arakelian Design of Multifunctional Assistive Devices with Various Arrangements of Gravity Compensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 S. D. Ghazaryan, M. G. Harutyunyan, Yu. L. Sargsyan, N. B. Zakaryan, and V. Arakelian

vii

viii

Contents

Gravity Balancing of Parallel Robots by Constant-Force Generators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 Giovanni Mottola, Marco Cocconcelli, Riccardo Rubini, and Marco Carricato

A Modularization Approach for Gravity Compensation of Planar Articulated Robotic Manipulators Vu Linh Nguyen and Chin-Hsing Kuo

Abstract A modularization approach for gravity compensation of planar articulated robotic manipulators is addressed in this chapter. The modularization is realized by installing a set of gear-spring modules (GSMs) on the manipulator so that the gravitational torques can be eliminated by the spring torques applied on its joints. The design concept and mathematical formulation for the gravity compensation of the GSM are presented. Then, the gravity compensation of the manipulator by using the GSMs is proposed. Evaluation criteria for the performance of the design are suggested. Last, selected examples of planar-type serial and quasi-serial manipulators are provided for illustrating the design concepts and their performances. Keywords Gravity compensation · Static balancing · Robotic manipulator · Zero-stiffness mechanism

1 Introduction The gravity compensation of robotic manipulators and mechanisms has been an attractive research theme in the last decades [1–3]. Gravity compensation aims to lessen the effect of gravity on the robot caused by the masses of its links and payload. A perfect gravity compensation can completely eliminate the gravitational torques at the robot joints, allowing the robot to maintain itself at any configuration with zero input torques from the actuators. While an approximate gravity compensation only decreases the gravitational torques, it is necessary to use a small effort to keep the robot stationary at a configuration. For low-speed robotic manipulation, gravity compensation becomes more beneficial for the robot because the gravitational torques contribute the most to its actuation torques [1]. Along with reducing the actuation V. L. Nguyen (B) Department of Mechanical Engineering, National Chin-Yi University of Technology, Taichung 411030, Taiwan e-mail: [email protected] C.-H. Kuo School of Mechanical, Materials, Mechatronic and Biomedical Engineering, University of Wollongong, Wollongong, NSW 2522, Australia © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 V. Arakelian (ed.), Gravity Compensation in Robotics, Mechanisms and Machine Science 115, https://doi.org/10.1007/978-3-030-95750-6_1

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V. L. Nguyen and C.-H. Kuo

torques, the gravity compensation of robotic manipulators can provide other benefits, such as reducing energy consumption, the sizes of the actuators, the structural compliance of the robot, and improving safety and dynamic response [2]. Gravity compensation of robotic manipulators is made by attaching store energy elements, mainly counterweights and springs, to the robot to cancel the gravitational torques caused by the link masses and payload [1, 4]. In static equilibrium, the potential energy of a robotic manipulator with gravity compensation is constant at any configuration. This condition is usually taken to determine the parameters of the energy elements and their attachment to the robot [5, 6]. In particular, counterweight methods aim to keep the total center of mass of the system (including the masses of the robot links, payload, and added counterweights) stationary during operation [7–9]. Spring methods exploit the elastic energy stored by the springs to cancel the variation of the potential energy of the robot [10–14]. As compared with counterweights, the use of springs is more preferred in robot design because it adds less mass and inertia to the robot [1–3]. However, the most challenging issues when using the spring design regard to the complex structure and large volume requirement for implementing the springs to the robot [1, 2, 4]. Various spring methods have been proposed for robotic manipulators, where the spring implementation can be achieved via the noncircular pulley and wire mechanism [15], cam-follower mechanism [16], Cardan gear mechanism [17], noncirculargear mechanism [18], or inverted slider-crank mechanism [19]. However, the abovedescribed mechanisms may undergo their own limitations, such as a large volume required to implement the springs, many auxiliary components added, low durability induced by the pulley and wire transmission, inadequate motion accuracy occurred, or friction. More importantly, they were only designed for a single degree-of-freedom (DoF) robotic manipulator. Successful spring designs for multi-DoF robotic manipulators can be noticed in [20–26]. Their generic design approach is to take advantage of multiple parallelogram linkages [20, 21] to decouple the gravity effect at the robot joints. Then, the gravity compensation at each joint can separately be performed as a single-DoF gravity compensation. The design approach is still workable for virtual parallelograms, such as multiple pulley-wire systems [22–24] and multiple bevelgear trains [25, 26]. Note that this design approach still has several limitations. For instance, the use of parallelogram linkages may limit the workspace and increase the total weight of the robot. Multiple pulley-wire systems often require a relatively large volume to implement the springs and wires and reduce the durability of the entire robot. Multiple bevel-gear trains may induce a cumbersome and bulky mechanical structure, which may significantly increase the total weight and inertia of the robot. To advance gravity compensation in robotics, this chapter presents a modularization approach for gravity compensation of multi-DoF planar articulated robotic manipulators by using the GSMs. This approach can remove the use of multiple parallelogram linkages and their virtual designs. The performance of the presented gravity compensation design is evaluated through numerical examples.

A Modularization Approach for Gravity Compensation …

3

2 The Basic Concept of Gravity Compensation Design This section gives a simple example to illustrate the gravity compensation design of a mechanical system. The determination of the parameters and the characteristics of the system with gravity compensation are also explained. Considering a single-DoF rotating link with a mass m located at point C, i.e., the mass center of the link, and a distance s from the revolute joint O. To compensate for the gravity effect of the rotating link, a zero-free-length (ZFL) or ideal spring with a stiffness coefficient k is attached to the link at point B, as shown in Fig. 1. In static conditions, the gravity compensation of the rotating link is to make either its total potential energy constant or total torque at joint O equal to zero for any rotation angle θ. The two above-described conditions will be taken to determine the stiffness coefficient of the spring for gravity compensation as follows. Constant-Energy Approach. Assume the origin of a coordinate frame (x, y) is placed at the center of the revolute joint O. The location of joint O is defined by a height h from the datum, which is used to measure potential energy. Then, the potential energy of the rotating mass, denoted W w , can be determined as: Ww = mg(h + s cos θ )

(1)

where g represents the gravitational acceleration (g = 9.81 m/s2 ). Recall that for a ZFL spring, the elastic force generated by the spring is linearly dependent on its total length [5]. Then, the potential energy of the spring, denoted W s , with an instantaneous length L can be expressed as: Ws =

 1 2 1  k L = k la2 + lb2 − 2la lb cos θ 2 2

(2)

y

Fig. 1 Gravity compensation design of a 1-DoF rotating link

C

A Fs

k β

la θ

lb

mg B s

x

O h Datum

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V. L. Nguyen and C.-H. Kuo

where l a and l b stand for the distances OA and OB, respectively. From Eqs. (1) and (2), the total potential energy of the system, denoted W total , is written as: Wtotal = Ww + Ws

 1  = mg(h + s cos θ ) + k la2 + lb2 − 2la lb cos θ 2  1  = (mgs − kla lb ) cos θ + mgh + k la2 + lb2 2

(3)

The total potential energy W total in Eq. (3) is constant for any joint angle θ if: ∂ Wtotal =0 ∂θ

(4)

Substituting Eq. (3) into Eq. (4) can obtain: ∂ Wtotal = (kla lb − mgs) sin θ = 0 ∂θ

(5)

equivalently k=

mgs la lb

(6)

The result above indicates that if the spring stiffness k is determined by Eq. (6), the total potential energy W total (Eq. (3)) is constant for any input angle θ. Zero-torque approach. Let T w denote the gravitational torque at joint O induced by the rotating mass m. The torque T w can be expressed as: Tw = −mgs sin θ

(7)

For a ZFL spring, the elastic force caused by the spring, denoted F s , along its longitudinal axis can be determined as: Fs = k L

(8)

where L is the length of the spring (L = AB). The elastic force F s induces a spring torque, denoted T s , at joint O as: Ts = OB × Fs  = lb Fs sin β

(9)

where OB and Fs represent the position vector from point O to B and the elastic force vector, respectively; ||.|| stands for the Euclidean norm of a vector. Substituting Eq. (8) into Eq. (9) can yield:

A Modularization Approach for Gravity Compensation …

5

Ts = lb k L sin β

(10)

Consider the triangle OAB, the following trigonometric relation can be derived as: sin β sin θ = L la

(11)

L sin β = la sin θ

(12)

or

By substituting Eq. (12) into Eq. (10), it can yield: Ts = kla lb sin θ

(13)

When dynamic torques are neglected, the total torque at joint O, denoted T total , is equal to zero for any joint angle θ, i.e.: Ttotal = Ts + Tw = 0 for any θ

(14)

Substituting Eqs. (7) and (13) into Eq. (14) can yield: (−mgs + kla lb ) sin θ = 0

(15)

As can be seen, Eq. (15) is satisfied for any input angle θ if the spring stiffness k in Eq. (6) is taken. Characteristics. The derived results above show that the constant-energy and zerotorque approaches yield the same spring stiffness k (Eq. (6)) for gravity compensation. The potential energies and torques of the single-DoF rotating link with gravity compensation are plotted in Figs. 2a and 2b, respectively. The figures illustrate that the total potential energy of the system W total is constant while the total torque T total is zero for any joint angle θ from 0 to 360°.

3 Gear-Spring Module (GSM) Concept In the previously illustrated example in Sect. 2, the gravity compensation of a 1-DoF rotating link was done with a ZFL spring. In practice, such a spring can be realized by increasing the initial tension of a non-ZFL (practical) spring via a pulley-wire mechanism or hiding the free length of a non-ZFL spring via a guiding element [5]. These two embodiments may cause friction and dimensional errors and need a practical spring with a large displacement, resulting in practical inconvenience. Thus, the use of a non-ZFL spring becomes more attractive in gravity compensation

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Fig. 2 Characteristics of a 1-DoF rotating link with gravity compensation: (a) potential energy curves and (b) torque curves

design. The latter work of this section will present a gravity compensation design with a non-ZFL spring, known as the GSM.

3.1 Kinematic Structure The kinematics of the GSM is illustrated in Fig. 3. The GSM is essentially a geared five-bar mechanism consisting of five links connected by three revolute joints, one gear joint, and one prismatic joint. Link 5 represents the main link that rotates about a pivot point O and carries a weight to be gravity compensated. A pair of spur gears are attached to the mechanism in which gear 1 (link 1) is fixed to the ground while gear 2 (link 2) is connected to the rotating link by a gear joint H. A slider (link 4) is placed along with the rotating link, and it is connected to a connecting rod (link 3) by a revolute joint B. The connecting rod is connected to gear 2 by a revolute joint A. To determine the DoF of a mechanism with joint constraint, the Kutzbach-Grübler criterion [27] can be used as: DoF = λ(n − j − 1) +



fi − f p

(16)

i

where λ, n, j, f i , and f p represent the DoFs of the space, the number of links, the number of joints, the degrees of relative motion permitted by joint i, and the number of passive DoFs in the mechanism, respectively. Applying Eq. (16) to the geared five-bar mechanism in Fig. 3 can yield: DoF = 3(5 − 5 − 1) + 5 − 1 = 1

(17)

A Modularization Approach for Gravity Compensation …

7

Mass m

Gravity

Spring k

Fr (4) Slider

C

φc Fs

B

mg

B (5) Rotating link (3) Connecting rod

φ0 ψ H

(2) Gear 2

α

Ts

r2

H θ

r2a

y x

φc

γ

A

O

r3

A

(1) Gear 1

ds

ψ T2

O

r1 (a)

(b)

Fig. 3 Kinematics of the GSM: (a) initial configuration and (b) instantaneous configuration

Equation (17) implies that the GSM has one DoF of motion.

3.2 Parameter Determination Let r 1 and r 2 denote the pitch radii of gears 1 and 2, respectively; let T s and T 2 represent the spring torques exerted on the rotating link and gear 2, respectively; let θ and α denote the joint angles of the rotating link and gear 2, respectively. Then, the gear transmission ratio, denoted ng , can be expressed as: ng =

r2 T2 θ = = r1 Ts α

(18)

Let m denote the equivalent mass of the GSM with a distance s from the mass center C to the pivot point O. Then, the gravitational torque induced by the weight, denoted T w , can be determined as: Tw = −mgs sin θ

(19)

The elastic forces generated by the spring along the slider axis, denoted F s , and along the connecting rod axis, denoted F r , can be calculated as: Fs = k(d0 + ds )

(20)

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V. L. Nguyen and C.-H. Kuo

Fr =

Fs cos ϕc

(21)

where d 0 , d s , and k represent the spring compression at the initial position, the slider displacement, and the stiffness coefficient of the spring, respectively. In Eq. (20), the slider displacement d s can be determined as: ds = s1 − s0

(22)

s1 = r3 cos ϕc − r2a cos(ψ + α)

(23)

where

s0 =

 2 r32 + r2a − 2r3r2a cos(ψ − ϕ0 )

ϕc = sin

−1



r2a sin(ψ + α) r3

α=

(24)



θ ng

(25) (26)

In Eqs. (22)–(26), s1 and s0 represent the distances HB at the initial position (θ = 0) and an instantaneous position (θ = 0), respectively; ϕ 0 and ϕ c represent the initial and instantaneous deflection angles of the connecting rod from the rotating link, respectively; r 2a , r 3 , and ψ stand for the distance HA, the length of the connecting rod AB, and the installation angle of gear 2, respectively. The elastic force F r along the connecting rod can cause a torque T 2 on gear 2 as: T2 = HA × Fr  = r2a Fr sin γ

(27)

where HA, Fr , and γ stand for the position vector from point H to A, the elastic force vector along the connecting rod, and the angle between HA and AB, respectively. As shown in Fig. 3, a trigonometric relation in the triangle HAB can be obtained as: sin γ sin(ψ + α) = s1 r3

(28)

By substituting Eqs. (20), (21), and (28) into Eq. (27), it can yield: T2 =

kr2a (d0 + ds )s1 sin(ψ + α) r3 cos ϕc

(29)

A Modularization Approach for Gravity Compensation …

9

From Eqs. (18) and (29), the spring torque T s on the rotating link can be written as: Ts =

T2 kr2a (d0 + ds )s1 sin(ψ + α) = ng n g r3 cos ϕc

(30)

As presented in Eq. (14), the sum of the spring torque and the gravitational torque at the pivot joint O should always be zero under gravity compensation. By substituting Eqs. (19) and (30) into Eq. (14), the gravity compensation condition can be expressed as: kr2a (d0 + ds )s1 sin(ψ + α) − mgs sin θ = 0 for any θ n g r3 cos ϕc

(31)

Because the joint angle θ is varied with the configuration of the GSM, it is difficult to find an explicit solution for Eq. (31). In this circumstance, some assumptions on the geometry and design parameters of the GSM will be made. First, assume that the free length of the spring is equal to its initial length when the GSM is situated at the initial position (θ = 0). This assumption leads to: d0 = 0

(32)

Next, the length of the connecting rod is much larger than the distance HA, i.e., r 3 > > r 2a . As seen in Fig. 3, one can have ϕ c ≈ 0, ϕ 0 ≈ 0, and r 3 ≈ HB. Accordingly, Eqs. (22) and (23) can be rewritten as: ds = r3 cos ϕc − r2a cos(ψ + α) −

 2 r32 + r2a − 2r3r2a cos(ψ − ϕ0 )

= −r2a cos(ψ + α) s1 = r 3

(33) (34)

Substituting Eqs. (26), (32), (33), and (34) into Eq. (31) can give:     θ θ kr2a −r2a cos ψ + sin ψ + − mgs sin θ = 0 ng ng ng

(35)

equivalently  2 kr2a 2θ sin 2ψ + − π − mgs sin θ = 0 2n g ng

(36)

Because the installation angle ψ and the gear transmission ratio ng are user-defined parameters, it is possible to simplify Eq. (36) as:

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V. L. Nguyen and C.-H. Kuo



2 kr2a − mgs sin θ = 0 4

(37)

when ψ and ng are defined as: π 2

(38)

ng = 2

(39)

ψ=

As can be seen, Eq. (37) is satisfied for any joint angle θ if the spring stiffness k is selected as: k=

4mgs 2 r2a

(40)

In sum, the gravity compensation condition of the GSM can be achieved when the spring stiffness k is calculated by Eq. (40) with d 0 = 0, ng = 2, ψ = π /2, and r 3 > > r 2a .

3.3 A Numerical Example Following the design idea of the GSM for gravity compensation, this section provides a numerical example for showing its performance. The parameters of the GSM are listed in Table 1. From these data, the gravitational torque T w , spring torque T s , and total torque T total are plotted in Fig. 4a. It shows that, with the compensation using the spring torque, the total torque approaches nearly zero throughout a range of joint angle θ = [0, 360]°. The maximum absolute value of the total torque is about 0.076 N-m, while that of the gravitational torque is 1.962 N-m. The obtained results indicate that the variation of the gravitational torque with the spring attached is much lower than without the spring. The reduction in the torque variation is about 96.1%. Moreover, the mass m of the GSM can be changed if the spring stiffness k is re-selected, and their relationship is linearly dependent, as illustrated in Fig. 4b. Table 1 Parameters of the gear-spring module (GSM) ψ (deg)

ng

r 1 (m)

r 2 (m)

r 2a (m)

r 3 (m)

s (m)

m (kg)

k (N/mm)

90

2

0.02

0.04

0.01

0.3

0.2

1

78.48

A Modularization Approach for Gravity Compensation …

11

Fig. 4 Results of the gear-spring module (GSM): (a) torque curves and (b) spring stiffness varying with the applied mass

4 Gravity Compensation of Robotic Manipulators This section presents the modularization approach for gravity compensation of multiDoF articulated robotic manipulators using the GSMs. In particular, this study focuses on planar serial and quasi-serial manipulators. Note that the main difference of a quasi-serial manipulator compared with its serial counterpart is that the quasi-serial manipulator undergoes kinematic parallelogram constraint. This constraint allows relocating the actuators at the elbow joints to the base joints. The inverse and direct kinematic models of these two types can be derived similarly. Therefore, the gravity compensation will only be illustrated via a general serial manipulator. Then, numerical examples are given to demonstrate the effectiveness of the design approach for both serial and quasi-serial manipulators. The performances of these two types are also compared.

4.1 Design Approach The gravity compensation design of a general multi-DoF planar serial manipulator using the GSMs is illustrated in Fig. 5. The manipulator is constructed by n consecutive links connected by revolute joints Oi (i = 1, 2,…, n) and integrated with n GSMs for gravity compensation. Each GSM i is attached to the manipulator such that the rotating link of the i-th GSM aligns with the longitudinal axis of link i. The first gear of the i-th GSM is centered at the revolute joint Oi and is fixed to link (i – 1), where link 0 stands for the ground.

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V. L. Nguyen and C.-H. Kuo Ts2

Gravity

O2

k1 A1 Ts1

Link 1

Tsn

A2 GSM 3 A3

GSM 2 B2

m1g

B1

θ1

θ2 H2

s1

H3

s2 Link 2

GSM 1 y O1

An

mng Link n

θ3

O3

θn Hn

Link n-1

k2 m2g

H1

On

kn-1

sn

GSM n kn Bn E

Ts3

x

Fig. 5 Gravity compensation design of a general multi-DOF planar serial manipulator using the GSMs

Let l i and θ i denote the length of link i and the joint angle of the manipulator at joint Oi , respectively. Then, the position of the end effector E is represented by (x E , yE ), which are determined as:

n   θi x E = l1 sin θ1 + l2 sin(θ1 + θ2 ) + ... + ln sin

(41)

1

n   θi y E = l1 cos θ1 + l2 cos(θ1 + θ2 ) + ... + ln cos

(42)

1

Gravitational Torques. For an articulated serial manipulator with n links, the gravitational torque at joint Oi , denoted T wi , is induced by not only the mass of link i but also the masses of the later links (i + 1, i + 2,…, n). The gravitational torque T wi is also accumulated by the gravitational torque at the later joint [27]. For example, the gravitational torque at joint On , denoted T wn , is expressed as: Twn

n   = −m n sn g sin θi

(43)

1

where mn and sn represent the mass of link n and the distance from the mass center of link n to joint On , respectively. For link (n – 1), the gravitational torque at joint On–1 , denoted T w(n–1) , is written as: Tw(n−1)

n−1   = −(m n−1 sn−1 + m n ln−1 )g sin θi + Twn

(44)

1

Substituting Eq. (43) into Eq. (44) can give: Tw(n−1)

n 

n−1    = −(m n−1 sn−1 + m n ln−1 )g sin θi − m n sn g sin θi 1

1

(45)

A Modularization Approach for Gravity Compensation …

13

Similarly, the gravitational torques at joints On–2 , On–3 ,…, and O1 can be determined in sequence. By combining all the derived gravitational torques, a general formula for determining the gravitational torque T wi at joint Oi is derived as: Twi = −

n 

m i si +

n 

i

  m j li g sin

i 

 θu

(46)

1

i+1

where j = i + 1, i + 2,…, n and u = 1, 2,…, i. Spring Torques. Considering joint Oi where GSM i is attached, the spring torque at this joint, denoted T si , should include the spring torque caused by the i-th GSM and the spring torque at the latter joint, i.e., joint Oi+1 . For example, the spring torque at joint On , denoted T sn , can be derived from Eqs. (30), (36), and (39) as: Tsn =

2 kn r2an sin(θn + 2ψn − π ) 4

(47)

where k n , r 2an , and ψ n stand for the stiffness coefficient of the spring n, the distance H n An , and the installation angle of the n-th GSM, respectively. For link (n – 1), the spring torque at joint On–1 , denoted T s(n–1) , is expressed as: Ts(n−1) =

2 kn−1r2a(n−1)

sin(θn−1 + 2ψn−1 − π ) + Tsn

4

(48)

where k n–1 , r 2a(n–1) , and ψ n–1 stand for the stiffness coefficient of the spring (n – 1), the distance H n–1 An–1 , and the installation angle of the (n–1)-th GSM, respectively. Then, substituting Eq. (47) into Eq. (48) can give: Ts(n−1) =

2 kn−1r2a(n−1)

4

sin(θn−1 + 2ψn−1 − π ) +

2 kn r2an sin(θn + 2ψn − π ) (49) 4

The spring torques at joints On–2 , On–3 ,…, O1 can similarly be determined in sequence. Then, the combination of all the derived spring torques can give a general formula for determining the spring torque T si at joint Oi as: Tsi =

n  ki r 2

2ai

i

4

sin(θi + 2ψi − π)

(50)

Gravity Compensation Condition. For gravity compensation at a given joint Oi , the sum of the spring torque T si and the gravitational torque T wi at this joint must always be zero. This condition is expressed as: Tsi + Twi = 0 for any θi

(51)

14

V. L. Nguyen and C.-H. Kuo

Then, substituting Eqs. (46) and (50) into Eq. (51) can give: n  ki r 2

2ai

i

4

sin(θi + 2ψi − π ) −



n 

m i si +

i

n 

  m j li g sin

i+1

i 

 θu

=0

1

(52) Equation (52) can be satisfied for any joint angle θ i if the two following conditions are simultaneously obtained, i.e.:

n   2  ki r2ai − m i si + m j li g = 0 4 i+1 i 

θi + 2ψi − π −

θu = 0

(53)

(54)

1

Equations (53) and (54) are rewritten as: 

 4 m i si +

n 

 m j li g

i+1

ki =

2 r2ai i 

ψi =

(55)

(θu ) − θi + π

1

2

(56)

From Eq. (56), one can see that the installation angle ψ i depends on the joint angles θ 1 , θ 2 ,…, θ i , and these joint angles are varied with the configuration of the manipulator. Practically, it is very difficult to obtain such a variable angle ψ i unless an actuator is taken to control it. In this circumstance, a more practical solution is to assume a desired configuration C Θ where the gravity compensation condition is satisfied. If the desired configuration C Θ is expressed by a set of joint angles Θ u , Eq. (56) can be rewritten as: i−1 

ψi =

( u ) + π

1

2

(57)

Equation (57) shows that the installation angles ψ i of the GSMs are only determined once the desired configuration C Θ is defined. As a result, the gravitational torques of the manipulator at other configurations may not completely be canceled. However, it could be a feasible design for approximate gravity compensation as long as a set of installation angles ψ i are appropriately chosen.

A Modularization Approach for Gravity Compensation …

15

In sum, the gravity compensation of multi-DoF planar serial manipulators using the GSMs can be achieved via the spring stiffnesses k i determined by Eq. (55) and the installation angles ψ i by Eq. (57) with design choices d 0i = 0, ngi = 2, and r 3i > > r 2ai .

4.2 Evaluation Criteria The initial design purpose for the gravity compensation of robotic manipulators is to reduce their actuation torques. For a specific task with an identified trajectory of the end effector of the manipulator, it is possible to evaluate the power and energy consumption to complete this task. This work allows having a comprehensive on the effectiveness of the proposed design approach for gravity compensation. Thus, criteria for evaluating the actuation torque, power, and energy reductions are all presented. Torque Reduction Rate. Let T mi and T msi denote the motor torques applied at joint Oi to maintain the manipulator at an instantaneous configuration when the GSMs are detached and attached, respectively. Under frictionless and static conditions, the motor torques T mi and T msi can be determined as: Tmi = −Twi

(58)

Tmsi = −Tsi − Twi

(59)

In Eqs. (58) and (59), T mi and T msi can also stand for the uncompensated and compensated torques at joint Oi , respectively. From Eqs. (58) and (59), to evaluate the effectiveness of the gravity compensation using the GSMs, the torque reduction rate at joint Oi , denoted δ ti , is defined as:   max(|Tmsi |) δti = 1 − × 100% max(|Tmi |)

(60)

Note that the torque reduction rate δ ti in Eq. (60) merely considers the peak torques of T mi and T msi for a series of joint angles θ i . These angles can be found from an operation task or designated values. Power Reduction Rate. Assume that the power losses caused by electrical heating and frictions in mechanical transmission are neglected. The power of the manipulator without the GSMs, denoted Pm , is calculated as: Pm =

n    Tmi θ˙i  1

(61)

16

V. L. Nguyen and C.-H. Kuo

where θ˙i represents the joint rate or angular velocity of the i-th actuator. Next, the power of the manipulator when the GSMs are attached, denoted Pms , is computed as: Pms =

n    Tmsi θ˙i 

(62)

1

Similarly, Pm and Pms can also be called the uncompensated and compensated power of the manipulator, respectively. From Eqs. (61) and (62), one can see that the powers Pm and Pms are dependent on the joint rates of the manipulator, and these joint rates can be found from a prescribed operation. In this situation, the peak power during operation is chosen to evaluate the power reduction of the manipulator. Accordingly, the power reduction rate, denoted δ p , is formulated as:   max(Pms ) × 100% δp = 1 − max(Pm )

(63)

Energy Reduction Rate. Similar to the derivation of the power reduction rate in Eq. (63), the energy reduction rate, denoted δ e , is computed as:  E ms × 100% δe = 1 − Em

(64)

where Em =

n 

E mi

(65)

E msi

(66)

Tmi θ˙i dt

(67)

Tmsi θ˙i dt

(68)

1

E ms =

n  1

t1 E mi = t0

t1 E msi = t0

In Eqs. (64)–(68), E m and E ms represent the energy consumption of the manipulator when the GSMs are detached and attached, respectively; E mi and E msi are the energy consumption of the actuator at joint Oi , respectively; t 0 and t 1 are the start time and end time of an operation, respectively.

A Modularization Approach for Gravity Compensation …

17

Table 2 Parameters of the 2-DoF planar serial manipulator m1 (kg)

m2 (kg)

s1 (m)

s2 (m)

l 1 (m)

l 2 (m)

5

3

0.5

0.4

1

0.8

Table 3 Parameters of the GSMs attached to the 2-DoF planar serial manipulator GSM i

ngi

r 1i (m)

r 2ai (m)

r 3i (m)

ψ i (deg)

k i (N/mm)

i=1

2

0.05

0.1

0.5

90

21.58

i=2

2

0.04

0.08

0.4

105, 112.5*

7.36

*105° and 112.5° for trajectories I and II, respectively

4.3 Illustrative Examples This section provides numerical examples to illustrate the effectiveness of the modularization approach for gravity compensation of multi-DoF articulated robotic manipulators using the GSMs. Specifically, a 2-DoF planar serial manipulator and its quasiserial counterpart are taken into account. These two manipulators are assumed to perform two pick-and-place operations, each characterized by a specific trajectory. The performances of the two manipulators are also compared. A 2-DoF Planar Serial Manipulator. The parameters of the manipulator and the GSMs are listed in Tables 2 and 3, respectively. The studied trajectories (P1 P2 P3 P4 and Q1 Q2 Q3 Q4 ) and the joint angles of the manipulator for tracking along these trajectories are illustrated in Figs. 6 and 7, respectively. The execution time for tracking along each trajectory is 8 s. The start point in each trajectory (i.e., point P1 in trajectory I and point Q1 in trajectory II) is chosen as a desired configuration. Based on Eqs. (55) and (57), the spring stiffnesses k i and the installation angles ψ i of the GSMs can be determined, as presented in Table 3. Figure 8 shows the motor torques of the manipulator along trajectory I. The peak uncompensated torque at joint θ 1 is about 46.8 N-m, while its compensatedcounterpart is 5.3 N-m. These results indicate that the gravity compensation can reduce the motor torque at joint θ 1 with a torque reduction rate of 88.7%. The peak uncompensated and compensated torques at joint θ 2 are 10.1 and 1.8 N-m, respectively, resulting in a torque reduction rate of 82.2% at this joint. For tracking along trajectory II, high torque reduction is also achieved via torque reduction rates of 92.6% and 70.5% at joints θ 1 and θ 2 , respectively, as shown in Fig. 9. A 2-DoF Planar Quasi-Serial Manipulator. The gravity compensation design of the manipulator using the GSMs is illustrated in Fig. 10. The installation of the GSMs on the quasi-serial manipulator is quite similar to its serial counterpart (see Fig. 6). The main difference is that, in the quasi-serial design, the first gear of GSM 2 is fixed to link 2 of the manipulator (rather than link 1 as the serial design in Fig. 6). With a parallelogram kinematic constraint (i.e., link 1 is parallel with link 4 and link 2 is parallel with link 3), the actuator at the elbow joint O2 can be relocated to the base joint O1 . As a result, two actuators are now situated at joint O1 and they are

18 Fig. 6 Pick-and-place operations with the 2-DoF planar serial manipulator: (a) trajectory I and (b) trajectory II

V. L. Nguyen and C.-H. Kuo

Gravity

60º 0.4 m

P3

P2

30º

1

0.2 m

P1 P4

y

Θ1 = 30º Θ2 = 120º

x (a)

0.2 m Q3

45º

Q4 2

0.4 m

45º Q2 y

Q1 Θ1 = 45º Θ2 = 90º

x (b)

Fig. 7 Joint angles of the 2-DoF planar serial manipulator: (a) trajectory I and (b) trajectory II

A Modularization Approach for Gravity Compensation …

19

Fig. 8 Motor torques of the 2-DoF planar serial manipulator tracking along trajectory I: (a) joint θ 1 and (b) joint θ 2

Fig. 9 Motor torques of the 2-DoF planar serial manipulator tracking along trajectory II: (a) joint θ 1 and (b) joint θ 2

decoupled, i.e., one drives link 1 while the other drives link 3. The masses of links 3 and 4 can be accumulated to those of links 2 and 1, respectively. The parameters of the quasi-serial manipulator and the GSMs attached to it are listed in Tables 4 and 5, respectively. The 2-DoF planar quasi-serial manipulator is also used for tracking along the trajectories, P1 P2 P3 P4 and Q1 Q2 Q3 Q4 shown in Fig. 6, and each takes 8 s. The joint angles of the manipulator during trajectory tracking are presented in Fig. 11. The start point in each trajectory (i.e., point P1 in trajectory I and point Q1 in trajectory II) is chosen as a desired configuration. The spring stiffnesses k i and installation angles ψ i for the quasi-serial manipulator can be determined similarly to those for its serial counterpart, and the results are listed in Table 5.

20

V. L. Nguyen and C.-H. Kuo

Fig. 10 Gravity compensation design of a 2-DoF planar quasi-serial manipulator using the gearspring modules (GSMs) Table 4 Parameters of the 2-DoF planar quasi-serial manipulator m1 (kg) m2 (kg) m3 (kg) m4 (kg) s1 (m) s2 (m) s3 (m) s4 (m) l 1 (m) l 2 (m) l 3 (m) 5

3

0.5

0.5

0.5

0.4

0.2

0.5

1

0.8

0.4

Table 5 Parameters of the GSMs attached to the 2-DoF planar quasi-serial manipulator GSM i

ngi

r 1i (m)

r 2ai (m)

r 3i (m)

ψ i (deg)

k i (N/mm)

i=1

2

0.05

0.1

0.5

90

22.56

i=2

2

0.04

0.08

0.4

90

5.52

Fig. 11 Joint angles of the 2-DoF planar quasi-serial manipulator: (a) trajectory I and (b) trajectory II

A Modularization Approach for Gravity Compensation …

21

The motor torques of the quasi-serial manipulator tracking along trajectory I are shown in Fig. 12. At joint θ 1 , the peak motor torque can reach up to 39.8 N-m, and it is reduced to 0.8 N-m with gravity compensation. Based on Eq. (60), a torque reduction rate of 98% can be obtained at this joint. The peak uncompensated and compensated torques at joint θ 2 are about 7.5 and 0.1 N-m, respectively, resulting in a torque reduction rate of 98.6% at this joint. Similarly, significant torque reduction rates at the actuation joints of the manipulator when tracking along trajectory II can also be achieved, being up to 98% and 97.3% at joints θ 1 and θ 2 , respectively, as shown in Fig. 13.

Fig. 12 Motor torques of the 2-DoF planar quasi-serial manipulator tracking along trajectory I: (a) joint θ 1 and (b) joint θ 2

Fig. 13 Motor torques of the 2-DoF planar quasi-serial manipulator tracking along trajectory II: (a) joint θ 1 and (b) joint θ 2

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V. L. Nguyen and C.-H. Kuo

Table 6 Performance comparison of the 2-DoF planar serial and quasi-serial manipulators Criteria Peak torque (N-m)

Peak power (W)

Energy consumption (J)

Uncompensated

Trajectory I

Trajectory II

Serial

Quasi-serial

Serial

Quasi-serial 48.1

56.9

47.4

57.4

Compensated

7.1

0.9

6.6

Reduction rate

87.5%

98.1%

88.5%

Uncompensated

7.81

5.46

8.9

Compensated

0.51

0.13

1.12

Reduction rate

93.5%

97.6%

87.4%

97.6%

Uncompensated

28.37

21.8

28.53

21.87

Compensated

1.64

0.36

2.54

Reduction rate

94.2%

98.3%

91.1%

1 97.9% 5.47 0.13

0.33 98.5%

Performance Comparison. The performances of the 2-DoF planar serial and quasiserial manipulators during trajectory tracking are now compared. The comparison results are presented in Table 6. Motor Torque Reduction. As shown in Table 6, the peak uncompensated torque of the serial manipulator tracking along trajectory I (56.9 N-m) is higher than that of the quasi-serial manipulator (47.4 N-m). This result can be explained by the decoupling of the gravitational torques at joint O1 in the quasi-serial manipulator shown in Fig. 10. In this design, the gravitational torque exerted on link 2 is transmitted to the base joint via links 3 and 4, rather than link 1 as the serial design shown in Fig. 6. The peak compensated torque of the quasi-serial manipulator is about 0.9 N-m, which is lower than that of the serial manipulator (7.1 N-m). There are a torque reduction rate of 87.5% for the serial manipulator and that of 98.1% for the quasi-serial manipulator, reflecting the better performance of the quasi-serial manipulator. Similar results can also be achieved when trajectory II is considered. Power and Energy Consumption. The power and energy consumption of the serial and quasi-serial manipulators during trajectory tracking are computed by Eqs. (61), (62), (65), and (66). The computed results are presented in Table 6, Figs. 14 and 15. As compared with the serial manipulator, the quasi-serial manipulator requires lower power and energy consumption for tracking and obtains higher power and energy reduction rates.

A Modularization Approach for Gravity Compensation …

23

Fig. 14 Power of the 2-DoF planar serial and quasi-serial manipulators: (a) trajectory I and (b) trajectory II

Fig. 15 Energy consumption of the 2-DoF planar serial and quasi-serial manipulators: (a) trajectory I and (b) trajectory II

5 Discussion The proposed GSM for gravity compensation can provide several advantages as compared with other similar designs, e.g., in [1, 19]. First, the GSM can offer a compact mechanical structure because all the auxiliary parts and the spring are arranged along the longitudinal axis of the rotating link. The compactness of the

24

V. L. Nguyen and C.-H. Kuo

mechanism can be enhanced with no space on the fixed base required for the spring attachment. This feature can make the GSM convenient for modularization on multiDoF robotic manipulators, as presented above. The GSM does not need an initial spring compression that makes its implementation to the manipulator considerably handy as compared with other designs, e.g., in [20, 21, 25]. Although the modularization approach using the GSMs cannot meet a perfect gravity compensation, its great favors lie in the practical implementation for robotic manipulators via a compact mechanical structure and less effort in the spring installation and computation. Besides, this approach also provides high performance of gravity compensation, as demonstrated via numerical examples above.

6 Conclusion This chapter presented a modularization approach for gravity compensation of planar articulated robotic manipulators by using the GSMs. The GSM was detailed via the kinematic structure, parameter determination, and performance evaluation. The design for the manipulator using the GSMs and their parameters for gravity compensation were presented. Criteria were also established to evaluate the performance of gravity compensation via torque, power, and energy consumption reductions. The effectiveness of the modularization approach was demonstrated via the 2-DoF planar serial and quasi-serial manipulators.

References 1. Arakelian, V.: Gravity compensation in robotics. Adv. Robot. 30(2), 79–96 (2016) 2. Carricato, M., Gosselin, C.: A statically balanced Gough/Stewart-type platform: conception, design, and simulation. ASME J. Mech. Robot. 1(3), 031005 (2009) 3. Gosselin, C.: Gravity Compensation, Static balancing and dynamic balancing of parallel mechanisms. In: Wang, L., Xi, J. (eds) Smart Devices and Machines for Advanced Manufacturing. Springer, London (2008).https://doi.org/10.1007/978-1-84800-147-3_2 4. Nguyen, V.L., Lin, C.Y., Kuo, C.-H.: Gravity compensation design of planar articulated robotic arms using the gear-spring modules. ASME J. Mech. Robot. 12(3), 031014 (2020) 5. Herder, J.L.: Energy-free systems: theory, conception and design of statically balanced spring mechanisms. Ph.D. Thesis, Department of Design, Engineering and Production, Delft University of Technology, Delft, The Netherlands (2001) 6. Arakelian, V., Briot, S.: Balancing of Linkages and Robot Manipulators: Advanced Methods With Illustrative Examples. Springer, Switzerland (2015) 7. van der Wijk, V.: Design and analysis of closed-chain principal vector linkages for dynamic balance with a new method for mass equivalent modeling. Mech. Mach. Theory 107, 283–304 (2017) 8. Kuo, C.H., Nguyen-Vu, L., Chou, L.T.: Static balancing of a reconfigurable linkage with switchable mobility by using a single counterweight. In: IEEE International Conference on Reconfigurable Mechanisms and Robots (ReMAR), Delft, Netherlands, June 20–22, pp. 1–6 (2018)

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9. Gosselin, C.M., Wang, J.: On the design of gravity-compensated six-degree-of-freedom parallel mechanisms. In: IEEE International Conference on Robotics and Automation (ICRA), Leuven, Belgium, May 20, pp. 2287–2294 (1998) 10. Nguyen, V.L., Lin, C.Y., Kuo, C.H.: Gravity compensation design of Delta parallel robots using gear-spring modules. Mech. Mach. Theory 154, 104046 (2020) 11. Kuo, C.H., Nguyen, V.L., Robertson, D., Chou, L.T., Herder, J.L.: Statically balancing a reconfigurable mechanism by using one passive energy element only: a case study. ASME J. Mechanisms and Robotics 13(4), 040904 (2021) 12. Kuo, C.H., Lai, S.J.: Design of a novel statically balanced mechanism for laparoscope holders with decoupled positioning and orientating manipulation. ASME J. Mech. Robot. 8(1), 015001 (2016) 13. Essomba, T.: Design of a five-degrees of freedom statically balanced mechanism with multidirectional functionality. Robotics 10(1), 11 (2021) 14. Tseng, T.Y., Lin, Y.J., Hsu, W.C., Lin, L.F., Kuo, C.H.: A novel reconfigurable gravity balancer for lower-limb rehabilitation with switchable hip/knee-only exercise. ASME J. Mech. Robot. 9(4), 041002 (2017) 15. Fedorov, D., Birglen, L.: Differential noncircular pulleys for cable robots and static balancing. ASME J. Mech. Robot. 10(6), 061001 (2018) 16. Takesue, N., Ikematsu, T., Murayama, H., Fujimoto, H.: Design and prototype of variable gravity compensation mechanism (VGCM). J. Robot. Mech. 23(2), 249–257 (2011) 17. Hung, Y.C., Kuo, C.H.: A novel one-DoF gravity balancer based on cardan gear mechanism. In: Wenger, P., Flores, P. (eds) New Trends in Mechanism and Machine Science. Mechanisms and Machine Science, vol 43. Springer, Cham (2017).https://doi.org/10.1007/978-3-319-441566_27 18. Bijlsma, B.G., Radaelli, G., Herder, J.L.: Design of a compact gravity equilibrator with an unlimited range of motion. ASME Journal of Mechanisms and Robotics 9(6), 061003 (2017) 19. Arakelian, V., Zhang, Y.: An improved design of gravity compensators based on the inverted slider-crank mechanism. ASME J. Mech. Robot. 11(3), 034501 (2019) 20. Lee, W.B., Lee, S.D. and Song, J.B.: Design of a 6-DOF collaborative robot arm with counterbalance mechanisms. In: IEEE International Conference on Robotics and Automation (ICRA), Singapore, May 29 – June 3, pp. 3696–3701 (2017) 21. Ahn, K.H., Lee, W.B., Song, J.B.: Reduction in gravitational torques of an industrial robot equipped with 2 DOF passive counterbalance mechanisms. In: IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), Daejeon, South Korea, October 9–14, pp. 4344–4349 (2016) 22. Nakayama, T., Araki, Y., Fujimoto, H.: A new gravity compensation mechanism for lower limb rehabilitation. In: IEEE International Conference on Mechatronics and Automation, Changchun, China, August 9–12, pp. 943–948 (2009) 23. Endo, G., Yamada, H., Yajima, A., Ogata, M., Hirose, S.: A passive weight compensation mechanism with a non-circular pulley and a spring. In: IEEE International Conference on Robotics and Automation (ICRA), Anchorage, AK, USA, May 3–7, pp. 3843–3848 (2010) 24. Lee, D., Seo, T.: Lightweight multi-DOF manipulator with wire-driven gravity compensation mechanism. IEEE/ASME Trans. Mechatron. 22(3), 1308–1314 (2017) 25. Kim, H.-S., Min, J.-K., Song, J.-B.: Multiple-degree-of-freedom counterbalance robot arm based on slider-crank mechanism and bevel gear units. IEEE Trans. Rob. 32(1), 230–235 (2016) 26. Min, J.-K., Kim, D.-W., Song, J.-B.: A wall-mounted robot arm equipped with a 4-DOF yawpitch-yaw-pitch counterbalance mechanism. IEEE Rob. Autom. Lett. 5(3), 3768–3774 (2020) 27. Tsai, L.-W.: Robot analysis: The Mechanics of Serial and Parallel manipulators: John Wiley & Sons. NY, USA, New York (1999)

Stiffness Modeling for Gravity Compensators Alexandr Klimchik

and Anatol Pashkevich

Abstract The article focuses on stiffness modeling of manipulators with gravity compensators. Particular attention is paid to the impact of different types of gravity compensators and gravity compensation algorithms on the manipulator behavior under external loading. The issues of model parameters identification from experimental data are also considered. The validity of presented study was tasted on real industrial robots, corresponding experimental results are presented. Keywords Robotics · Stiffness modeling · Gravity compensators · Industrial robots

1 Introduction Gravity compensators are an integral part of most robotic manipulators used in the industry. They are introduced in order to balance the robot mass and reduce payload on the actuators. However, this compensation definitely affects the stiffness properties of the manipulator and its behavior under the loading becomes more complex and may lead to additional non-linearities. Moreover, this non-linearity also affects on both manipulator and compensator parameters identification [1, 2]. In the majority of cases, gravity compensators are passive and may change their compensation ratio only because of manipulator configuration changes. This leads to non-homogeneity of gravity compensation within the manipulator workspace. A trade-off leads to compensation of the gravity in average, but the price for it is non-compensating all gravity forces in some areas while overcompensation in another areas. From that point of view design of gravity compensators and tuning their parameters plays an important role in the manipulator design [3–7].

A. Klimchik (B) Innopolis University, Innopolis, Russia A. Pashkevich IMT Atlantique, Nantes, France e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 V. Arakelian (ed.), Gravity Compensation in Robotics, Mechanisms and Machine Science 115, https://doi.org/10.1007/978-3-030-95750-6_2

27

28

A. Klimchik and A. Pashkevich

Presents of gravity compensators influence on the manipulator stiffness behaviors and should be properly addressed in the stiffness modelling. Robotic community distinguishes three main approaches for stiffness modeling of robotic manipulators: Virtual Joint Modelling (VJM), Matrix Structural Analysis (MSA) and Finite Element Analysis (FEA). Virtual Joint Modelling (VJM) represents manipulator as a set of rigid/perfect components and virtual springs taking into account compliance of corresponding elastic components [8–14]. This approach can be used both for linear and non-liner stiffness modeling [8, 15, 16], allowing to detect geometrical buckling in the manipulator configuration [17, 18]. VJM models mainly developed for strictly serial or parallel structures with linear stiffness behaviors in the virtual joints space. There are limited number of works dealing with non-linear elastic components [19– 21] and quasi serial structures integrating closed loops in the serial manipulators [22, 23]. The main advantages of this approach are computational simplicity and ability to integrate sophisticated 6 d.o.f. stiffness matrices [9, 24]. This method gives also a possibility to develop simplified stiffness models with a single d.o.f. describing aggregated elasticity of actuated joint and adjacent link bending [2, 25], which can be easily integrated in the robot controller. Finite Element Analysis (FEA) is a powerful tool allowing to take into account all link particularities, including non-homogeneity of material properties and joint/connection properties. It decomposes the structure on a number of finite components, whose properties are described by differential equations and connection and boundary conditions with corresponding aggregation matrices [26–30]. It should be noted that FEA does not requires to develop mathematical model to describe manipulator stiffness model behavior under the loading in the current configuration. However, for a new configuration it will require new computations, including tedious remeshing. From that point of view FEA is heavy and time consuming to be used in real time. Nevertheless researchers attempts to use it if computation time is not critical and for model validation [31, 32]. A trade off between the manipulator model description and computational complexity provides by the Matrix Structural Analysis (MSA) approach. It uses the same concepts as FEA, but operates with relatively large components having physical meaning for robot (links, joints, platforms, ets.). This method is widely used in structural mechanics [33, 34] and found several modifications in robotics [35–39]. Comparing to classical applications robots have multiple passive joints that should be properly described and the model should operate with full size 3D components. The main advantage of the MSA for robotics is ability to derive stiffness model for any complex structure with numerous closed loops. The most essential limitation of this approach is the fact that it still works only for linear stiffness modeling. However, there is no general procedure how to find static equilibrium for such complex structures. It should be mentioned that gravity compensators change stiffness behavior of robotic manipulator under the loading, induce new parameters and affects stiffness identification procedure. Some aspects for spring-based and pneumatic gravity compensators were studied before [40, 41]. These works dealt with linearization of stiffness behavior in the vicinity of several constant compensator configurations

Stiffness Modeling for Gravity Compensators

29

on the first stage and identification compensator parameters on the second stage. The results showed non-linear contribution of gravity compensator to the stiffness behavior and dependency of the compensator geometric and elastic parameters. In the case if robotic manipulator is equipped with the secondary encoders it is possible to identify joint elasticities from the joint measures [42] while for gravity compensator parameters externa measures from absolute measurement system are required. In the case if CAD model are available link stiffness parameters can be estimated from virtual experiments [24, 43]. However, to get sophisticated and statistically meaningful model it is required to adapt model parameters from the experimental study [44]. Currently, identification procedures are developed for serial and quasiserial structures, but there is no general procedure for parallel and hybrid structures. This topic is still under development of the research community.

2 Stiffness Modelling for Manipulators with Gravity Compensation There are three main approaches to compensate gravity in the industrial manipulators in order to reduce the impact of gravity on the manipulator position accuracy: • using mechanical gravity compensators with different working principles to compensate joint torque in the joint(s) (Fig. 1a–c); • using counterbalance and kinematics to reduce the impact of robot mass on the manipulator positioning accuracy (Fig. 1b, c); • using program approaches and advanced feed-back control to compensate the impact of gravity (Fig. 1c).

Fig. 1 Examples of industrial manipulators with different gravity compensators: a Kuka KR270 robot with spring-based mechanical gravity compensator, b ABB IRB 6600 robot with kinematic parallelogram and pneumatic mechanical gravity compensator, c Fanuc M-900iB/700 robot with the secondary encoder feedback control, counterbalance, kinematic parallelogram and spring-based mechanical gravity compensator

30

A. Klimchik and A. Pashkevich

All these approaches have their advantages and limitations. In practice, usually several of them are used together to improve robot positioning accuracy and reduce the impact on it own weight of robotic manipulator. Let us address them consequently in detail.

2.1 Mechanical Gravity Compensators There are two main types of mechanical gravity compensators commonly used in industrial manipulators to compensate the impact of link masses on the robot positioning accuracy: • Spring based gravity compensators (Fig. 2a); • Pneumatic gravity compensators (Fig. 2b). The influence of each type of compensator on the stiffness modeling will presented in details above. The mechanical structure and its equivalent model of typical spring-based mechanical gravity compensator is shown in Fig. 3a. This type of compensator incorporates a passive spring attached to the first and second links, which creates a closed loop that generates (and reduces) the torque applied to the second joint of the manipulator. This design allows us to limit the stiffness model modification by incorporating in it the compensator torque Mc and adjusting the virtual joint stiffness matrix Kθ that here depends on the second joint variable q2 only. The mechanical structure and principal components of pneumatic gravity compensator are presented in Fig. 4a; its equivalent model is shown in Fig. 4b. The mechanical compensator is a passive mechanism incorporating a constant cross-section cylinder

Fig. 2 Example of mechanical gravity compensators a spring-base gravity compensator at Yaskawa GP250 robot, b pneumatic gravity compensator at Kuka KR-120 robot

Stiffness Modeling for Gravity Compensators

31

Fig. 3 Example of spring-base gravity compensator and its equivalent model: Kuka KR-270 robot

Fig. 4 Pneumatic gravity compensator and its equivalent model: Kuka KR-210 robot

and a constant volume gas reservoir. The volume occupied by the gas linearly depends on the piston position that defines the internal pressure of the cylinder. It is clear that this mechanism can be treated as a non-linear virtual spring influencing on the manipulator stiffness behavior. It is worth mentioning that in general case the gas temperature has impact on the pressure inside the tank, which defines the compensating force. Nevertheless, one can assume that in the case of continuous or periodical manipulator movements the gas temperature remains almost constant, i.e. the process of the gas compression-decompression can be assumed to be the isothermal one. In the frame of the above presented manipulator models, the compensator is attached to the first and second links that creates a closed-loop acting on the second actuated joint. This particularity allows us to adapt the conventional stiffness model of the serial manipulator (with constant joint stiffness matrix Kθ ) by introducing the configuration dependent joint stiffness matrix Kθ (q) that takes into account the compensator impact and depends on the vector of actuated coordinates q. In this case, the Cartesian stiffness matrix KC of the robotic manipulator [14] can be presented in the following form KC = (Jθ · Kθ−1 (q) · Jθ−1 )−1

(1)

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A. Klimchik and A. Pashkevich

where Jθ is the Jacobian with respect to the virtual joint coordinates θ (in the case of industrial robots it is usually equivalent to the kinematic Jacobian computed with respect to actuated coordinates q). Thus, to obtain the stiffness model of the industrial robot with the pneumatic gravity compensator it is required to determine the nonlinear joint stiffness matrix Kθ (q) describing elasticity of both actuators and the gravity compensation mechanism. It should be mentioned that in the majority of works devoted to the stiffness analysis of the serial manipulators the matrix Kθ is assumed to be a constant and strictly diagonal one [14, 45, 46]. To find the desired matrix Kθ (q), let us consider the compensator geometry in detail. As follows from Fig. 3b and Fig. 4b, the compensator geometrical model contains three principal node points P0 , P1 , P2 , where P0 , P1 define the passive joint rotation axes and P2 defines the second actuated joint axis. In this model, two distances |P1 , P2 |, |P0 , P2 | are constants, while the third one |P0 , P1 | varies with the robot motions and non-linearly depends on the angle q2 . Below, this distances are denoted as follows: L = |P1 , P2 |, a = |P1 , P2 |, s = |P0 , P1 |. In addition, let us introduce parameters α, ϕ, ax and a y defining relevant locations of points P0 , P1 , P2 (see Fig. 3b and Fig. 4b). This allows us to compute the compensator length s using the following expression s 2 = a 2 + L 2 + 2 · a · L · cos(α − q2 )

(2)

which defines the non-linear function s(q2 ). For this geometry, the impact of the gravity compensator can be taken into account by replacing the considered quasi-serial architecture by the serial one, where the second joint stiffness coefficient is modified in order to include elasticity of both the actuator and compensator. To find relevant non-linear expression for this coefficient, let us present the static torque in the second joint M2 as a geometric sum of two components. The first of them is caused by the deflection δq2 in the mechanical transmission of the second actuated joint and can be expressed in a usual way as Mq2 = K q2 δq2

(3)

where K q2 is the stiffness coefficient, K q2 = 1/kq2 . The second component can be presented as MGC = FS · L · sin ϕ

(4)

where FS is the force generated by the gravity compensator. It is clear that sin ϕ can be computed from the triangle P0 P1 P2 using the sines theorem: sin ϕ = a / s · sin(α − q2 ) The latter allows us to express the torque M2 in the following form:

(5)

Stiffness Modeling for Gravity Compensators

M2 = K q2 δq2 + FS · L ·

33

a sin(α − q2 ) s

(6)

where both the force FS and the compensator length s depend on the joint variable q2 = q20 + δq2 . Hence, this mechanical design theoretically allows to balance the manipulator weight for any given configuration by adjusting the compensator parameters. However in practice the compensator parameters are selected once on the deign stage and remains constant. That leads to under-compensation and overcompensation of gravity effects in some areas. Spring-Based Mechanical Gravity Compensators. To find the force FS for springbased gravity compensator, let us use introduce the zero-value of the compensator length s0 corresponding to the unloaded spring. Under this assumption, the compensator force applied to the node P1 can be expressed as follows S · (s − s0 ) FS = K GC

(7)

S S = 1/k GC is the compensator spring stiffness. This allows us to compute where K GC the compensator torque MGC applied to the second joint S · (1 − s0 /s) · a · L · sin(α − q2 ) MGC = K GC

(8)

Upon differentiation of the latter expression with respect to q2 , the equivalent stiffness of the second joint (comprising both the manipulator and compensator stiffnesses) can be expressed as: S · a L · ηq2 K θ2 = K θ02 + K GC

(9)

where the coefficient ηq2 =

s0 · s



 aL 2 sin (α − q ) + cos(α − q ) − cos(α − q2 ) 2 2 s2

(10)

highly depends on the value of joint variable q2 and the initial preloading in the compensator spring described by s0 . To illustrate this property, Fig. 5 presents a set of curves η(q2 ) obtained for different values of s0 (the remaining parameters α, a, L correspond to robot KUKA KR-270. Hence, using expression (9), it is possible to extend the classical stiffness model of the serial manipulator by modifying the virtual spring parameters in accordance with the compensator properties. It is worth mentioning that the geometrical and elastostatic models of a heavy manipulator with a gravity compensator should include S , s0 for the presented case) that are some additional parameters (α, a, L and K GC usually not included in datasheets. For this reason, the following Sections focus on the identification of the extended set of manipulator parameters. Pneumatic Mechanical Gravity Compensators. To find the force FS for pneumatic gravity compensator, let us use the isothermal process assumption that yields the

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Fig. 5 Variation of the gravity compensator impact on the equivalent stiffness of the second joint

relation P · V = const, where P is the tank pressure, V = A · (s − s0 ) + V0 is the corresponding internal volume, A is the piston area, s0 is compensator link length corresponding to zero compensating force and V0 is the tank volume corresponding to the atmospheric pressure P0 = 1.01 · 105 Pa. These assumptions allow us to express the tank pressure as P=

P0 · V0 A · (s − s0 ) + V0

(11)

Taking into account that compensating force FS depends on the internal and external pressure difference and is computed as FS = (P − P0 ) · A

(12)

one can rewrite expression (6) as M2 = K q2 δq2 − P0

A · (s − s0 ) a · L · sin(α − q2 ) A · (s − s0 ) + V0 s

(13)

and present in a more compact form M2 = K q2 δq2 − P0 · L · a ·

sin(α − q2 ) s − s0 s s − sV

(14)

where a constant sV = V0 /A − s0 is the equivalent distance. Further, after computing the partial derivative ∂ M2 /∂q2 and using Eq. (2) for ∂s/∂q2 = a · L/s · sin(α − q2 ) the desired aggregated stiffness coefficient is presented in the form

(15)

Stiffness Modeling for Gravity Compensators

 P0 · L · a s0 − s V a · L · sin2 (α − q2 ) · s 2 (s − sV ) s − sV   2  s − s0 + −s · cos(α − q2 ) − a · L · sin2 (α − q2 ) · s

35

K 2 (q2 ) = K q2 −

(16)

which is obviously highly non-linear with respect to manipulator configuration (here, s is also a non-linear function of q2 ). Nevertheless, it allows us to compute an relevant stiffness coefficient K 2 for the equivalent serial chain and directly apply Eq. (16) to evaluate stiffness of the quasi-serial manipulator with pneumatic gravity compensator. It should be mentioned that in practice the compensator parameters s0 , sV and actuator stiffness coefficients K 1 , K 2 , K 3 , . . . , K 6 are usually not given in the robot datasheets, so they should be identified via dedicated experimental study.

2.2 Static and Kinematic Approaches for Gravity Compensation Among static and kinematic approach for gravity compensation the most commonly used are the following: • using counterbalance to the third and possible for the second link (Fig. 1b, c); • using kinematic parallelogram(s) to reduce moving mass by moving heavy motor(s) on the robot base (Fig. 1b, c). Let us consider on their impact on the manipulator stiffness behavior. Both sources of gravity compensation we can find in the FANUC M-900iB/700 robot (Fig. 1b). So let us describe the impact of static and kinematic approaches on the example of this manipulator. Its kinematic architecture and principle components are presented in Fig. 6. The manipulator contains an active kinematic parallelogram with three passive joints and two actuators on the same axis, a spring-based gravity compensator connected with the second actuator, and two mass-based compensators. Comparing to the pure-serial architectures, the considered manipulator contains two closed-loops that have impact on its force-deflection behavior. In this case, to conserve the conventional VJM approach it is reasonable to introduce non-linear virtual springs in the second joint, which takes into account particularities of this architecture. Relevant geometric parameters and notations are presented in Fig. 6. Stiffness Modeling of a Mass-Based Gravity Compensator. In the manipulator under study, there are also two mass-based gravity compensators. The first of them (m1 ) is located on extension of the parallelogram lower edge; it is aimed at reducing of the torque in the actuated joint #3. The second one (m2 ) is located at the upper parallelogram edge; it compensates the masses of the link #3 and robot wrist. It should be stressed that the second compensator is implemented using the masses of the robot wrist actuating motors.

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A. Klimchik and A. Pashkevich

Fig. 6 Architecture of FANUC M-900iB/700 manipulator and its principal parameters

Both of the mass-based compensators effect the end-effector location, but they do not influence the robot response to the external loading, i.e. the force-deflection behavior. Hence, these masses should be taken into account in stiffness modeling in order to determine the end-effector displacement and the joint deflections due to the applied wrench (caused by gravity forces, internal and external loadings). However, they do not effect identification of the manipulator elastostatic parameters since their impact is the same for both loaded and unloaded cases. The difficulties here may arise only if, due to the external loading, the vertical line passing the compensator mass center intersects the parallelogram passive joint. It can be shown that in the relevant configurations the force-deflection relation is discontinuous. For this reason, it is prudent to avoid corresponding measurement configurations in calibration experiments. Stiffness Model of a Kinematic Parallelogram. Now, let us consider particularities of stiffness modeling for closed-loops induced by kinematic parallelograms. Typical active parallelogram with two actuators and three passive joints is presented in Fig. 7.

Fig. 7 Active kinematic parallelogram of industrial robots

Stiffness Modeling for Gravity Compensators

37

It is assumed that the parallelogram is perfect and its geometry is defined by the parameters d2 and d P . The actuators drive the angles q2 , q3 and rotate the link #2 and link #3p respectively. From geometric point of view, compared to the conventional serial architecture, such actuators location does not affect the position and orientation of the link #3, since the angle of the motor #3 is directly included in the angle q3 . From the control point of view, driving this parallelogram-based mechanism requires recomputing of input signal for the third actuator only. Such solution improves the robot dynamics (since the heavy third motor is moved to the base to reduce inertial forces) and benefits from the elastostatic point of view as parallel chain increases the stiffness of the robot. Yet, the stiffness model of the parallelogram-based structure cannot be described similar to the conventional serial chain. To prove that conventional stiffness model with a 1-dof virtual spring in Axis #3 cannot describe elastostatic properties of the kinematic parallelogram, let us transform the joint compliance matrix k P = diag(k2 , k3 ) from a quasi-serial to an equivalent serial representation, i.e. from the space (q2 , q3 ) to the space (q2 , q3 ). For the parallelogram, the Jacobian matrix is presented as 

−d2 sin q2 −d3 sin q3 JP = d2 cos q2 d3 cos q3

 (17)

For the equivalent serial chain, corresponding matrix is 

−d2 sin q2 − d3 sin(q2 + q3 ) −d3 sin(q2 + q3 ) JS = d2 cos q2 + d3 cos(q2 + q3 ) d3 cos(q2 + q3 )

 (18)

Both models should lead to the same Cartesian stiffness matrix, i.e. −1  −1  = J S k S JTS J P k P JTP

(19)

whose solution allows us to find an equivalent joint compliance matrix T −T k S = J−1 S J P k P J P JS

(20)

After relevant substitutions of (12), (13) into (15) one can get a non-diagonal matrix   k2 −k3 (21) kS = −k3 k3 for which there is no physically meaningful counterpart in the frame of serial chain architecture. Hence, the obtained expression (16) for equivalent serial chain proves that it is not possible to define an equivalent 1-dof virtual spring in Axis #3 that takes into account the actuator #3 compliance for the parallelogram. Nevertheless, if the user

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prefers the serial model, it is required to use a non-diagonal stiffness matrix in the joint space. This restriction does not violate general idea of VJM approach and can be used in certain cases. However, to simplify the model in the joint space, it is easier to take into account the particularities of parallelogram structure by modifying the Jacobian matrix.

2.3 Algorithmic Approaches for Gravity Compensation Apart from mechanical level gravity compensation can be realized on the software level. Here we would like to mention two main strategies: • Off-line compensation based on the stiffness model (Fig. 8a); • On-line (real-time) compensation using advanced feedback control (Fig. 8b). Both approaches have their advantages and limitations. The first one requires additional experiments for stiffness model parameters identification and cannot be transfer to another manipulator without dedicated experiments. However it might be able to compensate both link and joint compliances. The second approach requires additional sensors, like secondary encoders on each joint to weather and compensate deviations in the manipulator joints and is not able to compensate deflections in the manipulator links. However this approach will efficiently work even with variable payload. Let us address both approaches in details. Off-Line Compensation Based on the Stiffness Model. Elastostatic model of a serial robot is usually defined by its Cartesian stiffness matrix, which should be computed in the neighborhood of loaded configuration. Let us propose numerical technique for computing static equilibrium configuration for a general type of serial manipulator. Such manipulator may be approximated as a set of rigid links and virtual joints, which take into account elastostatic properties (Fig. 9). Since the link weight of serial robots is not negligible, it is reasonable to decompose it into two parts (based on the link mass center) and apply them to the both ends of the link. All this loadings will be aggregated in a vector G = [G1 . . . Gn ], where Gi is the loading applied to the i-th node-point. Besides, it is assumed that the external loading F (caused by the interaction of the tool and the workpiece) is applied to the robot end-effector. Following the principle of virtual work, the work of external forces G, F is equal to the work of internal forces τθ caused by displacement of the virtual springs δθ n   T  G j · δt j + FT · δt = τTθ · δθ

(22)

j=1

where the virtual displacements δt j can be computed from the linearized geometrical ( j) model derived from δt j = Jθ δθ, j = 1..n, which includes the Jacobian matrices ( j) Jθ = ∂g j (q, θ) ∂θ with respect to the virtual joint coordinates.

Stiffness Modeling for Gravity Compensators

39

Fig. 8 Algorithmic approaches for gravity compensation: a Off-line compliance error compensation based on the stiffness model; b On-line gravity compensation based on the advanced feedback control Fig. 9 VJM model of industrial robot with end-point and auxiliary loading

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A. Klimchik and A. Pashkevich

So, expression (22) can be rewritten as n



  ( j) GTj · Jθ · δθ + FT · Jθ(n) · δθ = τTθ · δθ

(23)

j=1

which has to be satisfied for any variation of δθ. It means that the terms regrouping the variables δθ have the coefficients equal to zero. Hence the force balance equations can be written as τθ =

n 

( j)T

Jθ

· G j + Jθ(n)T · F

(24)

j=1

These equations can be re-written in block-matrix form as τθ = Jθ(G)T · G + Jθ(F)T · F

(25)

T T  where Jθ(F) = Jθ(n) , Jθ(G) = Jθ(1)T . . . Jθ(n)T , G = G1T . . . GnT . Finally, taking into   account the virtual spring reaction τθ = Kθ · θ, where Kθ = diag Kθ1 , . . . , Kθn , the desired static equilibrium equations can be presented as Jθ(G)T · G + Jθ(F)T · F = Kθ · θ

(26)

To obtain a relation between the external loading F and internal coordinates of the kinematic chain θ corresponding to the static equilibrium, Eq. (26) should be solved either for different given values of F or for different given values of t. Let us solve the static equilibrium equations with respect to the manipulator configuration θ and the external loading F for given end-effector position t = g(θ) and the function of auxiliary-loadings G(θ) Kθ · θ = Jθ(G)T G + Jθ(F)T F; t = g(θ); G = G(θ)

(27)

where the unknown variables are (θ, F). Since usually this system has no analytical solution, iterative numerical technique can be applied. So, the kinematic equations may be linearized in the neighborhood of the current configuration θi ti+1 = g(θi ) + Jθ(F) (θi ) · (θi+1 − θi );

(28)

where the subscript ‘i’ indicates the iteration number and the changes in Jacobians Jθ(G) , Jθ(F) and the auxiliary loadings G are assumed to be negligible from iteration to iteration. Correspondingly, the static equilibrium equations in the neighborhood of θi may be rewritten as

Stiffness Modeling for Gravity Compensators

41

Jθ(G)T · G + Jθ(F)T · Fi+1 = Kθ · θi+1

(29)

Thus, combining (8), (9) and expression for θ = Kθ−1 (Jθ(G)T · G + Jθ(F)T · F), the unknown variables F and θ can be computed using following iterative scheme −1



ti+1 − g(θi ) + Jθ(F) θi − Jθ(F) Kθ−1 Jθ(G)T Gi Fi+1 = Jθ(F) · Kθ−1 · Jθ(F)T

 (30) θi+1 = Kθ−1 Jθ(G)T · Gi + Jθ(F)T · Fi+1 The proposed algorithm allows us to compute the static equilibrium configuration for the serial robot under external loadings applied to any point of the manipulator and the loading from the technological process. In order to obtain the Cartesian stiffness matrix, let us linearize the force– deflection relation in the neighborhood of the equilibrium. Following this approach, two equilibriums that correspond to the manipulator state variables (F, θ, t) and (F + δF, θ + δθ, t + δt) should be considered simultaneously. Here, notations δF, δt define small increments of the external loading and relevant displacement of the end-point. Finally, the static equilibrium equations may be written as t = g(θ);

Kθ · θ = Jθ(G)T · G + Jθ(F)T · F

(31)

and t + δt = g(θ + δθ) T T



Kθ · (θ + δθ) = Jθ(G) + δJθ(G) · (G + δG) + Jθ(F) + δJθ(F) · (F + δF)

(32)

where t, F, G, Kθ , θ are assumed to be known. After linearization of the function g(θ) in the neighborhood of the loaded equilibrium, the system (31), (32) is reduced to equations δt = Jθ(F) δθ Kθ · δθ = δJθ(G) G + Jθ(G) δG + δJθ(F) F + Jθ(F) δF

(33)

which defines the desired linear relations between δt and δF. In this system, small variations of Jacobians may be expressed via the second order derivatives δJθ(F) = (F) (G) Hθθ · δθ, δJθ(G) = Hθθ · δθ, where (G) = Hθθ

n j=1

∂ 2 gTj G j ∂θ2 ;

 (F) Hθθ = ∂ 2 gT F ∂θ2

(34)

Also, the auxiliary loading G may be computed via the first order derivatives as δG = ∂G ∂θ · δθ. Further, let us introduce additional notation

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A. Klimchik and A. Pashkevich

(F) (G) Hθθ = Hθθ + Hθθ + Jθ(G)T · ∂G ∂θ

(35)

which allows us to present system (33) in the form 

δt 0



 =

0

Jθ(F)T

   δF Jθ(F) · δθ −Kθ + Hθθ

(36)

So, the desired Cartesian stiffness matrices KC can be computed as −1

KC = Jθ(F) (Kθ − Hθθ )−1 Jθ(F)T

(37)

Below, this expression will be used for computing of the elastostatic deflections of the robotic manipulator. In industrial robotic controllers, the manipulator motions are usually generated using the inverse kinematic model that allows us to compute the input signals for actuators ρ0 corresponding to the desired end-effector location t0 , which is assigned assuming that the compliance errors are negligible. However, if the external loading F is essential, the kinematic control becomes non-applicable because of changes in the end-effector location. It can be computed from the non-linear compliance model as tF = f −1 (F|t0 )

(38)

where the subscripts ‘F’ and ‘0’ refer to the loaded and unloaded modes respectively, and ‘|’ separates arguments and parameters of the function f (). Some more details concerning this function are given in [15, 18, 47, 48]. To compensate this undeterred end-effector displacement from t0 to tF , the target point should be modified in such a way that, under the loading F, the end-platform is located in the desired point t0 . This requirement can be expressed using the stiffness model in the following way 

F = f t0 |t0(F)

(39)

where t0(F) denotes the modified target location. Hence, the problem is reduced to the solution of the nonlinear Eq. (31) for t0(F) , while F and t0 are assumed to be given. It is worth mentioning that this equation completely differs from the equation F = f (t|t0 ), where the unknown variable is t. It means that here the compliance model does not allow us to compute the modified target point t0(F) straightforwardly, while the linear compensation technique directly operates with Cartesian compliance matrix [49]. To solve Eq. (39) for t0(F) , similar numerical technique can be applied. It yields the following iterative scheme

Stiffness Modeling for Gravity Compensators

 t0(F) = t0(F) + α · t0 − f −1 (F|t0(F) )

43

(40)

where the prime corresponds to the next iteration, α ∈ (0, 1) is the scalar parameter ensuring the convergence. More detailed presentation of the developed iterative routines is given in Fig. 10. Hence, using the proposed computational techniques, it is possible to compensate the main part of the compliance errors by proper adjusting the reference trajectory that is used as an input for robotic controller. In this case, the control is based on the inverse kinetostatic model (instead of kinematic one) that takes into account both the manipulator geometry and elastic properties of its links and joints. Implementation of the developed compliance error compensation technique is presented in Fig. 11. Real-Time Compensation Using Advanced Feedback Control. Schematic representation of manipulator joint with double encoders is presented in Fig. 12. Robotic manipulators with double encoders provide us with two options for control of the actuated axes: using feedback either from primary or secondary encoders. It is clear that the second approach is more attractive because it allows us to compensate directly the errors caused by the joint compliances. However, the link compliances are outside of such error compensation strategy. The latter is confirmed by experimental results presented in Fig. 13, where the secondary encoder feedback compensates less than 2/3 of the end-effector compliance errors. Moreover, the efficiency of the compliance error compensation based on the second encoder feedback for the worst case reduced

Fig. 10 Procedure for compensation of compliance errors

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Fig. 11 Implementation of compliance error compensation technique

Fig. 12 Schematic representation of manipulator joint with double encoders

down to 50% (Fig. 13). Results presented in Table 1 show that accuracy estimation based on the experimental set (configuration used for the parameters identification) gives slightly better results than analysis within entire robot workspace, while this difference is important for the maximum compliance errors. It is worth mentioning that similar results (i.e. compensation the joint compliances) can be achieved using the primary encoder only. In this case, the manipulator should be equipped with a force/torque sensor allowing to estimate external loading. Using this data and assuming that the manipulator stiffness model is known, it is possible to modify the actuator inputs to compensate the above-mentioned compliance errors. On the other side, the desired external force/torque measurement can be also done indirectly, using a combination of the primary and secondary encoders that provides the difference between the angles on the motor-side and on the link-side.

Stiffness Modeling for Gravity Compensators

45

Fig. 13 Efficiency of embedded and proposed compliance error compensation algorithms for entire robot workspace (results corresponds to the model of industrial robot Kuka KR120 R3900 under external loading 100 kg)

Table 1 Comparison of error compensation algorithms based on experimental set and within entire robot workspace

Compensation algorithm

End-effector errors, mm Mean

Std

Max

Primary encoder

4.03 (100%)

2.17 (100%)

9.52 (100%)

Secondary encoders

1.74 (43.1%)

0.93 (42.7%)

3.36 (35.2%)

Stiffness model 0.64 (15.7%)

0.33 (15.3%)

1.52 (15.9%)

Experimental set

Entire robot workspace Primary encoder

4.03 (100%)

2.28 (100%)

9.17 (100%)

Secondary encoders

1.81 (44.7%)

0.79 (44.8%)

4.63 (50.4%)

If the joint stiffness is known, this difference provides another way of the external force estimation. To compensate both joint and link compliances, the following strategy can be applied. First, using the difference between the primary and secondary encoder as well as the joint stiffness coefficients, it is estimated the external loading. Further, using the complete stiffness model (that includes both joint and link compliances), the actuator inputs are modified to compensate both types of the compliance errors. It should be mentioned that here the feedback relies on the primary encoder, while the secondary encoder is used to measure the elastic deflections. As follows the dedicated study, this control strategy allowed to compensate about 4/5 of the endeffector deflections, which is more than twice better compared to the embedded error compensation algorithm based on the secondary encoder feedback (see Fig. 14c).

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Fig. 14 End-effector deflections caused by external loading 100 kg for the robot Kuka KR120 R3900 with primary (a) and secondary (b) encoder feedback and c compliance error compensation based on stiffness model

More details concerning the efficiency of the proposed error compensation technique and comparison with the methods that are currently used in industry with the entire robot workspace are given in Fig. 14 and Table 1. Taking into account that enhanced stiffness model is rather difficult for industrial implementation, an alternative control strategy can be used. Its main idea is based on the proportionality of the errors caused by the joint and link compliances. This allows us to replace the conventional feedback coordinate q (2) from the secondary encoder by the linear combination q (2) +η·(q (2) −q (1) ) computed using both encoder readings. In this expression, the coefficient η = k L /k J considers the ratio between the link to joint compliances, which differs from joint to joint. As follows from the simulation study, this industry-oriented strategy is slightly worth compared to the above presented one, but the compensation degree is quite acceptable for practice: it is twice better than the error compensation algorithm used by the robot manufacturers (see Table 1 and Fig. 13 and 14c).

Stiffness Modeling for Gravity Compensators

47

Table 2 Comparison of error compensation algorithms based on single and double encoder schemes Scheme

Compensation algorithm

End-effector errors, mm Mean

Max

Build-in control algorithms (compensation based on joint feedback) Single encoder

Controller built-in compensation OFF

4.03 (100%)

9.17 (100%)

Double encoders

Controller built-in compensation ON

1.81 (44.7%)

4.63 (50.4%)

Control algorithms based on reduced stiffness model (compensation of joints deflections) Double encoders

Controller input adjustment by 1.81 (44.7%) ϕ = ϕ2 − ϕ1 ; Evaluation of the end-effector deflections using geometric model

Single encode and force Controller input adjustment by sensor ϕ = k J τ ; Evaluation of the end-effector deflections using stiffness model

1.81 (44.7%)

4.63 (50.4%)

4.63 (50.4%)

Control algorithms based on enhanced stiffness model (compensation of joints and links deflections) Double encoders

Controller input adjustment by 1.08 (19.5%) ϕ = k (ϕ2 − ϕ1 ); Evaluation of the end-effector deflections using geometric model

2.28 (24.7%)

Single encoder and force sensor

Controller input adjustment by ϕ = (k J + k L ) τ ; Evaluation of the end-effector deflections using stiffness model

2.28 (24.7%)

1.08 (19.5%)

The control strategies considered in this section are summarized in Table 2, which contains both conventional built-in algorithms and proposed in this work. It clearly shows that the existing compensation algorithm (embedded in the robot controller) allows reducing the robot positioning errors from 4.03 to 1.81 mm, while the proposed technique allows achieving further error reduction down to 1.08 mm. Its simplified version (easy for industrial implementation) ensures the same error reduction. Some control strategies ensure the same degree of the compliance error compensation. This fact confirmed by mathematical equivale between the model used for error compensation, while in the real industrial environment their efficiency may differ since they have different inputs and measurement noise will affect differently of the manipulator positioning accuracy. Hence, for the robot with double encoder, essential improvement of the robot accuracy can be achieved by simple modification of the feedback signal in the actuators. The presented study is based on the particular robot model, but the proposed advance control (error compensation) algorithms will have similar effects on the

48

A. Klimchik and A. Pashkevich

other robots also. It should be mentioned that the degree of compliance error compensation based on the second encoder feedback depends on the robot geometry and gearboxes compliances, i.e. by ratio η = k L /k J indicating link compliance impact in the total robot compliance. This hypothesis has been confirmed in the experimental studies with robots from different manufacturers. In all cases, the stiffness model-based compliance error compensation technique allowed to improve robot positioning accuracy compared to the built-in algorithm and was able to compensate more than 80% of compliance errors.

3 Identification of Stiffness Model Parameters for Manipulators with Gravity Compensators 3.1 Spring-Bases Mechanical Gravity Compensator In contrast to the serial manipulator that can be treated as a principal mechanism of the considered robots (whose geometry is usually defined in datasheets and can be perfectly tuned by means of calibration [50, 51]), geometrical data concerning gravity compensators are usually not included in the technical documentation provided by the robot manufacturers. For this reason, this Section focuses on the identification of the geometrical parameters for the described above compensator mechanism (see Fig. 3). Methodology. The geometrical structure of the considered gravity compensator is presented in Fig. 15. Its principal geometrical parameters are denoted as L, ax , a y , where ax = a · cos α, a y = a · sin α. As follows from the figure, the identification problem can be reduced to the determination of relative locations of points P0 and P1 with respect to P2 . Fig. 15 Geometrical parameters of the gravity compensator and location of the measurement points labeled with markers

Stiffness Modeling for Gravity Compensators

49

It is assumed that the measurement data are provided by the laser tracker whose “world” coordinate system is located at the intersection of the first and second actuated manipulator joints. The axes Y, Z of this system are aligned with the axes of joints #1 and #2 respectively, while the axis X is directed to ensure right-handed orthogonal basis. To obtain required data, there are several markers attached to the compensator mechanism (see Fig. 15). The first one is located at point P1 , which is easily accessible and perfectly visible (the center of the compensator axis P1 is exactly ticked on the fixing element). In contrast, for the point P0 , it is not possible to locate the marker precisely. For this reason, several markers P0i are used that are shifted with respect to P0 , but located on the rigid component of the compensator mechanism (these markers are rotating around P0 while the joint coordinate q2 is actuated). It should be noted that for the adopted compensator geometrical model (which is in fact a planar one), the marker location relative to the plane XY is not significant, since the identification algorithm presented in the following sub-section will ignore Z-coordinate. Using this setting, the identification problem is solved in two steps. The first step is devoted to the identification of the relative location of points P1 and P2 . Here, for different values of the manipulator  joint coordinates {q2i , i = 1, m}, the laser tracker provides the set of the vectors pi1 describing the points that are located in an arc of the circle. After matching these points with a circle, one can obtain the desired value of L (circle radius) and the Cartesian coordinates p2 of the point P2 (circle center) with respect to the laser tracker coordinate system. The second step deals with the identification of the relative P0   of points   location and P2 . Relevant information is extracted from two data sets pi01 and pi02 that are provided by the laser tracker while targeting at the markers P01 and P02 . Here, the points are matched to two circle arcs with the same center (explicitly assuming that the compensator model is planar), which yields the Cartesian coordinates p0 of the point P0 , also with respect to the laser tracker coordinate system. Finally, the desired values ax , a y are computed as a projection of the difference vector a = p2 − p0 on the corresponding axis of the coordinate system. As follows from the presented methodology, a key numerical problem in the presented approach is the matching of the experimental points with a circle arc. It looks like a classical problem, however, there is a particularity here caused  by  availability of additional data {q2i } describing relative locations of the points pi1 . This feature allows us to reformulate the identification problem and to achieve higher accuracy compared with the traditional approach. Identification Algorithm. The above presented methodology requires solution of two identification problems. The first one aims at approximating of a given set of points (with additional arc angle argument) with an arc circle, which provides the circle center and the circle radius. The second problem deals with an approximation of several sets of points by corresponding number of circle arcs with the same center. Let us consider them sequentially. To match the given set of points {pi } with additional set of angles {qi } with a circle arc, let us define the affine mapping

50

A. Klimchik and A. Pashkevich

pi = μ R ui + t

(41)

where ui = [cos qi , sin qi , 0]T denotes the set of reference points located on the unit circle whose distribution on the arc is similar to pi , μ is the scaling factor that defines the desired circle radius, R is the orthogonal rotation matrix, t is the vector of the translation that defines the circle center. It worth mentioning that such a formulation has an advantage (in the sense of accuracy) comparing to a traditional circle approximation and it is a generalization of Procrustes problem known from the matrix analysis. Using Eq. (41), the identification can be reduced to the following optimization problem F=

m i=1

(pi − μR ui − t)T (pi − μR ui − t) → min

μ,R,t

(42)

which should be solved subject to the orthogonality constraint R T R = I. After differentiation with respect to t, the latter variable can be expressed as t = m −1

m i=1

pi − μ m −1 R

m i=1

ui

(43)

That leads to the simplification of (42) to F=

T

 m





pi − μR ui pi − μR ui → min i=1

(44)

r,R

where pi = pi − m −1

m i=1

ui = ui − m −1

pi ;

m i=1

ui

(45)

Further, differentiation with respect to μ yields to μ=

m i=1

T



pi R ui



m

i=1

T

ui ui

(46)

So, finally, after relevant substitutions the objective function can be presented as  m −1  m 2 m   T

 T

T

F= pi pi − ui ui pi R ui → min i=1

i=1

i=1

R

(47)

where the unknown matrix R must satisfy the orthogonality constraint R T R = I. Since the matrix R is included in the second term only, the problem can be further simplified to

Stiffness Modeling for Gravity Compensators

F =

m i=1

51

  m

T

pi Rui = trace R

T

i=1



ui pi

→ max

(48)

R

and can be solved using SVD-decomposition of the matrix m



i=1

ui pi = U  VT

(49)

where the matrices U, V are orthogonal and  is the diagonal matrix of the singular values. Further, using the same approach as for the Procrustes problem, it can be proved that the desired rotation matrix can be computed as R = V UT

(50)

which sequentially allows to find the scaling factor μ defining the arc radius and the vector t defining the arc center.     The second problem aims at approximating of several point sets pi1 , . . . , pik by corresponding number of concentric circle arcs with the same center p0 . It should be noted that here the data set {qi } is not useful, since the required angles {βi } are not measured directly and cannot be computed without having exact compensator geometry. In this case, the objective function can be written in a straightforward way F=

m 

k j=1

i=1

R 2j −

T

2

j j pi − p0 pi − p0 → min

(51)

p0 ,R j

But it can be proved that this optimization problem does not lead to a unique solution (in fact, it gives the rotation axis passing through the desired center). For this problem, differentiation of the objective function F with respect to R 2j yields R 2j = m −1

m

i=1

j

pi − p0

T

 j pi − p0

(52)

which after substitution into (51) allows us to rewrite the problem in the following way F=

k j=1

2 m  j

j T

2 p0 pi − s i → min i=1

(53)

p0

where j

pi = pi − m −1

j

m l=1

j

pl ;

j

s i = pi pi − m −1 jT

j

m l=1

jT

j

pi pi

(54)

Further, after differentiation with respect to p0 , one can get the underdetermined system of linear equations

52

A. Klimchik and A. Pashkevich

 k j=1

m i=1



j jT

pi pi

 1 k m j j p0 = s p j=1 i=1 i i 2

(55)

whose solution p0 = pc + ξ n

(56)

is expressed via the rotation axis vector n and a point belonging to this axis pc (here ξ is an arbitrary scalar factor). To solve this ambiguity, an additional objective should be defined k j=1

R 2j → min

(57)

Rj

which leads to the following solution for the scalar parameter

k ξ = nT −pc + k −1 m −1

m

j=1

j

i=1

 (58)

pi

and for the vector   1 k m j j T −1 k m j j pc = p p s p j=1 i=1 i i j=1 i=1 i i 2

(59)

So, the desired arc center is expressed as follows k   p0 = I − n nT pc + k −1 m −1 n nT

j=1

m i=1

j

pi

(60)

It should be mentioned that practical application of the latter expression is essentially simplified by the adopted assumption concerning orientation of the reference coordinate system (see previous sub-section), the direction of the identified rotation axis is close to Z-direction. Hence, the developed algorithms allows us to identify the compensator geometrical parameters L, ax , a y that are directly related to the above mentioned rotation center points P0 , P2 and corresponding radii. Below they will be applied to the processing of the experimental data. Experimental Results. To demonstrate efficiency of the developed technique, the experimental study has been carried out. The experimental setup employed the robot KR-270 and the Leica laser tracker, which allowed us to measure the Cartesian coordinates of the markers attached to the compensator elements (see Figs. 3, 15). Six different manipulator configurations where considered that differed in the value of the joint angle q2 and three markers has been used. The experimental data are presented in Table 3. These data has been processed using the identification algorithm presenting in the previous sub-section. The obtained values for the parameters of interest L, ax , a y are

Stiffness Modeling for Gravity Compensators

53

Table 3 Experimental data for geometrical calibration q2 [deg]

P1 x, [mm]

−0.01

P01 y, [mm]

P02

x, [mm]

y, [mm]

x, [mm]

y, [mm]

−31.84

183.86

−872.10

−125.38

−813.50

−255.59

−30

−118.44

143.42

−872.30

−126.07

−813.33

−256.18

−60

−173.30

65.12

−872.50

−109.90

−825.09

−244.64

−90

−181.76

−30.14

−868.43

−78.20

−844.66

−219.04

−120

−141.45

−116.82

−858.90

−47.60

−859.43

−190.44

−145

−78.10

−165.47

−852.53

−33.68

−864.66

−176.01

Table 4 Geometrical parameters of gravity compensator

L, [mm]

ax , [mm]

ay , [mm]

Value

184.72

685.93

120.30

CI

±0.06

±0.70

±0.69

given in Table 4. It also includes the confidence intervals computed as ±3σ , where the standard deviation σ has been evaluated using the Gibbs sampling. In the next section, the obtained model will be used for some transformations required during elastostatic calibration. Elastostatic Parameters Identification In contrast to geometrical calibration, where the manipulator and compensator can be considered independently, in elastostatic calibration the corresponding equations cannot be separated and the model parameters should be identified simultaneously. This section gives general ideas of the developed methodology and relevant identification algorithms, and also presents experimental results validating the proposed technique. Methodology. In the frame of the adopted VJM-based modeling approach the desired stiffness parameters describe elasticity of the virtual springs located in the actuated joints of the manipulator, and also elasticity and preloading of the compensator spring. Let us denote them as kθ j , j = 1, 6 for the manipulator joint compliances and kc , s0 for the compliance and preloading of the compensator. To find the desired parameters, the manipulator sequentially passes through several measurement configurations where the external loading is applied to the special endeffector described in Fig. 16 (it allows to generate both forces and torques applied to the manipulator). Using the laser tracker, the Cartesian coordinates of the reference points are measured twice, before and after loading. To increase identification accuracy, it is preferable to use several reference points (markers) and to apply the loading of the maximum allowed magnitude. It is worth mentioning that in order to avoid numerical singularities, the direction of the external loading should not be the same for all experiments (in spite of the fact that the gravity-based loading is

54

A. Klimchik and A. Pashkevich

Fig. 16 End-effector used for elastostatic calibration experiments and its equivalent model

the most attractive from the practical point of view). Thus, the calibration experiments yield the dataset that includes values of the manipulator joint coordinates {qi }, applied forces/torques {Fi } and corresponding deflections of the reference points {pi }. Using these data, the elastostatic parameters of kθ j , j = 1, 6 and kc , s0 should be identified. Identification Algorithm. To take into account the compensator influence while retaining our previous approach developed for serial robots without compensators, it is proposed to include in the second joint an equivalent virtual spring with nonlinear stiffness depending on the joint variable q2 (see Eq. (9)). Using this idea, it is convenient to consider several independent parameters kθ2i corresponding to each value of q2 . This allows us to obtain linear form of the identification equations that can be easily solved using standard least-square technique. Let us denote the set of desired parameters k1 , (k21 , k22 . . .), k3 , . . . , k6 as the vector k and linearize Jθ(G)T · G + Jθ(F)T · F = Kθ · θ with respect to this vector. This allows us to present the relevant force displacement relations in the form ( p)

pi = Bi k ( p)

where matrices Bi

(61)

are composed of the elements of the matrix   T T Fi , . . . , Jni Jni Fi (i = 1, m) Ai = J1i J1i

(62)

that is usually used in stiffness analysis of serial manipulators [52]. Here, Jni denotes the manipulator Jacobian column, Fi is the applied external force, and superscript ‘(p)’ stands for the Cartesian coordinates (position without orientation). It is clear ( p) that transformation from Ai to Bi is rather trivial and is based on the extraction from Ai the first three lines and inserting in it several zero columns. Using these notations, the elastostatic parameters identification can be reduced to the following optimization problem F=

m i=1

( p)

( p)

(Bi k − pi )T (Bi k − pi ) → min

k j ,kc ,ρ0

(63)

Stiffness Modeling for Gravity Compensators

55

which leads to the following solution k=

 m 

−1 ( p)T ( p) Bi Bi

·

i=1

 m 

 ( p)T Bi pi

(64)

i=1

where the parameters k1 , k3 , . . . , k6 describe the compliance of the virtual joints #1, #3, … #6, while the rest of them k21 , k22 . . . present an auxiliary dataset allowing to separate the compliance of the joint #2 and the compensator parameters kc , ρ0 . Using Eq. (9), the desired expressions can be written as 

K θ02 K c s0 · K c

T

=

m q i=1

CiT Ci

−1 m q i=1

CiT K θ2i

 (65)

where m q is the number of different angles q2 in the experimental data,    Ci = 1 − a L · cos γi a L/s · a L/s 2 · sin2 γi + cos γi

(66)

here γi = α − q2i . Thus, the proposed modification of the previously developed calibration technique allows us to find the manipulator and compensator parameters simultaneously. An open question, however, is how to find the set of measurement configurations that ensure the lowest impact of the measurement noise. Design of Calibration Experiments. The main idea of the calibration experiment design is to select a set of robot configurations {qi } (and corresponding external loadings {Fi }) that ensure the best identification accuracy. The key issue here is the ranging of different plans in accordance with the prescribed performance measure. This problem has been already studied in the classical regression analysis [53– 57], however the results are not suitable for the elastostatic calibration and require additional efforts. In this work, it is proposed to use the industry oriented performance measure that evaluates the calibration plan quality. Its physical meaning is the robot positioning accuracy (under the loading), which is achieved after compliance error compensation based on the identified elastostatic parameters. It should be noted that usual approach (used in the classical design of experiments) evaluates the quality of experiments via covariance matrix of the identified parameters, which is does not have sense for our application. Assuming that each experiment includes the additive measurement error εi , the covariance matrix for the desired parameters k can be expressed as cov(k) =

m i=1

( p)T

Bi

( p)

Bi

−1 m  m −1 ( p)T ( p) ( p)T ( p) E Bi εiT εi Bi Bi Bi i=1

i=1

(67)

56

A. Klimchik and A. Pashkevich

Following also usual assumption concerning the measurement errors (independent identically distributed, with zero expectation and standard deviation σ 2 for each coordinate), the above equation can be simplified to cov(k) = σ 2

m

( p)T

i=1

Bi

( p)

Bi

−1

(68)

Hence, the impact of the measurement errors on the accuracy of the identified m ( p)T ( p) Bi Bi (in regression analysis it is parameters k is defined by the matrix i=1 known as the information matrix). It is evident that in practice the most essential is not the accuracy of the parameters identification, but the accuracy of the robot positioning achieved using these parameters. Taking into account that this accuracy highly depends on the robot configuration (and varies throughout the workspace), it is proposed to evaluate the calibration accuracy in a certain given “test-pose” provided by the user. For the considered application, the test pose is related to the typical machining configuration q0 and corresponding external loading F0 related to the technological process. Let us denote ( p) the mean square value of the mentioned positioning error as ρ02 and the matrix Ai ( p) (see Eq. (62)) corresponding to this test pose as A0 . It should be noted that that the proposed approach operates with a specific structure of the parameters included in the vector k, where the second joint is presented by several components k21 , k22 . . . while the other joints are described by a single parameter k1 , k3 . . . k6 . This motivates further re-arrangement of the vector k and replacing it by several vectors k j = (k1 , k2 j , k3 , . . . k6 ) of size 6 × 1. Using this notation, the above mentioned performance measure can be expressed as ρ02 =

m q j=1



( p)T ( p) E δk Tj A0 A0 δk j

(69)

where δk j is the elastostatic parameters estimation error caused bythe measurement  noise for q2 j . Further, after substituting δpT δp = trace δpδpT and taking into account that E(δk j δk Tj ) = cov(k j ), the performance measure ρ02 can be presented as   −1 m q m ( p) j ( p)T j ( p) ( p)T (70) Ai Ai A0 ρ02 = σ 2 trace A0 j=1

i=1

Based on this performance measure, the calibration experiment design can be reduced to the following optimization problem   −1 m q m ( p) j ( p)T j ( p) ( p)T → min trace A0 Ai Ai A0 j=1

subject to

i=1

{qi ,Fi }

(71)

Stiffness Modeling for Gravity Compensators

57

Fig. 17 Experimental setup for the identification of the elastostatic parameters

Fi  < Fmax ,

i = 1..m

(72)

whose solution gives a set of the desired manipulator configurations and corresponding external loadings. It is evident that its analytical solution can hardly be obtained and a numerical approach is the only reasonable one. More detailed description of the developed technique providing generation of the optimal calibration plans is presented in [58], where the problem of gravity compensator modeling had not been addressed yet. Experimental Results. The developed technique has been applied to the robot KR270. The parameters of interest were the compliances k j of the actuated joints and the elastostatic parameters kc , s0 of the gravity compensator. To generate elastostatic deflections, the gravity forces 250–280 kg have been applied to the robot end-effector (see Fig. 17). The Cartesian coordinates of three markers located on the tool (see Fig. 16) have been measured before and after the loading. To find optimal measurement configurations, the design of experiments has been carried out for five different angles q2 that are distributed between the joint limits. For each q2 three optimal measurement configurations have been found taking into account physical constraints that are related to the joint limits and the possibility to apply the gravity force (work-cell obstacles and safety reasons). The results of the calibration experiment design are presented in Table 5. Here, values of q1 have been chosen to ensure good visibility of the markers for the laser tracker. To ensure identification accuracy for each configuration, the experiments were repeated three times. In total, 405 equations were considered for the identification, from which 7 physical parameters have been obtained (because of page limit, the experimental data cannot be presented in the paper). The obtained experimental data have been processed using above presented identification algorithm. Corresponding values of the gravity compensator and the manipulator elastostatic parameters are presented in Table 6. It also includes the confidence intervals computed as ±3σ , where the standard deviation σ has been evaluated using Gibbs sampling. Using the obtained results it is possible to identify an equivalent

58

A. Klimchik and A. Pashkevich

Table 5 Optimal measurement configurations Joint angles, [deg] q1

q2

q4

q5

q6

−5.57

51.00

−97.52

−91.67

63.00

−12.22

−56.49

41.42

150.55

63.00

−47.98

−70.04

−61.55

177.16

79.20

−0.01

q3

33.00

129.69

−98.10

90.57

95.00

−107.01

109.95

−61.19

174.21

105.00

14.30

55.21

41.26 41.90

152.06

−98.260

−90.55

95.00

−25.24

44.54

−55.11

56.60

64.73

−129.65

144.80

104.49

−69.41

61.67

−6.33

41.53

−152.48

56.60

−41.00

−56.9

−152.97

−91.68

55.12

−143.00

−99.85

−32.64

110.31

−61.47

−6.29

−143.00

−72.01

129.65

−98.09

90.82

147.68

129.64

−97.90

90.99

−60.00

7.59

−110.09

−61.36

−174.09

−60.00

−52.00

−124.89

−41.62

27.78

133.00

−140

Table 6 Elastostatic parameters of robot Kuka KR-270

Parameter

Value

CI

kc , [rad × μm/N]

0.144

±0.031 (21.5%)

s0 , [mm]

458

±27 (5.9%)

k2 , [rad × μm/N]

0.302

±0.004 (1.3%)

k3 , [rad × μm/N]

0.406

±0.008 (2.0%)

k4 , [rad × μm/N]

3.002

±0.115 (3.8%)

k5 , [rad × μm/N]

3.303

±0.162 (4.9%)

k6 , [rad × μm/N]

2.365

±0.095 (4.0%)

non-linear spring k2 (q2 ) (see Fig. 18) that is used in the stiffness modeling of the manipulator with the gravity compensator. The identified joint compliances can be used to predict robot deformations under the external loading. The identification errors of the joint compliances vary from 1.3 to 4.9%, where the lowest errors have been achieved for the joints #2 and #3. The highest errors are in the joints #4–#6. So, higher precision has been achieved for the joints that are closer located to the manipulator base. This fact is due to the different joint errors contribution to the total positioning error, which have been minimized while design of calibration experiments. Comparatively low identification accuracy for the compensator spring is caused by limited number of different angles q2 . Figure 18 shows that due to the gravity compensator the equivalent compliance

Stiffness Modeling for Gravity Compensators

59

Fig. 18 Compliance of equivalent non-linear spring in the second joint

of the second joint has been decreased comparing to the compliance of the second joint of the corresponding serial manipulator.

3.2 Pneumatic Mechanical Gravity Compensator Identification Algorithm. Similar to spring-based gravity compensator case, in the frame of the VJM-based modelling approach developed for serial kinematic chains [14, 15] and adapted for the case of quasi-serial manipulators with pneumatic gravity compensators, the desired stiffness model parameters describe elasticity of the virtual springs located in the actuated joints of the manipulator, and also compensator parameters s0 , sV of defining preloading of the compensator spring and the equivalent distance for the tank volume V0 . In the frame of this model, let us denote the manipulator joint compliances as kθ j , j = 1, 6 and the compensator elastic parameters as s0 , s V . To find the desired set of elastic parameters, robot sequentially passes through several measurement configurations where the external loading is applied to the specially designed end-effector presented in Fig. 16 (it allows us to generate both forces and torques applied to the manipulator). Using the absolute measurement system the Cartesian coordinates of the reference points are measured twice, before and after loading. To increase identification accuracy, it is reasonable to have several markers on the end effector (reference points) and to apply the loading of the maximum allowed magnitude. It should be mentioned that to avoid singularities caused by numerical routines, the external force/torque directions should not be the same for all calibration experiments (while from the practical view point the massbased gravity loading is the most attractive). Thus, the calibration experiments yield the dataset that includes values of the manipulator joint coordinates {qi }, applied forces/torques {Fi } and corresponding deflections of the reference points {pi }.

60

A. Klimchik and A. Pashkevich

Using these data, it is required to identify the manipulator elastostatic parameters of kθ j , j = 1, 6 and gravity compensator parameters s0 , sV . To take into account the compensator influence while using classical approach developed for strictly serial manipulators without compensators [13, 59], it was proposed below to use in the second joint an equivalent virtual spring with nonlinear stiffness, which depends on the joint coordinate q2 (see Eq. (16)). Using this idea, it is convenient to consider several aggregated compliances kθ2i corresponding to each different value of angle q2 . This idea allows us to linearize the identification equations with respect to extended set of model parameters and that can be easily solved using standard least-square technique. Let us denote this extended set of desired parameters as k1 , (k21 , k22 . . .), k3 , . . . , k6 and collect them in the vector k. In this case for the elastostatic parameters identification we will obtain similar to previous case solution (64), where parameters k1 , k3 , . . . , k6 describe the compliance of the virtual joints #1, #3, …#6, while the rest of them k21 , k22 . . . present an auxiliary dataset allowing to separate the compliance of the joint #2 and the compensator parameters s0 , sV . Using Eq. (16), the desired optimization problem is written as  mq   P0 · L · a s0 − s V −1 k2i a · L · sin2 (α − q2i ) · − K q2 + 2 si − sV s (s − s ) V i i i=1 2   s0 − si + si2 cos(α − q2i ) + a L sin2 (α − q2i ) → min K q2 , s0 , sV si

(73)

where m q is the number of different angles q2 in the experimental data. It is obvious that Eq. (73) is highly non-linear and can be solved numerically only. Thus, the proposed modification of the previously developed calibration technique allows us to find the manipulator and compensator parameters. An open question, however, is how to find the set of measurement configurations that ensure the lowest impact of the measurement noise. For this purpose the design of experiments can be used [53, 54, 58]. Experimental Study. The developed technique was applied to the elastostatic calibration of robot Kuka KR-120. The parameters to be identified were the compliances k j of the actuated joints and the gravity compensator parameters s0 , sV . To generate deflections in the actuated joints, the gravity forces 140 kg were applied to the robot end-effector (see Fig. 19). The Cartesian coordinates of three markers located on the tool (see Fig. 16) have been measured before and after the loading. To find optimal measurement configurations for calibration, the design of experiments was used for six different angles q2 that are distributed between the joint limits [52, 60]. For each q2 from three to seven optimal measurement configurations were found, which satisfy joint limits and physical constraints related to the possibility carry out experiments. In total 31 different measurement configurations and 186 measurements were considered for the identification, from which 7 physical parameters were obtained. The obtained set of optimal measurement configurations

Stiffness Modeling for Gravity Compensators

61

Fig. 19 Experimental setup with industrial robot Kuka KR-120 for the identification of the elastostatic parameters

is given in Table 7. The obtained experimental data have been processed using the identification algorithm presented before. Identified values for the extended set of joint compliances (for 6 different angles q2 ) and their confidence intervals (CI) are presented in Table 8. As follows from these results, wrist compliances were identified with lower accuracy. The reason for it is smaller shoulder from the applied external forces comparing with manipulator joints. Relatively small accuracy of the first joint is due to a smaller number of measurements in the experiments in which the deflections were generated in the first joint (additional experiments with 5 different configurations with almost horizontal loading were used to generate deflections in Joint #1). Further, obtained compliances k21 … k26 were used to estimate pneumonic compensator parameters by solving optimization problem (73), which are presented in Table 9 and were used to build gravity compensator model (Fig. 20). The identified joint compliances can be used to predict robot deformations under the external loading.

3.3 Identification of Stiffness Model Parameters Using Double Encoders The robotic system under study is presented in Fig. 21. It consists of a robot, an end-effector location measurement unit, and equipment for external force/torque generation and estimation. In more details, the system includes the following components: • Robot manipulator with double encoders at each actuated joint; • Special end-effector equipped with several reference markers for position measurement; • Additional equipment allowing to generate and estimate external loadings;

62

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Table 7 Measurement configurations for elastostatic calibration of robot Kuka KR-120 Conf

q1

1-1

−35.0o

q2 0.0o

q3 0.0o

q4 0.0o

q5 0.0o

q6 −120.0o

1-2

−35.0o

0.0o

0.0o

90.0o

0.0o

−210.0o

1-3

−35.0o

0.0o

−30.6o

0.0o

30.6o

90.0o

1-4

−35.0o

0.0o

−90.0o

0.0o

90.0o

90.0o

1-5

−60.0o

0.0o

0.0o

90.0o

90.0o

−210.0o

1-6

25.0o

0.0o

0.0o

90.0o

−90.0o

30.0o

1-7

22.0o

0.0o

0.0o

0.0o

−90.0o

30.0o

2-1

−35.0o

−19.0o

−87.2o

0.0o

106.2o

270.0o

2-2

−20.0o

−19.0o

19.0o

−90.0o

90.0o

180.0o

2-3

−60.0o

−19.0o

19.0o

90.0o

90.0o

−180.0o

2-4

−17.5o

−19.0o

−62.6o

0.0o

81.6o

60.0o

3-1

−35.0o

−33.6o

−62.6o

0.0o

96.2o

270.0o

3-2

−35.0o

−33.6o

28.0o

0.0o

5.6o

90.0o

3-3

−25.0o

−33.6o

33.6o

−120.0o

90.0o

180.0o

3-4

−60.0o

−33.6o

33.6o

60.0o

90.0o

−180.0o

4-1

−35.0o

−90.0o

90.0o

0.0o

0.0o

−90.0o

4-2

−35.0o

−90.0o

90.0o

90.0o

0.0o

−20.0o

4-3

−70.0o

−90.0o

90.0o

90.0o

90.0o

−180.0o

4-4

55.0o

−90.0o

−90.0o

90.0o

−90.0o

−180.0o

4-5

−17.5o

−90.0o

60.0o

0.0o

30.0o

60.0o

4-6

145.0o

−90.0o

−90.0o

0.0o

0.0o

−110.0o

4-7

145.0o

−90.0o

−90.0o

90.0o

0.0o

20.0o

5-1

−35.0o

−104.6o

87.4o

0.0o

17.2o

90.0o

5-2

−35.0o

−104.6o

104.6o

0.0o

0.0o

−60.0o

5-3

−35.0o

−104.6o

104.6o

90.0o

0.0o

−30.0o

5-4

−35.0o

−104.6o

60.0o

0.0o

44.6o

270.0o

6-1

−35.0o

−120.0o

120.0o

90.0o

0.0o

−180.0o

6-2

−35.0o

−120.0o

84.5o

0.0o

35.5o

270.0o

6-3

−90.0o

−120.0o

120.0o

90.0o

90.0o

−180.0o

6-4

145.0o

−120.0o

−60.0o

−90.0o

0.0o

−180.0o

6-5

90.0o

−120.0o

−60.0o

−90.0o

90.0o

−180.0o

• Laser tracker for measuring the reference-point positions (such as Leica). To identify the stiffness model parameters, two types of experiments were carried out: with and without external loading, which can be either vertical or non-vertical one (almost horizontal).

Stiffness Modeling for Gravity Compensators Table 8 Elasto-static parameters of robot Kuka KR-120

Table 9 Compensator parameters of robot Kuka KR-120

63

Parameter

Value

CI

k1 , [rad × μm/N]

1.13

±0.15 (13.3%)

k21 , [rad × μm/N]

0.34

±0.004 (1.1%)

k22 , [rad × μm/N]

0.36

±0.005 (1.4%)

k23 , [rad × μm/N]

0.35

±0.005 (1.4%)

k24 , [rad × μm/N]

0.28

±0.007 (2.6%)

k25 , [rad × μm/N]

0.32

±0.011 (3.6%)

k26 , [rad × μm/N]

0.26

±0.007 (2.8%)

k3 , [rad × μm/N]

0.43

±0.007 (1.8%)

k4 , [rad × μm/N]

0.95

±0.31 (31.8%)

k5 , [rad × μm/N]

3.82

±0.27 (7.0%)

k6 , [rad × μm/N]

4.01

±0.35 (8.7%)

Parameter

Value

CI

k2 , [rad × μm/N]

0.35

±0.02 (5.7%)

s0 , [m]

1.20

±0.04 (3.3%)

sV , [m]

7.79

–

Fig. 20 Identified model of pneumatic compensator of Kuka KR-120

To use benefits of the double encoders, all experiments were repeated for two feedback modes in the joint position control, i.e. using either the primary or secondary encoders in the closed loop. It is worth mentioning that for a conventional robot with a single encoder (on the motor-side), each configuration allows to get two sets of experimental data corresponding to the unloaded and loaded modes [2, 25, 59, 61]. In contrast, robots with double encoders provide us the additional possibility to carry out the same experiments with joint feedback on the primary (motor-side) and secondary (link-side) encoders. In addition, in all experiments, it is possible to record

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Fig. 21 Experiment setup for elastic parameters identification

both values, obtained from the primary and secondary encoders. It is clear that this additional information gives us some benefits for identification of the manipulator stiffness parameters; it will be addressed in the following Section. Thus, for each robot configuration there are 4 types of the measurement modes corresponding to possible combinations of the manipulator loading and feedback control: U1: unloaded calibration experiment with primary encoder feedback (without external loading and without joint error compensation). It provides with the j following data: (i) the end-effector locations pU1,i , (ii) the data of the primary (1) (2) encoder qU1,i , (iii) the data of the secondary encoders qU1,i . U2: unloaded calibration experiment with secondary encoder feedback (without external loading and with joint error compensation). It provides with the following j (1) , data: (i) the end-effector locations pU2,i , (ii) the data of the primary encoder qU2,i (2) (iii) the data of the secondary encoders qU2,i . L1: loaded calibration experiment with primary encoder feedback (with external loading Fi and without joint error compensation). It provides with the following j (1) , data: (i) the end-effector locations pL1,i , (ii) the data of the primary encoder qL1,i (2) (iii) the data of the secondary encoders qL1,i . L2: loaded calibration experiment with secondary encoder feedback (with external loading Fi and with joint error compensation). It provides with the j following data: (i) the end-effector locations pL2,i , (ii) the data of the primary (1) (2) encoder qL2i , (iii) the data of the secondary encoders qL2,i .

Stiffness Modeling for Gravity Compensators

65

It should be stressed that in order to increase the calibration accuracy, the measurements for all 4 modes should be executed without moving the robot. It allows us to reduce the influence of the robot limited repeatability on the experimental data. Besides, to reduce the impact of the error compensation, the calibration experiments must be carried out in the following order: U1, L1, L2, U2. The latter is motivated by the possible existence of some hysteresis in on/off switching of the compliance error compensation algorithm based on the secondary encoder feedback. In the presented study for the experimental part model of industrial robot KUKA KR120 R3900 with its true geometric parameters and joint limits is used. Its stiffness model parameters (simplified stiffness model) were obtained in [59], which will be used here as an estimate for the joint compliances. Link stiffness models are approximated by hollow beams with cross-sections closed to real robot geometry. Stiffness Model Parameters Identification from End-Effector Measurements. In this case, the desired stiffness parameters are identified by approximating the linear functions describing force-deflection relations for each manipulator configurations. The identification technique remains the same for robots with both single and double encoders. The relevant procedure includes two steps [52]. At the first step, the 1 2 3 , utool , utool parameters are identified using the differbase pbase , ϕbase and tool utool ence pg,i between the coordinates extracted from the classical direct geometric transformations and the experiment results in the unloaded mode:  m −1  m   jT j  jT   1 2 3 pbase ; ϕbase ; utool ; utool ; utool = Ai Ai Ai pg,i i=1

(74)

i=1

At the second step, the stiffness parameter vector k is identified using measured j end-effector deflections pe,i caused by the external loading (for each measurement point) ⎛ k=⎝

m  n  i=1 j=1

⎞−1 ⎛ j ( p)

Bi

T

j ( p) ⎠

Bi

⎝

m  n 

⎞ j ( p)

Bi

T

pe,i ⎠ j

(75)

i=1 j=1

j ( p)

j

Here Ai and Bi are the observation matrices, that are described in details j in our previous work [52]. It is worth mentioning that the matrix Ai depends on j ( p) depends also on the applied the geometric parameters only, while the matrix Bi external loading Fi j Bi

j jT j jT = J1i J1i Fi , . . . , Jni Jni Fi ; j

j

 j Bi

=

j ( p)



Bi j (ϕ) Bi

where the vectors J1i , ..., Jni are the columns of kinematic Jacobian.

(76)

66

A. Klimchik and A. Pashkevich

Fig. 22 A two-link serial system with a passive joint

It should be mentioned that for robots with double encoders in order to produce different sets of the stiffness parameters the above relations can be used in several ways. All possible cases of the compliance errors and their physical meaning for four measurement modes are presented in Fig. 22. For example, the difference between the measurements in the loaded mode using the feedback based on the primary encoders (L1) and secondary encoders (L2) depends on the joint compliances, the gravity loading, and external loading. Hence, these data allow us to identify the joint compliances only if the robot link masses and their mass centers are known. The experimental results for the robot with double encoders allow the user to estimate the joint k J and link k L compliance as well as their sum k J +L using the following expressions ⎛ kJ = ⎝

m  n 

⎞−1 ⎛ j ( p)

Bi

T

j ( p) ⎠

Bi

i=1 j=1

⎝

m  n 

⎞ j ( p)

Bi

T

(pL2i + pU2,i − pL1i − pU1,i )⎠ j

j

j

j

i=1 j=1

(77) ⎞−1 ⎛ ⎛ ⎞ m  m  n n   T T j ( p) j ( p) j ( p) j j kL = ⎝ Bi Bi ⎠ ⎝ Bi (pL2,i − pU2,i )⎠ i=1 j=1

k J +L

⎛ ⎞ ⎞−1 ⎛ m  m  n n   T T j ( p) j ( p) j ( p) j j =⎝ Bi Bi ⎠ ⎝ Bi (pL1,i − pU1,i )⎠ i=1 j=1

(78)

i=1 j=1

(79)

i=1 j=1

It should be mentioned that here only one component of link compliance is taken into account that corresponds to bending around the adjacent axis joint. It should be

Stiffness Modeling for Gravity Compensators

67

also stressed that for all identification Eqs. (4)–(6) the observation matrices are the same and depend on the external loading applied to the manipulator end-effector. Hence, the above-presented methodology allows us to estimate separately the joint and link compliances and evaluate their impact on the end effector positioning accuracy. It is worth mentioning that the identification procedure developed for conventional industrial robots with single encoders [25, 59] allowed the user to identify the sum of joint and link compliances only, i.e. k J +L . This important contribution (separate identification of the link and joint compliances) will be effectively used below for the on-line error compensation. Stiffness Model Parameters Identification from Joint Encoders Measurements. For the robots with double encoders, there are additional data for the identification of the stiffness model parameters: primary and secondary encoders readings. In this case, the basic equation for the identification can be written as follows k J qi = JiT · Fi

(80)

where qi is a vector of angular deflections due to the external force Fi applied to the end-effector, and Ji is kinematic Jacobian for the ith measurement configuration. It is clear that here only the joint compliance can be identified using the usual least square expression kJ =

 m  i=1

 JiT · Fi · qiT

m 

−1 qiT · qi

(81)

i=1

It should be stressed that if link masses are not negligible (and provoke essential compliance errors) definition of qi should be clarified. In this case, it is prudent to apply equations based on differences of the secondary encoder measurements for (2) (2) − qU1 . experiments L1 and U1, i.e. qi = qL1

4 Conclusions Gravity compensation is one of the most important issues affecting manipulator positioning accuracy in industrial applications. There are three main approaches to reduce the impact of robot mass on positioning accuracy: integrating mechanical compensators, employing special kinematics and counterbalances, and using algorithmic approaches. In modern industrial manipulators usually, several types of gravity compensation are used. All these types of gravity compensation were discussed in detail with particular attention to the manipulator behavior prediction under the loading and model parameters identification. It should be mentioned that gravity compensators essentially affect stiffness modeling and induce additional closed loops in the manipulator kinematic structure, that cannot be ignored in the

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model and creates nonlinearity in the manipulator behavior. All theoretical results were confirmed by experimental studies that dealt with pneumatic and spring-based gravity compensators, kinematic parallelograms and robots with double encoders. Acknowledgements This work was Supported by Russian Scientific Foundation (Project number 22-41-02006).

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60. Klimchik, A., Caro, S., Pashkevich, A.: Optimal pose selection for calibration of planar anthropomorphic manipulators. Precis. Eng. 40, 214–229 (2015) 61. Dumas, C., Caro, S., Cherif, M., Garnier, S., Furet, B.: Joint stiffness identification of industrial serial robots. Robotica 30(04), 649–659 (2012)

Multi-DOF Counterbalancing and Applications to Robots Jae-Bok Song, Hwi-Su Kim, and Won-Bum Lee

Abstract Most robot arms use expensive motors and speed reducers to provide torques sufficient to support the robot mass and payload. If the gravitational torques due to the robot mass and/or payload can be compensated by some means, the robot would need much smaller torques for its operation, which can save energy and enables the use of cheaper actuator modules. To this end, counterbalance mechanisms (CBMs) or passive gravity compensators, which can fully or nearly compensate for the gravitational torques due to the robot mass and/or payload, have been developed so far. CBMs can be implemented by various mechanisms, such as wire-type, geartype, and link-type mechanisms. Since most robot arms have 2 or 3 pitch joints, which are subjected to gravity, multi-DOF counterbalancing techniques are also required to fully compensate for the gravitational torques regardless of the robot’s configuration. In this chapter, the principle of CBM and multi-DOF counterbalancing are discussed in detail. In addition, some counterbalance robot arms are presented to demonstrate their performance of gravity compensation. Simulation and experimental results show that the CBMs effectively decrease the torque required to support the robot mass and payload, thus allowing the prospective use of low-cost motors and speed reducers for high performance robot arms. Keywords Counterbalance mechanisms · Gravity compensators · Multi-DOF counterbalancing · Counterbalance robots

J.-B. Song (B) · W.-B. Lee Korea University, Seoul, Korea e-mail: [email protected] W.-B. Lee e-mail: [email protected] H.-S. Kim Korea Institute of Machinery and Materials, Daejeon, Korea e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 V. Arakelian (ed.), Gravity Compensation in Robotics, Mechanisms and Machine Science 115, https://doi.org/10.1007/978-3-030-95750-6_3

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1 Introduction Most conventional robot arms are equipped with expensive speed reducers and highperformance motors, which are used to produce high torques that allow the arms to withstand the load due to their own weight and motion. Cheap motors and speed reducers do not provide sufficient torques to support the arm mass and to accelerate it. Therefore, innovative strategies are needed to reduce the torque required for robot arms to achieve high performance even with cheap components. Note that the word “low-cost” means that the proposed robot arms can achieve performance (e.g., payload) similar to conventional robot arms more cheaply. To deal with these problems, several types of counterbalance mechanisms (CBMs) have been proposed and developed as a means to effectively compensate for the torques required to operate a robot arm. In the case of industrial robots, a heavy counterweight is often attached to the opposite side of the robot arm to balance the gravitational torque due to the manipulator mass [1]. However, such a large external mass not only increases the total mass of a robot, which results in an increase in the torque required for acceleration and deceleration, but can only be applied to the first pitch joint (e.g., the shoulder joint). To cope with this limitation, several types of CBMs using springs were developed [2–11]. These studies show that springs can be used to provide a counterbalancing torque instead of a heavy external mass. In addition, a robot arm called “the counterbalance robot arm” was developed, and several tests were conducted to investigate its effectiveness. Most robot arms are of 6 or 7 DOFs and 2 or 3 DOFs are subjected to gravity. Simple and independent implementation of two or three 1-DOF CBMs at the pitch jonits under the influence of gravity does not offer perfect counterbalancing. Therefore, a new concept of multi-DOF counterbalancing is required for application to the robot arms having multiple joints. Various types of multi-DOF counterbalancing techniques have been proposed for perfect counterbalancing [6–11]. Counterbalancing techniques can be implemented in real robot arms. Although multi-DOF counterbalancing is desirable for perfect cancellation of gravity, some counterbalance robot arms are equipped with only a single CBM because multiDOF counterbalancing requires linking the reference planes of each CBM and thus results in complicated mechanisms. The required capacity of the motors and speed reducers can be greatly decreased since the gravitational torques due to the robot mass are compensated for by the counterbalance mechanisms (CBMs). Therefore, compared to conventional robot arms, the counterbalance robot arm can achieve the same performance (such as payload) using cheaper components. The rest of the chapter is organized as follows. Various types of CBMs are introduced in Sect. 2. Multi-DOF counterbalancing techniques are discussed in Sect. 3. Section 4 presents some examples of counterbalance robot arms. Finally, the conclusion is presented in Sect. 5.

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2 Counterbalance Mechanisms The highest torques required for the motion of robot arms are gravitational torques and inertial torques. The Coriolis and centrifugal effects are not great unless the robot moves at high speeds. The inertial torque, which accelerates or decelerates the robot, is usually small, provided that the robot moves at reasonable speeds and accelerations. A gravitational torque occurs due to the masses of the robot and payload, but most of the gravitational torque is caused by the robot mass, which is much greater than the payload mass for most robots. This phenomenon can be seen in Fig. 1, in which the results of a simulation show the relatively small dependence of the required torque on the payload mass. It is therefore clear that counterbalancing the gravitational torque due to the robot mass can minimize the torque required at each joint. In this section, the basic concept of a counterbalance mechanism (CBM) (also called a passive gravity compensator) is presented. Counterbalancing based on springs can be realized by various mechanisms, such as tension spring-type, wiretype, link-type, and gear-type mechanisms. The wire-type, link-type and gear-type CBMs are based on compression springs. In the following sections, their structures and principles of operation are discussed in detail.

Fig. 1 Simulations of the torques required to operate a typical 6 DOF robot arm: a simulation model and condition, b torques required to operate joint 2 and c joint 3 with a payload of 0–2 kg between 0 and 90%, at 90°/s

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2.1 Tension Spring-Type Counterbalance Mechanisms Consider a single link having a pitch joint that is parallel to the ground, as shown in Fig. 2a [7]. The gravitational torque T g due to the mass of a link is given by Tg = mglc sin θ

(1)

where m and l are the mass and length of the link, lc is the distance from the joint axis to the link center of mass, and θ is the angular displacement of the link from the reference plane (RP), which is a virtual plane perpendicular to the ground. To maintain the link at a desired angular position, the compensation torque T c of the same magnitude as the gravitational torque (i.e., T c = T g ) must be provided against the gravitational torque. This compensation torque can be easily supplied by a motor and its magnitude and direction is controlled by the motor control system. Another way of offering this compensation torque is a passive gravity compensator or a counterbalance mechanism (CBM), which is a pure mechanical system including an elastic body, such as a coil spring of a gas spring. For a 1-DOF link shown in Fig. 2a, a CBM can be easily designed by installing a tension spring with a proper stiffness k between the RP and the link to generate a counterbalancing torque that cancels the gravitational torque. It is obvious that this simple installation of a spring can only offer a fixed compensation torque, but cannot provide a variable compensation torque. Therefore, some mechanisms are needed to automatically adjust the magnitude of the compensation torque so that it can cancel the gravitational torque that changes as a function of the link position, as shown in Fig. 2b where T c = T g . Let us analyze the counterbalancing systems in terms of energy. Assume that the spring is installed between point A on the RP and point B on the link so that OA = OB = a for convenience of computation in Fig. 2a. The spring is further assumed to be a zero-length spring whose length is 0 when it is undeformed. Then the spring force F s becomes

Fig. 2 Tension spring-type CBM: a concept model, and b torque as a function of θ

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Fs = k s = 2k a sin

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θ 2

(2)

where k is the spring stiffness and s is the tension length of a spring. When the link rotates by θ , the compensation torque is described by Tc = Fs d = 2ka sin

θ θ θ θ · a cos = 2ka 2 sin cos = ka 2 sinθ 2 2 2 2

(3)

where d is the shortest distance from the joint O to the spring. Perfect counterbalancing is obtained by equating the gravitational torque T g in Eq. (1) to the compensation torque in Eq. (3) as follows: m g lc = k a 2

(4)

When the CBM is designed to satisfy Eq. (4), the gravitational torque and compensation torque achieve torque equilibrium, and thus the link can be maintained at any rotation angle without the help of a motor. Let us analyze the counterbalancing system in terms of energy conservation. When the link is in an upright vertical position (i.e., θ = 0), the potential energies E gravity due to gravity and E spring due to the spring are both zero and thus their sum is also zero. When the link rotates by θ , the potential energy can be described by E gravity = mglc (cosθ − 1) E spring =

θ 1 2 k s = 2ka 2 sin2 2 2

(5) (6)

Note that both potential energies are set to zero for θ = 0, but E gravity is negative while E spring is positive for θ = 0. As q increases, E gravity decreases and E spring increases. The sum of both energies for perfect counterbalancing is obtained by E gravity + E spring

  2 θ = ka cos θ − 1 + 2 sin 2   2 2 θ 2 θ 2 θ − sin − 1 + 2 sin =0 = ka cos 2 2 2 2

(7)

When the CBM is designed to satisfy Eq. (6), the sum of potential energies is also preserved. That is, as the center of mass of the link is lowered, E gravity is converted into E spring without loss, thereby conserving energy and achieving torque balance.

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2.2 Wire-Type Counterbalance Mechanism The concept model of a wire-type CBM, which consists of a compression spring and a wire, is shown in Fig. 3 [7]. Note that a compression spring is used instead of a tension spring shown in Fig. 2. The idlers and spring holder are fixed to the link. One end of the wire is attached to the wire anchor point A that is fixed to the base and the other end is attached to the spring block. The reference plane (RP), which is perpendicular to the ground, coincides with the line connecting the center of joint O and the wire anchor point A. Let the angle between the wire and the link be φ, as shown in Fig. 3b. Then the following trigonometric relations hold: sinφ sinθ sin(θ + φ) = = a c b

(8)

where a, b, and c are defined in Fig. 3b. Note that a and b are constants while c changes with the rotation of the link. When the link is rotated, the spring is compressed by the spring block pulled by the wire. The compression length s of the spring is given by s = c − (b − a)

(9)

The distance between the wire anchor point A and the idler point B is (b − a) for θ = 0, but it increases to c for the posture shown in Fig. 3a. The wire tension F caused by spring compression is then described by F = k{s0 + s} = k{s0 + c − (b − a)}

Fig. 3 Wire-type CBM: a concept model, and b geometry and applied force

(10)

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Fig. 4 Comparison of gravitational and compensation torque of the wire-type CBM

where k is the spring stiffness, and s0 is the initial compression length of the spring. Noth that even when the link is in an upright position (i.e., θ = 0), the spring is compressed by s0 . The shortest distance d from the joint O to the wire is given by d = a sin{π − (θ + φ)} = a sin(θ + φ) = b sin φ

(11)

The compensation torque T c is then computed as Tc = Fd = k{s0 + c − (b − a)} · b sin φ =

abk [{s0 + c − (b − a)} sin θ ] (12) c

where the relation of c sin φ = a sin θ in Eq. (8) is used in this derivation. If s0 is set to (b − a), Eq. (12) is reduced to Tc = ab k sin θ

(13)

Perfect counterbalancing is obtained by equating the gravitational torque T g in Eq. (1) to the compensation torque in Eq. (13) as follows: a bk = mglc

(14)

If the design parameters a, b, and k are selected to satisfy Eq. (14), perfect counterbalancing is possible, as shown in Fig. 4.

2.3 Link-Type Counterbalance Mechanisms Wire-type CBMs discussed in Sect. 2.2 are easy to analyze and available for perfect counterbalancing, but it is difficult to apply them to real robots due to durability problems caused by wires. Link-type CBMs cope with this problem by using the link-based mechanisms instead of wire-based ones in compressing the spring [8]. Consider a 1-DOF robot link that is connected to the structure of a slider-crank mechanism with a compression spring in Fig. 5a. As the link rotates, the connecting

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Fig. 5 Link-type CBM: a concept model, and b geometry and applied force

rod located between the link and the slider moves the slider to compress the spring. The compression length s of the spring is given by s = (a + c) − b

(15)

where a, b, and c are defined in Fig. 5b. Note that a and c are constants while b changes with the rotation of the link. Successive applications of Eq. (8) yields sin θ cos φ + cos θ sin φ cos θ sin φ sin(θ + φ) =c = c cos φ + c sin θ sin θ sin θ cos θ sin θ = a cos θ + c cos φ (16) = c cos φ + a sin θ

b =c

Substitution of Eq. (15) into (16) gives s = (a + c) − (a cos θ + c cos φ)

(17)

The restoring force F s from the compressed spring becomes Fs = k(s0 + s)

(18)

where k is the spring stiffness. Since the component of the force F acting on the connecting rod along the link is equal to the spring force F s , the following relation holds: F=

Fs k (s0 + s) = cos φ cos φ

(19)

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The shortest distance d from the joint O to the connecting rod is given by d = a sin{π − (θ + φ)} = a sin(θ + φ)

(20)

The compensation torque T c is then computed as Tc = d F = a k{s0 + (a + c) − (a cos θ + c cos φ)}

sin(θ + φ) cos φ

(21)

Perfect counterbalancing is obtained by equating the gravitational torque T g in Eq. (1) to the compensation torque in Eq. (21) as follows: mglc sin θ = a k{s0 + (a + c) − (a cos θ + c cos φ)}

sin(θ + φ) cos φ

(22)

Of course, it is theoretically possible to adjust the design parameters a, c, k, and s0 so that Eq. (22) is satisfied, but numerous trial and error iterations are needed to find an appropriate combination of design parameters. To cope with this, let us simplify the compensation torque through the following assumption: ca

(23)

If this assumption is satisfied, the angle f becomes small and thus cosφ ≈ 1 and sinφ ≈ 0. Therefore, Eq. (21) is simplified to Tc ≈ a k(s0 + a − a cos θ ) sin θ

(24)

If s0 is set to (c − a), Eq. (24) is further reduced to Tc ≈ a k(c − a cos θ ) sin θ ≈ ack sin θ

(25)

Therefore, Eq. (22) is simplified to a c k ≈ mglc

(26)

If the design parameters a, c, and k are selected to satisfy Eq. (26), approximate counterbalancing can be achieved. Figure 6 shows the results of the simulation which compare the gravitational torque with the compensation torques generated from the CBMs with different parameters for a 1 DOF robot arm of m = 4 kg and lc = 200 mm. In this simulation, the length of the connecting rod was increased to c = 60, 80, and 120 mm while the length of the crank was fixed at a = 40 mm. As shown in the figure, it is seen that as c increases compared to a, the compensation torque graph approaches the sinusoidal shape of the gravitational torque. It is possible to completely cancel the gravitational torque only when the assumption of Eq. (23) is satisfied, which is impossible to

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Fig. 6 Simulation of a link-type CBM with different lengths of the connecting rod

12

a = 40 mm

Torque (Nm)

10

c = 80 mm

c = 60 mm

c = 120 mm

8

Tg

6 4 2 0

0

90 Displacement (deg)

180

Fig. 7 Gear-type CBM

implement, so there is always a compensation error in the link-type CBM. However, as shown in the figure, since the maximum error is only approximately 4% of the maximum gravitational torque even when c = 3a, sufficient gravity compensation can be performed even if Eq. (23) is not fully satisfied. Therefore, if the design parameters are appropriately selected to satisfy the assumption of Eq. (23) in addition to the condition of Eq. (26), the link-type CBM show the performance of imperfect but effective gravity compensation.

2.4 Gear-Type Counterbalance Mechanism A gear-type CBM shown in Fig. 7 consists of a compression spring, a link, two gears,

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and rollers. The link and rollers are fixed to gears 1 and 2, respectively. Gear 1 is meshed with gear 2 through a gear ratio of 1:2. The spring block is constrained to move in the horizontal direction through a LM guide. As the link rotates by θ , gear 2 rotates by θ/2 in the opposite direction, and the roller located at a distance R from the rotation center of gear 2 pushes the spring block and thus compresses the spring. Then the compression length s of the spring becomes s = R sin

θ 2

(27)

and the restoring force F of a spring is given by F = k s = k R sin

θ 2

(28)

where k is the spring stiffness. The restoring force F acting on the roller induces a restoring torque T r applied to gear 2 as follows: Tr = F R cos

θ k R2 θ θ = k R 2 sin cos = sinθ 2 2 2 2

(29)

This restoring torque is transmitted to gear 1 and serves as the following compensation torque T c Tc =

k R2 Tr = sinθ 2 4

(30)

Perfect counterbalancing is obtained by equating the gravitational torque T g in Eq. (1) to the compensation torque in Eq. (30) as follows: k R2 = mglc 4

(31)

If the design parameters k and R are selected to satisfy Eq. (31), perfect counterbalancing is possible, as shown in Fig. 4.

2.5 Comparison of Various Types of CBMs In the previous sections, four types of CBMs based on tension spring-type, wiretype, link-type, and gear-type mechanisms were discussed in detail. The wire-type,

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Fig. 8 Compression and tension springs

Compression spring

Tension spring

Table 1 Comparison of four types of CBMs Advantages

Disadvantages

Tension spring-type

- Easy and intuitive design - No buckling - No need for spring guides

- Multiple springs needed for sufficient compensation torque - Relatively large volume

Wire-type

- Perfect counterbalancing - Small volume and weight

- Wire durability problem - Difficult to adjust spring preload

Link-type

- Good durability - Compact and slim structure

- Imperfect counterbalancing

Gear-type

- Perfect counterbalancing - Good durability

- Relatively large volume and weight - Need for gear lubrication

link-type and gear-type CBMs are based on compression springs while the tension spring-type CBMs utilize tension springs. Figure 8 show the photos of compression and tension springs. In general, compression springs can have greater stiffness than tension springs. Due to the characteristics of the tension spring, it is difficult to increase the magnitude of the compensation torque because the maximum force and deformation are small for the same coil diameter compared to the compression spring. To cope with this, it is necessary to use multiple tension springs in parallel, so the volume of the CBM increases. However, since no buckling occurs in the tension spring, it does not need a spring guide that must be used on the compression spring to prevent buckling from occurring during the compression process. Table 1 lists the advantages and disadvantages of four types of CBMs.

3 Multi-DOF Counterbalancing Most robot arms are of 6 or 7 DOFs and 2 or 3 DOFs are subjected to gravity. Simple and independent implementation of two or three 1-DOF counterbalance mechanisms (CBMs) presented in Sect. 2 at the links under the influence of gravity does not offer perfect counterbalancing. Therefore, a new concept of multi-DOF counterbalancing is required for application to the robot arms having multiple joints, as shown in Fig. 9. Various types of multi-DOF CBMs applicable to continuous pitch joints are discussed in this section [7, 8, 10]. Multi-DOF counterbalancing is possible for the structures including a roll joint (e.g., roll-pitch-pitch, roll-pitch-yaw, etc.) [11], but its principle of operation is much more complicated than the structure having

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Fig. 9 Two cases of a 2-DOF linkage composed of two pitch joints

only pitch joints. The key element of multi-DOF counterbalancing is how to link the reference planes at each joint. Several types of reference plane linking methods will be discussed in this section. Although link-type CBMs are mainly explained in connection with multi-DOF counterbalancing in the section, the same principle can be used for wire-type and gear-roller type CBMs.

3.1 Principle of CBMs for Continuous Pitch Joints Multi-DOF counterbalancing applicable to the robot arm with continuous pitch joints are presented in this section. Floor-mounted robot arms usually have a RPP (i.e., rollpitch-pitch) structure in the lower body. The roll joint is not affected by gravity, while the two pitch joints are affected by gravity. Therefore, multi-DOF counterbalancing for continuous pitch joints is the most basic and widely used, and much research has been studied to realize it in the real robots. Consider a 2-DOF linkage composed of continuous pitch joints shown in Fig. 9, where RPs 1 and 2 are the reference planes for links 1 and 2, respectively. Note   that RP 2 is set along link 1. The gravitational torques Tg1 and Tg2 at each joint are described by    = m 1 glc1 sin θ1 + m 2 g l1 sin θ1 + lc2 sin(θ1 + θ2 ) Tg1 = (m 1 glc1 + m 2 gl1 ) sin θ1 + m 2 glc2 sin(θ1 + θ2 )  Tg2 = m 2 glc2 sin(θ1 + θ2 )

(32) (33)

where m1 is the mass of link 1, l c1 is the distance from the center of rotation of link 1 to the center of mass of link 1, m2 is the mass of link 2, and lc2 is the distance from the center of rotation of link 2 to the center of mass of link 2, l1 is the length  of link 1, and θ1 and θ2 are the rotation angles of joint 1 and joint 2, respectively.

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Fig. 10 CBMs installed at 2-DOF linkage composed of two pitch joints

The second term of Eq. (32) is identical to Eq. (33). As can be seen in Eqs. (32) and (33), the gravitational torques are dependent on both joint angles. Generally speaking, the gravitational torque of each joint is affected by the joint angles of not only the corresponding joint but also other joints. Consider the 2-DOF linkage composed of two continuous pitch joints in Fig. 10 to understand the problems with multi-DOF counterbalancing. As shown in Fig. 10a, assume that the reference plane is set in the preceding link. That is, RP 1 for link 1 is set at the base link, which is link 0, and RP 2 for link 2 is set at link 1. CBM 1 is installed between RP 1 and link 1, and CBM 2 is installed between RP 2 and link 2.  In the first case shown in Fig. 10b, link 2 rotates by θ2 , while link 1 remains fixed.  In this case, CBM 2 is activated to compensate for Tg2 caused by link 2. However, in   Eq. (32), if θ2 changes by the rotation of link 2, Tg1 also changes. Compensation for  this change in Tg1 requires of the action of CBM 1, but CBM 1 cannot be activated because of no movement of link 1. To solve this problem, the rotation of link 2 should   not affect Tg1 of link 1 so that Tg1 is expressed as a function of only θ1 . To this end,  Tg2 compensated by CBM 2 must be transmitted to the ground through a separate path without being delivered to link 1. In the second case shown in Fig. 10c, links 1 and 2 rotate together by the identical  rotation angle θ1 . In this case, Tg2 is affected by the change of θ1 , but because of no relative rotation of link 2 with respect to link 1, CBM 2 cannot be activated and thus  Tg2 cannot be properly compensated. To solve this problem, RP 2 of link 2 must be set so that it does not move with link 1 and always maintains a direction perpendicular to the ground. Combining the above two solutions leads to the following two conditions: Condition 1) Condition 2)

The reference plane of each CBM must be independent of the motion of other links and must always be kept perpendicular to the ground. The compensation torque in each CBM must be transmitted to the ground through a separate path without affecting other joints.

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RP 1 RP 2 Driving joint

Pulley 2

RP 2

Belt Driven link

Base

RP 1

y x

Base (a)

Pulley 1 (Fixed) (b)

Fig. 11 Parallelogram structure that links reference planes: a four-bar linkage, and b belt-pulley mechanism

Condition 1 can be realized in various ways. Among them, the parallelogram structure based on a four-bar linkage shown in Fig. 11a is the most basic and several mechanisms are based on it. As shown in the figure, the driven link is always perpendicular to the ground regardless of the rotation angles of the driving joint. Another parallelogram structure can be realized by the belt-pulley mechanism shown in Fig. 11b. Pulley 1 is fixed to the base and it does not rotate. Suppose RPs 1 and 2 are set at pulleys 1 and 2 so that both are perpendicular to the ground. Although the link rotates, both RPs remain perpendicular to the ground at any link configurations, thereby satisfying condition 1. Condition 2 can be satisfied by both mechanisms in Fig. 11. In what follows, condition 2 associated with the belt-pulley mechanism in Fig. 12 will be discussed. CBM 2 generates the compensation torque Tc2 to cancel Tg2 . Since CBM 2 is installed at the pulley, the reaction torque of Tc2 is created at pulley 2 and transmitted to pulley 1 through the belt and finally to the ground. Note that the torque Tb due to the belt tension is equal to Tc2 in magnitude. Since Tc2 is not delivered to link 1, the gravitational torque Tg1 are dependent only on the joint angle θ1 as follows: Tg1 = (m 1 glc1 + m 2 gl1 ) sin θ1

(34)

Note that Tg1 is decoupled with link 2, but it still includes the effect of the mass of link 2. Additionally, since RP 2 is always perpendicular to the ground, the gravitational torque Tg2 can be computed as a function of θ2 as follows: Tg2 = m 2 glc2 sin θ2

(35)

As shown in Eqs. (35) and (36), when the reference plane linking device based on the parallelogram structure is applied, the gravitational torque acting on each joint

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Fig. 12 Parallelogram structure for multi-DOF counterbalancing in two pitch joints

is expressed as a function of only the rotation angle of the corresponding joint. The CBMs other than the link-type CBM can be applied to continuous pitch joints for multi-DOF counterbalancing similarly.

3.2 Parallelogram Structure for Continuous Pitch Joints The parallelogram structures for linking the reference planes can be implemented in several ways including the belt-pulley mechanism, gear mechanism and 4-bar linkage. Figure 13 shows various types of the reference plane linking mechanism based on the belt and pulley. Timing belt-pulley, steel wire-pulley, chain-sprocket structure, and so on can be used, and in all three cases, idlers to maintain proper tension must be installed inside the link. The timing belt-pulley structure of Fig. 13a allows the use of standard products for both the belt and pulley, so it has the advantage of having a relatively low design difficulty and adequate strength [5, 7]. However, since the timing belt is less rigid than the metal link and the reference plane may be slightly inclined, the tension of the idlers or the preload of the CBM must be adjusted accordingly. The steel wire-pulley structure of Fig. 13b is advantageous in that it has a relatively high degree of freedom in design and component arrangement, but it has a disadvantage that installation and maintenance are difficult, and special care should be taken not to damage the wire [1]. The chain-sprocket structure of Fig. 13c allows the use of standard products, so it has low design difficulty and high strength and rigidity. However, the weight is somewhat heavier than the other two types, and there is a disadvantage in that backlash occurs according to the direction of the torque applied to joint 2.

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Idler

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RP 2

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CBM 1 RP 1 2

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(c)

(b)

(a)

Fig. 13 Parallelogram structures based on belt-pulley mechanism: a timing belt-pulley type, b steel wire-pulley type, and c chain-sprocket type

The two types of parallelogram structure based on gears are shown in Fig. 14. Figure 14a shows a structure that maintains parallel between the reference planes by succussive spur gears. It has high reliability, low design difficulty, and very high strength and stiffness compared to its volume, but it should be noted that as the number of gears increases, backlash accumulates and the overall weight increases as well. Figure 14b is based on bevel gears [8]. Compared to the spur gear structure, there is an advantage that the reference planes can be efficiently linked even when the

Joint 2

RP 2

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RP 2 CBM 2

CBM 1

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2 1

1

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(b)

Fig. 14 Parallelogram structures based on gears: a spur gear type, and b bevel gear type

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2

1

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1

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(b)

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RP 1

Output link RP 1, 2

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CBM 1

Joint 1 1 2

Link 1 (c)

Input link

Link 2

1

Link 1

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Joint 1 (d)

Fig. 15 Parallelogram structure based on 4-bar linkage: a basic type, b internal linkage type, c double 4-bar linkage type, and d external linkage type

distance between the joints is great, but there is also a disadvantage that the design is complicated and the space efficiency is not good due to the bevel gear arrangement. Figure 15 shows various types of reference plane linking mechanism based on the 4-bar linkage. Figure 15a is the most basic type of parallel 4-bar linkage [1, 6], in which RPs 1 and 2 and a pair of support links constitute a parallel 4-bar linkage structure, and two support links play the role of the robot’s body. This structure is the simplest to design and manufacture and is used when there is no need to seal the space between joints 1 and 2. In Fig. 15b, RP 1, RP 2 and one support link, and link 1 itself constitutes a parallel 4-bar linkage. Since it is a structure that wraps between joints 1 and 2 with link 1, there is no fear of foreign substances getting caught between the 4-bar linkage, and the spring of the CBM can be built into the link. However, due to the characteristics of a 4-bar linkage structure, the devices of Fig. 15a and 15b cannot support an external load by forming a dead point when a pair of adjacent links is in a straight line. That is, since the robot cannot approach the point where θ1 becomes 0◦ or 180◦ , the angle of joint 1 is limited to approximately 20◦ < θ1 < 160◦ . Figure 15c is a double parallelogram linkage structure implemented with link 1 and two support links inside to solve this problem [4]. The rotation points of the two support links are distributed at right angles based on the joint center, so that when one support link is located at the dead point, the other support link supports the load. According to this principle, the double parallelogram linkage structure has the advantage that no angle limitation occurs in joint 1. However, it has to be precisely machined so that the lengths of link 1 and the two support links are completely equal, and there is a disadvantage that backlash or over-constraint may occur in RP 2 even with a slight machining error and assembly error. Figure 15d shows the structure of arranging the driving part of joint 2 coaxially with joint 1 and transmitting the driving torque of

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joint 2 to link 2 through an external 4-bar linkage [3]. The CBMs of joints 1 and 2 use the same reference plane, and the 4-bar linkage transmits both the driving torque of joint 2 and compensation torque of CBM 2 to link 2. This structure has the advantage of high rigidity in the process of transmitting the compensation torque and driving torque because it is possible to manufacture each link thickly and robustly because the 4-bar linkage is mounted on the outside of the robot. Due to the nature of the 4-bar linkage structure, the angle of link 2 with respect to link 1 is limited to less than 180◦ , but considering the workspace of the robot, there is no need for link 2 to move to the back of link 1. The biggest disadvantage of this structure is that it is difficult to apply for safety reasons to robots that need to contact humans, such as collaborative robots or service robots because the 4-bar linkage is exposed outside, thus causing jamming accidents. As discussed before, in multi-DOF counterbalancing for continuous pitch joints, various methods can be used for linking the reference plane through a parallelogram structure. The designer should properly determine which method to select for the target robot by considering the characteristics of each device.

4 Counterbalance Robot Arms Multi-DOF counterbalancing can be implemented in real robot arms. Among many counterbalance robot arms that have actually been implemented, two examples are introduced in this section because all counterbalance robots exhibit similar performance. The effect of multi-DOF counterbalancing is discussed in simulation and experiments. The required capacity of the motors and speed reducers can be greatly decreased since the gravitational torques due to the robot mass are compensated for by the counterbalance mechanisms (CBMs). Therefore, compared to conventional robot arms, the counterbalance robot arm can achieve the same performance (such as payload) using cheaper components.

4.1 6-DOF Robot Arm with a Single CBM In this section, a 6-DOF material handling robot, which is equipped with a single gear-type CBM explained in Sect. 2.4, is presented. As shown in Fig. 16, this robot handles high-payload objects, such as tires of car bumpers, to help the worker perform assembly tasks. Its weight, payload and reach are 120, 25 kg and 1.80 m, respectively. This robot has a single gear-type CBM at the pitch joint 2, as shown in Fig. 17. To increase the spring stiffness, six springs are arranged in two rows, three per row. In the figure, the two meshing gears are exposed, but in reality the two gears are sealed by a case, and the case is filled with grease. The robot is ceiling-fixed, with a maximum power of 200 W for each joint. A motor of at least 1500 W is required to lift the weight of 150 kg, which is the sum of

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Joint 2 (Pitch)

Joint 2P (Passive)

Joint 4 (Yaw) Joint 5 (Roll)

Parallelogram linkage CBM

Joint 3 (Yaw) Joint 6 (Roll)

(a)

(b) Fig. 16. 6-DOF material handling robot arm: a schematic, and b photo

Fig. 17 Gear-type CBM used in the robot arm: a uncompressed springs and b compressed springs

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Joint angle (°)

Fig. 18 Required torque of joint 2 with CBM

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45

0 0

1

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With 20 kg payload

300

Joint torque

Torque (Nm)

250 200 - 78 %

150 100

Compensation torque

50 0 0

Required torque 1

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3

4 5 Time (s)

6

7

8

the robot weight of 120 kg and the payload of 25 kg. However, thanks to the built-in CBM, this robot can do this with only motors of up to 200 W. It consists of 6 active joints and 1 passive joint (joint 2P), all of which are revolute joints, as shown in Fig. 16a. The vertical movement of the robot is determined by combining the active pitch joint 2 with the passive joint 2P through the parallelogram linkage structure. In this joint configuration, it is possible to mount the CBM on the base link because the robot can determine the up and down position with only one pitch joint 2 while maintaining the posture parallel to the ground. Figure 18b shows the joint torque according to the joint motion of Fig. 18a. The torque required to operate the robot arm is significantly reduced by the compensation torque created by the CBM. Theoretically, complete gravity compensation is possible and thus the required torque can be almost zero when moving slowly, but a certain amount of motor torque is required to overcome friction existing in the robot joints.

4.2 6-DOF Robot Arm with Dual CBMs In this section, a 6-DOF collaborative robot, which is equipped with the link-type CBMs explained in Sect. 2.3, is presented. Its weight, payload and reach are 19.5, 5 kg and 0.95 m, respectively. The volume of the developed CBM and related components were minimized so as to be embedded inside the robot. Also, as shown in Fig. 19b, a joint module is constructed with a hollow shaft motor, a harmonic drive, a brake, an incremental encoder and a joint torque sensor.

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Joint 3 CBM 1

CBM 2

CBM 1

Joint 2

Joint 2

Joint Joint 4

Joint module

Joint 5 Joint 1

Joint 6

Joint 1 (b)

(a)

Fig. 19 Design of 6-DOF collaborative robot: a structure, b joint module, and c side sectional view of joint 2

To determine the CBM parameters, the arm parameters that are obtained from a CAD design were substituted into (35) and (36) to compute τg2 and τg3 with the subscripts 1 and 2 replaced by 2 and 3, respectively. Figure 20 shows the comparison of the gravitational torque and compensation torque. The design parameters, which are listed in Table 2, are primarily selected to minimize the size of the CBMs. As

Fig. 20 Simulation results of the CBM: a joint 2, and b joint 3

Table 2 Design parameters of link-type CBMs Joint

CBM

Design parameters a (mm)

c (mm)

k (N/mm)

s0 (mm)

Joint 2

CBM 1

52

120

7.84

90

Joint 3

CBM 2

48

150

6.54

66

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shown in Fig. 20, τc2 and τc3 can nearly cancel τg2 and τg3 , respectively, which significantly reduces the torques required to operate a robot. The performance of the proposed CBM and the collaborative robot equipped with two CBMs was verified based on several simulations. RecurDyn, which is a dynamics simulation software, was used in these simulations, while the dynamic model of the robot was generated from the CAD file of the robot. In these simulations, the magnitude of the joint torques and the energy consumptions that are required to perform a certain task were employed as indicators of the gravity compensation performance. Figure 21 shows the simulation conditions, and the simulations were conducted with two types of collaborative robots: without CBMs (i.e., τc2 = τc3 = 0) and with CBMs. Simulations were carried out based on three square paths in xy, yz, and zx-plane with one side being 0.3 m long, as shown in Fig. 21a–21c. The trapezoidal velocity profile was used to generate the paths shown in Fig. 21d, and the velocity and acceleration of the end-effector were set to 0.45 m/s and 1.3 m/s2 , respectively. It was assumed that a mass of 5 kg was attached at the end-effector as a payload. Figure 22 shows the results of the simulations discussed in Fig. 21. In Fig. 22, τ2 and τ3 are the torques required at joints 2 and 3 to perform the tasks, respectively. As shown in the figure, τ2 and τ3 were significantly reduced by the use of the CBMs. The torques became close to zero when the robot was operated at a constant velocity, while they increased up to 23 and 29 Nm when it was accelerated or decelerated.

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m P3 z P4 00 m 3 y (c)

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(a) P1

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P1 0.6

P2

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P4

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0

1

Fig. 21 Simulation conditions: a task 1 in the xy-plane, b task 2 in the yz-plane, c task 3 in the zx-plane, and d velocity profile

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Fig. 22 Simulation results for a joint 2 and b joint 3

This is because the inertial torques required for acceleration and deceleration should be created regardless of counterbalancing. However, the magnitude of this inertia torque is smaller than that of the gravitational torque except for special cases such as emergency stop. Furthermore, the magnitude of the inertia torque can be reduced by adjusting the acceleration profiles of a robot. As a result, it is possible to operate the proposed collaborative robot even with relatively low torques with the aid of CBMs. The maximum value of the joint torque required is important in determining the capacity of a motor and a harmonic drive in the design process of a robot. Therefore, their capacity can be lowered with CBMs installed. Table 3 shows the maximum values of τ2 and τ3 without CBMs and with CBMs. As can be seen in this table, the maximum values of τ2 and τ3 were significantly reduced by the introduction of CBMs in the proposed collaborative robot. Especially, the maximum value of τ2 can

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Table 3 Maximum values of joint torque Task

Joint torque

Maximum value (Nm)

Percentage reduction

Without CBM

With CBM

t2

64

23

64.1%

t3

52

29

44.2%

Task 2

t2

85

22

74.1%

t3

60

29

51.7%

Task 3

t2

86

23

73.2%

t3

56

21

62.5%

Task 1

be reduced by up to 74.1% when the robot performed task 2 in Fig. 21. Therefore, the collaborative robot can guarantee the performance to conduct its tasks even though it is equipped with low-capacity motors and harmonic drives. The motors and harmonic drives at joints 2 and 3 were selected based on the maximum values of the joint torques listed in Table 4. The angular velocity and acceleration required at each joint of the robot were calculated based on the simulation conditions shown in Fig. 21. The maximum angular velocities of joints 1, 2 and 3 were set to 120°/s, so the robot’s tip velocity does not exceed the maximum value recommended by ISO/TS 15,066. The acceleration was set to 400°/s2 , which enables the robot to reach its maximum speed in 0.3 s. Table 4 lists the capacity of the selected motors and harmonic drives at joints 2 and 3. As listed in the table, a motor of 300 W or higher should be used for joint 2 for a robot without CBMs, whereas a motor of 60 W or higher (possibly 100 W) motor is sufficient to provide the peak torque required with CBMs. Note that the UR-5 model of a payload of 5 kg of Universal Robots use a 400 W motor for joint 2. However, the proposed collaborative robot uses only 100 W motor and subsequently low-torque harmonic drive for an identical payload. Another benefit from the used of low-power motor is associated with collision safety which is a main requirement for collaborative robots. In other words, a robot equipped with low-power motors causes much less damage to humans and environments with accidental collision than that with high-power motors. Another advantage of the proposed collaborative robot is energy saving. To verify this feature, the amount of energy E consumed at each joint to perform the three tasks in Fig. 21 was calculated based on joint torque τ j as follows: Table 4 Capacity of motors and harmonic drives Joint

Speed reduction ratio of H/D

Peak torque of H/D

Max. power of motors

w/o CBM

w/ CBM

w/o CBM

w/ CBM

Joint 2

1/160

261 Nm

64 Nm

300 W

60 W

Joint 3

1/160

123 Nm

64 Nm

120 W

60 W

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Table 5 Energy consumption at joints 2 and 3 Task

Joint

Energy consumption (J) Without CBM

Task 1

Percentage reduction

With CBM

Joint 2

80.2

29.9

Joint 3

44.8

17.6

60.7%

Task 2

Joint 2

199.4

33.0

83.5%

Joint 3

126.6

50.2

60.3%

Task 3

Joint 2

111.4

17.0

84.7%

Joint 3

80.4

17.4

78.4%

1  |τ j (i)|θ (i) 0.5 i=0

62.7%

n

E=

(36)

where i is the discrete time index, n is the final time, and θ is an infinitesimal angular displacement of the joint. We assumed that the energy efficiency of the harmonic drive was 50% and the energy consumed by the controller were negligible. As listed in Table 5, the calculated energy consumed at joints 2 and 3 were significantly reduced with CBMs. In particular, the energy consumed at joint 2 can be reduced by up to 84.7% with the aid of CBM when the robot performs task 3 in Fig. 21. Therefore, much energy can be saved for the proposed collaborative robot.

5 Conclusion Counterbalance mechanisms (CBMs), which can fully or nearly compensate for the gravitational torques due to the robot mass and/or payload, have been discussed in this chapter. CBMs can be implemented by various mechanisms, such as tension-spring type, wire-type, gear-type, and link-type mechanisms. Each type has advantages and disadvantages, so the CBM type should be decided according to the charactersitcs of the applications. Since most robot arms have 2 or 3 continuous pitch joints, which are subjected to gravity, multi-DOF counterbalancing techniques are also required to fully compensate for the gravitational torques regardless of the robot’s configuration. The key element of multi-DOF counterbalancing is how to link the reference planes at each joint. The parallelogram structures are the most commonly used for this purpose. Some counterbalance robot arms have been presented to demonstrate their performance of gravity compensation. Simulation and experimental results show that the CBMs effectively decrease the torque required to support the robot mass and payload, thus allowing the prospective use of low-cost motors and speed reducers for high performance robot arms.

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References 1. Lacasse, M.-A., Lachance, G., Boisclair, J., Ouellet, J., Gosselin, C.: On the design of a statically balanced serial robot using, remote counterweights. In: International Conference on Robotics and Automation, pp. 4174–4179. IEEE (2013) 2. Morita, T., Kuribara, F., Shiozawa, Y., Sugano, S.: A novel mechanism design for gravity compensation in three dimensional space. In: International Conference on Advanced Intelligent Mechatronics, pp. 163–168. IEEE/ASME (2003) 3. Ulrich, N., Kumar, V.: Passive mechanical gravity compensation for robot manipulator. In: International Conference on Robotics and Automation, pp. 1536–1541. IEEE (1991) 4. Nakayama, T., Araki, Y., Fujimoto, H.: A new gravity compensation mechanism for lower limb rehabilitation. In: International Conference on Mechatronics and Automation, pp. 943–948. IEEE (2009) 5. Koser, K.: A cam mechanism for gravity-balancing. Mech. Res. Commun. 36(4), 523–530 (2009) 6. Cho, C.H., Kang, S.C.: Static balancing of manipulator with hemispherical work space. In: International Conference on Advanced Intelligent Mechatronics, pp. 1269–1274. IEEE/ASME (2010) 7. Kim, H.S., Song, J.B.: Multi-DOF counterbalance mechanism for a service robot arm. IEEE/ASME Trans. Mechatron. 19(6), 1756–1763 (2014) 8. Kim, H.S., Min, J.K., Song, J.B.: Multi-degree-of-freedom counterbalance robot arm based on slider-crank mechanism and bevel gear units. IEEE Trans. Robot. 32(1), 230–235 (2016) 9. Ahn, K.H., Lee, W.B., Song, J.B.: Reduction in gravitational torques of an industrial robot equipped with 2 DOF passive counterbalance mechanisms. In: International Conference on Intelligent Robots and Systems, pp. 4344–4349. IEEE (2016) 10. Lee, W.B., Lee, S.D., Song, J.B.: Design of a 6-DOF collaborative robot arm with counterbalance mechanisms. In: International Conference on Robotics and Automation, pp. 3696–3701. IEEE (2017) 11. Lee, W.B., Moon, B.Y., Kim, T.J., Song, J.B.: Wall-mounted robot arm equipped with 3-DOF roll-pitch-pitch counterbalance mechanism. In: International Conference on Intelligent Robots and Systems, pp. 3571–3576. IEEE (2019)

Series Parallel Elastic Actuator: Variable Recruitment of Parallel Springs for Partial Gravity Compensation Furnémont Raphaël, Glenn Mathijssen, Tom Verstraten, Bram Vanderborght, and Dirk Lefeber

Abstract In several applications, the gravitational load is dominant and requires high torque or force from the actuation. This load can be reduced using gravity compensaters which can provide a counter-torque/force passively. For applications with several degrees of freedom, providing perfect compensation can be extremely challenging and thus methods to provide partial compensation are used. The SeriesParallel Elastic Actuators use compliant elements, placed in series and parallel which can be tensioned separately. Through this recruitment, they can provide partial gravity compensation for complex tasks. Two Series parallel Elastic Actuators will be presented: the iSPEA which combines one motor which can recruit an arbitrary number of springs and the +SPEA which uses several units composed of a motor and a spring. It will be shown that both actuators can reduce the peak and RMS value of the actuation torque through partial gravity compensation. Keywords Series-parallel elastic actuator · Optimal control · Redundant actuation · Intermittent mechanism · Partial gravity compensation

1 Introduction Robotic platforms generally require high torque and low-speed actuation which leads to the use of electrical motors with large transmissions. Gravitational loads often are dominant in the torque profile and mechanisms such as gravity compensators reduce the torque requirements. This provides several benefits (lower consumption, smaller actuators, increased safety) as some drawbacks (added inertia for counterweights, increased mechanical complexity, limited range of motion). Research mainly focuses on compensators using compliant elements (e.g. springs) and is still progressing (a review can be found in [1]). Amongst recent progress, we can mention compensators for multiple degrees of freedom (DOF) such as a F. Raphaël (B) · G. Mathijssen · T. Verstraten · B. Vanderborght · D. Lefeber Vrije Universiteit Brussel, Brussels, Belgium e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 V. Arakelian (ed.), Gravity Compensation in Robotics, Mechanisms and Machine Science 115, https://doi.org/10.1007/978-3-030-95750-6_4

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compensator for roll-pitch rotations [2] or compensators for various payloads [3–9]. Several works have also proposed methods to implement compensators for multiple DOF systems: [10] proposes a method to implement gravity compensators for multiDOF systems but with the limitation that they all need to be in a plane perpendicular to gravity and these mechanisms rely on parallelogram structures which limit the range of motion. [11] proposes a method to implement one DOF compensators (with a spring or a counterweight) for open and/or closed kinematic chains. [12] proposes a method to implement the minimum number of compensators (considering 1, 2, 3 or 4 DOFs compensators) and [13] designed an articulated arm with 5 DOFs for which all degrees of freedom subjected to gravity are compensated. Nonetheless, compensators become very challenging to design once systems with many DOFs are considered. For example, the systems from [13] and [10] require a specific kinematic configuration. This is particularly evident when looking at [14] where 18 compensators (plus diverse gear/belt mechanisms) need to be implemented to statically balance a 4 DOFs arm. The complexity stems from the dependence of the gravitational torque, for a given joint, to the succeeding DOFs. If one considers a 2DOFs pendulum (with joint angles q1 and q2 ), the torque at the joints 1 (τ1 ) and 2 (τ2 ) has the following form: τ2 = T2 cos(q2 ) τ1 = T1,1 cos(q1 + q2 ) + T1,2 cos(q2 )

(1)

T1,1 , T1,2 , and T2 are constant and we can observe that the torques are trigonometric functions of the joints angle. This means that the compensators need to create a torque which is also a trigonometric function of the joint angles and also of the sum of joint angles for τ1 . This trend worsens as the number of DOFs increases. An approach to mitigate this problem is to design simpler compensators that provide imperfect compensation (actually, most gravity compensators referenced so far are not “perfect” due to neglected elements such as pulley radii) with the general idea to reduce the amplitude of the required torque. For example, various works such as [15–17] have proposed parallel spring design principles for robots with cyclic (and known) trajectories. In line with this idea, the Series-Parallel Elastic Actuators (SPEA) are actuators where the level of compensation, provided by parallel springs, can be varied to lower the torque requirements of the actuator, and hence achieve a system with lower electrical consumption. We will first explain the general concept behind the SPEA in Sect. 2 and detail two different implementations of the actuator called intermittent SPEA (iSPEA) and plus-SPEA (+SPEA) in Sects. 3 and 4. Both actuators will be compared in Sect. 5 and a conclusion will be provided in Sect. 6.

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2 Concept The SPEA concept is directly inspired by biological muscles. Biological muscles are composed of motor units. A motor unit consists of a motor neuron that is connected to muscle fibers through an axon. The motor neuron can activate all the motor fibers at once (ON-OFF behavior) and, when activated, the muscle fibers contract. Motor units differ in strength and the muscle is made of a parallel and series arrangement of them. According to Henneman’s size principle, motor units are orderly recruited from the weakest to the strongest [18]. As such, weak motor units are used to lift a light object, while stronger motor units are also being recruited when heavier objects are lifted. The concept led to the development of two different topologies: the intermittent and plus Series-Parallel Elastic Actuators and various prototypes, designed according to these two principles, are depicted in Fig. 1.

2.1 Intermittent Series-Parallel Elastic Actuator The iSPEA concept is depicted in Fig. 2. A motor unit will be represented by a DC motor plus transmission and a spring. A single motor is used to recruit the springs one by one. When a motor unit is activated (pre-tensioning spring), it is in series with the motor. Once it has been fully activated (tensioned spring), it is locked, and the motor will recruit a new spring. Furthermore, the total load Fout is carried by all

Fig. 1 iSPEA and +SPEA actuators. The iSPEA relies on one motor and intermittent mechanism to tension several springs while the +SPEA uses several motors in series with springs [19]

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Fig. 2 The SPEA schematic shows that the output force equals the sum of every layer while the motor force is only the force of the second layer [10]

the springs while the motor is only in series with one of them. The motor thus only carries a fraction of the load (Fload = F2 on Fig. 2) while the output force builds up as more springs are recruited. To variably recruit the springs, the motor has to be able to decouple itself from one spring when it is locked and couple itself to a new spring. This is realized with intermittent mechanisms placed in parallel. We can notice that although we cannot lock our muscles, this feature is also present in nature. Some animals, such as oysters, have a ‘Catch’ state which is a mechanical state that can last up to several hours and is characterized by high force maintenance and resistance to stretch during extremely slow relaxation [21] (this state is familiar to people who tried to open oysters, mussels, etc.). This is a very important feature of the SPEA as once blocked, a unit is seen as a parallel spring from the output. As such, the unit can passively deliver a torque without consuming any energy (which is the core principle of PEA).

2.2 Plus Series-Parallel Elastic Actuator Biological muscles mentioned previously, are also redundant as they are composed of motor units which themselves are a collection of muscle fibers. Inspired by this redundancy, we developed the plus Series-Parallel Elastic Actuator (+SPEA) [22]. Two different approaches have been considered: one with actuators with continuous output and a second one with actuators having discrete output but we will only focus on the first approach in this chapter. This first approach is depicted in Fig. 3 and uses electrical motors which provide a continuous output. This actuator is composed of several SEAs (motor units), placed in parallel, actuating one common joint. Additionally, each motor is also equipped with a nonbackdrivable element (a holding brake on the current setup) which allows blocking the motors and thus the springs. The difference with the iSPEA is that the tension of each spring can be controlled independently. It can be decided whether all motors are

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Fig. 3 +SPEA schematic. The three main components are the motor (and gearbox), the non-backdrivable element, and the elastic element [23]

used together to deliver high power, for example, or if some units are being passive, delivering high torque, while other units set the desired output motion.

3 iSPEA Working Principle In this section, we will give a more technical description of the iSPEA and show how the springs of the actuator can provide partial gravity compensation. We will present the last version of the iSPEA depicted in Fig. 4. This iSPEA has 8 springs actuated by a single motor. The springs are not tensioned all at once but in a specific order which depends on an intermittent mechanism. The intermittent mechanism has two specific parts. The first one is a self-closing mechanism depicted in Fig. 5. This mechanism is a modification of an existing compliant actuator called the MACCEPA [25]. The motor is directly coupled to the motor arm (red) while the output of the system is connected to the point “b”. Point “b” is connected to point “c” through a spring and point “c” belongs to a tensioner (green part) which can move inside of a guide (blue). On the left side of Fig. 5, we can see that the motor arm is decoupled from the output and, as it rotates anti-clockwise, it will be in contact with the tensioner (the tensioner has a cavity that can accommodate the motor arm) and then be coupled to the output (as it is the case on the right side of Fig. 5). By stacking multiple layers on top of each other and placing several motor arms, de-phased from each other, along the same axis, it is possible to use a single motor and tension springs individually. This tensioning, called recruitment sequence, can be done in a specific order defined by the design of the self-closing mechanism (dimensions of the guide, tensioner, etc.) and dephasing between the different motor arms. To avoid jamming of the mechanism after a full rotation (thus having the motor arm hitting the tensioner on its sides), a cylindrical cam has been added. This cylindrical cam, depicted in Fig. 6, allows one motor arm not only to rotate around point “a” from Fig. 5 but also to move upward/downward (as depicted in the lower part of

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Fig. 4 An iSPEA consists of a single motor, which can recruit and lock subsequent springs in parallel through an intermittent mechanism. The motor in a MACCEPA-based iSPEA can lock subsequent springs through a cylindrical cam mechanism [24]

Fig. 5 Intermittent MACCEPA when the tensioner (green) and the motor arm (red) are (de-)coupled [20]

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Fig. 6 The PolyJet 3D printed cylindrical cam mechanisms have a 4–5–6–7 cam profile. The rising cam and cam followers allow the motor arm to subsequently rotate in the plane and transverse along the axis for 360° axis rotation [24]

Fig. 6). As such, a single motor arm can be used for several layers without jamming. Thanks to these two mechanisms, an arbitrary number of springs can be recruited with only one motor. A schematic of one layer is depicted in Fig. 7. k is the stiffness of the linear spring, B is the distance between point “a” and “c”, C is the distance between “a” and “b”, P is the pretension of the spring (which, if an additional motor is used, can be varied), F0 is the initial tension of the spring (since an extension spring is used). ω is the position of the lever arm, ϕ is the equilibrium position of the layer, ϕend is the maximum equilibrium position (reached when a tensioner is at one of the two Fig. 7 Schematic and nomenclature of the intermittent MACCEPA with a guide (blue) and tensioner (green). The motor arm (red) is disconnected from the spring if |ω| > ϕ end [20]

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ends of the guide),  is the output position, and α is the deviation of the output from the equilibrium angle (hence α =  − ϕ). In first approximation, one has: ⎧ ω ≤ −ϕend ⎨ −ϕend if ϕ(ω) = ω if −ϕend < ω < ϕend ⎩ ϕend if ϕend < ω

(2)

Hence, ϕ(ω) = ω when the motor arm is in contact with the tensioner and the position of the tensioner is equal to ±ϕend when it has reached an end of the guide meaning the motor arm is decoupled from it. The torque seen from the output can be computed as follows:  To = k B(ω)C sin α 1 + 

P − C + B(ω) + F0 /k



B(ω)2 + C 2 − 2B(ω)C cos α

(3)

Interestingly, one can notice that if ϕend = π/2, Eq. (3) is similar to the torque provided by the classical one DOF gravity compensator [26] although this does not mean that the actuator can perform well for gravitational loads since multiple layers are combined. When considering multiple layers, each layer provides a torque To,i (Bi (ω), αi ), and the total torque is the summation of all contributions. If one has q layers, then: To,S P E A =

q i=1

To,i (Bi (ω), αi ) =

q i=1

 k Bi (ω)C sin αi

1+ 



P − C + Bi (ω) + F0 /k Bi (ω)2 + C 2 − 2Bi (ω)C cos αi

(4) In first approximation, one can also state that the motor torque can be expressed as: Tl,S P E A =

 A

To,i (Bi (ω), αi )

(5)

We can see that Eqs. (5) and (4) differ from the summation index. The summation index A from Eq. (5) concerns the layers which are being actuated, hence the layers for which the lever arms are in contact with their tensioners. This is not the case in Fig. 7 where ω > ϕend and the motor arm is thus decoupled. If we use the indexes Bl to denote the layers from which the tensioners are locked on the left side of their guide (such as the tensioner from Fig. 7) and Br to denote the layers from which the tensioners are locked on the right side of their guide, one can rewrite Eqs. (5) and (4) as: To,S P E A = Tl,S P E A =

Bl

To,i (Bi (ω), αi ) +

To,i (Bi (ω), αi ) + Br To,i (Bi (ω), αi ) A A To,i (Bi (ω), αi ) (6)

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We can also infer that:    To,i (Bi (ω), αi )+ To,i (Bi (ω), αi ) To,S P E A = Tl,S P E A − Bl B





Tactive

(7)

T passive

Eq. (7) explains the basic principle of the iSPEA and how it can provide partial gravity compensation. We have split the output torque into two contributions: Tactive which is the torque loading the motor and T passive which is the torque provided by the locked tensioners. We denote these torques as passive and active simply because Tactive is the torque actively felt by the motor which will lead to power consumption while T passive is the torque passively delivered by the locked springs and can be seen as a parallel spring (hence this torque is provided for “free”) and will provide the partial gravity compensation. T passive and Tactive do not necessarily have the same sign and thus that |Tactive | < To,S P E A  (which is desired) but also that  it is possible    |Tactive | > To,S P E A although the goal is to use the motor to lock   tensioners on the right or left sides of their guide such that |Tactive | < To,S P E A  is achieved most of the time. To give a better feeling of how the actuator works, Fig. 8 depicts a blocked output experiment. Simply put, the output position is set to zero ( = 0°) and all tensioners are initially locked on one side of their guides and the motor will successively unlock each tensioner, one by one, and lock them to the other side of their guide. We can observe that the motor torque Tl,S P E A has a repeating profile over ω which is normal since it is the profile a single layer repeated 8 times. To,S P E A , on the other hand, goes from −30 to 30 Nm in a linear/staircase fashion. To,S P E A possesses also the same profile repeated 8 times with the exception that the curve is shifted each time one tensioner has been locked. This can be observed from Eq. (7): Tactive is repeating, as mentioned before, while the value of T passive is constant when a tensioner is actuated Fig. 8 Tmotor is up to 8 times lower than Toutput , which reaches up to 30 nm. The modeled and measured results are comparable which confirms the model is reliable [24]

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but switches to different values as the number of tensioners locked on the right or left change.   In this particular case, we have |Tactive | < To,S P E A  (as Tactive = Tmotor ) for all ω (except a very narrow region around ω = 0) but this is not true once the output is moving. Nonetheless, the blocked output gives a good idea of the working principle: we see that the motor torque remains bounded between ±5 Nm while the output torque can build up to 30 Nm of amplitude.

3.1 Case Study: iSPEA Driven Warehouse Robots This section provides a case study of warehouse robots. The robot which will be studied in simulation is an articulated warehouse robot arm, of which the shoulder and elbow joints are of particular interest since they are subject to gravitational load. Both joints will be simulated to be driven by iSPEAs. The main idea is to simulate a realistic warehouse task in which 3 packages are picked by the gripper and dropped at 3 locations with a different height on a warehouse shelf. The series of pictures in Fig. 9 shows the consecutive steps of picking and placing three packages. The software V-REP is used for the simulations [27]. The V-REP’s motion planning module for kinematic chains is used to generate trajectories that have a defined start and end position. By combining several of these calculations subsequently, a trajectory that allows to stack the three packages on the rack is obtained. The motion planning module accounts for the manipulator kinematics, manipulator joint limits, manipulator self-collisions, and collisions between manipulator and obstacles. Obstacles in our simulation environment are the robot’s base, the rack, and the shelf where the packages are initially stacked. The idea here is to use the ‘simplified’ short trajectories and simulate the motor requirements, Tmotor and, ωmotor for the shoulder and elbow joint (more details can be found in [24]).

Fig. 9 The warehouse robot is equipped with a gripper of 1 kg and subsequently picks up 3 rectangular packages of 1 kg which are placed at different heights in a rack [24]

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The task is performed at a low speed and as such most of the joint torques are due to gravity. Although we will not distinguish between the two torques and as such the passive torque will compensate the output torque (hence both static and dynamic components), the dominant term is still the gravitational load. To assess the effectiveness of the gravity compensation, we will use two metrics which are given by: 

V1 =

max|Tactive | max|To |

V2 =

1 tf



tf

2 ∫t0 Tactive dt 1 tf

tf

∫t0 To2 dt

(8)

V1 is the ratio between the maximum of the active and output torques and is thus the reduction of peak torque on the motor through the passive torque and V2 is the ratio of the RMS values of the active and output torques and is the reduction of torque over the whole task. Figure 10 depicts the motor trajectory and torque (decomposed into passive and active torques) for the shoulder joint. One can observe, similar to Fig. 8, that the motor/active torque remains bounded between ±5 Nm while the output torque can go as high as 20 Nm in amplitude (and thus V1 = 0,24). We also represented the torque over the joint angle in Fig. 11. One can observe that the output torque is not only dependent on the shoulder joint angle but also that the passive torque provides a good compensation of the output torque since we can observe that the active torque remains close to 0 Nm over the whole joint angle range which is also reflected by the fact that V2 = 0,22 meaning that a good reduction of the RMS value has been achieved.

Fig. 10 Motor trajectory and active/passive torques of the iSPEA for the shoulder joint. One can see that T passive matches T o well which leads to a low load on the motor (T act ive )

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Fig. 11 Torque-angle characteristic of the task of the shoulder joint

Figure 12 depicts the motor trajectory and torque (decomposed into passive and active torques) for the elbow joint. Although we see that the passive torque compensation of the output torque is less effective, it still leads to an active torque not exceeding ±5 Nm while the output torque can reach 10 Nm of amplitude (V1 = 0,45). The fact that the active torque remains bounded between ±5 Nm can also be seen looking at V2 = 0,28 which demonstrates that the actuator still provides an acceptable level of gravity compensation. The values of V1 and V2 are also given in Table 1 for clarity.

Fig. 12 Motor trajectory and active/passive torques of the iSPEA for the elbow joint

Series Parallel Elastic Actuator … Table 1 V 1 and V 2 for the tasks considered in the case study using an iSPEA

4

113

Joint

V1

V2

Shoulder

0,2452

0,2252

Elbow

0,4549

0,2857

+SPEA Working Principle

We will now show how the +SPEA, depicted in Fig. 13, can also provide partial gravity compensation. The current +SPEA has 4 units/layers which are composed of a motor, holding brake, transmission, and spring except for the last unit which has a stiff connection (equivalent to a spring with infinite stiffness). The stiff unit allows having precise position control while the compliant units (units with springs) can provide the gravity compensation. The basic principle of the actuator is to use the compliant units to set a certain tension in the springs and then lock them. They then deliver torque as parallel springs. Indeed, when a spring is locked, so is the motor of the associated unit and thus the unit becomes passive. Now comes the question of what tension and when to use the brake? The +SPEA is a redundant/over-actuated actuator (Multi-Input Single Output system) and there are infinite input signals which can lead to the same output. As such, a strategy needs to be devised to control the actuator according to a specific goal. The goal will be to achieve minimum electrical losses while performing a specific

Fig. 13 +SPEA prototype. Each unit is composed of a motor, holding brake, transmission (gearbox + bevel gear), and spring [23]

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Fig. 14 Schematic of the +SPEA with gearboxes (G), motors (M), and brakes (B). The position of the stiff unit is the position of the output (θ q = θ o ) [22]

task and simulations will show that this can be achieved by using the parallel springs to compensate gravity. Figure 14 depicts the schematic of the +SPEA. θi and Ti (i is the index of the unit considered) are the motor position and load after the transmission. Compliant elements of stiffness K i are placed between the output, whose position is given by θo , and the transmission of motors 1 to q − 1 with q the number of units (here q = 4). The total torque exerted on the output axis is To . The brakes are mounted on the motor axes and their control variables (which are Booleans) are denoted γi . The total torque on the output can be expressed as follows: To −

q−1 

K i (θi − θo ) − Tq = 0

(9)

i=1

K i (θi − θo ) = Ti is the torque provided by the compliant unit i and we see thus that the output torque is the sum of the torques of the compliant units plus the last stiff unit which provides a torque Tq . The dynamics of the different units are described as:

(10)

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Jm is the inertia of the motor, n is the speed ratio of the transmission, ν is the viscous friction, kt is the torque constant, Ii the current, Cη,i the efficiency function of the transmission, ηi the maximum efficiency of the transmission, and R is the resistance of the windings. γi Ti is the load torque felt by the motor. One can see that the load torque is zero nCηi when γi = 0. This happens when the holding brake is locking the unit and the axis is not moving (hence θ¨i = θ˙i = 0). The consumption of the compliant unit indeed becomes zero in this case and the unit is now passive. To ensure that the speed (and acceleration) is zero when switching the state of the brake, an additional kinematic constraint is considered: (1 − γi )θ˙i = 0

(11)

We can see that for γi = 1, any velocity θ˙i satisfies Eq. (11) while θ˙i = 0 if γi = 0. If we consider the voltage of the motors and brake state  variables being the control variable of the systems u = U1 · · · Uq γ1 · · · γq−1 (γq = 1 and is not a control variable because locking the stiff unit would lock the whole system) while the output of the system is the output position θo , we see that we have indeed a MISO system and that multiple control variable combinations can lead to the same output position. The goal of the actuator is to achieve minimum consumption. If the electrical consumption is denoted JE , we have: tf

JE = ∫ t0

q  (Ui Ii )dt

(12)

i=1

t0 and t f are the initial and final times of the considered task. We will justify in Sect. 4.1 why we use the electrical consumption instead of metrics more closely related to the gravity compensation such as V1 or V2 . The problem of finding the control set u, for a given task (hence if (To (t), θo (t)) are given functions), can be written under the compact form as an optimal control problem (OCP) [28]: min JE u

 x˙ = f (x, u, t) c x, x0 , x f , t, t0 , t f ≤ 0

(13)

  JE is the cost function, x˙ = f (x, u, t) (with x˙ = x˙1 · · · x˙q ) represents the   dynamics of the actuator (see Eq. (10)) and c x, x0 , x f , t, t0 , t f is a function where all (in-)equality constraints and boundary conditions of the problem are encompassed (limit on the current, maximum spring deflection etc..). By solving this equation, for a given task, the control effort u can be found (more details can be found in [22]).

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Table 2 Parameters of the +SPEA Parameter

Value

Units

Parameter

Jm

9,31e−7

kgm2

η

n

327

/

ν

1,7e−6

Nms/rad

kt θ˙max

0,0135

Nm/A

8,16

αmax

2,61

Table 3 V 1 and V 2 for the tasks considered in the case study using a +SPEA

Units

K

Value   0,65 0,65 0,65   10,5 10,8

R

0,341



rad/s

Umax

24

V

rad

Inom

4,2

A

Joint

V1

V2

Shoulder

0,7789

0,5275

Elbow

0,7572

0,7050

/ Nm/rad

4.1 Case Study: +SPEA Driven Warehouse Robots We will reconsider the warehouse robot studied in Sect. 3.1 but with +SPEAs actuating the shoulder and elbow joints. We will consider a +SPEA with two motors for the elbow joint while we will consider a +SPEA with three motors for the shoulder joint to match the torque requirements of both joints. The parameters of the +SPEA are given in Table 2. The two +SPEA use the same motors and only the characteristics of the springs between the +SPEA actuating the shoulder and elbow joints will slightly differ. We have increased the deflection of the springs of the +SPEA compared to the actual value of the +SPEA (αmax = 2,61 instead of αmax = 0,87) due to the large range of joint motion. This is not unrealistic to consider such large deformations as spiral springs can have deflections up to three turns (18 radians). Figure 15 depicts the motor angles, torques and brake variable of the +SPEA obtained when solving Eq. (13) for the elbow joint. We are looking first at the elbow joint because it uses a +SPEA with only two motors and the results are easier to interpret. We see that the first unit is active between t ∈ [4:7] s, t ∈ [15:19] s, and, t ∈ [37:41] (see γ1 ) meaning that the spring of the first unit is initially locked (the unit is passive) but at t = 4 s the equilibrium angle of the spring is changed and the spring is locked again at t = 7 s and so on. This idea of a locked/passive unit can also be understood when looking at the position of the first unit θ1 : the position is constant when γ1 = 0. When looking at the torques, one needs to understand that the motor of the first unit is consuming power only between t ∈ [4:7] s, t ∈ [15:19] s, and, t ∈ [37:41] s and that the torque provided outside of these time intervals come at no cost since the spring is locked and acts as a parallel spring. θ1 does not vary much and the first unit is passive during much of the task meaning that the motor of the first

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Fig. 15 +SPEA motor trajectories, torques, and brake variable for the task of the elbow joint

unit is not often used to deliver power. This is one limitation of the +SPEA concept: once a unit is passive, the motor connected to it cannot be used which reduces the power the actuator can deliver. The spring of the passive unit can exchange power with the output, but this is limited by the maximum energy that can be stored in the spring. To give a better feeling, We can once more split the output torque into 2 contributions: the passive torque T passive and active torque Tactive which are expressed as follows: Tactive = T passive =

q−1 i=1 q−1

K i (θi − θo )γi − Tq (14) K i (θi − θo )(1 − γi )

i=1

The active torque is the torque that will lead to power consumption while the passive torque (delivered by the compliant units when passive) is the torque that will provide partial gravity compensation. These torques are depicted in Fig. 16. We can observe that the active torque has a maximum amplitude close to maximum the output torque (V1 = 0,77) meaning that the peak torque of the actuator still needs, in general, to match the peak torque of the application. We see that the amplitude of the active torque is reduced generally but this is not true on the entirety of the trajectory which also matches the fact that V2 = 0,52. We can also have a look at the torque angle characteristics depicted in Fig. 17. The passive torque, which is only due to one unit here, has a fixed slope with respect to θo (the slope being −K 1 ). The passive torque forms a set of lines that are shifted upward/downwards with respect to each other. This is logical since, when a unit is passive, the motor angle is constant and one has T passive =

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Fig. 16 Active/passive torques of the +SPEA for the elbow joint

Fig. 17 Torque-angle characteristic of the task of the elbow joint

  q−1 q−1 K i θi (1 − γi )− i=1 K i (1 − γi ) θo with i=1 K i θi (1 − γi ) the constant contribution (hence why the curves of the passive torque are shifted) which is adjusted, by the motor, when the unit is active. We can thus see the fundamental idea of the +SPEA: its ability to provide partial gravity compensation is defined by the number of compliant  units and their stiffness since the slope is q−1 −K − γ dT passive /dθo = (1 ) i i . By using several units with different stiffi=1 ness, various torque profiles can be approximated. This principle was also used in [29] where several linear springs were used to approximate a non-linear spring profile.

q−1 i=1

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This also shows that the ability of the +SPEA to provide partial gravity contribution is limited here since there is only one compliant unit. Figure 18 depicts the brake variables, motor angles, and torques of the +SPEA obtained when solving Eq. (13) for the shoulder joint. At some time intervals, one unit is passive while the other is active (see t ∈ [6 : 17] s) but it is also possible for both units to be passive (see t ∈ [22 : 28] s) or active (see t ∈ [28 : 33] s). One can note that both compliant units are active when large variations of torque and position occur (see t ∈ [28 : 33] s). As large mechanical power is required and the springs of the passive units cannot inject the required power, the motors of the compliant units need to be used.

Fig. 18 Motor trajectory and active/passive torques of the +SPEA for the shoulder joint

Fig. 19 Torque-angle characteristic of the task of the shoulder joint

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In essence, the compliant units will generally be passive when high torques but low power is needed and active when high power is required. As for the elbow joint, the peak of torque is reduced (V1 = 0,75) but to a lesser extent and V2 = 0,7 which gives only a small reduction of the RMS value. This can be well understood when looking at the torque angle characteristics depicted in Fig. 19. The passive torque can now switch between two linear curves which are also shifted from each other as was the case in Fig. 17. The slope is either −K 1 or −2K 1 (since K 1 ≈ K 2 we do not distinguish which unit is passive) depending on whether one or two units are passive. For θ > 0◦ , we see that the slope of the output torque (dTo /dθo ) is positive which is a slope that cannot be provided by the passive units and also explains why V2 and V1 are high. This is not a limitation of the +SPEA itself but of the compliant elements used which are spiral springs. By using units with different torque profiles (such as the units of the iSPEA), better results could be achieved. An important remark concerns the cost function used to solve the OCP: one may argue that it would be wiser to use V1 or V2 since the goal is to achieve the best gravity compensation possible. This is a valid remark and the cost function can be selected as seen fit by the user but since the overall goal of gravity compensation is to reduce the electrical consumption of the actuation, using the electrical power seems a better choice. Using the electrical consumption accounts for the energy required to tension the springs for example which can be important in case of large accelerations. Furthermore, the main goal was to show that the actuator can perform partial gravity compensation which was demonstrated. The values of V1 and V2 are also given in Table 3 for clarity.

5 Discussion We have presented two SPEAs with distinct working principles and showed how they can provide partial gravity compensation to reduce the torque loading the actuator. In our previous works [22, 24], we have also shown how these actuators can achieve lower electrical consumption compared to equivalent stiff actuators (which are traditionally used in robotics) but we have not discussed how both actuators compare to each other. Making a fair comparison between both actuators is difficult because the number of design variables of the iSPEA is very large and hard to compare with the design variables of the +SPEA. A simple example is the torque–angle characteristics of the layers of both actuators: the iSPEA presented in this chapter have complex nonlinear torque angle characteristics (see Eq. (3)) while the layers of the +SPEA have linear springs and we have explained how this lead to better values for V2 (and V1 by extension) in the case study from Sects. 3.1 and 4.1. We will thus discuss what are the main advantages and disadvantages of each actuator and thus for which applications one might perform better than the other. As mentioned, the recruitment sequence of the iSPEA is imposed by its design and can

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thus be optimized for a specific task to achieve low consumption but finding a design which performs well for a large set of tasks is challenging and thus the iSPEA might not be suitable for applications where high reconfigurability is required. This is less likely to be an issue for the +SPEA since every task is presented as an optimization problem with a new trajectory minimizing the given objective. Although this does not assure that the cost function is low, it still offers a variable of adjustment towards this goal. As such, the +SPEA seems more suitable for applications where the tasks might often change. Although not discussed in this chapter, the +SPEA has also been made modular [30] and this might also be done for the iSPEA which makes both actuators more versatile and easier to manufacture. The motor of the iSPEA is always in series with the output and as such the actuator can at least provide the rated power of the motor (plus the power from the locked springs) while the +SPEA cannot provide the power of all the motors if some units are passive. Since the advantage of the actuator comes from the reduced power consumption which can be achieved by appropriately using its passive units, the + SPEA cannot generally perform well for tasks with a continuous high output power requirement and is more suited for tasks with high torque but low average power which is not a problem for the iSPEA. The iSPEA is a compliant actuator which is more challenging to control. Furthermore, during the locking of the tensioner, the motor arm will stop being in contact with the tensioners as they move towards the end of the guides inside of which they will be locked. It means that during a small phase of the actuation, the tension of the springs is not controlled, and the locking of the springs also causes undesirable shocks. This can be solved by using a different intermittent mechanism though, but overall, the iSPEA would be more suitable for tasks where precise positioning is not required. The +SPEA, on the other hand, is a stiff actuator and it is thus possible to achieve good position accuracy. Unfortunately, to find the control variables for a given task, an optimal control problem needs to be solved which is very challenging for online control and it is thus preferable to use the +SPEA for tasks where the trajectories are known in advance. The +SPEA is also an over-actuated actuator giving it the potential to be fault-tolerant meaning that if one unit fails, the actuator can still perform the task although there will be a degradation of performance. In summary, the +SPEA is most likely best suited for tasks with high torque, low average mechanical power, and that are known in advance while the iSPEA will be best suited for applications with little variations in the tasks and where position accuracy is not required.

6 Conclusion In this chapter, we have discussed how the SPEA concept can be declined into actuators able to provide partial gravity compensation. We have seen that two implementations of it, the iSPEA and +SPEA, can both reduce the peak torque seen by the actuator as the RMS value of the torque for arbitrary tasks. As these actuators rely on

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compliant elements which can be used as parallel springs, they can perform well for gravitational load and, although they cannot provide perfect gravity compensation, they can be used for applications with a high dynamic load too or applications where the static loads are very complex functions of the joint angles as seen in Sects. 3,1 and 4.1. The SPEA concept itself is very general and can be implemented in many ways (this was pointed in Fig. 1 where many SPEAs can be seen) and offer thus many possibilities. We have also discussed weak points of both iSPEA and +SPEA and the main challenge of the SPEA is to find a way to tension springs in a desired fashion (as in the +SPEA) but without losing the power the motors can provide (as in the iSPEA) which could bring best of both worlds.

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Design, Optimization and Control of a Cable-Driven Robotic Suit for Load Carriage Yang Zhang

and Vigen Arakelian

Abstract In recent years, exoskeletons show impressive results in the rehabilitation, field works, and military. However, their rigid structures lead to a lot of drawbacks like additional inertia and incompliant user interfaces. Comparing to the exoskeleton, exosuit is a wearable suit made by soft material and actuated by cable or pneumatic force, which is inherently compliant and much lighter. In this paper, an exosuit has been designed for assisting people while carrying heavy loads. For this exosuit, soft polyethylene braid-style cables are coupled with rigid frames that link with user’s body through anchors and attachment points, which function as auxiliary muscles to apply force on both upper arm and forearm. The weight of the cables is negligible and adds nearly zero inertia to users. The proposed exosuit can function in two operation modes: passive and active modes. Static analysis and optimization are conducted for minimizing the cable tension and force exerted on user. Dynamic simulation shows that the proposed controller has good performance in active mode of the exosuit and experiments on the mannequin test bench validate its capability of holding loads with different postures in passive mode. Keywords Exosuit · Wearable robotics · Upper limb · Dynamic modelling · Control

1 Introduction The wearable robotic devices, also known as exoskeleton, have strong advantages given their unique features such as their outstanding physical performance, exceeding that of humans, and their mobility. As a result, attempts to adopt these devices in the rehabilitation and the industrial field provide a huge benefit for their users [1]. Y. Zhang · V. Arakelian (B) Team MECAPROCE, INSA-Rennes, Rennes, France e-mail: [email protected] LS2N UMR 6004, Nantes, France Y. Zhang Department SSE, HEI-Junia, Lille, France © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 V. Arakelian (ed.), Gravity Compensation in Robotics, Mechanisms and Machine Science 115, https://doi.org/10.1007/978-3-030-95750-6_5

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Assistive robots have applications in the industrial field as well as for patients and the elderly with mobility impairments. In the industrial field, work-related diseases of muscle and skeletal system are prevalent among physically demanding labors. Despite the industrialization has already been proceeding for hundreds of years, the number of occupational diseases, as well as cumulative trauma disorders (CTD) caused by overwork, is still significant nowadays. They are caused by ergonomic factors of the work environment, such as physical overload, compulsive working postures, the local stiffness of definite muscle groups and an adverse microclimate [2]. When workers carry loads, the gravity and dynamic force of the load will induce quite significant moments to the elbow, shoulder and trunk joint which will lead to the fatigue and even injury of muscles like the bicep, anterior deltoid, spinal extensor etc. [3]. Many researches in the field of wearable assistive robots have been conducted for augmenting the physical ability of human and minimizing work-related fatigue [4, 5]. The Berkeley Lower Extremity Exoskeleton (BLEEX) is powered by an internal combustion engine which is located in the backpack [12]. The hybrid engine delivers hydraulic power for locomotion and electrical power for the electronics. The exoskeleton is actuated via bidirectional linear hydraulic cylinders. BLEEX consumes an average of 1143 Watts of hydraulic power during level-ground walking, as well as 200 Watts of electrical power for the electronics and control. In contrast, a similarly sized, 75 kg human consumes approximately 165 W of metabolic power during level-ground walking. Instead of using powered components where actuator, control system, and power supply make the exoskeleton very bulky and heavy, some designs have been made with passive power sources like springs. By example, EXHAUSS offers a range of exoskeletons, each intended to relieve and protect operators, for various constraints related to handling or carrying loads or tools [6]. In these designs, springs are often used for gravity balancing of carrying loads. However, the support mechanical system is relatively heavy, and the handling is not easy. Additionally, the extra inertia effects must be compensated by users and the misalignment of the joints may cause the discomfort or even dangerous injuries. For solving the previous problems, several soft, suit-like exoskeletons (also known as “exosuit”) have been proposed in recent years. Because of using soft materials like fabrics and cables, therefore, much less additional inertial effect is imposed to wearer’s movement. Additionally, since there is not any rigid joint in exosuits, so it has no joint misalignment problem. Several exosuits have been designed for upperlimb rehabilitation [7, 8] and power augmentation [9, 10]. In these exosuits, cables are anchored on user’s body by light-weight bracelets or flexible suit and winding actuators are used for tensioning cables and provide moment on the upper-limb joint. However, for the mentioned upper-limb exosuits, despite of the advantages of lightweight and compliant, they still some drawbacks like actuators are rather mounted on the user’s body but are mounted on the fixed frame or on the ground, which makes them unportable.

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In order to reduce the moment generated by loads during transporting, in this paper, an exosuit for load carriage has been proposed. In this exosuit, polyethylene braid-style cables function as auxiliary muscles to augment the power of human. The exoskeleton has two function mode: passive and active. In passive mode, a locking mechanism will keep cables in tension and holding user’s position during load carriage. And in active mode, actuators will provide suitable cable tension while user is carrying load and moving his arm. The paper is organized as follows. At first, the design concept of the exosuit for carrying heavy load is presented. Then, the kinematic and dynamic model of exosuit-user coupled system is constructed in the third section. In the fourth section optimizations are made for reducing cable tensions and the workspace or exosuit is analyzed. Next, a controller is designed for the active mode of exosuit and simulations are carried out for evaluating its performance. Experiments on a mannequin test bench and results show in the sixth section. Conclusions are made in the last section.

2 Design Concept In this study, a soft robotic suit to assist users for carrying heavy loads is proposed. This robotic suit is planned to have three features: 1) lightweight: the mass of the suit adds little extra inertia hence user’s movement will not be impeded while wearing the robotic suit; 2) compliant: the interface between user and suit is compliant so user will not be hurt due to the installation error or system failure; 3) passive mode: the robotic suit can work in a passive mode where no actuation forces are need.

2.1 Development of the Robotic Suit’s Rigid Frame The rigid frame of the robotic suit consists of five parts: a main frame which fixed on the user’s body, two upper arm bracelets which fixed with the user’s upper arm, and two forearm bracelets which fixed with the user’s forearm. The whole rigid frame is illustrated in Fig. 1. In the main frame of the robotic suit, an adjustable plastic belt is at the bottom of the frame, and when users wear the suit, it fastens on the user’s waist as a fixing component of the suit. Two shoulder pads are linked with the belt through two adjustable curved beams which can adapt to users with different height, and between these two beams, six horizontal beams are fixed between them to increase the whole structural strength. In order to create a comfortable interface between the shoulder pad, soft foam materials are stuck at the back of the shoulder pads.

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Fig. 1 Rigid frame of the robotic suit

All the four bracelets have a similar structure which consist of two rings a curve pad linked with the rings. The rings are adjustable which can adapt the size of the user’s arms. The pads can avoid force concentration since they increase the contact surface between the bracelet and the user’s arm. Similar to the should pad, soft foam materials are also used at the interface between the bracelet and the user’s arm for enhancing comfortability. It should be also noticed that on all the bracelets and shoulder pads, there are some pins combined with some small rings on them, and these are the cable anchors and cable attachment points. We define an anchor is where the cable starts from, the cable is fixed with the small ring. And an attachment point is where the cable only passes through the small ring. For the shoulder pads and upper arm bracelets, the pins are fixed on them, and for the forearm bracelets, the pins are inserted into the groove on the rings which allow the pins have a translational movement along the rings. Thus, the pronation/supination movement of the forearm will not change the position of the anchors and attachment points on the forearm bracelets. The positions of attachment points and anchors are optimized in the previous research for minimizing the cable tension [11].

2.2 Coupling of Cables with the Rigid Frame When humans want to move their arms, their muscles contract for generating tension forces; then the forces are transmitted to the skeletons through tendons hence moments will be generated on the articulations for moving their limbs. In the proposed robotic suit, the cables are used for similar functionality as the human muscles where the cable tension is the muscle’s contract force. The cables can transmit forces on the user’s arm through arm bracelets and with accurate control

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Fig. 2 3D model of the robotic suit

of the tensions of the cables, a desirable assistive performance can be obtained by users during load carriage. The cables for the robotic suit are the polyethylene braid-style cables fabricated by Caperlan which is originally used for fishing. The diameter of the cable is only 0.4 mm and the weight is extremely light. The cable can hold up a load up to 34 kg, while in the meantime only have a tiny deformation. Figure 2 illustrates a 3D model where a user wears the proposed robotic suit where cables are coupled with the rigid frame and the batteries, control system and motors are mounted on the rigid frame. For each of the user’s arm, two cables are started from the anchors on forearm and upper arm, respectively; then pass through the attachment points on the arm and shoulder and then through the inside of the hollow beam which links the shoulder pad and the belt; finally end at spool on the motor which is mounted on the belt.

2.3 Robotic Suit Operation When humans want to move a load from one place to another place, normally the transportation can be separated into three phases. The first phase is to let the hands approach the load, then grab it and hold it with a comfortable posture. Then the phase is to hold the load and walk to the final destination where we want to put the load. The last phase is a discharge prosses in which we adjust our arms to adapt to the place where we want to put the load at, for example, a shelf. (a)

Passive mode

It can be seen that during the second phase of the load transportation, one stays at a quasi-fixed posture, and the moments generated on the arm’s joints are mainly due to gravity and stay relatively constant. Therefore, in this phase, we can let the all the

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Fig. 3 Cable locking mechanism

cables in tension and stop the movement of the cables, then the movement of user’s arm will be constrained, and the moment induced by the gravity will be transmitted on the shoulder. Thus, the fatigue of the muscle on the arms due to holding load will be relived. Instead of letting the motors generating torque to hold the position of the cable which needs a lot of energy if the distance from the start point and destination is too far, a cable winding and locking mechanism (shown in Fig. 3) was designed. This mechanism includes a cable spool, a torsional spring, and a ratchet mechanism. The cable spool links with the torsional spring and the ratchet. The torsional spring can provide a suitable passive moment for the spool to keep the cable always in tension while users moving their arms. And the ratchet mechanism is used as a locking mechanism to stop the movement of the cable. When wearers pick up the load and hold it in a comfortable position, the locking mechanism can be engaged by pressing the controller on their hands. Then the pawl moves down and enters the gap between two teeth of the ratchet. Consequently, the spool is stopped from moving and the passively generated cable tension can be used for compensating the gravity of the load. In contrast, when the user is not carrying loads, the pawl will lift and the spool can freely rotate. (b)

Active mode

Unlike the second phase of the load carriage where user’s arms stay at a fixed posture and only constant forces are needed for compensating the gravitational effect, in the first and the third phase of the load carriage, wearers need to hold the load and move it to the expected position. Therefore, in these phases, variable forces will be required since the dynamic effect will occur, and the forces utilized to compensate the gravitational effect will vary as the user changing their arms postures. In the proposed design, four DC motors are linked the cable spools providing active forces to the four cables, respectively. The torques of the motors are controlled through an on-board closed-loop controller. The pose data obtained by the potentiometers mounted on the joints of the arm can be used by the on-board controller.

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For the safety of the users, when robotic suit works in active mode, the maximum force provided to the cable is limited to 150 N for all the cables. Additionally, there is also a safety button for cutting off the power source if any emergency happens.

3 Modelling of the Coupled System 3.1 Kinematic Modelling The user carrying the load while wearing the robotic suit can be in analogy with a coupled system including a 2-DOF planar mechanical structure with the load at the end effector and cables attached on the manipulator for exerting forces. A planar diagram of a user wearing the exosuit system is shown in Fig. 4. In this diagram, two cables start from the anchors on the forearm and upper arm respectively and end at the cable winding and locking mechanisms fixed on the hip. The cable arrangements are as following: cable 1 starts form anchor on point 1, pass through attachment points on point 2, point 4, point 5, point 6, point 7, point 8; cable 2 starts form anchor on point 3, pass through attachment points on point 4, point 5, point 6, point 7, point 8. It is assumed that the deformation of the cable is very tiny and negligible. As shown in Fig. 4, two local coordinate systems X 1 O1 Y 1 and X 2 O2 Y 2 are fixed on the upper arm and forearm, respectively. The general coordinate of point i on the global coordinate system can be calculated by: Fig. 4 Planar diagram of the distribution of anchors and attachment points while user wearing the robotic suit

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⎡

⎤  xi 0 Trans · pi1 pi = ⎣ yi ⎦ = 0 1 1 1 1 Trans · 2 Trans · pi 1

(1)

where, ⎡

⎤ cos(3π/2 + θ1 ) − sin(3π/2 + θ1 ) 0 0 ⎣ sin(3π/2 + θ1 ) cos(3π/2 + θ1 ) 0 ⎦ 1 Trans = 0 0 1 ⎡ ⎤ cos θ2 − sin θ2 lar 1 ⎣ Trans = sin θ2 cos θ2 0 ⎦ 2 0 0 1 and p1i and p2i stand for the local general coordinates of point i in local coordinate systems X 1 O1 Y 1 and X 2 O2 Y 2 respectively, l tr is the length of the trunk and lar is the length of the upper arm. Thus, the global coordinate of point i is:  ci =

xi yi

 (2)

For both two cables, they all have a fixed length through attachment points 5, 6, 7, and 8 since all these points are fixed on the user’s trunk. Let us define this fixed length as l0 . And as shown in Fig. 4, the local general coordinates of point 1, 2, 3, and 4 are p11 = [a1 , c1 , 1], p12 = [a1 −b1 , d 1 , 1], p23 = [a2 , c2 , 1], p24 = [a2 −b2 , d 2 , 1], and the global coordinate of point 5 is c5 = [b3 , a3 ]. Then the lengths of cable 1 and cable 2 can be calculated by: 

l1 = f 1 (θ1 , θ2 ) l2 = f 2 (θ1 )

(3)

Then, using the time derivative of Eq. (3), the relationship between the cable velocity and the angular velocity of the user’s joints can be obtained by:     θ˙1 l˙1 = J v θ˙2 l˙2

(4)

where l˙1 and l˙2 denote the cable velocities. The detailed information of function f 1 (θ 1 ,θ 2 ), f 2 (θ 1 ) and matrix Jv can be found in Appendix.

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3.2 Dynamic Modelling The general dynamic equations of the coupled system can be obtained from the Lagrangian formulation where the generalized coordinate is q = [θ1 , θ2 ]T . Lagrange’s equation can be written in the form of kinetic energy K, potential energy V and generalized forces or torques Q in the following form: d dt



˙ ∂ K (q, q) ∂ q˙

−

˙ ∂ K (q, q) ∂ V (q) + = Q ∂q ∂q

(5)

The kinetic energy of the coupled system combines three parts: kinetic energy of the upper arm K 1 , kinetic energy of the forearm K 2 , and kinetic energy of the load K 3 . They can be calculated by: K1 = K2 =

1 1 2 ˙2 θ1 Iua θ˙12 + m ua lcua 2 2

(6)

2 1  2   1  2 ˙2 θ1 + lc2f a θ˙1 + θ˙2 + 2lua lc f a cosθ2 θ˙12 + θ˙1 θ˙2 I f a θ˙1 + θ˙2 + m f a lua 2 2 (7)

 2   1 2 ˙2 2 K 3 = m load lua (8) θ1 + lcload θ˙1 + θ˙2 + 2lua lcload cosθ2 θ˙12 + θ˙1 θ˙2 2

where I ua and I fa are the moment of inertias of upper arm and forearm respectively; mua , mfa and mload are the masses of upper arm, forearm, and load respectively; lua is the lengths of upper arm; lcua is the distances between the center of mass of upper arm to the shoulder joint, lcfa and l cload are the distances between the centers of mass of forearm and load to the elbow joint. In this study, the elastic and damping effects of cable are neglected, and each cable is assumed to be a force element. Therefore, the potential energy of the system is only due to the gravitational effect and the total potential energy is:   V = −m ua glcua (cos θ1 − 1) + m f a g lua (1 − cos θ1 ) + lc f a (1 − cos(θ1 + θ2 )) + m load g[lua (1 − cos θ1 ) + lcload (1 − cos(θ1 + θ2 ))] (9) where g is the gravity acceleration.

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In the coupled system, the generalized torques are induced by the cable tension trough anchors and attachment points on the user’s arm. It is obvious that the torques have a relationship with the geometrical distribution of the anchors and attachment points. The moments on the elbow and shoulder joint generated by cable 1 and 2 can be calculated by:   cable1 = v ∗O1 1 × v 12 + v ∗O1 2 × (v 21 + v 24 ) + v ∗O1 4 × (v 42 + v 45 ) · T1 Mshoulder cable1 Melbow = [v ∗O2 1 × v 12 + v ∗O2 2 × (v 21 + v 24 )] · T1 cable2 Mshoulder = [v ∗O2 3 × v 34 + v ∗O2 4 × (v 43 + v 45 )] · T2 cable2 Melbow =0

(10)

where, v ∗i j = c j − ci c j − ci  vi j =   c j − ci  ci is the global coordinate of point i. The total moments on shoulder and elbow joints induced by the cable tensions can be calculated by: 

cable Mshoulder cable Melbow



 =

cable1 cable2 + Mshoulder Mshoulder cable1 cable2 Melbow + Melbow



 = JM

T1 T2

 (11)

where,  JM =

v ∗O2 1 × v 12 + v ∗O2 2 × (v 21 + v 24 ) + v ∗O2 4 × (v 42 + v 45 ) v ∗O2 3 × v 34 + v ∗O2 4 × (v 43 + v 45 ) 0 v ∗O3 1 × v 12 + v ∗O3 2 × (v 21 + v 24 )



Through substituting Eq. (6)–(9) into Eq. (5), the general dynamic equation of the coupled system in terms of generalized coordinates can be obtained as the following form:  T ˙ q˙ + G(q) = J M T1 T2 M(q)q¨ + C(q, q)

(12)

˙ is the matrix of Coriolis and where M(q) is the inertia matrix of the system, C(q, q) centripetal terms, G(q) is the vector of gravity terms. The detailed form of Eq. (12) can be found in Appendix.

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4 Optimization and Work Space Analysis 4.1 Static Simulations and Optimization When people carry an object at a relatively fixed position, their arms undertake the load generally due to the gravitational effect of the arm. The gravity of object and arm generate moments on the shoulder and elbow joints which must be compensated by our muscle force. The moment on the shoulder and elbow joints induced by the gravitational force is G(q) term in Eq. (12) and it can be calculated by:     gravit y Mshoulder = m ua glcua sin θ1 + m f a g lua sin θ1 + lc f a sin(θ1 + θ2 ) + m load g lua sin θ1 + lcload sin(θ1 + θ2 )  gravit y Melbow = m f a lc f a + m load lcload g sin(θ1 + θ2 )

(13)

The average body segment parameters of an adult male with 172.68 cm height and 63.97 kg weight are given in Table 1 [13]. Using these parameters, the gravity moments on waist, shoulder, and elbow joint when carrying a 10 kg load with different postures (θ 1 ∈ [0°, 45°], θ 2 ∈ [0°, 90°]) are shown in Fig. 5. It is shown that when carrying the load with different postures, the gravitational moments on the joints also vary and these moments must be compensated by the muscle force.

Table 1 Body segment parameters of an adult male

Mass (kg)

Length (m)

Location of COM (m)

Upper arm

2.07

0.364

0.182

Forearm

1.7

0.299

0.149

Fig. 5 Gravitational moment on the shoulder (a) and elbow (b) joint when carrying a 10 kg load with different postures

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When a user wears the exosuit which is in passive mode, the cable locking mechanism is engaged, and the gravitational effect of the load and arm can be compensated by cables and distributed on the shoulder. According to Eq. (11), the cable tensions for compensating the gravitational effect can be solved by: 

gravit y

T1 gravit y T2



 =

−1 JM

gravit y



Mshoulder gravit y Melbow

(14)

and the force applied on the shoulder by cables is: gravit y

gravit y

Fshoulder = v54 (T1

gravit y

+ T2

)

(15)

The elements in the matrix JM are determined by the geometrical distribution of anchors and attachment points. To investigate the cable tensions during the usage of the exosuit in passive mode, at first, we chose the initial positions of anchor and attachment points as following: a1 = 0.3 m, b1 = 0.1 m, c1 = 0.08 m, d 1 = 0.08 m, a2 = 0.26 m, b2 = 0.1 m, c2 = 0.1 m, d 2 = 0.1 m, a3 = 0.65 m, b3 = 0.1 m, c3 = 0.1 m, d 3 = 0.1 m and e3 = 0.1 m. Figure 6 shows the cable tensions when the user carries a 10 kg load with different arm postures, where the maximum tensions are 130 N and 711 N for cable 1 and 2, respectively. It should be noticed that, for the maximum tension in cable 2, the limit of the max tolerable tension has already been passed. Besides, for some postures, the tension of cable 2 is negative which is not permissible. Another thing must be concerned with is the force exerted on the shoulder by cables, since a large portion of force has been redistributed on the shoulder. Figure 7 shows the force on the shoulder while the user carrying a 10 kg load with different postures. It can be seen the maximum force on the shoulder reaches up to 1355 N, and this amount of force will be harmful for users.

Fig. 6 Cable tensions when carrying a 10 kg load with different postures

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Fig. 7 Force on the user’s shoulder when carrying a 10 kg load with different postures

It is obvious that the initial locations of the anchors and attachment points are not suitable for our robotic suit since the large cable tension and the large pressure on the shoulder. Therefore, in order to have a suitable arrangement of anchors and attachment points, their locations must be optimized. In the optimization, we take Matlab fmincon function as our optimization algorithm and the parameters which are going to be optimized are a1 , b1 , c1 , d 1 , a2 , b2 , c2 , d 2 , a3 , b3 , c3 , d 3 , and e3 (as shown in Fig. 4). For the possible postures of the user, we choose the arm configurations as θ1 ∈ [0◦ , 15◦ , ..., 45◦ ] and θ2 ∈ [0◦ , 10◦ , ..., 90◦ ] which gives us 40 possible postures. In this optimization, we are going to minimize the total cable tensions and force exerted on the shoulder when the user carrying a 10 kg load with these 40 possible postures. Another thing has to be concerned is the constraints. In our problem, there is two kinds of constraints which are: (1) (2)

For all the configurations, cables must be always in tension i.e., cable tensions are always positive. The space constraints of the positions of anchors and attachment points. Hence the problem in this optimization can be concluded as: min

a1 ,b1 ,...,e3

   θ1

θ2

gravit y T1

+

gravit y T2

+

gravit y Tshoulder



 +P

(16)

subject to : 0.2 ≤ a1 ≤ 0.3; 0.01 ≤ b1 ≤ 0.1; 0.08 ≤ c1 ≤ 0.13; 0.08 ≤ d1 ≤ 0.13; 0.16 ≤ a2 ≤ 0.26; 0.01 ≤ b2 ≤ 0.1; 0.1 ≤ c2 ≤ 0.15; 0.1 ≤ d2 ≤ 0.15; 0.55 ≤ a3 ≤ 0.7; −0.1 ≤ b3 ≤ 0.1; 0.05 ≤ c3 ≤ 0.15; −0.1 ≤ d3 ≤ 0.1; 0.05 ≤ e3 ≤ 0.15.

where P is a very large value to penalize the goal function when any of the cable tension is negative for all configurations. After applying fmincon function in Matlab, the obtained optimal parameters are the followings: a1 = 0.23 m, b1 = 0.1 m, c1 = 0.11 m, d 1 = 0.08 m, a2 = 0.26 m, b2 = 0.01 m, c2 = 0.14 m, d 2 = 0.15 m, a3 = 0.7 m, b3 = −0.1 m, c3 = 0.15 m, d 3 = 0 m and e3 = 0.05 m. With these parameters, the cable tensions with respect to

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Fig. 8 Cable tensions when carrying a 10 kg load with different postures after optimization

Fig. 9 Force on the user’s shoulder when carrying a 10 kg load with different postures after optimization

different postures are shown in Fig. 8. It can be seen that after optimization the cable tensions have been drastically reduced, with maximum values of 65 N and 114 N for cable 1 and cable 2 respectively. Meanwhile, as shown in Fig. 9, the force exerted on the shoulder has also been generally diminished, with the maximum value of 304 N. Therefore, after optimization, the exosuit is more suitable and comfortable to wear.

4.2 Workspace Analysis Unlike the traditional exoskeleton in which the actuation forces are transmitted through rigid links to the user, in the proposed robotic suit, cables are used as the transmission media. Therefore, limited by the characteristic of the cable, only tension forces can be transmitted. While the determination of the workspace of the exosuit, the physical limitation of the cable must be considered. Hence, we can define the workspace of the exosuit as a set of joint angles [θ i 1, θ i 2] which fulfill the following condition: gravit y  i θ1 , θ2i

Tk

≥ 0 (k = 1 and 2)

(17)

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Fig. 10 Workspace of the propose exosuit

For investigating the workspace of the proposed robotic suit, the arm configurations which satisfy the condition described in Eq. (17) were found taking the optimal positions of the anchors and attachment point which was obtained in the previous part. Figure 10 shows all the possible arm configuration of the exosuit. The workspace comprises most part of the real configurations of the human arm except for some part when the upper arm is in extension (the humerus is rotated out of the plane of the torso so that it points backward) or upper arm has large flexion angle.

5 Controller Design and Dynamic Simulation When users want to pick up the load on the ground or put the load on a high shelf, their posture must be changed to adjust the vertical level of the load. Therefore, unlike the robotic suit works in passive mode, a controller must be needed for providing the different actuating force with respect to different postures. In addition, the dynamic effect of the user’s arm and load must be taken into consideration. In the proposed design, a closed loop control based on a PD controller and a feedforward term for gravity compensation is utilized for controlling the active force. Potentiometers are mounted on the user’s arms for the measurement of elbow and shoulder joint angles. It should be noticed that, for the controlled output forces, only positive values can be provided since only tension forces can exist in the cables. The final controlled active force can be expressed as the following:      T  −1 Tactive = max JM (θ) Kp (θd − θ) + Kd θ˙ d − θ˙ + G(θd ) , 0 0

(18)

where θ d and θ˙ d are the desired joint angles and joint velocities; Kp and Kd are the diagonal matrices consist of proportional and derivative gains of the PD controller respectively. A block diagram of the proposed controller is shown in Fig. 11. It can be seen that a prescribed desired state of the system is given as the input. A feedforward loop which is comprised of the gravity compensation to eliminate the effect of gravitational force.

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Fig. 11 Block diagram of the PD controller with gravity compensation

A feedback loop consists of a PD controller which uses the measured real state of the system to generate appropriate dynamic forces which actuate the system to the desired state. Additionally, in order to let simulation model be more realistic to the working environment, an extra term has been added on the system dynamic model representing nonlinear external disturbances (e.g., vibration during walking, etc.). These disturbances are modeled as random white noises in the proposed model. For generating a smooth trajectory between the initial position and final desired position, a trajectory generation method based on sinusoidal function has been utilized. The trajectory of a joint can be represented by the following equations:  q(t) =

qi +

q f −qi 2

 t  sin τ π − π2 + 1 (t ≤ τ ) q f (t > τ )

(19)

where qi is the initial position, qf is the final desired position and τ is the duration of the sinusoidal trajectory. Based on the dynamic model described in the previous section, a Simulink model of the proposed system has been built. Dynamic simulations of the system have been carried out via the Simulink model for validating the effectiveness of the controller and the exosuit’s operation in active mode. And for demonstrating the system having the capability of coping with various trajectories, four different trajectories have been chosen based on previously proposed trajectory generation method. The parameters of these trajectories are shown in Table 2. Table 2 Parameters of the trajectories

1 2 3 4

qi1 (deg) 0 0 45 90

qf1 (deg) 0 20 0 0

τ1 (s) 0 1.11 2.5 5

qi2 (deg) 0 0 45 0

qf2 (deg) 90 70 90 90

τ2 (s) 5 3.89 2.5 5

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Fig. 12 Comparison of the actual and desired trajectories

In Fig. 12, the solid lines show the variations of joint angles during simulation and the dash lines shows the desired input trajectories to follow. It can be noticed that, for all four trajectories, the robotic suit shows high tracking performance with the help of the proposed PD controller. The robot suit follows the prescribed trajectories quite accurately even under the external random disturbances. The cable tensions (which is also the active force in robotic suit system) during the different trajectory tracking are shown in Fig. 13. It can be seen that the cable tension fluctuates during the simulations, which is a response to the external disturbance controlled by PD controller. In most of the situations, the cable tension is under 120 N which is a quite safe value since the cable can hold the tension up to 340 N.

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Fig. 13 Cable tensions for the tested trajectories

6 Experimental Validation To evaluate the performance of proposed passive exosuit, a testbench was fabricated as shown in Fig. 14. The test bench was converted from a mannequin where spherical joints were added on the elbow joints and shoulder joints. To test the performance of the exosuit when a user is in different postures, several postures have been chosen in this experiment. For the shoulder extension angle, we chose 0° and 30°, and for the forearm flexion angle 0°, 30°, 60°, and 90° were chosen. And for each posture, we put no load, 5 kg load, and 10 kg load respectively. Then the cable tensions were measured. When the experiments were conducted with different postures and different load, with the help of the exosuit, the mannequin test bench can hold the load steadily in the designated postures. The change of the cable tensions with respect to the change of forearm flexion angle when shoulder extension angle is 0° and 30° are shown in Fig. 15 and Fig. 16 respectively.

Design, Optimization and Control …

Fig. 14 Mannequin test bench (a) and the added joints (b, c) Fig. 15 Cable tensions with respect to the change of forearm flexion angle (shoulder extension angle is 0°)

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Fig. 16 Cable tensions with respect to the change of forearm flexion angle (shoulder extension angle is 30°)

7 Conclusion A robotic suit has been proposed which intends to assist its user in the carriage of heavy load in this paper. The proposed robotic suit presents a symbiosis of two systems: a rigid support frame and a cable system. The cable system is coupled with the rigid support frame providing assistive force to its users during load carriage and it is made up of high-intensity polyethylene cables, which are very light and add almost zero inertia to users. A cable locking mechanism has been designed in order to keep cables in tension and to stop the movement of the cables. The robotic suit has two operation modes: passive and active modes. When wearers hold the load at a fixed posture, the cable locking mechanism is engaged to stop the movement of the cable and the robotic suit works in the passive mode where no energy is needed. When users change their postures while holding the load, the robotic suit works in the active mode providing variable assistive forces with respect to the movement. The kinematic and dynamic models of the robotic suit system have been built. Static simulation has been carried out via the built model and an optimization of the position of the anchors and attachment points of the robotic suit has been made by using fmincon function in Matlab for minimizing the cable tension and force exerted on the shoulder. The workspace of the robotic suit has been analyzed for showing the possible posture while wearing the exsosuit. Dynamic model of the exosuit-human coupled system has been built. A PD controller with gravitational force compensator was designed. Dynamic simulations of the robotic suit in trajectory tracking have been carried out via Matlab Simulink. All the simulations showed that the proposed controller had good performances during working in active mode and the maximum cable tension is around 120 N which is much lower than the maximum supportable tension. In order to evaluate the performance of the robotic suit in a more realistic environment and validate the simulation results, a mannequin test bench has been fabricated

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and tested. The test results showed that with the help of the robotic suit, the mannequin can hold different weights of loads steadily in different postures. In the future study, a prototype of the proposed exosuit with actuators integrated into a jacket plans to be developed. Then tests in the actual working scenario with exosuit in active mode will be conducted.

Appendix This appendix shows the detailed form of the kinematic and dynamic equation of the coupled system Sect. 3. The detailed form of the Eq. (3) is:     d cos(θ1 + θ2 ) + (a1 − b1 ) sin(θ1 + θ2 ) − d2 cos θ1 − (a2 − b2 − lar ) sin θ1 2 + f 1 (θ1 , θ2 ) =   1 2 + (a1 − b1 ) cos(θ1 + θ2 ) − d1 sin(θ1 + θ2 ) − (a2 − b2 − lar ) cos θ1 + d2 sin θ1   (a3 + a2 cos θ1 − b2 cos θ1 − d2 sin θ1 )2 + (b3 − d2 cos θ1 − a2 sin θ1 + b2 sin θ1 )2 + b12 + (c1 − d1 )2 + l0

 f 2 (θ1 ) =

[a3 + (a2 − b2 )cosθ1 − d2 sinθ1 ] + [b3 − d2 cosθ1 − (a2 − b2 )sinθ1 ] + 2

2



(20) b22

+ (c2 − d2 ) + l0 2

(21) The Jacobean matrix of the exosuit is:  v v  j j12 Jv = 11 v v , j21 j22

(22)

where: 1 − b1 )(a2 − b2 − lar ) + d1 d2 ]sinθ2 + [d1 (a2 − b2 − lar ) − d2 (a1 − b1 )]cosθ2 v = [(a j12  [d1 cos(θ1 + θ2 ) + (a1 − b1 )sin(θ1 + θ2 ) − d2 cosθ1 − (a2 − b2 − lar )sinθ1 ]2 + [(a1 − b1 )cos(θ1 + θ2 ) − d1 sin(θ1 + θ2 ) − (a2 − b2 − lar )cosθ1 + d2 sinθ1 ]2 (a3 b2 − a3 a2 + b3 d2 )sinθ1 − (b3 a2 − b3 b2 + a3 d2 )cosθ1 v = jv =  j11 21 [a3 + (a2 − b2 )cosθ1 − d2 sinθ1 ]2 + [b3 − d2 cosθ1 − (a2 − b2 )sinθ1 ]2 v j22 =0

The detailed form of dynamic equation of the coupled system Eq. (12) is:

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  2  ⎧ cable 2 2 2 +I 2 Mshoulder = m ua lcua ⎪ ua + m f a lua + lc f a + 2lua lc f a cosθ2 + I f a + m load lua + lcload + 2lua lcload cosθ2 θ¨1 ⎪

   ⎪   ⎪ ⎪ 2 ⎪ + m f a lc2f a + lua lc f a cosθ2 + I f a + m load lcload + lua lcload cosθ2 θ¨2 − m f a lua lc f a sinθ2 2θ˙1 θ˙2 + θ˙22 ⎪ ⎪ ⎪    ⎨ −m load lua lcload sinθ2 2θ˙1 θ˙2 + θ˙22 + m ua glcua sinθ1 + m f a g lua sinθ1 + lc f a sin(θ1 + θ2 ) ⎪ ⎪ +m ⎪

  load g[lua sinθ1 + lcload sin(θ1 + θ2 )]  ⎪ ⎪ cable = m ⎪ 2 ⎪ Melbow lc2f a + lua lc f a cosθ2 + I f a θ¨1 + m f a lc2f a + I f a + m load lcload θ¨2 ua ⎪ ⎪ ⎩   2 ˙ + m f a lc f a + m load lcload lua sinθ2 θ1 + m f a lc f a + m load lcload gsin(θ1 + θ2 )

(23)

References 1. Zhang, Y., Arakelian, V., Le Baron, J.: Design concepts and functional particularities of wearable walking assist devices and power-assist suits – a review. In: Proceedings of the 58th International Conference of Machine Design Departmants (ICDM 2017), pp. 436–441, Prague (2017) 2. Enoka, R., Duchateau, J.: Muscle fatigue: what, why and how it influences muscle function. J. Physiol. 586(1), 11–23 (2008) 3. Jaworski, Ł., Karpi´nski, R., Dobrowolska, A.: Biomechanics of the upper limb. J. Technol. Exploit. Mech. Eng., 2(1), 56–59 (2016) 4. Kazerooni, H.: Exoskeletons for human power augmentation. In: 2005 IEEE/RSJ International conference on intelligent Robots and Systems, pp. 3459–3464. IEEE, Edmonton (2005) 5. Marcheschi, S., Salsedo, F., Fontana, M., Bergamasco, M.: Body extender: Whole body exoskeleton for human power augmentation. In: 2011 IEEE international conference on robotics and automation, pp. 611–616. IEEE, Shanghai (2011) 6. Exhauss Homepage. http://exhauss.com/fr_produits.htm. Accessed 15 Apr 2021 7. Mao, Y., Agrawal, S.: Design of a cable-driven arm exoskeleton (CAREX) for neural rehabilitation. IEEE Trans. Rob. 28(4), 922–931 (2012) 8. Jin, X., Aluru, V., Raghavan, P., Agrawal, S.K.: The effect of CAREX on muscle activation during a point-to-point reaching task. In: 2015 IEEE International Conference on Rehabilitation Robotics (ICORR), pp. 73–78. IEEE, Singapore (2015) 9. Lessard, S., et al.: Crux: A compliant robotic upper-extremity exosuit for lightweight, portable, multi-joint muscular augmentation. In: 2017 International Conference on Rehabilitation Robotics (ICORR), pp. 1633–1638, IEEE (2017) 10. Samper-Escudero, J.L., Giménez-Fernandez, A., Sánchez-Urán, M.Á., Ferre, M.: A cabledriven exosuit for upper limb flexion based on fibres compliance. IEEE Access 8, 153297– 153310 (2020) 11. Zhang, Y., Arakelian, V.: Design of a passive robotic exosuit for carrying heavy loads. In: 2018 IEEE-RAS 18th International Conference on Humanoid Robots (Humanoids), pp. 860–865. IEEE, Beijing (2018) 12. Zoss, A., Kazerooni, H., Chu, A.: On the mechanical design of the Berkeley Lower Extremity Exoskeleton (BLEEX). In: 2005 IEEE/RSJ International Conference on Intelligent Robots and Systems, pp. 3465–3472. IEEE, Edmonton (2005) 13. Drillis, R., Contini, R., Bluestein, M.: Body segment parameters. Artif. Limbs 8(1), 44–66 (1964)

Tool Compensation for a Medical Cobot-Assistant Juan Sandoval

and Med Amine Laribi

Abstract A collaborative robot, i.e. cobot, is proposed by the CoBRA Team of Pprime Institute to serve as a medical cobot-assistant for different applications, as for Doppler sonography, minimally invasive surgery or craniotomy. Since the cobot requires a whole knowledge of its dynamic model to ensure an optimal performance in terms of accuracy, stability and safety during human–robot interaction, the inertial parameters of the medical tools attached to its end-effector have to be known and integrated to the robot controller. However, the inertial parameters of the medical tools are generally unknown and may vary at each use. This chapter presents the method and procedure set up to identify the inertial parameters of the metical tools used by the proposed medical cobot-assistant. The proposed procedure is based on least-squares estimation, comparing the joint torques provided by the cobot to execute the same trajectory for two different conditions: with and without the tool attached to the end-effector. We experimentally validate the procedure on the cobot-assistant platform. Keywords Collaborative robots · Model identification · Inertial parameters · Dynamic compensation · Medical cobots

1 Introduction The use of Collaborative Robots, i.e. Cobots, is growing rapidly, particularly in the field of medicine. Actually, cobots emerge as a suitable solution to assist medical experts in their gestures. For instance, the work of surgeons is being revolutionized by cobots, helping them to improve the accuracy and comfort during surgical procedures while reducing the operating time. One can cite the Stryker Mako robot used for total J. Sandoval (B) · M. A. Laribi Department of GMSC, Pprime Institute CNRS, ENSMA, University of Poitiers, UPR 3346 Poitiers, France e-mail: [email protected] M. A. Laribi e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 V. Arakelian (ed.), Gravity Compensation in Robotics, Mechanisms and Machine Science 115, https://doi.org/10.1007/978-3-030-95750-6_6

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knee arthroplasty [1] or the Kuka medical robot proposed for several applications [2]. In the teleoperation scheme shown in Fig. 1, a cobot is used as an ultrasound probe holder for doppler sonography and controlled by a vascular doctor through a haptic interface [3, 4]. Thus, the cobot reproduces the movements performed by the end-effector of the interface. Since vascular doctors use different types of probes according to the type of examination, e.g. carotids, abdominal aortas or lower limbs, the weight and form of each probe is different and the inertial parameters are generally unknown. Fixing the probe on the cobot, at his end effector, will add an unknown additional mass leading to wrong gravity compensation. This latter will alter the trajectory generation as well as the force feedback at the haptic interface. As a result, behavioral issues may arise while performing the exam. In addition, this problem is not limited to cobot-assisted doppler sonography but is encountered in other medical applications, such as cobot-assisted laparoscopy and robotic-assisted craniotomy. In the first case, an autonomous robot-assistant solution for laparoscopic surgery has been proposed for freeing up surgeons from the laparoscope motion control. The laparoscope, whose inertial parameters are unknown, is held by a cobot and oriented around the incision point so that a permanent visual feedback of the instruments’ tips is guaranteed (Fig. 2) [5]. In the case of craniotomy application, a drilling tool is held by a cobot while a surgeon tele-operates the cobot movements. This is the case of the teleoperated robotic platform using a 3-DoF master device and a cobot as the slave device proposed in [6, 7], as shown in Fig. 3. Despite the fact that torque-controlled robots equipped with joint torque sensors or a 6-axis force sensor are able to identify the payload through a specific integrated

Fig. 1 Cobot-assisted doppler sonography

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Fig. 2 Robot-assistant platform for laparoscope-holder during laparoscopic surgical procedures

Fig. 3 Platform for tele-operated craniotomy robot (experiments on cadaver)

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approach, e.g. Kuka IIWA, in some cases, robot functionalities don’t include this kind of software or procedure to compensate the gravity effects. This is the case of the Franka Emika robot used by the CoBRA team of the Pprime Institute for the development of a medical cobot-assistant for the aforementioned applications [3–7]. Therefore, after having studied the main existing payload estimation methods based on the proprioceptive sensors data, a suitable method has been implemented based on least-squares estimation. This chapter presents three experimental study cases validating the effectiveness of the implemented method. The chapter is organized as follows. Section 2 details the identification method used to estimate the inertial parameters of the medical tools. In Sect. 3, we present different case studies to experimentally validate the method in medical applications. Finally, a brief conclusion about the obtained results is provided.

2 Estimation of the Inertial Parameters With the aim of characterizing the tool held by a manipulator, 10 inertial parameters have to be identified: mass (M L ), center of mass (C oM = C oM x , C oM y , C oM z ) and the parameters of the symmetric inertial matrix (I X X , I Y Y , I Z Z , I XY , I Y Z , I X Z ). Several payload identification methods have been proposed in the literature, based on measures of joint motion and torque sensors or of a 6-axis force sensor mounted at the end-effector of the robot. These methods are generally based on least-squares estimation [8–12]. A simple method consists in measuring the joint torques for a same trajectory twice, with and without the tool attached to the robot. Then, the difference between the torques measured in the two conditions is used to estimate the inertial parameters of the tool [8]. Besides not needing a 6-axis force sensor, this method is particularly advantageous since it avoids the estimation of the overall dynamic parameters of the robot, considerably minimizing the calculation time. The aforementioned method, suitable for collaborative robots equipped with joint torque sensors, will be explained below and proposed as a solution to identify the inertial parameters of different medical tools mounted in a 7-DoF Franka Emika cobot as part of medical assistive platforms.

2.1 Identification Modeling Let define the dynamic model of a rigid-body n-DoF serial manipulator can be defined as follows, ˙ q˙ + g(q) T0 = M(q)q¨ + C(q, q)

(1)

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151

With q ∈ n as the vector of joint positions, M(q) ∈ n×n is the inertia matrix, ˙ q˙ ∈ n the generalized centrifugal and Coriolis forces and g(q) ∈ n C(q, q) the vector of gravitational forces. Moreover, the joint torques vector is defined by T 0 ∈ n . Equation (1) can also be written in linear form as, ˙ q) ¨ ·ϕ T0 = K (q, q,

(2)

where ϕ ∈ 10n×1 is a vector containing the inertial parameters of the robot’s links ˙ q¨ ) ∈ n×10n is called the observation matrix, also known as regressor and K (q, q, matrix. In order to include an additional payload attached to the robot’s end effector into the dynamic model, the total torque vector T ∈ n can me written in function ˙ q¨ ) ∈ n×10(n+1) and vector of inertial of an overall observation matrix K m (q, q, 10(n+1)×1 , or decomposed, yielding, parameters ϕ m ∈  T = K m (q, q, ˙ q) ¨ · ϕm = T0 + K L (q, q, ˙ q) ¨ · ϕL

(3)

where ϕ L ∈ 10×1 is a vector containing the inertial parameters of the payload and ˙ q¨ ) ∈ n×10 represents the observation matrix of the payload. Therefore, K L (q, q, from Eq. (3), the inertial parameters of the payload can be expressed as, ϕ L = K LP+ · [T P − T0P ]

(4)

Equation (4) shows that the inertial parameters of a payload can be estimated based on the difference between the measured joint torques when the payload is mounted at the end-effector, i.e. T P ∈ n P×1 , and the measured joint torques before charging the robot, i.e. T 0P ∈ n P×1 , during the execution of the same trajectory, with P as the number of points. It should be mentioned that this method supposes negligible the difference between the motions reproduced in the two conditions as well as the generated friction forces [8–10].

2.1.1

Calculation of the Observation Matrix

As presented in Eq. (4), in order to estimate the 10 unknown inertial parameters of ˙ q¨ ) has to be defined. This matrix depends the tool, the observation matrix K L (q, q, on the generalized coordinates and their time derivatives and is related to the actuator forces through the dynamic parameters. According to [10], the components of the total observation matrix K m are represented by the following equation.  K mi j =

zi 0

 Fj

· i X j · Aj

(5)

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With the sub-indices i=1, . . .n and j= 1, . . ., (n+1) for the joints and links, respectively. Moreover, z i ∈ 3×1 denotes the joint rotational axis and matrix A j ∈ 6×10 is defined as,  Aj =

        03×1   −S d¨0 j   L w˙ j + L w j · S w j 03×6 d¨0 j S w˙ j + S w j · S w j

(6)

where d¨ 0 j = v˙ j − g is the acceleration of the link reference frame including the account of the gravity g, whereas w j ∈ 3×1 is the link angular velocity. S(∗) is the skew-symmetric operator and L(∗) is a 3 × 6 matrix defined as follows for a 3D   vector a = a x , a y , a z . ⎛

⎞ 0 −az a y S(a) = ⎝ az 0 −ax ⎠ −a y ax 0 ⎞ ⎛ a x a y az 0 0 0 L(a) = ⎝ 0 ax 0 a y az 0 ⎠ 0 0 ax 0 a y ax

(7)

(8)

Fj

Concerning i X j ∈ R6×6 in Eq. (5), it is called the spatial force transform matrix and is calculated in terms of the special transform j X i ∈ R6×6 , i

Fj

Xj ≡

i

Xj

−T

≡

j

Xi

T

(9)

Where,  j

Xi =

j  j Ri j 0j3×3 S pi Ri Ri

 (10)

with j Pi ∈ 3×1 and j Ri ∈ 3×3 denoting the position and orientation of frame i with respect to frame j . Therefore, the observation matrix of the robot is defined by, ⎛

[z 1 0]T 1 X 1F1 A1 [z 1 0]T 1 X 2F2 A2 ⎜ 0 [z 2 0]T 2 X 2F2 A2 ⎜ Km = ⎜ .. .. ⎜ ⎝ . . 0 0

F(n+1) · · · [z 1 0]T 1 X n+1 An+1 T 2 F(n+1) · · · [z 2 0] X n+1 An+1 .. .. . .

F(n+1) · · · [z n 0]T n X n+1 An+1

Where the last column represents the observation matrix of the payload,

⎞ ⎟ ⎟ ⎟ ⎟ ⎠

(11)

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⎛

F(n+1) An+1 [z 1 0]T 1 X n+1 T ⎜ [z 2 0] 2 X F(n+1) An+1 n+1 ⎜ KL = ⎜ .. ⎝ . F(n+1) An+1 [z n 0]T n X n+1

153

⎞ ⎟ ⎟ ⎟ ⎠

(12)

3 Study Cases As aforementioned, an proper inertial compensation of the tool is essential to guarantee a suitable performance of a manipulator. This issue is a priority when using collaborative robots, since torque-controlled approaches are implemented to enhance comanipulation strategies as hand-guiding, where a proper gravity compensation of the robot’s body and tool is needed. In this section, three study cases are presented, in the context of medical robotic platforms, in order to verify the relevance of compensating the inertial effects of tools, as shown in Fig. 4. The collaborative robot used for these medical cobot-assistants is Franka Emika, a 7-DoF torque-controlled robot whose weight is 18 kg for a maximum payload of 3 kg. It is equipped of joint torque sensors in all axes, allowing to estimate the external torques produced by the interaction with the environment. It is important to mention that a wrong tool compensation produces misleading estimation of external forces. The Denavit-Hartenberg (DH) parameters of the Franka Emika robot are presented in Table 1. In order to facilitate the calculation of the inertial parameters, we have developed a Graphical User Interface (GUI) receiving the robot data recorded with and without

(a)

(b)

(c)

Fig. 4 Study cases. Identification of three different tools mounted at the end-effector of the Franka Emika robot. a Hand gripper, b Ultrasound probe for Doppler sonography, c Laparoscope

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Table 1 DH parameters of the Franka Emika robot Joint

θi (rad)

di (m)

ai (m)

αi (rad)

1

q1

0.333

0

0

2

q2

0

0

−π/2

3

q3

0.3160

0

π/2

4

q4

0

0.0825

π/2

5

q5

0.3840

−0.0825

−π/2

6

q6

0

0

π/2

7

q7

0

0.0880

π/2

Flange

0

0.1070

0

0

Fig. 5 Graphical user interface (GUI) developed at Prime Institute to calculate the inertial parameters of the tool

attaching the tool, providing the inertial parameters of the tool. The developed GUI is also able to export a file readable by the robot controller to automatically consider the identified parameters in the robot model. A screenshot of the GUI is shown in Fig. 5.

3.1 Fist Case—Hand Gripper The first presented study case concerns the identification of the hang gripper (HG) provided by the constructor. The main advantage of performing this first test is that the inertial parameters of the HG are known and a comparison can be done with the obtained results.

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According to the identification method detailed in Sect. 2, the same robot trajectory has to be reproduced two times, with and without the tool attached to the robot. Then, the delivered joint torques as well as the joint motion data are introduced in Eq. (4) to obtain the inertial parameters. A suitable trajectory has been defined to move and rotate the tool along the Cartesian directions. For sake of simplicity, only the three last joints of the robot have been moved subsequently, as it can be observed in Fig. 6. The positions of the remain joints were fixed at q 1 = q 4 = −135◦ , q 2 = −45◦ and q 3 = 0◦ . Table 2 presents the obtained parameters for the HG as well as the theoretical parameters provided by the constructor. Although these results are assumed to be very accurate and the errors negligible, two other ways exist to validate them. First, the identified parameters have been included in the robot controller and a gravity compensation mode has been activated to test the hand guiding, where the joint

Fig. 6 Trajectory executed for the identification of the inertial parameters of the tools

Table 2 Inertial parameters identified for the HG.

Parameter

A priori

Identified

Error

M L (kg)

0.730

0.7289

0.0011

CoMx (m)

−0.010

0.0081

0.0181

CoM y (m)

0

0.0007

0.0007

CoMz (m)

0.030

0.0288

0.0012

I X X (kg · m 2 )

0.001

0.0086

−0.0076 −0.0241

· m2)

0

−0.0241

I X Z (kg · m 2 )

0

−0.0164

IY Y (kg · m 2 )

0.0025

IY Z (kg · m 2 )

0

0.0200

0.02

I Z Z (kg · m 2 )

0.0017

0.0015

0.0002

I X Y (kg

0.0065

−0.0164 −0.004

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torques only compensate the gravitational effects produced by the own load of the robot. In fact, robot perfectly compensates the load of the tool regardless of the joint configuration imposed by the operator. A second way to validate these results is to observe the estimation of the external torques before and after the tool compensation. Therefore, Fig. 7 shows the estimated external torques of each joint during the execution of the same trajectory used for the identification phase (Fig. 4), for three different configurations: when the tool is attached before and after the compensation and when the tool is not yet attached. It is essential to mention that in the ideal case, when the tool is not attached (“not loaded”), the external torques should be zero. However, this is not the case due to the uncertainties of the mechanical system, namely due to the frictional forces acting on the robot joints. This phenomenon can also explain the differences in joint 1 between the external torques estimated with and without attaching the tool, regardless of the application of the tool compensation (Fig. 7). In fact, joint 1 is not subject to considerable strain for this movement, as proved by the torque intensity values provided, lower than 0.2 Nm. For the joints subject to considerable strain, one can assume that more the estimation of the external torques is near from the “not loaded” configuration more the compensation is accurate. As illustrated in Fig. 7, the external torques of joint 2 when the robot is not loaded are coincident with the external torques of the robot when it is compensated. In the case of joint 3, the external torques when the robot is not loaded are also very coincident with the external torques of the robot when it’s compensated, despite the lower intensity of the torques. Joint 4 is the one most subject to considerable strain, thus a wrong gravity compensation is more visible in this joint, as proved by Fig. 7. The external torques when the robot is not loaded are slightly coincident with the external torques of the robot when it is compensated, the values varying from 0.3 to 0.6 Nm. As for the values of the external torques when the robot is loaded, they vary from 2.5 to 4.5 Nm. Similar to the above joints, the external torques of joints 5, 6 and 7 when the robot is not loaded are near from the values estimated when the robot is already compensated. These results validate the effectiveness of the employed identification method and confirm the results obtained through the comparison of Table 2 and the hand guiding tests mentioned above.

3.2 Second Case—Doppler Sonography In the context of a cobot-assisted Doppler-sonography (Fig. 1), a cobot holds an ultrasound probe and reproduces over the patient’s body the motion orders given by the angiologist. This system enhances immediate diagnostics while guaranteeing a comfortable pose for the angiologist. Since the shape and inertial parameters of each ultrasound probe is different, according to the type of probe and the constructor, it

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Fig. 7 External torques estimated for the first study case

157

158 Table 3 Inertial parameters identified for the second study case

J. Sandoval and M. A. Laribi Parameter

Identified value

M L [kg]

0.8701

CoMx [m]

0.0011

CoM y [m]

0.0028

CoMz [m]

0.0486

I X X [kg · m 2 ]

0.0047

· m2]

−0.0063

I X Z [kg · m 2 ]

0.0062

IY Y [kg · m 2 ]

0.0024

· m2]

−0.0011

I Z Z [kg · m 2 ]

0.8701

I X Y [kg

IY Z [kg

is essential to identify the inertial parameters when a new probe is mounted to the robot. Unlike the first case study, in the second case study the theoretical inertial parameters of the tool are not provided and are unknown. Therefore, similar to the last validation of the first case, a proper way to validate the results is by incorporating the identified inertial parameters into the controller of the robot and extracting the external torques during the execution of a trajectory (the same trajectory executed for the payload identification phase). For this case, the tool is composed by the HG, a case fixed to the fingers of the HG and the ultrasound probe. After executing the trajectory two times (with and without the tool), P = 1340 points have been recorded. Table 3 presents the identified inertial parameters. As described above, to prove that these results allow a suitable tool compensation and therefore a proper gravity compensation of the robot, the external torques were estimated as in the first study case. Then, a suitable behavior after the compensation suggests measuring external torques values near to the ones measured when the robot is “not loaded” by the tool. Figure 8 shows the external torques of the robot’s joints before compensation, after compensation and when the robot is not loaded. The obtained results are very similar to the ones found in the first study case. In fact, the first joint not supports a significant strain for this movement. This can be observed through the torque intensity values which is lower than 0.2 Nm. One can consider for the joints with high strain request that more the external torques estimation is closer to the “not loaded” configuration more the compensation is accurate. For instance, the joint 2 presents an overlapping behaviors as reported in Fig. 8 for the external torques when the robot is loaded and is compensated. Further and in the case of joint 3, the coincidence between the external torques can be noted when the robot is not loaded and compensated and this without being affected by the lower intensity of the torques. The inaccurate gravity compensation is more noticeable on the joint behavior when this latter is subject to considerable strain and this is the case of joint 4 as

Tool Compensation for a Medical Cobot-Assistant …

Fig. 8 External torques estimated for the second study case

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shown on Fig. 8. The external torques values corresponding to the not loaded robot and compensated robot vary from 0.3 to 0.6 Nm, and they behaviors are slightly coincident. As for the values of the external torques when the robot is loaded, they vary from 3 to 5.4 Nm. The values of the external torques of joints 5, 6 and 7, equivalent to the above joints, present little changes over time when the robot is not loaded and when the robot is already compensated.

3.3 Third Case—Laparoscope-Holder System This last study case introduces the use of a robot-assistant platform for laparoscopeholder in minimally-invasive surgery (MIS), as the one shown in Fig. 2. In this application, a robot assists surgeons during MIS training, holding a laparoscope, or surgical camera, and autonomously orienting it to constantly focus towards the instruments tips. As for the previous study cases, a proper tool compensation is needed since a torque-controlled approach is implemented, including the calculation of gravity compensation torques. After executing the trajectory, a total of 1447 points are recorded. In this case, the tool is composed by the HG, a plastic case and the laparoscope, as shown in Fig. 4(c). Similarly to the previous case, the inertial parameters of the tool are unknown. Therefore, a suitable way to evaluate the tool compensation is to verify the external torques measured by the robot once the tool compensation has been integrated to the robot controller. Table 4 shows the inertial parameters identified after using the identification method. It is important to mention that the tool compensation has been tested by activating a hand-guiding mode and verifying that the robot compensates its own load regardless of the joint configuration manually imposed by the operator. Moreover, Fig. 9 shows the external torques measured for this case during the execution of the trajectory Table 4 Inertial parameters identified for the third study case

Parameter M L (kg)

Identified value 1.0302

CoMx (m)

0.0199

CoM y (m)

−0.0134

CoMz (m)

0.0137

I X X (kg · m 2 )

0.0170

I X Y (kg · m 2 )

−0.0096

I X Z (kg · m 2 )

−0.0013

IY Y (kg · m 2 )

0.0056

IY Z (kg · m 2 )

0.0035

· m2)

0.0042

I Z Z (kg

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Fig. 9 External torques estimated for the third study case

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of Fig. 5 for three different situations: when the tool is not attached (not loaded), when the tool is attached but not compensated and when the tool is attached and compensated. The obtained results are very similar to the ones found in the first and second study case. The external torque value for the joint 1 is less than 0.3 Nm, which indicated that this joint is not subject to considerable strain for the considered movement. For the joints subject to considerable strain, one can assume that more the estimation of the external torques is near from the “not loaded” configuration more the compensation is accurate. ones observe in Fig. 9 and in the case the not loaded robot, the external torques of joint 2 are overlapping with the external torques of the robot when it is compensated. In the case of joint 3, the external torques when the robot is not loaded are also very coincident with the external torques of the robot when it’s compensated, despite the lower intensity of the torques. During the considered movement, the joint 4 supports an important external torque as shown on Fig. 9. We can conclude that this joint is the most subject one to considerable strain. As a consequence, a visible bad gravity compensation is observed. The external torques when the robot is not loaded are somewhat coincident with the external torques of the robot when it is compensated, with the values between 0.1 to 0.6 Nm. In regards to the values of the external torques, they vary from 4 to 6.2 Nm when the robot is loaded. Comparable to the above joints, the external torques of joints 5, 6 and 7 in case of not loaded robot are near from the values estimated when the robot is already compensated.

4 Conclusion In this chapter, we dealt with the identification of the inertial parameters of the tools held by a medical cobot-assistant. This platform, using a 7-DoF Franka Emika robot, serve as assistant for different applications, such as for doppler sonography, minimally invasive surgery and neurosurgery. A procedure of identification has been implemented based on least-squares estimation, comparing the joint torques provided by the cobot to execute a same trajectory for two different conditions: with and without the tool attached to the end-effector. The main advantage of this method is that the estimation of the overall dynamic parameters of the robot is avoided. The identification procedure has been validated experimentally in the medical cobot-assistant for three different tools, ensuring an optimal performance in terms of accuracy, stability and safety during human–robot interaction. Acknowledgements This work was supported by the University of Poitiers and by the CNRS through the International Research Project RACeS. This work was also sponsored by the French government research program “Investissements d’avenir” through the Robotex Equipment of Excellence (ANR-10-EQPX-44).

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References 1. Grau, L., et al.: Robotic arm assisted total knee arthroplasty work flow optimization, operative times and learning curve. Arthroplast. Today 5(4), 465–470 (2019) 2. Su, H., Sandoval, J., Vieyres, P., Poisson, G., Ferrigno, G., De Momi, E.: Safety-Enhanced Collaborative Framework for Tele-Operated Minimally Invasive Surgery Using a 7-DoF Torque-Controlled Robot, vol. 16, pp. 1–9. Springer, Berlin (2018) 3. Sandoval, J., Laribi, M.A., Zeghloul, S., Arsicault, M.: On the design of a safe human-friendly teleoperated system for doppler sonography. Robot 8(29), 11 (2019) 4. Sandoval, J., Laribi, M.A., Zeghloul, S.: A low-cost 6-DoF master device for robotic teleoperation. In: Kuo, C.H., Lin, P.C., Essomba, T., Chen, G.C. (eds.) Robotics and Mechatronics. ISRM 2019. Mechanisms and Machine Science, vol. 78. Springer, Cham (2020). https://doi. org/10.1007/978-3-030-30036-4_43 5. Sandoval, J., Laribi, M.A., Faure, J.P., Brèque, C., Richer, J.P., Zeghloul, S.: Towards an autonomous robot-assistant for laparoscopy using exteroceptive sensors: feasibility study and implementation. IEEE Robot. Autom. Lett. 6(4), 6473–6480 (2021) 6. Essomba, T., Sandoval, J., Laribi, M.A., Wu, C.-T., Breque, C., Zeghloul, S., Richer, J.P.: Torque reduction of a reconfigurable spherical parallel mechanism based on craniotomy experimental data. Appl. Sci. 11, 6534 (2021) 7. Essomba, T., Hsu, Y., Sandoval Arevalo, J.S., Laribi, M.A., Zeghloul, S.: Kinematic optimization of a reconfigurable spherical parallel mechanism for robotic-assisted craniotomy. ASME. J. Mech. Robot. 11(6), 060905 (2019) 8. Khalil, W., Gautier, M., Lemoine, P.: Identification of the robot payload inertial parameters of industrial manipulators. IEEE Int. Conf. Robot. Autom. 2, 4943–4948 (2007) 9. Khalil, W., et al.: Identification experimentale des paramètres inertiels de la charge d’un robot Stäubli RX90. Arch. Ouvert. HAL 9, 1–6 (2009) 10. Siciliano, B., Khatib, O.: Springer Handbook of Robotics. Springer-Verlag, Heidelberg (2007) 11. Stürz, Y.R., Affolter, L.M., Smith, R.S.: Parameter identification of the KUKA LBR iiwa Robot including constraints on physical feasibility. IFAC-PapersOnLine 50(1), 6863–6868 (2017) 12. Gaz, C., Flacco, F., De Luca, A.: Identifying the dynamic model used by the KUKA LWR: a reverse engineering approach. In: Proceedings of International Conference on Robotics and Automation, pp. 1386–1392 (2014)

Design of Statically Balanced Assistive Devices S. D. Ghazaryan, M. G. Harutyunyan, Yu. L. Sargsyan, and V. Arakelian

Abstract The assistive devices such as active and passive exoskeletons and movable orthoses, designed to rehabilitate, support and reinforce the human musculoskeletal system functions, are performed by various linkages. The intended use of these mechanisms forms biomechanical systems of the device and human body that generally work in static and quasi-static modes. The gravity balancing of such systems plays a significant role in these mechanisms optimal design and application process. In a static work regime, the balancing of the system is realized by the use of counterweights, springs and auxiliary links. However, the use of springs is more desirable since they provide force balancing and have a small mass. The present study discloses methods and approaches to the static balancing of the linkage mechanisms applicable to assistive devices. However, they are universal and can be applied not only for assistive devices designing, but also, in general, for any manipulating devices with linkage rotating links. Keywords Assistive device · Exoskeleton · Movable orthoses · Gravity balancing · Spring

1 Introduction The most of plane and spatial manipulating mechanisms of robots are linkage systems, whose links are connected by rotational kinematic pairs and work mainly as swinging links. Different types of their presentation are given in Fig. 1. S. D. Ghazaryan (B) · M. G. Harutyunyan · Yu. L. Sargsyan National Polytechnic University of Armenia (NPUA), Yerevan, Armenia e-mail: [email protected] M. G. Harutyunyan e-mail: [email protected] Yu. L. Sargsyan e-mail: [email protected] V. Arakelian MECAPROCE, INSA-Rennes, Rennes, France LS2N/ECN UMR 6004, Nantes, France © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 V. Arakelian (ed.), Gravity Compensation in Robotics, Mechanisms and Machine Science 115, https://doi.org/10.1007/978-3-030-95750-6_7

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(a)

(b) Y

Y

x1

x1

y1

s1

y1

1

m1g

l1 O

X

(c)

1

φ

O

l1 X

(d)

Fig. 1 A manipulator: a—design, b—kinematic diagram, c—rotating link (1-DOF), d—mechanical model of a rotating link

Depending on the purpose, manipulators can be presented by several links or sections, but when designing, modeling and balancing they can be divided and analyzed section by section, for example, by a separate rotating link with one degree of freedom (1-DOF) and reduced mass at its end, replacing subsequent links/sections (Fig. 2). Then it can be reconfigured back by those separate sections into an integral manipulator. In the static and/or a quasi-static mode of operation the gravitational forces and their moments act on manipulating mechanisms. Gravitational balancing of the links of these mechanisms is one of the important research tasks and plays a significant role in their optimal design [1–16]. Many methods of system balancing are known [1–5], however, special requirements to small size, portability and low energy consumption for robotic mechanisms narrow the scope of their applications. The static balancing of manipulators is usually achieved by counterweights, springs or additional mechanisms (Fig. 3) [1–16, 26–39]. Implementing the masses balancing, the counterweights reduce the links gravity centers’ distances from their rotation axes, in particular, can equate them to zero. The springs, implementing the power balancing, realize the compensation of the gravity moments due to the gravity centers distances of system links from their

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y2

Y

Y

x1

x1 2

1

y1

l1

O

l2

2

x2

m3g 1

s1

y1

φ m1g

X

O

l1 X

(b)

(a) Fig. 2 A mechanical model of the mechanism’s separate section

(a) [1]

(b) [2]

(c) [2]

Fig. 3 Balancing of the mechanisms: a—by counterweights, b—by springs, c—by springs combined with additional links

rotation axes, then the spring stiffness is calculated from the system static balancing conditions. Additional parallel mechanisms bring the mass center of the system to a point that changes its position according to a given law. The tension and compression cylindrical springs have priority among the methods of links static balancing, thanks to their ability to develop great forces with small sizes and masses [1–16, 26–39]. For static balancing of a portable biomechanical system, as well as for other plane mechanisms, it is preferable to use cylindrical springs that provides power balancing (Fig. 4). The main area of application of the balancing solutions by using elastic elements is the design of rehabilitation and assistive devices, which support to carry the weight of a person’s body and its segments during movement (Fig. 5): the weight of the leg during walking, the weight of the arm during lifting weights and the weight of the body during sit-to-standing.

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(a) [2]

(b) [3]

(c) [4]

(d) [5]

(e) [6]

(f) [7]

(g) [8]

(h) [9]

(i) [10]

(j) [11]

(k) [12]

(l) [13]

(m) [14]

(n) [14]

(o) [15]

(p) [16]

Fig. 4 Spring static balancing devices

Exoskeletons and orthoses (subclass of rehabilitation devices) are the duplicating mechanical and/or electromechanical (mechatronic) devices intended for rehabilitation, support, amplification, limitation of the human musculoskeletal system functions [17–25]. Superimposed on the human body, they form biomechanical (biomechatronic) systems, mostly operating in static and quasi-static modes.

2 Static Balancing of Systems For the gravity balancing of the mechanical system with 1-DOF can be used: swinging link and reduced mass at its free end (see the model in Fig. 2b and Fig. 6), the triangular

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(a) [17]

(b) [18]

(c) [19]

(d) [20]

(e) [21]

(f) [22]

(g) [23]

(h) [24]

(i) [25]

Fig. 5 Rehabilitation and assisting devices: a–c—leg orthoses, d–f—leg exoskeletons, g–i—arm exoskeletons Fig. 6 Swinging link balancing

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Fig. 7 Cylindrical tension spring

d

l0 D

lH

lk

connection scheme (see Fig. 3b) of a cylindrical tension spring (Fig. 7). The static balancing is performed from the equilibrium condition of the moments of power acting on the links of the system [26–28, 37–39]. The unbalanced moment can be expressed as follows: Mu = Mg1 + Mb ,

(1)

where Mg1 is the moment of gravity, Mb is the balancing moment of the spring. To eliminate the unbalanced moment, the following condition must be met: [m 1 s1 + m 2 A r + m 3l1 ]g sin ϕ − FS l O B r sin ϕ/l S = 0,

(2)

where m 1 is the total mass of thigh and duplicating link 1, l1 is the length of the link 1, ϕ is the positioning angle of the link 1, r = l O A is the distance of point A from point O, l O B is the distance of point B from point O, l S = l AB is the current length of balancing spring, ψ is the positioning angle of the balancing spring, m 2 is the mass of the balancing spring, m 2 A is the mass-part of balancing spring coming to the link 1 at point A, s1 = l O S1 is the distance of mass center of the thigh and the link 1 S1 from point O, FS = F0 + k(l S − l0 ) is the elastic force of balancing spring, F0 is the pre-load/initial force of balancing spring, k is the spring stiffness coefficient, is the initial length of balancing spring, (see Fig. 7), l0 = l K + 2l H is the length of the spring active part, l K is the length of balancing spring hook. From the homogeneity condition, it is allowed that s1 = 0.5l1 and m 2 A = 0.5m 2 . From the geometry of the system, it can be written that the current length of the spring is expressed as: lS =

 l O2 B + r 2 − 2l O B r cos ϕ.

(3)

Then, partial balancing of the swinging link can be achieved due to the energy balance of the system, which can be expressed as: ϕ

ϕ

ΔE = ∫ϕif Mg dϕ + ∫ϕif Mb dϕ = 0, or:

(4)

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ϕ

ΔE = ∫ϕif (0.5m 1l1 + m 2 A r + m 3l1 )g sin ϕdϕ    F0 − kl0 ϕf k+ l O B r sin ϕ dϕ = 0, − ∫ϕi lS

(5)

where ϕi and ϕ f are the initial and final values of the rotation angle of the link 1. Integrating expression (5) and making ΔE = 0, we get: ϕ

ΔE = (kl O B r − (0.5m 1l1 + m 2 A r + m 3l1 )g) cos ϕ|ϕif ϕ f   2 2 − (F0 − l0 k) l O B + r − 2l O B r cos ϕ  = 0 ϕi

or: (0.5m 1 l1 + m 2 A r + m 3l1 )g cos k=

ϕ ϕ|ϕif

 ϕ l O B r cos ϕ|ϕif + l0 l O2 B

ϕ f   2 2 + F0 l O B + r − 2l O B r cos ϕ  ϕi ϕ f = 0.  2  + r − 2l O B r cos ϕ  ϕi

(6) All of the above is true in the case of using a standard linear cylindrical tension spring (Fig. 8). Its characteristics are shown in Fig. 8a, such a spring is conventionally called a spring of non-zero-free length. A special but laborious technological process makes it possible to obtain springs with different power characteristics (see Fig. 8b), where F0 = kl0 , and, in a particular case, a spring of zero-free length (see Fig. 8c), where F0 = kl0 [2, 37–39]. The use of the springs with zero-free length makes it possible to simplify the task, perform exact static balancing and to find the required spring stiffness coefficient from the condition of equilibrium of the moments of gravitational and balancing forces. In this case, the unbalanced moment can be expressed as follows: Mu = (0.5m 1l1 + m 2 A r + m 3l1 )g sin ϕ − kl O B r sin ϕ.

(7)

Under the condition Mu = 0, we get:

Fig. 8 Characteristics of cylindrical springs: a and b—with non-zero-free lengths, c—with zerofree length

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k = (0.5m 1l1 + m 2 A r + m 3l1 )g/l O B r.

(8)

The review of the works devoted to the problem of rotating links balancing by cylindrical springs shows that in equations, modeling the balancing, the mass of the spring is often determined approximately or included in the masses of the connected links [1–15]. The problem can be solved by using an iterative approach or a preliminary estimation of the used springs. However, to achieve the exact static balancing at the stage of designing, it is necessary to identify the relationship between the force and geometric characteristics of the used spring (see Fig. 7), i.e., the relationship between the stiffness coefficient and mass of the spring. k = Gd 4 /8D 3 n a ,

m 2 = ρ2 L W π d 2 /4,

(9)

from here we get: m 2 = ρ2 π L W



D 3 nk/2G,

(10)

where G is the Shear modulus calculated from the material’s elastic modulus E and Poisson ratio ν: (G = E/2(1 + ν)), d is the diameter of the wire that is wound into a helix, D is the mean diameter of a helix, n is the total number of coils, ρ is the material density of spring wire, L w is the length of spring wire: L w = π D(n/cos α) + 2L H ,

where α = tan −1 ( p/π D),

(11)

L H is the length of spring hook, which depends on its type and can be calculated from parameter L H (see Fig. 7) (several spring ends with prescribed heights are given in Table 1), p is the distance from center to center of the wire in adjacent active coils. On substituting expression (10) in (8), we obtain k=

  1  0.5m 1l1 + m 3l1 + 0.5rρπ L W D 3 nk/2G g, lO Br

(12)

from which we determine the stiffness coefficient:  2  2 k = −0.5q ± (0.5q) − u ,

(13)

where q=−

ρgπ L W  3 (0.5m 1l1 + m 3l1 )g , D n/2G and u = − 2l O B lO Br

(14)

taking into account that −0,5q ±



(0,5q)2 − u > 0.

(15)

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Table 1 Spring ends with prescribed heights

A) Half loop: = {0.55 . . .0.8}

A

B, C) Full loop: = {0.8 . . . 1.1}

B

D, E) Double twisted full loop: =

D

E

G, H) Raised hook: 1.2 < < 30

G

C F) Inside Full l.: = {1.05 . . . 1.2}

F L) Small eye: 2 < < 30 .

H

L

Numerical Example 1.—Static Balancing by Cylindrical Springs. Let us consider the balancing of the 1-DOF biomechanical system (Fig. 6) firstly by nonzero-free length spring and then by zero-free length spring. With the following system parameters [40]: m th = 8 kg; m 1 = 9 kg; m 2 = 1 kg (estimated); m 3 = 3.5 kg; l1 = 0.4 m; s1 = 0.2 m; r = 0.3 m; l O B = 0.15 m; ϕi = π/3; ϕ f = π ; we get the results presented in Fig. 9—dependence of k from F0 on the range of F0 ∈ [150 N − 200 N] (see Eq. 6); and Fig. 10—comparison of balancing moments Mb (Fo ) on the mentioned range of F0 with gravitational moment Mg (see Eq. 2). Now let us balance the same system by using zero-free length spring and taking into account its mass (see expression (12)). The parameters which characterize the selected spring are the following: G = 81·109 N/m2 , ρ = 7800 kg/m3 , D = 0.04 m, d = 0.004 m, by considering 0.4 ≤ D/d ≤ 15, l H = 0.12 m, n = 44 and L w = 5.9 m. In this case, we obtain k = 730 N/m, F0 = 190 N, m 2 = 0,5 kg.

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Fig. 9 Dependance of k from F 0

Fig. 10 System partial balancing results

These obtained values allow us to realize exact static balancing of the 1-DOF biomechanical system.

3 Considering the Links Deformation During the System Static Balancing The next objective is to consider the connection of a mechanical system to a biological object and the creation of a 1-DOF biomechanical system. Then, we have to optimize links parameters of the rehabilitation device by considering the methods of connecting the balancing springs to the rehabilitation device, as well as the rehabilitation device to the human limb, taking into account the acting forces and links deformations (Fig. 11) [29, 30, 37, 38]. Fig. 11 Balancing of the 1-DOF biomechanical system

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For this scheme, the gravitational and unbalanced moments and the coefficient of spring stiffness are expressed as follows: Mg = (m 1 s1 + m 0 A r + m 2 A r + m 3l1 )g sin ϕ,

(16)

Mu = (m 1 s1 + m 0 A r + m 2 A r + m 3l1 )g sin ϕ − kl O B r sin ϕ = 0,

(17)

 k = m 0 A r + m 1 s1 + 0.5rρπ L W D 3 nk/2G + m 3l1 g/rl O B .

(18)

For a cylindrical tension spring, its maximum working tension length is set, based on which the spring attachment points to the swinging link are determined. With the selected balancing scheme (see Fig. 11), it is convenient for mathematical expressions to take the value r = l O B , but in this case, the spring must lengthen twice from its initial length which is not desirable for the extension springs. The best option is to select r = 2l O B . By setting the required parameters of the biomechanical system it is possible to determine the required coefficient of spring stiffness and calculate the values of all forces and moments of forces acting in the plane of the mechanism. Until now, the mechanism links were considered as perfectly rigid bodies, but, in reality, they are elastic. Thus, due to the action of the aforementioned forces, their deformations take place: the bending over the entire length of the rotating link, reaching its maximum at the end of the link, and the torsion in cross-sections, reaching maximum at the connection points of the mechanism with biological segments. Providing the rotating link’s rigidity and strength conditions, we determine its minimum dimensions (m 1 → min), and, consequently, the minimum values for the mass and stiffness coefficient of the spring ( k|min m 1 →min ).

3.1 Bending Calculation of the 1-DOF System’s Swinging Link We define the reaction R t at the point O from the following condition: Σ F t = R t + kl O B sin ϕ − (m 0 A + m 1 + m 2 A + m 3 )g sin ϕ = o.

(19)

Taking into account the expression (17), we write: R t = (m 3 (l1 − r ) − m 1 (r − s1 ))gr −1 sin ϕ.

(20)

In the case of bending, the deformations from the acting forces in the x coordinate sections will be determined by the following expression:

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  y = (θ0 x + R t x 3 (6E Jz )−1 − m 1 gx 4 24E Jz l1 )−1 sin ϕ  I  + (m 1 s1 + m 3l1 )g(x − r )3 (6E Jz r )−1 sin ϕ  , II

(21)

where Jz is the moment of inertia of the section, E is the elasticity modulus, θ0 is the initial angle of link 1 rotation due to the bending deformation. It follows from the expressions (20) and (21) that the bends do not depend on the spring parameters, but only on the values of m 1 and m 3 . We assume that at the spring and leg connection point A the displacements are equal to zero: y|x=r = θ0 r + (m 3 (l1 − r ) − m 1 (r − s1 ))gr 2 (6E Jz )−1 sin ϕ − m 1 gr 4 (24E Jz l1 )−1 sin ϕ = 0.

(22)

From this condition, we define   θ0 = 0.25m 1 r 2 + 4l1 (r − s1 ) l1−1 − m 3 (l1 − r ) (6E Jz )−1rg sin ϕ.

(23)

Substituting (23) into (21), provided that s1 = 0.5l1 , we obtain  ⎤ − m 3 r 2 x + (m 3 − m 1 )x 3   ⎦(24E Jz )−1 g sin ϕ. y=y=⎣ −3m 1 x 4 /4l1 I 3 + (2m 1 + 4m 3 )(x − r ) I I ⎡   25m 1 16

(24)

We get the maximum value of y at the point x = l1 , when ϕ is equal to 900 and if r = 0.75l1 , so we can write that   215m 1 ymax = y| gl13 (36Jz )−1 . (25) m3 − x = l1 128 sin ϕ = 1 Under the following condition: ymax ≤ 0.01l1 ,

(26)

for a link with a rectangular section, height a and width b, when a = 4b, we can write 0.03Ea 4 /l12 g + 215ρa 2 l1 /128 − 4m 3 ≥ 0, when Jz = a 4 /48, m 1 = ρa 2 l1 /4. (27) Solving this quadratic equation, we obtain the values of a, b and m 1 .

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3.2 Torsion Calculation of the 1-DOF System’s Swinging Link At the point A the leg gravity force creates a torque Mx1 around the axis x1 (Fig. 12), and for the formed tangential stress τ , we can write τ = M1x1 r/Wk ,

(28)

where Wk is the polar section modulus of link determined by the following formula: Wk = αab3 = αa 4 /64.

(29)

M1x1 = m 0 A gh 1 sin φ,

(30)

Mx1 max = M1×1 |sin φ = 1 = m 0 A gh 1 ,

(31)

τ max ≥ Mx1 max r/Wk = 64m 0 A gh 1r/αa 4 ,

(32)

From Fig. 12, we have:

where h1 is the distance between the x-axes of the leg and the link 1, coefficient α depends on the ratio a/b and is determined experimentally, m 0 A is the part of the leg mass m 0 located at the point A and determined as: m 0 A r = m 0 s0 , h 1 is the distance between the x-axis of a leg and the link 1. When a = 4b ª and α = 0.282, we have:  (33) a ≥ 4 64m 0 A gh 1r (0.282τ max)−1 , and obtain the values of a, b and m 1 . However, both conditions (27) and (33) must be satisfied: Fig. 12 Link torsion determination

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Table. 2. Calculation results for bending and torsion cases Cases

a (mm)

b (mm)

m1 (kg)

F 0 (N)

k (N/m)

m2 (kg)

Bending

10

2.5

0.03

178

685

0.51

Torsion

50

12

0.7

186

715

0.52



 4 2 0.03Ea /l1 g + 215ρa 2 l1 /128 − 4m 3 ≥ 0,    a ≥ 4 64m 0 A gh 1r (0.282τ max)−1 . a=4b

(34)

To solve this system of equations two groups of parameter values are introduced: variables ρ, τ max, E, a, and constants g, l1 , m 0 A , h 1 , m 3 . The problem of minimizing the mass m 1 and the spring stiffness coefficient k are solved:

m 1 g, l1 , r, m 0 A , h 1 , m 3 , ρ, τ max, E, a → min,

(35)

k(m 0 A , m 1 , m 2 A , m 3 , s1 , r, l O B , l1 , g)|min m 1 (g,l1 ,r,m 0 A ,h 1 ,m 3 ,ρ,τ max,E,a )→min . (36) Numerical Example 2.—Bending and Torsion Determination. Balancing is considered with the same system parameters as presented above. If link 1 is made of aluminum, for which E = 7 ∗ 1010 N/m2 , ρ = 2700 kg/m2 , τmax = 0.5 ∗ 108 Pa, m 0 A = 4.85 kg then, h 1 = 0.1 m fulfilling the conditions (34), we get the results presented in Table 2.

3.3 Torsion Reducing by Constructive Changes If to leave unchanged the spring connection point, but remove link 1 parallel to its working plane at a certain distance h 2 from the leg, the difference (Fst − m 0 A g sin ϕ) in the spring forces will create a counter-torque (Fig. 13), and the Eq. (30) will take the following form:  M2x1 = Fst − m 0 A g sin ϕ h 2 − m 0 A g(h 2 + h 1 ) sin ϕ. Fig. 13 Link torsion compensation

(37)

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Substituting Eq. (17), we get:   M2x1 = (m 1 s1 + m 3l1 )h 2 r −1 − m 0 A h 1 g sin ϕ.

(38)

In a certain change zone of h 2 , we can provide the condition |M2x1 | ≤ M1x1 that will lead to a decrease in the value of a, and at the point h 02 we will have M2x1 equal to zero [30, 37, 38]: h 02 = m 0 A h 1r/(m 1 s1 + m 3l1 )

(39)

However, due to the presence of the bending, we can set the minimum value of a (amin ) and calculate the required values of M2x1 and h 2 . τ2 = M2x1 r/Wk ,

(40)

τ2 Wk /r ≥ M1x1 max,

(41)

M2x1 max = τ2 Wk /r,

(42)

|M2x1 ||sinϕ=1 = M2x1 max,

(43)

h 2 max = (m 0 A h 1rg + τ Wk )/(m 1 s1 + m 3l1 )g,

(44)

h 2 min = (m 0 A h 1rg − τ Wk )/(m 1 s1 + m 3l1 )g.

(45)

In the range h 2 min ≤ h 2 ≤ h 2 max, |M2x1 | takes values that satisfy the bending condition, and at the point h 2 = h 02 , when M2x1 = 0, the torsion effect can be ignored. Numerical Example 3—Torsion Compensation. If we remove link 1 within h 2 = [0.0484...0.0486] m, we can achieve a reduction in the link 1 torsion, and the consideration of the bending conditions will be enough only. At h 2 = 0.0485 m, the torsion of link 1 is fully compensated. The results of strength and stiffness calculations of the links showed that the requirements to the links’ minimum dimensions are more rigid in the bending case than in the torsion case. The optimal zones for fixing the spring to the device links and the human leg have been proposed, and, as a consequence, a significant reduction in the dimensions and the masses of device and a decrease in the stiffness coefficient and mass of the balancing spring are reached.

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4 Arrangement of Balancing Springs To minimize the weight of the balancing system, the arrangement of balancing springs was done (Fig. 14) [31, 37, 38]. This allowed varying the choice of working and connecting zones of used springs for obtaining total or partial exclusion of their mass’s components from the balancing equations and thereby reducing the values of spring stiffness coefficients required for balancing. Variant A. By dividing the balancing spring into two equal parts, connecting them by a metal cable and arranging, as shown in Fig. 14a, it will become possible to reduce in the balancing equations the gravitational moment of the balancing spring, depending on its position and mass, and, as a consequence, reduce also the required stiffness coefficient. Here, the balancing is achieved by non-zero-free length springs, but reduced to the characteristics of a zero-free length spring (let’s call it—pseudo zero-free length). We assume that  Mg = m sys1 ssys1 + m 22 s22 + m 3l1 g sin ϕ,

(46)

FS = F01 + k1 (0.5l S − l01 ) + m 21 g + F02 + k2 (0.5l S − l02 ),

(47)

where s22 is the distance of the gravity center from the point O of the spring second half, k1 , m 21 , F01 , l01 and k2 , m 22 , F02 , l02 are the stiffness coefficients, masses, initial tensions and initial lengths of the corresponding parts of the spring. From the condition of equivalence of these parts, we write: F01 = F02 = F0.1.2 , k1 = k2 = k12 , l01 = l02 = 0.5l0 , m 21 = m 22 = m 2.1.2 , (48) FS = 2F0.1.2 + k1.2 (l S − l0 ) + m 2.1.2 g.

(49)

If these springs are prepared and installed with the condition that F0.1.2 = 0.5(k1.2 l0 − m 2.1.2 g),

(a) [31] Fig. 14 Arrangement of balancing springs

(b) [31]

(50)

(c) [31]

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then we will have a spring with pseudo zero-free length, so we can write: Mb = FS l O B r sin ϕ/l S = k1,2 l O B r sin ϕ.

(51)

However, in this variant, s22 is not constant: s22 = 0.25l S , hence  Mg = m sys1 ssys1 + 0.25m 22 l S + m 3l1 g sin ϕ.

(52)

Since l S has its constant and variable components: l S = l0 + Δl S , then we write:  Mg = m sys1 ssys1 + 0.25m 22 (l0 + Δl S ) + m 3l1 g sin ϕ.

(53)

If the variable component of l S is rather small and can be ignored, then we can assume that the exact static balancing of system takes place. However, it is more appropriate to take into account its average value 0.5Δl Smax , then  Mg = m sys1 ssys1 + 0.25m 22 (l0 + 0.5Δl Smax ) + m 3l1 g sin ϕ, Mu =

(54)

 m sys1 ssys1 + 0.25m 22 (l0 + 0.5Δl Smax ) + m 3l1 g − k1,2 l O B r sin ϕ. (55)

Assuming Mu = 0 and taking into account expression (10), we obtain the value of k from the following quadratic equation: 

3 n 22 k1,2 /2G l O B r k1,2 − 0.25ρπ L W 22 (l0 + 0.5Δl Smax )g D22  − m sys1 ssys1 + m 3l1 g = 0.

(56)

Variant B. Here, both springs are reciprocally installed along to link 1, as shown in Fig. 14b, and the common gravity center of the springs connecting point O is mutually compensating and remains always constant. The sum of the gravitational moments of the system can be written as  Mg = m sys1 ssys1 + 0.25m 2.1.2 (r + l O B ) + m 3l1 g sin ϕ.

(57)

When using the zero-free length springs, we have FS = 2F0.1.2 + k1.2 (l S − l0 ) = k1.2 l S , when 2F0.1.2 − k1.2 l0 = 0.

(58)

The unbalanced moment of the system will take the form  Mu = m sys1 ssys1 + 0.25m 2.1.2 (r + l O B ) + m 3l1 g sin ϕ − k1,2 l O B r sin ϕ. (59) Assuming Mu = 0 and taking into account expression (10), we obtain the value of k from the following quadratic equation:

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Table. 3. Calculation results for balancing springs arrangement Variant

n

d (m)

D (m)

D/d

L W (m) l01.2 (m)

k (N/m)

F01.2 (N)

m 21.2 (kg)

A

22

0.0032

0.04

12.66

3.015

0.132

716.048

186.03

0.183

B

22

0.0032

0.04

12.66

2.14

0.132

715.693

185.94

0.183

C

22

0.0031

0.04

12.68

2.14

0.132

711.111

184.752

0.182

 3 l O B r k1,2 − 0.25ρπ L W 2.12 g(r + l O B ) D2.12 n 2.1.2 k1,2 /2G  − m sys1 ssys1 + m 3l1 g = 0.

(60)

Variant C. Now both springs are reciprocally installed along the OB, as shown in Fig. 14c, and only their gravitational forces participate in the balancing expressions. In this case, the sum of the moments of gravitational forces is written as  Mg = m sys1 ssys1 + m 3l1 g sin ϕ.

(61)

Again, the pseudo zero-free length springs are used: FS = 2F0.1.2 + k1.2 (l S − l0 ) + 2m 2.1.2 = k1.2 l S , when 2F0.1.2 + 2m 2.1.2 − k1.2 l0 = 0.

(62)

The unbalanced moment of the system will take the form  Mu = m sys1 ssys1 + m 3l1 g sin ϕ − k1,2 l O B r sin ϕ.

(63)

Assuming Mu = 0, we obtain the value of k from the following linear equation:  k1,2 = m sys1 ssys1 + m 3l1 g/rl O B .

(64)

Numerical Example 4.—Spring Arrangement. Balancing is performed with the same system parameters presented above, and the calculation results are shown in Table 3.

5 The Counterweight Correction of the Rotating Link’s Spring Unbalance The combined application of the mentioned above approaches can significantly lighten devices and make them easily portable. However, as already mentioned, the devices under consideration, due to their design features, require the use of special zero-free length springs (Fig. 8). With such springs, the solution is by far the best in terms of portability of the designed system, but their production is associated

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with several complex and often expensive technologies, and this also raises the question of springs’ non-universality. Therefore, it is proposed to use common cylindrical springs combined with corrective counterweights to achieve the exact static balancing (Fig. 15) [32, 33, 37–39]. From the triangle OBA (see Fig. 15) we have: the spring current length l S is determined by the formula (3), but under the particular conditions that the spring ends are connected to the system at equal distances from the rotation axis (hip joint) of the biomechanical system and the formed triangle is isosceles (l O B = r ), lS =

  r 2 + r 2 − 2r 2 sin ϕ = r 2(1 − sin ϕ) = 2rsin0.5ϕ.

(65)

The desired law of change for the spring balancing moment is achieved during leg positioning: Mu = ((0.5m 1 l1 + 0.5m 2 r + m 3l1 )g − krl O B ) sin ϕ − (F0 − kl0 )r cos 0.5ϕ. (66)

(a) [32, 33]

(b) [32, 33]

(c) [33]

(d) [33]

Fig. 15 The counterweight correction of spring unbalance of the rotating link

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Y

x1

x1 3

3 B

lOB

y1

A

ψ

s1 φ

2

B

2 m3g 1

lOB

m2Ag m1g

r

O

(a) [34]

φ

µr

A

ψ

s1

m3g 1

m2Ag

r

m1g

l1 X

mCg

y1

l1

O

X mCg

µ lOB

µ lOB

(b) [34]

Fig. 16 Universal counterweight correction of the spring unbalance of the swinging link

Adding to the system a two-armed lever (see Fig. 15a, b) centered on the rotation axis of the 1-DOF biomechanical system, with one arm connected by kinematic pairs to the center of the spring and with a counterweight at the end of the other arm, it is possible to achieve compensation for the partial unbalance of the biomechanical system: Mu = Mg − Mb1 + Mb2 = ((0.5m 1l1 + 0.5m 2 r + m 3l1 )g − krl O B ) sin ϕ −(F0 − kl0 )r cos0.5ϕ + m C gl OC cos0.5ϕ.

(67)

The constructive views a) and b) in Fig. 16 depend on the values of the initial parameters of the balancing spring. Therefore, when m C gl OC = (F0 − kl0 )r , we obtain an equation in the form of (7). For medical devices, one of the most important issues is the ability to adjust the dosage of loads and limits of movement, which requires the system wide replacement of the balancing springs. The proposed scheme is very convenient for such replacements: it is universal and allows eliminating the unbalance by choosing the values of the counterweight mass and its lever. Alternative constructive realizations of the above-given method of partial unbalance counterweight compensation for the 1-DOF biomechanical system statically balanced by a non-zero-free length spring are also proposed (see Fig. 15c, d). It should be noted that regardless of the designed structure perfection, the most exact static balancing is also ensured by taking into account in the calculations the dependence between mass and stiffness of used cylindrical springs [26]. The described balancing hybrid system (cylindrical tension-compression spring plus counterweight) allows us to propose a universal counterweight correction of the spring static balance of any “swinging link” with one degree of freedom (1-DOF), providing exact balance (Fig. 16a, b) [34, 37–39]. In the proposed new designs, the ends of the balancing spring can be connected to the system at any required distance from the rotation axis of the biomechanical

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system. Here it is important only to preserve the proportions between the pantograph links, so that the corrective mass will move according to the law of sin ψ, and not of cos0.5ϕ, as in the previous particular case, herewith, in the final position of the swinging link (ϕ = 1800 ) the balancing spring will lengthen by l O Bsmax , and not by lsmax .

6 Static Balancing of Systems with 2-DOF 6.1 The Static Spring Balancing of the Leg Biomechanical System with 2-DOF In Fig. 17 the biomechanical system of a leg and a 2-DOF rehabilitation device is presented, here zero-free length cylindrical springs 4 and 5 are statically balancing the system from gravity forces [35, 37, 38]. It should be mentioned that spring 4 is a tension spring, and spring 5 is a compression spring. Both are connected to link 2 at point A, which is chosen from the following condition: M = m 3 + m 2 + m 1 s1 /l1 , sm = (m 2 s2 + m 3l2 )/M,

(68)

where l1 is the length of duplicating link 1 (of thigh), s M the distance of point A from point D, M is the mass of the rotating links of the biomechanical system reduced to point A, m 1 and m 2 are the united mass of correspondent segment of leg and its duplicating link: m 1 = m t +m L1 , m 2 = m sh +m L2 , m t is the mass of thigh, m L1 is the Fig. 17 To the static balancing of leg 2-DOF biomechanical system

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mass of duplicating link 1, m sh is the mass of shank, m L2 is the mass of duplicating link 2, s1 is the distance of the center mass of thigh and its duplicating link from point O, s2 is the distance of the center mass of shank and its duplicating link from point D, l2 is the length of duplicating link 2 (of the shank). From the condition of homogeneity, it is allowed that s1 = 0.5l1 , s2 = 0.5l2 . The foot is considered as a point mass with value m 3 = m f in the end of link 2. And the static balancing is realized from the condition of constancy of potential energy of the biomechanical system: P = Pg + PS1 + PS1 = const,

(69)

where Pg is the potential energy of links reduced at point A: Pg = −[M + 0.5(m 4 + m 5 )]gl O A sin β,

(70)

where m 4 = m S1 and m 5 = m S2 are the mass of the applied springs, β is the angle XOA. The springs are prepared with the same stiffness coefficient k, but since their origins are different, the potential energies of springs can written as: 2 2 , PS2 = −0.5kl S2 , PS1 = 0.5kl S1

(71)

where l S1 and l S2 are the current lengths of springs 4 and 5. From united resolution of expressions (69)–(71) we obtain: 2 2 − 0.5kl S2 . P = −[M + 0.5(m 4 + m 5 )]gl O A sin β + 0.5kl S1

(72)

From the triangle BAO we can see that 2 = l O2 B + l O2 A + 2l O B l O A sin β, l S1

(73)

where l O B is the distance from point B to point O: P = (kl O B − Mg − 0.5g(m 4 + m 5 ))l O A sin β + 0.5kl O2 B .

(74)

The potential energy rests constant, if the following condition is achieved: kl O B − Mg − 0.5g(m 4 + m 5 ) = 0.

(75)

By using the non-linear dependence (2) between the stuffiness coefficient and mass of the spring we write:     √ kl O B − Mg − 0.5g k ρ4 π L W 4 D43 n 4 /2G 4 + ρ5 π L W 5 D53 n 5 /2G 5 = 0, (76)

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where G 4 and G 5 are Shear modulus, D4 and D5 are mean helix diameters, n 4 and n 5 are numbers of coils, L W 4 and L W 5 are wire lengths, ρ4 and ρ5 are material densities of spring 4 and spring 5, accordingly. By solving this quadratic equation we obtain the value of k. The Numerical Example 5—Static Balancing of System with 2-DOF. The calculations are done in the case of the following values of system parameters [40]: m th = 8 kg, m sh = 3.1 kg, m 3 = m f = 1 kg, m L1 = 1.3 kg, m L2 = 1.3 kg, l1 = 0.4 m, l2 = 0.4 m, l O B = 0.15 m, s1 = 0.5 l1 = 0.2 m, s2 = 0.5 l2 = 0.2 m, ρ4 = ρ5 = 7800 kg/m3 , G 4 = G 5 = 81 · 109 N/m2 , D4 = D5 = 0.04 m by considering 0.4 ≤ D/d ≤ 15, ϕ ∈ [π/3; π ]. When m 1 = 9.3 kg, m 2 = 4.4 kg, M = 10.05 kg, sm = 0.127 m, and for steel-wire springs: n 4 = 63, n 5 = 70, L W 4 = 8.0447 m, L W 5 = 8.8 m, from the condition of static balancing of biomechanical system it is obtained that k = 715.058 N/m, l04 = 0.354 m, l05 = 0.355 m, F04 = 253.844 N, F05 = 253.185 N, m 4 = 0.8435 kg, m 5 = 0.9583 kg. The results obtained for the 2-DOF rehabilitation device are the same as for the 1-DOF case.

7 Static Balancing of Arm Biomechanical Systems 7.1 The Static Spring Balancing of the Arm Biomechanical System with 1-DOF Experience in the development of balanced rehabilitation devices for the leg has created prerequisites for proposing balanced rehabilitation devices for the arm [36– 38]. The applied approaches and methods are the same as in the leg balancing cases, so the balancing is again done by using zero-free length springs. The equations have the same expressions, but the denotations are different. The proposed 1-DOF arm rehabilitation device consists of the forearm movement duplicating link 1 and spring 2 attached to that link at point A at distance r from point O (Fig. 6). The value of the spring stiffness coefficient is determined from the conditions of static balancing of the biomechanical system. The second end of the spring 2 is connected with Y -axes at the fixed-point B, in distance l O B from point O. In the free end of link 1, in distance l1 from point O, there is a load-link 3 with mass m 3 (the value of mass m 3 can be determined corresponding to the value of the mass of non-considered elements of the system reduced to the point D). In this case, computational equations are the same as for the leg case, with the difference that the forearm and duplicating link 1 total mass m 1 and the distance of the center of total mass from point O s1 are determined as: m 1 = m farm + m L1 ,

m 1 s1 = m farm sfarm + m L1 s L1 ,

(77)

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where m farm is the mass of forearm, sfarm is the distance of the mass center of the forearm from point O, m L1 is the mass of duplicating link 1, s L1 is the distance of the mass center of the link 1 from point O. Here, we assume that links are homogeneous. The zero-free length cylindrical spring is again used for balancing, and the non-linear dependence (10) between mass and stiffness coefficients of spring is used. The unbalanced moment of system Mu is expressed by Eq. (7), and by setting its constant part equal to zero we achieve the systems balance and determine the value of k from Eq. (12). The Numerical Example 6—Static Balancing of Arm System with 1-DOF. The calculations are done for the following values of the system parameters [40]: m farm = 1.25 kg, m L1 = 0.6 kg, m 1 = 1.85 kg, m 3 = 0.7 kg; l1 = 0.3 m, r = 0.2 m, l O B = 0.1 m, s1 = 0.15 m, ϕi = 600 , ϕ f = 1800 , and for steel-wire spring: G = 81 · 109 N/m2 , ρ = 7800 kg/m3 , D = 0.03 m, L H = 0.047 m, n = 36, L W = 3.4885 m, by considering 0.4 ≤ D/d ≤ 15. The following results are obtained: l0 = 0.1732 m, k = 243.94 N/m, consequently F0 = 42.24 N, m 2 = 0.1034 kg, d = 0.003 m, D/d = 10. The results obtained for the upper extremity rehabilitation devices have the same interpretation as for lower extremity rehabilitation devices.

7.2 The Static Spring Balancing of the Arm Biomechanical System with 2-DOF In Fig. 18, the biomechanical system of arm and 2-DOF rehabilitation device is presented with zero-free length cylindrical springs 4 and 5, statically balancing the system from gravity forces [36–38]. Spring 4 is a tension spring, and spring 5 is a Fig. 18 To the static balancing of arm 2-DOF biomechanical system

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compression spring. Both are connected to link 2 at point A, which is chosen from condition (68). Here l1 is the length of duplicating link 1 (of the forearm), sm is the distance of point A to point D, M is the mass of the rotating links of the biomechanical system reduced to point A, m 1 and m 2 are the united masses of a correspondent segment of arm and its duplicating link: m 1 = m farm + m L1 , m 2 = m wr + m L2 , m farm is the mass of forearm, m L1 is the mass of duplicating link 1, m wr is the mass of wrist, m L2 is the mass of duplicating link 2, s1 is the distance of the center mass of forearm and its duplicating link from point O, s2 is the distance of the center of masses of wrist and its duplicating link from point D, l2 is the length of duplicating link 2 (of the wrist). The hand is considered as a point mass with the value m 3 = m hand in the end of link 2. The static balancing is realized from the condition of constancy of potential energy of the biomechanical system (69). The springs are made with the same stiffness coefficient k the value of which is determined from the quadratic Eq. (76). The Numerical Example 7—Static Balancing of the Arm System with 2-DOF. The calculations are done for the following values of the system parameters [40]: m farm = 1.25 kg, m wr = 0.5 kg, m 3 = m h = 0.5 kg, m L1 = 0.6 kg, m L2 = 0.6 kg, s1 = 0.5l1 = 0.15 m, s2 = 0.5l2 = 0.15 m, l1 = 0.3 m, l2 = 0.3 m, l O B = 0.1 m, D4 = D5 = 0.03 m, G 4 = G 5 = 81 · 109 N/m2 , ρ4 = ρ5 = 7800 kg/m3 , by considering 0.4 ≤ D/d ≤ 15, ϕ ∈ [π/3; π ]. When m 1 = 1.85 kg, m 2 = 1.1 kg, M = 2.225 kg, sm = 0.101 m, and for steel-wire springs: n 4 = 58, n 5 = 66, L W 4 = 5,5627 m, L W 5 = 6,2228 m. From the conditions for static balancing of the biomechanical system it is obtained: k = 240.37 N/m, l05 = 0.265 m, l04 = 0.265 m, F04 = 63.733 N, F05 = 63.643 N, m 4 = 0.218 kg, m 5 = 0.248 kg.

8 Conclusions The application of zero-free length springs allows realizing the exact balancing of gravity forces of the system in static work regime due to the spring force special characteristics. By considering the non-linear dependence between the force and geometry parameters of spring, it is possible to find the exact value of the spring mass approximately defined previously and determine the value of spring stiffness coefficient from the balance conditions. Furthermore, it is shown how to take into account the elasticity of the biomechanical system elements to be balanced. The various examples illustrate the proposed design methods for static balancing.

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26. Arakelian, V., Ghazaryan, S.: Gravity balancing of the human leg taking into account the spring mass. In: Proceedings of the 9th International Conference on Climbing and Walking Robots (CLAWAR), Brussels, Belgium, 12–14 September 2006, pp. 630–635 (2006) 27. Arakelyan, V., Ghazaryan, S.: Design of balancing devices for robotic rehabilitation. In: XVIII International Internet - Conference of Young Scientists and Students on Problems of Machine Science (MIKMUS 2006): Conference Materials, 27–29 November 2006, p. 56 (2006) 28. Ghazaryan, S., Harutyunyan, M., Arakelyan, V.: Design of rehabilitation systems and optimization of their parameters. In: XIX International Internet - Conference of Young Scientists and Students on Problems of Machine Science (MIKMUS 2007): Conference Materials, 05–07 November 2007, p. 93 (2007) 29. Ghazaryan, S., Harutyunyan, M., Arakelyan, V.: Spring static balancing and optimization of parameters of the biomechanical system. In: Annual Scientific Conference of SEUA: Collection of Materials, Yerevan, 19–23 November 2007, vol. 1, pp. 406–410 (2008) 30. Ghazaryan, S.: Gravity balancing and optimization of the parameters of the biomechanical system of a human leg and the rehabilitation device. In: Proceedings of Engineering Academy of Armenia (PEAA), Yerevan, Armenia, vol. 5, no.1, pp. 119–122 (2008) 31. Ghazaryan, S., Harutyunyan, M., Arakelyan, V.: The optimization of parameters of the biomechanical system using spring composition during static balancing. In: Annual Scientific Conference of SEUA: Collection of Materials, Yerevan, 19–23 November 2007, vol. 1, pp. 410–414 (2008) 32. Ghazaryan, S.D., Harutyunyan, M.G., Arakelyan, V.H., Glazunov, V.A.: The exact static balancing of mechanotherapeutic devices. In: XXI International Innovation-Oriented Conference of Young Scientists and Students on Modern problems of Machine Science (MIKMUS2009): Conference Materials, 16–18 November 2009, p. 70 (2009) 33. Sargsyan, S.A., Ghazaryan, S.D., Arakelyan, V.H., Harutyunyan, M.G.: The Counterweight correction and regulation of spring balance of rehabilitation devices. In: Proceedings of Engineering Academy of Armenia (PEAA), Yerevan, Armenia, vol. 7, no. 2, pp. 338–341 (2010) 34. Ghazaryan, S.D., Sargsyan, S.A., Harutyunyan, M.G., Arakelyan, V.H.: The universal counterweight correction of swinging link spring balance. In: Proceedings of Engineering Academy of Armenia (PEAA), Yerevan, Armenia, vol. 8, no. 1, pp. 152–155 (2011) 35. Arakelian, V., Ghazaryan, S.: Improvement of balancing accuracy of robotic systems: application to leg orthosis for rehabilitation devices. Int. J. Mech. Mach. Theory 43(5), 565–575 (2008) 36. Ghazaryan, S.D., Harutyunyan, M.G., Arakelyan, V.H.: Design of an arm rehabilitation device. In: Annual Scientific Conference of SEUA: Collection of Materials, Yerevan, 17–21 November 2008, vol. 1, pp. 312–315 (2009) 37. Ghazaryan, S.D.: The analysis and optimization of the biomechanical system of human extremities and portative devices of rehabilitation of their functions. Thesis of Dissertation for obtaining the scientific grade “Doctor of Engineering” in specialization “Apparatus, systems, products for medical importance”: State Engineering University of Armenia (Polytechnic) and National Institute of Applied Science – Rennes of France, Yerevan, Armenia, 20 January 2009, p. 133 (2009). (Scientific advisers: prof. Harutyunyan M.G, prof. Arakelyan V.G.) 38. Ghazaryan, S.D., Harutyunyan, M.G., Arakelyan, V.H.: Design aspects of human movement assistance - rehabilitation means. In: Proceedings of the 1st International Conference MES 2018/IPM-2018 Mechanical Engineering Solutions. Design, Simulation, Testing and Manufacturing, Yerevan, Armenia, 17–19 September 2018, MES-2018-25, pp. 71–80 (2018)

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39. Ghazaryan, S.D., Harutyunyan, M.G., Arakelian, V.H.: Actual aspects of manipulation mechanism’s swinging links spring balancing. In: XXVI International Scientific and Technical Conference “Mechanical Engineering and Technosphere of the XXI Century”, Sevastopol, 23–29 September 2019, pp. 450–454 (2019) 40. Begun, P.I., Shukeylo, Yu.A.: Biomechanics: Textbook for High Schools. Polytechnic, SPb (2000). 463 pp.

Design of Multifunctional Assistive Devices with Various Arrangements of Gravity Compensation S. D. Ghazaryan, M. G. Harutyunyan, Yu. L. Sargsyan, N. B. Zakaryan, and V. Arakelian

Abstract When designing portative assistive linkage devices such as active and/or passive exoskeletons and movable orthoses the static gravity balancing of links from acting gravitational forces and their moments becomes essential for assuring their portability. The increasing of degree of mobility of devices increases their functionality, but it increases also the number of necessary balancing units. From this point of view, the use of springs is more desirable, since they provide a force balancing with small increasing of moving masses. Moreover, the application of regulating mechanisms that control the action-direction of springs and, by this, allow the use of one spring for different functional purposes can be a solution to the above problems. In this paper, the number of constructive schemes of the multipurpose portable movable orthoses and exoskeletons for assisting to human different locomotion functions: walking, sitting, holding and lifting, are proposed, their static balancing methods and approaches, comparative numerical analysis, advantages and disadvantages are given. In addition, the proposed approaches and solutions are universal and can be used in the design of assistive devices, as well as in industrial manipulators. Keywords Assistive device · Exoskeleton · Movable orthoses · Gravity balancing · Spring

1 Introduction Depending on the purpose, the active and/or passive exoskeletons and movable orthoses can operate in static/quasi-static and dynamic modes, and as linkage mechanisms can have one or more degrees of mobility (DOF). To statically balance links S. D. Ghazaryan · M. G. Harutyunyan (B) · Yu. L. Sargsyan · N. B. Zakaryan National Polytechnic University of Armenia (NPUA), Yerevan, Armenia e-mail: [email protected] S. D. Ghazaryan e-mail: [email protected] V. Arakelian MECAPROCE, INSA-Rennes, Rennes, France LS2N/ECN UMR 6004, Nantes, France © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 V. Arakelian (ed.), Gravity Compensation in Robotics, Mechanisms and Machine Science 115, https://doi.org/10.1007/978-3-030-95750-6_8

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from acting gravitational forces and their moments different balancing systems are used [1–5]. In linkages, every degree of freedom needed its own balancing system, hence the increase of the device mobility required to increase the number of such systems. Therefore, these can falter the portability of the device. However, the application of the springs, providing force balancing and having small masses [6–9], and the regulating mechanisms, regulating the action-direction of springs and allowing to use one spring for different functional purposes, can assure back the portability. In this paper, a number of constructive schemes of the multipurpose portable movable orthoses and exoskeletons for assisting to human different locomotion functions are proposed, their static balancing methods and approaches, comparative numerical analysis, advantages and disadvantages are given [10–32]. Moreover, the proposed approaches and solutions are universal and can be used not only in the design of assistive devices, but also for linkage manipulators in general.

2 Design of Multifunctional Assistive Devices 2.1 Design of the Exoskeleton—Assistant for Human Sit-to-Stand The design conditions of the multifunctional assistive devices presuppose the possibility of their reconfiguration for different modes of operation, while it becomes necessary to balance the system in each of the configurations associated with individual modes of operation. The authors proposed a project of an exoskeleton-assistant for sit-to-stand for patients with musculoskeletal disorders without applying much effort on their part (Fig. 1). The reduction of the patient’s motor efforts is ensured by spring static balancing [11]. Wherein, the linear cylindrical tension and compression springs are used, which develop great forces at their small sizes and masses, which make exoskeletons portable and applicable for both physiotherapy and permanent use. The biomechanical system has the following structure: on each side of a person are attached the same mechanical three-link orthoses (Fig. 1a) connected with the back and the corresponding leg: with the thigh and shin, copying movements in the sagittal plane, herein the hip joint, which has 3-DOF, is duplicated by a rotational kinematic pair of the orthosis. Figures show the designations for elements of the orthosis attached to one side of a person, due to the symmetry, the other side can be mirrored. By adding gear 5 with a flexible link (Fig. 1a), for example, a chain link, the connection is provided between the angles of rotation of the links 2—thigh and 3—shin. To ensure reliable grip and stability during sit-to-stand, shin 3 and foot 4 must be fixed. With regard of this connection, the biomechanical system can be considered having 1-DOF during sit-to-stand, performing the movement in the sagittal plane.

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A

2

ϕ

(a) [11]

A

r1

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4

7 l0 lsp 6 r2

5

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r1

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3

3 4

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B 2

1

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1

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3 C

(b) [11]

4

(c) [11]

Fig. 1 Schemas of the sit-to-stand exoskeleton modes: a—with 2-DOF, b—with 1-DOF, c— balanced

The moment of the gravitational forces of the links Mgs0 relative to point B: the knee joint axis of the exoskeleton (see Fig. 1), is determined as: Mgs0 = Mg2 + Mg1 ,

(1)

where Mg1 is the moment of the gravitational force of link 1, equal to: Mg1 = 0.5 m 1 l2 g sin ϕ,

(2)

Mg2 is the moment of the gravitational force of link 2, equal to: Mg2 = 0.5 m 2 l2 g sin ϕ,

(3)

where m 1 is the mass of the torso, head and arms with link 1, m 2 is the mass of the thigh with link 2, l2 is the length of link 2, ϕ is the positioning angle of link 2, the generalized coordinate of the exoskeleton. After substituting (2) and (3) into Eq. (1), we obtain Mgs0 = 0.5(m 2 + m 1 )l2 g sin ϕ.

(4)

Static Balancing of the System. To balance the biomechanical system, a linear cylindrical tension spring 7 with an initial length lsp is attached to the exoskeleton link 1 at the distance l F using a cable with length l T and a cable winding roller 6 with radius r2 (Fig. 3c). The spring stiffness is calculated from the condition of static balance of the system [7–9]: the equality to zero of the sums of moments of gravitational forces and the balancing spring force relative to point B.

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The moment Mgs of the gravitational forces of the links relative to point B is determined in the form (see Fig. 1c) Mgs = Mg0 + Mg6,7 ,

(5)

where Mg6,7 is the moment of gravitational forces of links 6 and 7, equal to: Mg6,7 = (m 6 + m 7 )l2 g sin ϕ,

(6)

where m 6 is the mass of the roller 6, m 7 is the mass of the spring 7. After substituting Eqs. (4) and (6) into (5), we obtain: Mgs = (0.5(m 2 + m 1 ) + m 6 + m 7 )l2 g sin ϕ.

(7)

The moment of force Mb of the balancing spring 7 will be: Mb = FS r2 ,

(8)

where FS is the elastic force of the balancing spring, defined as: FS = F0 + k(l S − l0 ) = F0 + kl S ;

(9)

where k is the stiffness coefficient, l0 and l S are the initial and current values of the working length, l S is the elongation; F0 is the initial force of the spring. Elongation l S is defined as: l S = r2 ϕ,

(10)

and, for reasons of fatigue strength, its maximum value is limited by the condition: l Smax = r2 ϕmax ≤ 0.4l0 . If we use a non-zero-free length spring of form F0 = 0, or bring the spring to the characteristic F0 = m 7 g, taking into account the springs own weight, in view of (9) and (10), Eq. (8) can be converted to the form: Mb = kl S r2 = kr22 ϕ.

(11)

The unbalance of the system is estimated by the difference in moments from the gravitational forces of the links and the balancing force of the spring: Mus = Mgs − Mb .

(12)

After substituting (7) and (11) into Eq. (12), we get: Mus = (0.5(m 2 + m 1 ) + m 6 + m 7 )l2 g sin ϕ − kr22 ϕ.

(13)

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Equating Mus to zero, taking into account the dependence of the mass of a cylindrical spring on its stiffness [7, 8], the value of the stiffness coefficient of the balancing spring is determined from the nonlinear equation: kr22 ϕ − ρπ L W



D 3 nk/2Gl2 g sin ϕ − ((m 2 + m 1 )/2 + m 6 )l2 g sin ϕ = 0 (14)

where D is the average diameter, n is the number of turns, ρ is the density of the material; G is the shear modulus, L W is the spring wire length. Numerical Example 1.—Static Balancing During Sitting. The balancing of a human-exoskeleton biomechanical system was performed at the following values of its parameters [10]: patient weight 60 kg; m 1 = 39 kg; m 2 = 10 kg; l2 = 0.4 m; m 3 = 4 kg; m 6 = 0.55 kg; r1 = 0.3m; r2 = 0.08 m; ϕmax = 120◦ ; ρst = 7800 kg/m3 ; l F = 0.08 m; l0 = 0.42 m; lsp = 0.48 m; lTSmax = 0.168 m; 0 G = 81 · 109 N/m2 ; D = 0.05 m; p = Dtg25  = 0.019 m is the spring pitch, n = r ound[(l0 )/ p] = 23; l W = π lsp − l0 /2 + Dπ n/ cos γ = 2.96, where γ = ar ctg( p/(π D)) = 8.45◦ . Graphs Mgs and Mb at the various values of k are built (Fig. 2), the corresponding obtained (Table 1), the diameter of the spring wire d is determined values of F0 , m 7 are √ 4 by formula: d = 8D 3 nk/G, and meeting the condition 4 ≤ D/d ≤ 15. Fig. 2 Dependence of Mgs and Mb on the generalized coordinates of the exoskeleton at various values of k

Table 1 Parametrs of spring for different values of k

k2 N/m

7000

10000

13000

m 7 kg

0.789

0.692

0.579

F0 N

7.733

6.782

5.675

dm

0.0066

0.0062

0.0057

D/d

6.07

6.48

7.08

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Fig. 3. Schema of cam roller

−α +α

Fig. 4 Schema of balanced 1-DOF assistive device of a leg

At the selected value of the spring stiffness coefficient k, the unbalance of the system is estimated in a certain range of variation of the exoskeleton’s generalized coordinate ϕ. Exact Static Balancing. If we design the profile of the winding roller (Fig. 3) so that the following condition is realized: l S r2S = r02 sin ϕ,

(15)

where r0 is the initial value and r2S is the current value of winding roller’s radius, then the exact static balancing can be achieved and the value of k will be determined from the following quadratic equation: kr02 − ρπ L W



D 3 nk/2Gl2 g − (0.5(m 2 + m 1 ) + m 6 )l2 g = 0.

(16)

For example, the function of the form r2S = r0 cos 0.5 ϕ2 satisfies the given condition, and we can write:

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l S = ∫ r2S d ϕ = ∫ r0 cos0.5 ϕd ϕ = 2r0 sin0.5 ϕ,

(17)

l S r2S = 2r02 sin0.5 ϕ cos0.5 ϕ = r02 sin ϕ.

(18)

With changing the angle ϕ ∈ [0°; 120°] the radius r0 changes from 0 to 0.5 (see Fig. 3). Hence, if the springis attached  at a distance l F = 0.75 r0 from the exoskeleton’s link 1 and on thigh lsp + l T (see Fig. 3c), then the angle α, formed by the deflection of the spring and  cable  from the vertical, will change in the interval   α ∈ ±ar ctg r0 /4 lsp + l T ; 0 . Numerical Example 2. Complete Static Balancing During Sitting. The balancing of the biomechanical system is performed using a cam roller with r0 = 0.08 m and the initial data of example 3.1. We get: k = 15777 Nm, m 7 = 0.89 kg, F0 = 8.66 N , α ∈ [0.885°; 0], αmax = 0.885°. Ignoring the change in angle α, the direction of the spring can be assumed to be vertical. Sit-to-Stand Ensuring Without Significant Effort. By allowing the human torso to have a little freedom of movement relative to the designed statically balancing assistive device (or by simple movements of the hands or head), it is possible to change the center of gravity of the biomechanical system in the sagittal plane. It allows one to break the static balance and to provide a person sitting or standing without much effort from his side. Design of a Multifunctional Exoskeleton for Human Walking and Sitting. In this part, a design of a multifunctional exoskeleton-assistant for walking and sitting for patients with musculoskeletal disorders is proposed. The device due to spring static balancing does not require a significant effort from patients during its application [12, 13]. The concept (Fig. 5) has been drawn up by combining the human lower limb assisting 1-DOF device (Fig. 4) and the human sit-to-stand device (see Fig. 1c). lF

1

Fig. 5 Schemas of balanced exoskeleton for walking and sitting: a—walking mode, b—sit-to-stand mode

D

lsp

D

6

ϕ r1

1

l0 8

5

ψ O

2

6

ϕ A

C

ϕ

3 4

(a) [12]

r2 r1 7

5

ψ O

2

A

C

ϕ

3

B

4

(b) [12,13]

B

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Static Balancing of the Biomechanical System During Walking. To balance the biomechanical system during walking, a linear cylindrical spring 6 with the initial length lsp1 is attached to link 2 of the exoskeleton at the distance l OC and to link 1 of the exoskeleton at a distance l O D from the hip joint axis (see Fig. 5). The spring stiffness is calculated from the condition of static balance of the system [6–8]: the equality to zero of the sums of moments of gravitational forces and the moment of force of balancing spring relative to the hip joint rotation axis (point O). The moment Mgw of the gravitational forces of the links relative to the point O and the moment Mbw1 of the force of balancing spring 6 are determined in the same way as in the previous chapters. Let us write only the final expression for the stiffness coefficient k1 of the balancing spring:  k1l OC l O D − ρ S1 π L W 1 D13 n 1 k1 /8G 1 l OC g − (m 2 /2 + m 3 + m 4 )l2 g,

(19)

where m 2 is the mass of the thigh with link 2, l2 is the length of link 2, m 3 is the mass of the shin with link 3, m 4 is the mass of the foot with link 4, D1 is the average diameter, n 1 is the number of turns, ρ S1 is the density of the material, G 1 is the shear modulus, L W 1 is the length of the spring wire. Numerical Example 3. Exact Static Balancing During Walking. The balancing of the human-exoskeleton biomechanical system during walking was performed with the following parameters [10]: patient weight 60 kg, m 2 = 10 kg; m 3 = 4 kg; m 4 = 1.3 kg; l2 = 0.4 m; l OC = 0.3 m; l O D = 0.15 m; l01 = 0.26; r1 = 0.03 m; D1 = 0.03 m; p1 = D1 tg150 = 0.08 m—pitch of the spring turns, n 1 = r ound[(l01 )/ p1 ] = 33; l W 1 = π D1 (2 + n 1 /cosγ1 ) = 3.3 m,γ1 = ar ctg( p1 /π D1 ) = 6.62◦ , ρs1 = 7800 kg/m3 ; G 1 = 81 · 109 N/m2 ; The values of k1 = 903 N/m, F01 = 234.6 N , m 6 = 0.18 kg weredetermined; then the spring wire diameter d1 was calculated by the formula: d1 =

4

8D13 n 1 k1 /G 1 = 0.003 m.

Ensuring the Human Stand Up and Full Stride Movements Without Significant Effort. As it was mentioned above, by allowing the human torso to have a simple movement by hands and/or head, it is possible to change the center of gravity of the whole biomechanical system in the sagittal plane, to break its static balance and provide a person’s sitting or standing process without much effort from his side. And if we integrate an artificial self-adjusting knee joint into the designed device, the system will provide extension of the knee joint and ejection forward of the shin during walking (Fig. 6). This will change the static balance and the links will rush downward, under the influence of gravitational forces, thereby providing a full step process. It is important to adjust the on time locking and unlocking functions of the artificial joint. Again, the static balancing of the human-exoskeleton system with any degree of accuracy can be performed using a lever system, springs attached to levers and cams by means of flexible binding, as shown in the example of a simplified 1-DOF system (see Fig. 5). The exoskeleton-assistant allows that a person with limited locomotor abilities could easily perform sitting and standing movements in everyday life.

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1

Fig. 6 Ensuring of the full step

D 6

ϕ r1

5

ψ O

2

A

ϕ

C 3

B

4

2.2 Design Concepts of Quasi-Static Balanced Multipurpose Exoskeletons The authors propose a unified design concept for a range of portable quasi-statically balanced multipurpose exoskeletons to expand the functionality of the earlier proposed devices. Illustrative examples of their implementation are given. Some original schemes of multipurpose statically balanced devices have been designed. They include a combined exoskeleton-chair (Fig. 7) and an exoskeletonwheelchair (Fig. 8) [14]. On their basis, the authors patented an active exoskeleton that transforms into a wheelchair (Fig. 9) [15]. lF

Fig. 7 Project of an exoskeleton-chair with drop-down legs

1

5

B

7 l0 lsp 6 r2 r1 A

2

8

3 C

4

D

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Fig. 8 Project of an exoskeleton-wheelchair with autonomous wheels

1

lF

2 B

7 l0 lsp r1 6 r2

5

9 A

3 4

C

8

10

Fig. 9 Active exoskeleton wheelchair with drives

Multipurpose Exoskeleton-Chair. By adding in the above presented scheme on Fig. 1c setting-movable link-leg 8 (see Fig. 7), the exoskeleton can perform an additional function and serves as a chair. In terms of human safety, the suggested scheme is reliable in cases of accidental loss of balance during sitting and standing up. To balance this model of the exoskeleton, the moment of the gravitational forces of additional links should be taken into account, i.e., add the mass m 8 to Eqs. (5) and (7) and recalculate the value of the balancing spring stiffness. Numerical Example 4. Static Balancing of the Exoskeleton-Chair. Quasi-static balancing of the human-exoskeleton-chair system is performed for m 8 = 0.3 kg and the same values of its other parameters as in Numerical example 1, resulting in k1 = 16035 N/m, m 7 = 108 kg.

Design of Multifunctional Assistive Devices …

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Multipurpose Exoskeleton Wheelchair. By adding in the above-presented scheme on Fig. 1c an autonomous supporting driving wheel 9 and small wheels 10 on each side of the exoskeleton we enable the device to perform also the function of a wheelchair (see Fig. 8). In this exoskeleton-wheelchair, the moments of gravitational forces of the additional links remain unchanged during sitting and standing up. Again, as in the previous case, their masses should be taken into account in Eqs. (5) and (7) to achieve the balance of the system. In terms of human safety, the suggested scheme is also reliable in cases of accidental loss of balance during sitting and standing up. After standing up, the autonomous driving wheels 9 can also serve as manual actuators during walking, similar to the mode of a mechanical wheelchair when it moves. Numerical Example 5. Static Balancing of the Exoskeleton-Wheelchair. Quasistatic balancing of the human-exoskeleton-chair system is performed for m 8 = 0.8 kg and the same values of its other parameters as in Numerical examples 1 and 2, resulting in k1 = 16348 N/m, m 7 = 109 kg. The Active Multipurpose Exoskeleton-Wheelchair. The authors received a patent for the active multipurpose exoskeleton-wheelchair shown in Fig. 9. By combining the above multifunctional exoskeletons a new exoskeleton is designed (see Figs. 7 and 8) and hip and knee drives are installed on each side of the exoskeleton to ensure walking and two rear-wheel-motors to provide movement in the wheelchair mode. In such a design, it is more expedient to use for the implementation of static balancing spiral tape springs installed together with drives in the area of the hip and knee nodes [16, 17].

2.3 Design of Portable Assistive Multifunctional Devices Design of Portable Assisting Multifunctional Devices with Improved Functional Characteristics. Schemes of exoskeletons and orthoses, as well as their different constructive arrangements proposed by the authors earlier [6–8] are realized with the use of metal springs and/or counterweights for exact balancing of the biomechanical system. They emphasized the problems of mathematical modeling and considered theoretical aspects of balancing in general. In the schemes for system balancing during walking and/or sitting [11, 14], two different springs were alternately used to obtain the needed multidirectional forces of the springs during these two actions. So, the ensuing developments of the authors were aimed to find new constructive solutions and to create new conceptual schemes of passive, but multifunctional and portative exoskeleton-assistants for sitting, standing up and walking, focused on patients with musculoskeletal disorders, which do not require significant physical efforts during operation (Figs. 10, 12, 14, 15, 16, and 17) [18–20]. One of the main approaches is to ensure the balance of the device by using the smallest number of balancing elements. To provide the necessary balancing forces,

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Fig. 10 Schema of a portable orthosis with an adjusting roller: a—during walking, b—during sitting 3

4

2

1

B

φ

A

2

6

5 О r1

3

4

1

B A

φ

6

5 О r1

C

C lsp

lsp

l1

D

l1

D

(a)

(b)

it is foreseen to use in the developed schemes universal elements able to operate in all configurations: springs, rubbers, actuators with high efficiency and minimum dimensions and masses. A decrease in the number of used elements is achieved both by using switches and regulators of the balancing force’s direction and dosage, and by using retainers and special rollers [18, 19], cams [19] or semi-rollers [20–23], which serve for the exact balancing of the system (see Figs. 10, 12, 14, 15, 16, and 17). Portable Walking and Sitting Orthosis with a Roller—Partial Balancing. Twolink adjustable orthosis (see Fig. 10) with telescopic links 1 and 2, is inserted into the provided pockets of special orthopedic shorts. The balancing elastic element 3 is installed along to link 1, and its free end is retracted from the roller 5 by means of cable 4 [18]. Roller 5 has positional holes for fixing with link 2 by clamp 6. By the roller 5 rotation, clockwise or counterclockwise, the function of elastic element 3 is changed, and the system works as an assistant for human walking or sitting. Positioning holes on the roller 5 are made individually, depending on the diameter of the roller itself, the patient’s weight, walking and sitting angles, the type and allowable stroke of the elastic element, and the dosage of rehabilitation loads. It should be noted that this scheme does not consider additional mechanisms for providing convenient control and positioning of the balancing system. These solutions can be mechanical: superposition rollers, reduction gears, and mechatronic: controllers, sensors, small-sized electric motors, actuators. They will be presented in the next schemes. Below, the calculation of the parameters of elastic element 3 is made from the condition of the system balance. The gravitational moment during walking is defined as: Mgw = (0.5 m 1 + m C D )l1 g sin ϕ,

(20)

where m 1 is the mass of the thigh segment with the orthosis, the mass of element 3 is neglected, m C D is the mass of the leg and foot segments, l1 is the length of link 1, g is the gravitational constant, ϕ is the positioning angle of link 1. The gravitational moment during sitting is defined as:

Design of Multifunctional Assistive Devices …

Mgs = (0.5 m H − m C D − 0.5 m 1 )l1 g sin ϕ,

205

(21)

where m H is the mass of a human’s body. Based on the well-known empirical formulas of biomechanics, we can assume that Mgs = 3Mgw [10]. The balancing moment during walking is defined as: Mbw = Fsw r1 = (F0w + klsw )r1 = (ϕ0w + ϕ)kπr12 /180,

(22)

where r1 is the radius of the roller 5, k is the stiffness coefficient, lsw is the working elongation, Fsw is the developed force of the elastic element 3 during walking, and F0w is the force of its pre-tension, provided by winding the cable 4 on the roller 5 at an angle ϕ0w :F0w = ϕ0w kπr1 /180. The moment from F0w will be equal to M0w = ϕ0w kπr12 /180.

(23)

As it can be seen, this moment is constant and depends only on the value of ϕ0w . After the appropriate substitutions, we get the unbalance of the system during walking: Muw = Mgw − Mbw = (0.5m 1 + m C D )l1 g sin ϕ − (ϕ0w + ϕ)kπr12 /180 → 0. (24) Having previously set the initial positioning angle ϕ pw of link 1 by varying the an appropriate elastic element for a given combinations of k and ϕ0w   ,we can select variation interval of ϕ ∈ ϕ pw ; 180◦ . As already mentioned, for humans sitting from a standing position, the balancing system should be released by using the fixator 6. Roller 5 will rotate counterclockwise in half a turn, but the pre-tension F0w of element 3 may still retain, if cable 4 is wound on roller 5 at an angle ϕ0w . Further, by the human efforts, roller 5 again should be turned counterclockwise to the required angle and be fixed with link 2 using the retainer 6 (see Fig. 10b). In this case, the system works as a sitting assistant. The balancing moment during sitting is defined as: Mbs = Fss r1 = (F0s + klss )r1 = (ϕ0s + ϕ0w + ϕ)kπr12 /180.

(25)

where lss is the working elongation, Fss is the developed force of the elastic element 3 during sitting, and F0s is the force of its pre-tension, provided by the preliminary winding of the cable 4 on the roller 5 at an angle ϕ0s : F0s = (ϕ0s + ϕ0w )kπr1 /180. At that, the moment from F0s is equal to M0s = (ϕ0s + ϕ0w )kπr12 /180. Here, the moment is also constant and depends only on ϕ0w and ϕ0s . After substitutions, we get the unbalancne of the system during sitting:

(26)

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Mus = Mgw − Mbw = (0, 5m 1 + m C D )l1 g sin ϕ − (ϕ0w + ϕ)kπr12 /180 → 0. (27) Numerical Example 6—Portable Walking and Sitting Orthosis—Partial Balancing. As an illustration, the balancing during walking of a human-orthosis biomechanical system (see Fig. 10) was performed with the following values of its parameters [10]: patient weight 60 kg, m 1 = 8.5 kg, m C D = 3.5 kg, l1 = 0.4 m, ϕ pw = 60°, r1 = 0.05 m. Thus, we get: Mgw max = 30.7 Nm, and k max = 78222 N/m at ϕ0w = 0°. Graphs Mgw (ϕ), Mbw (ϕ) and Mgs (ϕ),Mbs (ϕ) are built (Fig. 11). It can be imagined countless combinations of k and ϕ0w , ϕ0s , which may partially satisfy the problem requirements, however, in all these solutions and in all given   ranges ϕ ∈ ϕ pw ; 180◦ there is no exact continuous balancing solution during walking, as during sitting. However, despite the simplicity of the design, this scheme can be successfully applied in all cases when the exact balancing during walking and sitting is not principal.

Fig. 11 Balancing variant: a—during walking and b—sitting with different parameters of k, ϕ0w and k, ϕ0w , ϕ0s , respectively

Fig. 12 Schema of a portable orthosis with an adjusting cam: a—during walking, b—during sitting (please pay attention to the arrangement of the cable joining the spring with the cam 5’)

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The results also showed that after the roller 5 was moved to the sitting position (see Fig. 11b) at the beginning of the sitting, the balancing moment exceeds the gravitational, so, it is maybe undesirable for sitting, but this effect can be increased by increasing the value of ϕ0s and be applied to fix the patient in a standing position. In this case, we will have a new function for this portable orthosis—assistant for human walking, sitting- standing and holding in a standing position. Portable Walking and Sitting Orthosis with Cam: Complete Balancing. To achieve the goal, the roller 5 can be replaced with a cam 5’ with a given profile r S = r0 cos0.5ϕ (Fig. 12), where r0 is the initial value (for simplicity we accept that r0 = r1 ); r S is the current value of the radius of the cam [18, 19]. The balancing moment during walking is defined as: Mbw = Fsw rs = F0w rs + klsw rs = (ϕ0w πr1 /180 + lsw )krs .

(28)

Here, the function of F0w is equivalent to the previous one, and the moment of force that it creates will be equal to. M0w = ϕ0w π kr1rs /180 = ϕ0w π kr1 .cos0.5 ϕ/180.

(29)

When F0w = 0, i.e., ϕ0w = 0, the balancing moment during walking is written as: Mbw = Fsw rs = klsw rs ,

(30)

lsw = ∫ rs d ϕ = ∫ r0 cos0.5 ϕ d ϕ = 2r0 sin0.5 ϕ,

(31)

lsw rs = 2r02 sin0.5ϕ cos0.5 ϕ = r02 sin ϕ

(32)

where lsw is defined as:

hence:

and we can get the exact static balancing and determine the value of k from the equation of the system balance: Muw = (0.5m 1 + m C D )l1 g sin ϕ − kr02 sin ϕ = 0,

(33)

Consequently: k = (0,5m 1 + m C D )l1 g/r02 . The balancing moment during sitting is determined as in the previous case.

(34)

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Numerical Example 7. Portable Walking and Sitting Orthosis: Complete Balancing. The calculation of the stiffness coefficient of element 3 is done at the same values as in the previous numerical example. We get a value of k = 12300 N/m. Graphs Mgw (ϕ), Mbw (ϕ) and Mgs (ϕ), Mbs (ϕ) are built (Fig 13). Here exact balancing during walking takes place in the entire given range ϕ ∈ ϕ pw ; 180◦ , but the required stiffness coefficient is maximum. Providing a pretension ϕ0w = 0, it is possible to reduce its value, but there will be no exact balancing. During sitting, we can imagine many combinations with the value of k and the variable angle of rotation ϕ0s of the roller 5’. Here, during sitting, we have partial balancing in all given range ϕ ∈ [0; 120◦ ], but certainly the system can always serve as an assistant for sitting. More, by changing the value of ϕ0s before sitting down we can always hold the patient in a standing position, within the appropriate range. length of cable 4, it should be equal to lt =   As for the minimum ϕ0s + 180 − ϕ pw πr1 /180. With the already specified values ofϕ0w , ϕ pw and,lt = 7πr1 = 0.19 m, the maximum length of the elastic element 3 is:le = l1 − lt − r1 = 6 0.16 m, i.e., its relative elongation is (le + lt )/le = 2.19. In case, if the value of le is too small to develop the required effort, a constructive solution, as an arrangement of the balancing system, can be proposed. For example,

Fig. 13 Exact balancing during walking using a cam: with different parameters of: a—k and ϕ0w , and b—k, ϕ0w and ϕ0s , respectively

Fig. 14 Schema of a portable orthosis with an adjusting semi-roller: a—during walking, b—during sitting (please pay attention to the arrangement of the cable joining the spring with the cam-roller 5)

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Fig. 16 A new scheme of a 2-DOF assistive device with a semi-roller regulating mechanism and additional parallel link: a—during walking, b—during sitting. Here 1, 2 and 3 are telescopic links, 4 is an elastic element, 5 is a cable, 6 is an adjusting mechanism, 7 is a retainer, 8 is a flexible element, 9 is a parallel link, 10 is a retainer (please pay attention to the arrangement of the cable joining the spring with the cam-roller 5)

connect the elastic element not from the knee, but the hip side of link 1, and pull the cable in the opposite direction to the knee end then return through the small additional roller to the roller 5, thereby reducing the relative elongation of the elastic element from 2.19 to 1.5 times. Exact Balancing of the Portable Walking and Sitting Orthosis with a SemiRoller. To achieve the goal, the cam-roller 5 (see Fig. 12) can be replaced with a semi-roller 5 (Fig. 14), and the desired distance between OA and OB can be structurally performed [20, 28]. Here, during walking, we have a classic case of balancing, which has been presented many times along with calculations and results. The semi-roller 5 has positional holes for fixing with link 2 through retainer 6. If we assume that the distances OA and OB are equal and equal to the radius r1 of the semi-roller 5, then even though we have the maximum value for spring stiffness

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Fig. 17 Multifunctional assistive device

k again, this variant of construction will become more ergonomic compared to the previous ones [see 18, 19]. The balancing moment during walking can be written as: Mbw = Fsw rs = F0w r1 cos0.5 ϕ + kr12 sin ϕ = π ϕ0w kr12 cos0.5 ϕ/180 + kr12 sin ϕ.

(35)

Here, F0w is equivalent to the considered cases, and its moment will be equal to: M0w = ϕ0w π kr1rs /180 = ϕ0w π kr1r0 cos0.5 ϕ/180.

(36)

IfF0w = 0. i.e., ϕ0w = 0, the balancing moment during walking can be write as Mbw = kmax r12 sin ϕ.

(37)

We can get the exact static balance and determine the maximum value of k from the balance equation of the system, it will be the same as in the previous case: Muw = (0.5m 1 + m C D )l1 g sin ϕ − kmax r12 sin ϕ = 0.

(38)

kmax = (0.5m 1 + m C D )l1 g/r12 .

(39)

Hence:

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Numerical Example 8. Portable Walking and Sitting Orthosis with a SemiRoller—Exact Balancing. The initial parameters and calculation results for the walking and sitting cases are similar to the results of Numerical Example 7. Once again, it is important to say that in addition to walking and sitting assisting, the proposed devices can hold the human body in a standing position, like the previous devices. But certain disadvantages of this device have also been identified. For example, despite the simplicity of the design and the ability to provide exact balancing during walking, this device can ensure only partial balancing during sitting, like the previous devices. However, this device with a regulating semi-roller can be considered as the most optimal among the recent solutions (see Fig. 14). Based on the scheme of this balancing assistive device and experience in rehabilitation devices designing [7, 8], new schemes of multipurpose assisting two-movable devices (Figs. 15 and 16) for walking, sitting and holding a human body in a standing position with improved functional characteristics have been proposed [21, 22, 28]. A Two-Movable Assistant Device with Flexible Elements. Schema of a multipurpose assistant device with 2-DOF and additional flexible elements, designed for walking, sitting and fixing a human in a standing position is suggested (see Fig. 15) (see [21, 28]). Here, an adjustable orthosis is fixed on a human lower back and hip with the help of belts. The elastic element 3 is installed along to link 2 and through the cable retracted to the semi-roller 5, which has positional holes for fixing and de-fixing to the link 1 by means of the retainer 6. Positioning holes on the roller 5 are individual, depending on the semi-roller’s diameter, patient’s weight, walking and sitting angles, an allowable stroke of the elastic element, and the dosage of rehabilitation loads. By turning the roller 5 clockwise or counterclockwise the purpose of the elastic element 3 is changed, and the system works as an assistant for human walking or sitting. The introduction into the system of a group of flexible plates 7, which are attached at one end to link 2, and the other free end to the human shin, provides a static balancing of the shin and foot in the sagittal plane during walking, helps to reduce the unbalance during sitting. In contrast to the balancing moment created by the elastic element 3 in the hip joint area, the balancing moment created by the flexible plates 7 in the knee joint area does not need to be redirected when walking or sitting, since the work of these plates is directed accordingly for both cases. With specific design data, it is easy to calculate the parameters of the device and provide a numerical example, but similar calculations were given in previous works [18–20]. We just add that the device has several advantages: portability, adjustability, small size, high efficiency, versatility, ergonomics and low cost. In addition to knee flexible elements, an additional parallel telescopic link can also be introduced into the structure to ensure convenient control and positioning of the balancing system. This new design is illustrated in Fig. 16 [22, 28]. In this design, when the length of the additional link 9 is fixed by means of the latch 10, it remains parallel to link 2, thereby reducing DOF of the structure and providing a comfortable and smooth positioning of the human body during sit-to-stand. During

212 Fig. 18 Assistive device for walking and sitting with an adjusting slider regulator: a—during walking; b—during sitting. 1, 2, 3—device’s links, 4—balancing spring, 5—metallic cable, 6—regulating slider, 7—additional telescopic link used when sitting

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walking, there is no need for link 9, and it can be released. The design approach to the multipurpose balancing devices for assisting the musculoskeletal system of a human is quite universal and can be used in the design of not only assistive but also other portable rehabilitation devices of a similar purpose. Based on the latter scheme (see Fig. 16), the authors have proposed and patented a multipurpose passive exoskeleton (Fig. 17) [23]. An Assistive Device with Slide Controller. Further search for an optimal solution led to a scheme of a portable passive balancing exoskeleton for human walking and sitting, in which, using an adjustable slider mechanism, the device can be easily switched to each of its modes of operation (Fig. 18) [24, 28]. To transfer the device to sitting mode: a person, being in a standing position, can easily move the slider to the lowest point and make the system change the direction of balancing, thus partially balance the body during sitting. The distance of the slider positioning point regulates the desired limits of the system partial unbalance. Here, the maximum spring tension in walking mode is the initial tension for sitting. It is this effect that is used to approximate balancing during sitting. Exact static balance during sitting can also be ensured, however, this requires an increase of the spring force in the arm, which is unwanted when designing small-sized assistive devices. Let us now consider a constructive schema of the assistive device for human walking and sitting. It is a three-link linkage mechanism connected by rotational kinematic pairs, having balancing cylindrical tension springs and worn on each side of the human body. The lever system is similar to those presented earlier [22, 23], but here, to convert the balancing spring’s action and forces directions, the system is equipped with an adjusting slide mechanism 6. In each mode of operation, the system can be considered as a 1-DOF swinging link with reduced mass at its free end, i.e., where the thigh link 2 swings relative to the hip joint O at a certain angle ϕ ∈ [0◦ ; 180◦ ]. The shin together with the foot are represented by a concentrated mass, or the thigh link swings relative to the knee joint D, and the mass of one arm and half of the mass of the torso and head are reduced to link 1. As in the previous case, to ensure a smooth sit-to-stand mode, an

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additional telescopic link 7 was introduced, which connects links 1 and 3 when fixed and converts the two-movable system to a one-movable one. To balance the system, we use a linear non-zero-free length cylindrical tension spring, the stiffness coefficient of which is selected from the condition of exact balancing of the leg segments in the walking mode. The spring is fixed along to thigh link 1 from the knee side, and to its free end a cable 5 is attached, which, passing through point A, is retracted to the regulating slider 6 (point B) and fixed behind it (see Fig. 18). Here, the value of the stiffness coefficient is regulated by the ratio of the AOB triangle’s sides OA and OB, and if they are not equal, then the cable must be taken out of slider 6 and fixed at the distance equal to their difference (OA-OB). In this case, the power characteristics of the balancing spring become close to the characteristics of a zero-free length spring [1–9]. With this structure, the leg is indifferently balanced over the entire range of angle change. The maximum force developed by the spring in the walking mode is achieved in the standing position of a person. To transfer the device to the siting mode, exactly and only in the human standing position, the slider moves freely to point C, then forcing the system to change the direction of balancing and partially balance the human body during sitting. Design of the Assistive Device with a Slide Controller. The constructive design of the proposed device has been developed [25, 28]. Compared to the previous ones, this scheme does not require a high stiffness spring to balance the system, but it does require a fairly large stretch range. In the future, constructive solutions with the spring arrangement will be presented to eliminate this flaw. Reinforcing Device with a Slide Controller. As it was mentioned, the exoskeletons suggested by the authors are passive, and their activation requires the application of drives, actuators, sensors and controllers. Active assistive devices are able not only to rehabilitate but also to reinforce the human locomotors’ functions. But such devices will certainly depend on the power supply. When designing assistive devices for industrial purposes, the issue of power supply for drives and actuators is less critical than when designing assistive devices for military purposes. For this reason, the authors proposed a new design of a passive device for military purposes using slider regulators for springs balancing forces direction and roller regulators for springs preliminary tension arrangement (Fig. 19) [26, 28]. The device can provide human balancing and reinforcing during walking, running and sitting, hold a person in the required position and partially reduce the loads acting on the musculoskeletal system. The balancing and reinforcing of the system are carried out in the sagittal plane by using a group of cylindrical compression springs. Unlike the previous schemes, this design requires small spring strokes and small dimensions of the regulating mechanisms, allows the variations of balancing forces. Design and Modeling of an Assistive Device. An analysis of the above-described developed devices revealed their advantages and disadvantages: they have a simple design and can provide an exact static balancing of the human leg during walking,

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Fig. 19 Example of an exoskeleton. a—in walking, running mode; b—in the sitting-standing mode. 1, 2, 3, 4—adjustable links of the device, 5—additional telescopic link, 6—retainer of additional link, 7—balancing spring, 8—metallic cable, 9—operating mode slider regulator, 10—spring pre tensioning roller, 11—retainer of roller, 12—elastic elements

but they do not always provide an exact balance of the human body during sit-tostand, however, this allows holding a person in a standing position. To reach the exact balancing during sit-to-stand in the designed devices with spring directivity regulator, it becomes necessary to lengthen the balancing springs, which is undesirable for the springs operating in tension. Therefore, when designing, it is very important to correctly model the device with the calculation of the individual conditions of its use, justification of the type of the regulator-roller, semi-roller, cam, slide or combined; type of the spring-tape, plate, flat, cylindrical in tension or compression; zones of installation and operation of the selected type of spring. In this section, it is suggested a new scheme of a passive portable exoskeleton oriented on its use for human balanced walking and sitting (Fig. 20) [27, 28]. A spring with a reduced stiffness coefficient and dimensions is used here. Ease of switching the device to the desired operation mode is provided. However, such a scheme, in Fig. 20 Assistive devices for walking and sitting with an adjusting slider regulator and: a—in walking mode, b—in sitting mode: 1–3—links, 4—balancing spring, 5—metallic cable, 6—slider regulator, 7—additional telescopic link, 8—retainer

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Fig. 21 Equivalent schema of assistive devices for walking and sitting with an adjusting slider: a—in walking mode, b—in sitting mode: 1–3—links, 3—reduced mass, 4—spring, 5—cable, 6— directional adjusting slider, 7—additional telescopic link used when sitting, 8—retainer

contrast to the previous ones, requires a significant lengthening of the spring, which creates some issues related to its installation, dimensions and links, permissible swing angles. For this reason, in the new scheme, the working zone of the spring has been changed and the mathematical modeling of the device has been performed to obtain technical characteristics that ensure the portability of the device. Let us consider a constructive schema of an assistive device for walking and sitting. This is a three-link flat lever mechanism connected by rotational kinematic pairs (links 1–3), having balancing cylindrical tension spring 4 and worn on each side of the human body. Due to its symmetry, the device is shown only from one side (see Fig. 21). For the possibility of converting the action direction of the springs, cables 5 drive them to the regulating slide mechanisms 6. In each mode of operation, the system can be simplified and represented as a 1-DOF swinging link with a reduced mass at its free end (Fig. 21), i.e., where thigh  link 2 swings relative to the hip joint O at an angle ϕ ∈ 00 ; 1800 . The shin with the foot are considered as a concentrated mass 3 , or the thigh link swings relative to the knee joint D, and the mass of one arm and half of the body and head weights are reduced to the mass of link 1 . Let us proceed to the presentation of the mathematical model of the exoskeleton. The moment from gravitational forces in the walking mode is determined in the form Mgw = ((0.5m 2 + m 3 )l2 + m 4 S4 )g sin ϕ,

(40)

where m 2 is the mass of link 2, m 3 is the reduced mass of the shin, foot and link 3, m 4 is the mass of the spring 4, l2 is the length of the link 2; S4 is the distance of the mass center of the spring 4 from the point O, g is the gravitational constant. The gravitational moment in sitting mode is defined as Mgs =

 0.5m 2 + m 1 l2 + m 4 S4 g sin ϕ,

(41)

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where S4 is the distance of the mass center of the spring 4 from point D, m 1 is the reduced mass of link 1, arms and half the mass of the body and head. As already mentioned, based on the experience of designing assistive devices [6– 9, 11–26], as well as the well-known empirical relationships for the biomechanics of the human body [10], it can be argued that the gravitational moment from the leg segments are related as Mgs ≈ 3Mgw . The balancing moments and forces, respectively, and the stiffness of the balancing springs must correspond to this ratio. Therefore, exact balancing with a cylindrical spring during sitting is fundamentally possible, but it is not desirable to have a spring three times cumbersome than during walking. To balance the system, a linear cylindrical tension non-zero-free length spring is used, its stiffness coefficient is selected from the condition of exact balancing of the leg segments in walking mode (see Figs. 18, 20 and 21). The spring is fixed along to thigh link from the knee side and retracted by cable through point A to the regulating slider (point B) and fixed behind it, as shown in Fig. 21. Again, the value of the stiffness coefficient is regulated by the ratio of the AOB triangle’s sides OA and OB, and if they are not equal, then the cable must be taken out of slider 6 and fixed at the distance equal to their difference (OA-OB). In this case, the power characteristics of the balancing spring become close to the characteristics of a zero-free length spring [1–9]. With this structure, the leg is indifferently balanced over the entire range of angle change ϕ ∈ [0◦ ; 180◦ ]. The maximum force developed by the spring in the walking mode is achieved in the standing position of a person. To transfer the device to the mode of sitting, exactly and only in the human standing position the slider moves freely to point C, then forcing the system to change the direction of balancing and partially balance the human body during sitting. It should be noted, that the maximum spring tension in the walking mode serves as the initial tension for the sitting mode. This effect is used to approximate balancing the gravitational moments acting during sitting. It is possible to achieve exact static balance during sitting, but it will be necessary to increase the OB arm by three, which is undesirable when designing a small-sized assistive device (Fig. 22). The force developed by the balancing spring 4 (see Fig. 22) during walking is equal to Fsw = Fs0w + klsw = Fs0w + k(lsw − ls0w ) = klsw + (Fs0w − kls0w ),

(42)

where k is the stiffness coefficient, Fs0w is the pre-tensioning force of the spring during walking, lsw is the elongation, lsw is the working elongation, ls0w is the initial length of the spring 4. During walking in the upper extreme position of link 2, the initial distance between the points AB will  be determined by the difference l AB0 = z − x, the current distance—l AB = z 2 + x 2 − 2zx cos ϕ and the maximum—l ABmax (see Fig. 22a). Accordingly, if pre-tension the spring by the difference l S0W = z − x, then the initial force will be provided FS0W = k(z − x), the spring force will be determined as

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Fsw = klsw .

(43)

The balancing moment during walking is defined as Mbw = Fsw z cos β =

Fsw zx sin ϕ = kzx sin ϕ. lsw

(44)

As it can be seen, the gravitational Mgw and balancing Mbw moments will change according to identical laws, thus, the unbalance of the system during walking Muw = 00 , so then k = ((0.5m 2 + m 3 )l2 + m 4 S4 )g sin ϕ/x z.

(45)

The maximum spring force is determined as Fsw max = Fs0w + 2kx = k(x + z).

(46)

By moving the slider from the upper position to the lower one (see Fig. 22b), we get a smooth descent, if z ≥ y. At the beginning of the sit, the maximum spring force will decrease by k(z − y) and the initial tension will be equal to Fs0s = Fsw max = k(x + z),

(47)

and the force developed by the spring 4 during sitting will be determined as Fss = Fs0s + k(lss − ls0s ) = klss + k(x + y).

(48)

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where lss is the working elongation, ls0s is the initial length of the spring 4 during sitting. From Fig. 23 we have lss =



z 2 + y 2 − 2zycosϕlss .

(49)

The balancing moment during sitting is defined as  Mbs = Fss zy sin ϕ/lss = kzy sin ϕ (1 + (x + y)/ z 2 + y 2 − 2 zy cos ϕ). (50) As it can be seen, the gravitational Mgs and balancing Mbs moments will only partially change according to identical laws, so then we write

Fig. 23 Balancing results during sitting: a—when x = y = z; b—when x = z and; y < z; c— and d—when x > z > y and reduced values of y

Design of Multifunctional Assistive Devices …

0.5m 2 + m 1 l2 + m 4 S4 g sin ϕ − kzy sin ϕ  −kzy(x + y)sin ϕ/ z 2 + y 2 − 2zy cos ϕ. Mus =

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(51)

Numerical Example 9. Modeling of an Assistive Device for Walking and Sitting. For the human-exoskeleton biomechanical system, the calculation of the acting moments and forces during walking and sitting (see Figs. 21 and 22) was performed with the following values of its parameters [10]. The patient’s height l H = 1.7 m, and his weight M H = 60 kg, l2 = 0.4 m, the length of the thigh (link 2), m 2 = 9 kg the mass of the thigh (with the link 2), m 2 = 3.5 kg reduced mass of the shin and foot (with link 3). As can be seen from formula (45), the required value of the spring stiffness coefficient depends on the values of z and x, therefore, we will consider three cases when, x = z x = zx = z, x < zx < z and x > z (provided that z ≥ y): a) x = z = 0.06 m and y = z, then, with this design, the maximum elongation of the spring is lmax . If we take a spring steel wire with a density ρ = 7800 kg/m3 , shear modulus G = 81 · 109 N/m2 , diameter d = 0.003 m and make a cylindrical spring with the minimum allowable ratio D/d = 4, the average diameter of the spring coil D = 0.012m with the number of turns n = 47, we get the following values for spring parameters: L W = 1.846 m, m 4 = 1.018 kg, l0 = 0.165 m, k = 10098 N/m. The chosen spring and design can provide exact static balancing of the leg during walking, but during sitting, the balancing moment exceeds the gravitational moment of the body (Fig. 23a). If we reduce the value of y, let us say to y = 0.05 m, then the balancing moment will partially balance the gravitational moment of the body (Fig. 23b). then the maximum elongation of the spring will be equal lmax . b) When x < z, either the required spring stiffness k or the maximum spring elongation lmax will increase. c) When x > z, we take the following values x = 0.1m, z = 0.06m and y = 0.05 m, then, with such a design, the maximum elongation of the spring will be equal to lmax . If we take a spring steel wire with the diameter d and make a cylindrical spring with the minimum allowable ratio D/d = 4.8, the average diameter of the spring coil D = 0.012 m with the number of turns n = 50, we get the following values for spring parameters: L W = 1.96 m, m 4 = 0.75 kg, l0 = 0.15 m, k = 5925.1 N/m. The chosen spring and design can provide exact static balancing of the leg during walking and partial balancing of the body during sitting (see Fig. 23c). With a decrease in the value of y, say to y = 0.04 m, the balancing moment will decrease, however,by continuing in this way, if necessary, it will be possible to minimize its excess over the gravitational moment of the body at the beginning of the sit (Fig. 23d). With this approach, the maximum spring elongation will also decrease and will be equal to lmax , which is more consistent with the design.

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To the Selection of the Working Zones of the Balancing Springs of the Assistive Device. However, if structural calculations do not allow obtaining a compact device due to the required large spring travel, then it is possible to arrange the springs and carry out the output of the springs to more convenient operating zones and implement subsequent remote (from outside) balancing using a metal cable [24], as shown in the assistive device new schema in Fig. 25 [29, 30]. Here, in the area of the human back, compression cylindrical springs or a group of springs 4 are installed, and for the possibility of converting the action direction of springs, the cables 5 are attached to their movable ends, which are brought through the regulating slider mechanisms 6 to point A on the thigh link 2. In walking mode, the system has one degree of mobility and performs a balanced swing at an angle ϕ ∈ [00 ; 1800 ] in the sagittal plane relative to the hip joint O. The shin with the foot can be considered as a concentrated mass suspended at the end of link 2. However, it is permissible to use other elastic elements to balance the shin and foot, as in the previous device schemes (see Figs.15, 16, 17, 18 and 19). In the sitting mode, the thigh link swings relative to the knee joint D, and half of the mass of the remaining body segments can be reduced to the mass of link 1 (see Fig. 24b). It should be noted that an additional telescopic link 7 was added to the system to impose a constraint on the relative movement of links 1 and 3 and to reduce the degree of mobility of the system, so that the system has one degree of mobility during human sitting [11–18, 22–28]. We will only add that, in this case, it is necessary to fix the length of the telescopic link 7 with a retainer 8, which was freely movable during walking. A low stiffness cylindrical tension–compression spring can be inserted into this link to improve the smoothness of movement, which can only aid the system when walking and sitting. In this case, taking into account the spring masses in the balancing equations is not necessary; it should be calculated only when determining the total mass of the device. The maximum force developed by the spring in the walking mode is achieved in the standing position of a person. To transfer to the sitting mode, the slider can freely move to point C (and back: to B) only in this particular position, and, forcing the Fig. 24 Exoskeleton for walking and sitting with a cylindrical compression spring and a slider regulator: a—in walking mode, b—in sitting mode: 1, 2 and 3—links of the exoskeleton, 4—spring, 5—cable, 6—slider, 7—additional telescopic link used when sitting, 8-retainer

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Fig. 25 Moments due to the gravitational forces during sitting

system to change the direction of its action, balance the human body during sitting. It should be noted that the maximum spring tension in the walking mode serves as the initial tension for the sitting mode. This effect is used to balance the gravitational moment acting when sitting down. Let us consider the acting moments and forces during walking and sitting. Here, in contrast with schemes of the above-described devices, it is not necessary to consider the balancing of the mass of spring during walking, so the gravitational moment acting on the biomechanical system can be written as (see Fig. 24a): Mgw = (0.5m 2 + m 3 )l2 g sin ϕ,

(52)

where m 2 is the mass of the thigh segment with the orthosis, m 3 is the summary mass of the shin and foot segments with the orthosis, l2 - the length of the link 2, g is the gravitational constant. The force developed by linear compression spring during walking is defined as: Fsw = klsw

(53)

where k is the stiffness coefficient, lsw is the working extension of the spring 4. The balancing moment of the spring during walking is defined as: Mbw = Fsw l O A cos βw = Fsw l O A l O B sin ϕ/lsw = kl O A l O B sin ϕ.

(54)

After the substitutions, we get the unbalanced moment of system during walking: Munbw = ((0.5m 2 + m 3 )l2 g − kl O A l O B )sin ϕ.

(55)

It can be seen that for the certain values of the components in this equation, the exact static balancing during walking can be achieved.

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Moment due to gravitational forces during sitting can be written as (see Fig. 24b):  Mgs = 0.5m 2 + m 1 + m 4 l2 g sin ϕ,

(56)



where m1 is the sum of a person’s head and torso half-mass and one arm mass. The force developed by liner compression spring during sitting is defined as: Fss = Fs0w + klss = k(l OC + l O B ) + klss ,

(57)

where lss is the further (during sitting) working extension of the spring 4,l OC is the distance from point O to point C. The balancing moment of the spring during sitting is defined as: M B S = Fss l O A cos βs = Fss l O A l OC sin ϕ/lss = (k(l OC + l O B )/lss + k)l O A l OC sin ϕ

(58)

After substitutions, we get the unbalanced moment of the system during sitting:  Munbs = 0.5m 2 + m 1 + m 4 l2 g sin ϕ − (k(l OC + l O B )/lss + k)l O A l OC sin ϕ. (59) It is obvious that during sitting the approximate static balancing can be only achieved, until using a spring (or group of springs) with a variable stiffness coefficient. Numerical Example 10. Determination of Acting Moments. As an illustration, the simulation of the acting moments during walking and sitting for the humanexoskeleton biomechanical system (see Fig. 25) is performed with the following values of its parameters [10]: the patient weight 60 kg, m 1 = 18 kg, m2 = 8.5 kg, m3 = 3.5 kg, l2 = 0.4m, l O A = 0.1 m, l OC = 0.06 m, l O B = 0.07 m. Through this, we get: lmax = lswmax + lssmax = 0.3 m, Mgwmax = 31 Nm and kmax = 4430 N/m, m4 = 0.5 kg. Graphs Mbs (ϕ), Mgs (ϕ), Munbs (ϕ) are shown in Fig. 25. From the obtained results, it can be confirmed that this scheme provides exact static balancing of the biomechanical system during walking and only approximate balancing during sitting. In addition, the unbalanced moment during sitting can be partially regulated by moving the arms forward and/or by changing the position of the human torso gravity center. The results also showed that at the beginning of the sit the counterbalancing moment exceeds the gravitational moment, which may not be desirable for sitting, but this effect can be applied for holding the patient in a standing position. In this case, this exoskeleton can become useful for humans walking, standing and sitting. Compared with the previous schemes, this solution does not require a high spring stiffness to balance the system. But, to ensure the required fairly large stretch range,

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the spring is taken out of the balancing zone and works separately with the help of a metal cable. The results also confirmed that such an exoskeleton can also be offered as a portable machine for training the muscles of the leg, back and abdominals.

2.4 Design of the Portable Assistive Reconfigurable Device Portable Passive Balanced Assistive Device for Industrial Purposes. The experience of designing the above-described assistive devices [6–9, 11–30], such as balanced movable orthoses and exoskeletons for the lower and upper extremities has led to the creation of a reconfigurable portable passive-balanced device for industrial purpose, designed to hold heavy and uncomfortable objects and tools in the desired position (Fig. 26) [31]. Here, telescopic links are used to adjust the geometry of the device, as well as coil springs and force adjusting mechanisms for convenient use of the device. This “local” scheme can be integrated with one of the previously designed schemes of assistive devices for walking and sitting [e.g. 26, 28] to create a new “global” exoskeleton. Let us consider the construction: a lumbar rigid link 1 with weight distribution pillows 1 is attached to the operator’s lower back by using a belt and/or a vestcorset (see Fig. 26). Then, on this link 1 on the right for right-handers, or the left for left-handers, through the h1 hinge, telescopic links are attached: 2–2—horizontally (lateral support) and 3–3’—vertically (elbow support), an elbow pad 4 is placed on link 3’. Such a support structure has the right to individual existence, and can find application not only in industry, but also in military affairs to support arms and weapons or ammunition, and also in medicine for supporting an injured or operated arm. If

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necessary, pad 4 can be easily removed from the elbow support and assistance can be stopped. By attaching pantograph 5 (can also be telescopic) with an adjustable telescopic link 6 to the lateral support through the h2 hinge, the system can be reconfigured. Link 6 can have various gripping-attachments designed for the required application. To balance the pantograph 5 by cylindrical springs, the previously repeatedly presented methods can be applied [6–9, 28]. The pantograph can swing in balance in the angle range ϕ ∈ [0◦ ; 180◦ ], and if necessary, its lowering can be limited using the lock 8. In the device, use of a linear non-zero initial length cylindrical tension spring is provided. The free end of this spring is retracted to a roller regulator by manse of a metallic cable, this connection allows to vary the force of balancing spring. Portable Passive Balanced Assistive Device for Military Purposes. In this section, authors focused on the design of reconfigurable, portable and passivebalanced local exoskeletons designed to hold a firearm. There are three main firearmshooting positions: standing, kneeling and prone. When shooting from standing and kneeling positions, the shooter should hold the firearm with both hands to ensure the retention and stability of the firearm. In the case of one-handed shooting, often right hand, the shooter must exert sufficient efforts to overcome the gravity of the firearm, as well as to compensate for its knockback. To ensure required gripping and firearm stability, it is necessary to use a ground-supported stand for fixed-position shooting, or so-called “third hand”—a local stabilizer for weapons with support on the shooter’s back for mobile shooting. In a local shooter-weapon-support system, it is necessary to provide the stability and balance of the weapon in the profile plane for the targeting fire. Let us consider the structural design of a reconfigurable passive-balancing device intended to hold a firearm when firing with one hand from standing, kneeling and prone positions (Fig. 27) [32]. In this device, the telescopic links are used to adjust its geometry and the cylindrical springs are applied for power balancing these links [6– 9]. The cylindrical zero-free length springs are used here, and from the system potential energies constancy condition, the ‘indifferent balancing’ is ensured throughout the entire swing range of both swinging links of the assisting device. To create a new universal (“global”) exoskeleton, the developed scheme (”local”) can be synthesized with previously proposed schemes of assistive devices for walking and sitting (Figs. 18, 19, 20 and 21) [26, 28]. The lumbar rigid link 1 with weight distribution pillows 1’ are attached to the operator’s lower back using a belt and/or a vest-corset. In the sleeve, first with a square, and then a circular section of the hole of link 1 on the right, a pin of link 2 is horizontally inserted, respectively, with round and square sections (see Fig. 27a and 27b). This prevents the action of torque relative to the axis a-a’ when the finger of link 2 is fully inserted. Link 3 is connected to the vertical end of link 2 through the hinge p1 , which allows the entire subsequent structure to rotate relative to the vertical part of link 2. The authors previously [6–8] have published the rest part of the structure,

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balancing methods and modeling. Thus, the links 4 and 5 are telescopic to provide them with the necessary dimensions, according to the individual parameters of the operator and the firearm. Various attachments-grips 6 or hangers can be installed on the link 5’ through the hinge O5 , specially designed for the required purpose. As mentioned above, the balancing is carried out by the linear cylindrical tension 7 and compression 8 springs with zero-free lengths. This system provides exact static balancing of links 4 and 5 over the entire range of the rotation angle ϕ4,5 . It is possible to limit the link 4 rotation by using latch 9, if necessary. If we make the telescopic link 4 from a strong, but unilaterally flexible material, or give it the necessary curvature (see Fig. 27c), we can ensure a significant increase of the firing sector for the weapon mounted on the proposed device. Installation of the additional loop p2 on the link 4’ (see Fig. 27d) can be another option. The developed structure can provide a balanced grip of the weapon when shooting from standing and kneeling positions. The pin of link 2 must be not fully inserted in case of shooting from the prone position, and the system will be able to rotate about the axis a-a’. The determination of optimal dimensions of the structure, allowing for a convenient and continuous transition from one position to another, can be a problem.

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The device can be used not only in military applications, for supporting human hands, weapons or ammunition, but also in industry, for maintaining tools or heavy objects and in medicine, for supporting an injured or already operated human hand.

3 Conclusions Proposed designs of statically balanced multifunctional portative adjustable assistive devices can assure not only simple assistance to human locomotion functions during walking and sitting, but hold also the human body in a standing position, lift heavy and uncomfortable objects. It becomes possible due to use of balancing systems and regulating mechanisms that conduct the balancing of the biomechanical system with any degree of accuracy. Evaluative choice of the types of springs or group of springs and their working areas, balancing systems: roller, cam, semi-roller and slider; has provided the ease of switching modes and ensured several advantages of suggested devices: portability, adjustability, easy dosage and small size relative to their initial ones [11–13], as well as high efficiency, versatility, ergonomics and low cost. Moreover, the applied approaches to the design of balanced devices are quite versatile and can be used in the design of not only assistive devices but also other linkage based manipulation systems.

References 1. Agrawal, S., Fattah, A.: Reactionless space and ground robots: novel designs and concepts studies. Mech. Mach. Theory 39(1), 25–40 (2004) 2. Agrawal, A., Agrawal, S.K.: Design of gravity balancing leg orthosis using non-zero-free length springs. Mech. Mach. Theory 40(6), 693–709 (2004) 3. Fattah, A., Agrawal, S.K.: Design and modeling of classes of spatial reactionless manipulators. In: Proceedings of the 2003 IEEE International Conference on Robotics and Automation. Taipei, Taiwan, 14–19 September 2003, pp. 3225–3230 (2003) 4. Agrawal, S., Fattah, A.: Gravity-balancing of spatial robotic manipulators. Mech. Mach. Theory 39(12), 1331–1344 (2004) 5. Agrawal Sunil, K.: Fattah abbas gravity balancing of a human leg using an external orthosis. In: IEEE International Conference on Robotics and Automation. -Roma, Italy, 10–14 April 2007, pp. 3755–3760 (2007) 6. Arakelian, V., Ghazaryan, S.: Improvement of balancing accuracy of robotic systems: application to leg orthosis for rehabilitation devices. Int. J. Mech. Mach. Theory 43(5), 565–575 (2007) 7. Ghazaryan, S.D.: (Scientific advisers: prof. Harutyunyan M.G, prof. Arakelyan V.G.). The analysis and optimization of the biomechanical system of human extremities and portative devices of rehabilitation of their functions. Thesis of Dissertation for obtaining the scientific grade “Doctor of Engineering” in specialization “Apparatus, systems, products for medical importance,” pp. 133. State Engineering University of Armenia (Polytechnic) and National Institute of Applied Science – Rennes of France. - Yerevan, Armenia, 20 January 2009

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8. Ghazaryan, S.D., Harutyunyan, M.G., Arakelyan, V.H.: Design aspects of human movement assistance - rehabilitation means. In: Proceedings of the 1st International Conference MES-2018/IPM-2018 Mechanical Engineering Solutions. Design, Simulation, Testing and Manufacturing, 17–19 September 2018, pp. 71–80. MES, Yerevan (2018) 9. Ghazaryan, S.D.: Harutyunyan, M.G.: Arakelian, V.H.: Actual aspects of manipulation mechanism’s swinging links spring balancing. In: XXVI International Scientific and Technical Conference “Mechanical Engineering and Technosphere of the XXI Century”, 23–29 September 2019, pp. 450–454. Sevastopol (2019) 10. Begun, P.I., Shukeylo, Yu. A.: Biomechanics: Textbook for High Schools, 463 pp. - SPB: Polytechnic (2000) 11. Ghazaryan, S.D., Sargsyan, S.A., Harutyunyan, M.G., Arakelyan, V.H.: The design of exoskeleton-assistant of human sit-to-stand. In: Proceedings of National Academy of Armenia and SEUA (Polytechnic), Yerevan, Armenia, vol. LXIV, pp. 121–128 (2011) 12. Sargsyan, S.A., Ghazaryan, S.D., Arakelyan, V.H., Harutyunyan, M.G.: The design of exoskeleton-assistant of a human walk and sit-to-stand. In: Proceedings of National Academy of Science of Armenia and SEUA (Polytechnic), vol. LXIV, pp. 343–349. Yerevan, Armenia (2011) 13. Harutyunyan, M.G., Ghazaryan, S.D., Sargsyan, S.A., Arakelyan, V.H.: Balancing of walk and sit-to-stand assistant exoskeleton. In: Mechanical Engineering and Technosphere of the XXI Century, Proceedings of the XVIII International Scientific Conference in Sevastopol, 12–17 September 2011 4 vol, T-4, pp 37–40. Donetsk National Technical University, Donetsk (2011) 14. Ghazaryan, S.D., Harutyunyan, M.G., Arakelyan, V.H.: Design concepts of quasi-static balanced multipurpose exoskeletons. Proc. Eng. Acad. Armenia 16(2), 46–50 (2013) 15. Harutyunyan, M., Zakaryan, N., Ghazaryan, S., Sargisyan, S.: Exoskeleton/Invention patent No. 2747 A, 25 June 2013 16. Zakaryan, N., Ghazaryan, S., Harutyunyan, M.: Static and dynamic balancing of an exoskeleton by means of spiral springs and drivers. In: Proceedings of SEUA, Collection of Scientific Articles, Version 2, pp. 362–368 (2013). 17. Harutyunyan, M., Zakaryan, N., Ghazaryan, S., Sargisyan, S.: Assist device for walking/Invention patent No. 2721A – 25 March 2013 18. Ghazaryan, S.D., Harutyunyan, M.G.: Design of a portable orthosis for walking and sitting of a man. In: Proceedings of SEUA, Collection of Scientific Articles, N2, pp. 435–439. Yerevan (2018) 19. Ghazaryan, S.D., Harutyunyan, M.G.: Design of constructions of a portable orthosis for walking and sitting of a man. In: Proceedings of SEUA. “Mechanics, Machine Science, Mechanical Engineering, N2, pp. 43–52. Yerevan (2017) 20. Ghazaryan, S.D., Harutyunyan, M.G.: The design of multi-purpose portable movable orthosis. ROMANSY 2018. In: 22nd CISM IFToMM Symposium on Robot Design, Dynamics and Control, 25–28 June 2018, pp. 296–303. Rennes, France (2018) 21. Ghazaryan, S.D., Harutyunyan, M.G., Zakaryan, N.B., Sargsyan, Yu. L.: Construction of an assistant device with flexible elements for walking and sitting of a human. In: Proceedings of SEUA, Collection of Scientific Articles. N2, pp. 305–309. Yerevan (2020) 22. Ghazaryan, S.D., Zakaryan, N.B., Harutyunyan, M.G., Sargsyan, Yu. L.: Designing multipurpose balanced assistive devices of a human locomotors system. In: Proceedings of NPUA. “Mechanics, Machine Science, Mechanical Engineering, N2, pp. 49–58. Yerevan (2018) 23. Zakaryan, N.B., Ghazaryan, S.D., Harutyunyan, M.G., Sargsyan, Yu. L., Shahinyan, S.S., Shahazizyan, B.H.: Multi-purpose exoskeleton/Invention Patent N0. 3299 A, 12 February 2019 24. Harutyunyan, M.G., Ghazaryan, S.D., Zakaryan, N.B., Shahinyan, S.S.: Passive balancing solution for a human walking and sitting assisting exoskeleton. Adv. Technol; Mech. Eng. Syst. 65(2), 95–99 (2018). 25. Shahazizyan, B.H., Ghazaryan, S.D.: Development of a new construction of an assisting robotic device with improved technical characteristics. In: Proceedings of SEUA, Collection of Scientific Articles, 26–30 November 2019, pp. 316–323. Yerevan (2019).

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26. Shahazizyan, B.H., Ghazaryan, S.D., Harutyunyan, M.G.: Design of military purpose robotic exoskeleton devices. In: 16th Scientific Conference on 25th Anniversary of Military University after V. Sargsyan Yerevan, 30–31 May 2019. 27. Ghazaryan, S.D., Harutyunyan, M.G., Zakaryan, N.B.: Modeling of the portable exoskeleton for a human walking and sitting passive balancing. In: Proceedings of NPUA. “Mechanics, Machine Science, Mechanical Engineering”.N2, pp. 50–61. Yerevan (2019) 28. Ghazaryan, S.D., Zakaryan, N.B., Harutyunyan, M.G., Sargsyan, Yu. L., Verlinski S.V: Developing balancing methods of biomechanical systems applied to the design of reconfigurable exoskeleton-assistants with improved functional characteristics. In: Proceedings of NPUA. “Mechanics, Machine Science, Mechanical Engineering,” N1, pp. 55–68.Yerevan (2020) 29. Ghazaryan, S.D, Harutyunyan, M.G., Zakaryan, N.B., Arakelian, V.: Design concepts for human walking and sitting wearable exoskeletons. In: Intelligent Technologies in Robotics, 21–23 October 2019, Moscow, pp. 63–71 (2019) 30. Harutyunyan, M.G., Ghazaryan, S.D., Zakaryan, N.B., Sargsyan, Yu. L., Verlinski S.V.: Creation of a new exoskeleton with the possibility of reconfiguration for implementation of various locomotor functions. In: Proceedings of the XXVII International Scientific and Technical Conference “Machine-Building and Technosphere of XXI Century”, pp. 32–35. Donetsk-Sevastopol (2020) 31. Ghazaryan, S.D., Harutyunyan, M.G., Zakaryan N.B., Sargsyan, Yu. L.: Portable passive balanced assistive device for industrial purpose. Progress. Technol. Syst. Mech. Eng. 73(2) 42–46 (2021) 32. Ghazaryan S.D., Harutyunyan, M.G., Zakaryan, N.B., Sargsyan, Yu. L.: Portable assistive device for military purpose. In: Proceedings of the First International Scientific and Technical Conference “Current Problems of Science and Technology”, 20–22 May 2021, Sarapul (2021)

Gravity Balancing of Parallel Robots by Constant-Force Generators Giovanni Mottola, Marco Cocconcelli, Riccardo Rubini, and Marco Carricato

Abstract This Chapter reviews the literature on gravity balancing for parallel robots by using so-called constant-force generators. Parallel robots are formed by several kinematic chains connecting, in parallel, a fixed base to a moving end-effector. A constant-force generator is a mechanism that is able to exert, at a given point, a force having constant magnitude and direction. Gravity balancing of serial robots is a well established technique; conversely, application in parallel robotics is controversial. Indeed, the addition of gravity-balancing mechanisms to a parallel robot may worsen its dynamic behavior, as shown in some referenced works. In this Chapter, we introduce a taxonomy of constant-force generators proposed so far in the literature, including mass and spring balancing methods, toghether with more niche concepts. We also summarize design considerations of practical concern.

1 Introduction Parallel kinematic manipulators (PKMs, also called parallel robots) are actuated mechanical systems where the end-effector (EE) is connected to the fixed frame by several independent kinematic chains, acting “in parallel”, whereas conventional serial architectures employ only one chain of links connected in series. PKMs offer a number of interesting features over serial alternatives, such as increased stiffness and accuracy, higher dynamic capabilities (in terms of both reachable velocities and G. Mottola · M. Cocconcelli · R. Rubini University of Modena and Reggio Emilia, Via G. Amendola 2, 42122 Reggio Emilia, Italy e-mail: [email protected] M. Cocconcelli e-mail: [email protected] R. Rubini e-mail: [email protected] M. Carricato (B) University of Bologna, Viale Risorgimento 2, 40136 Bologna, Italy e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 V. Arakelian (ed.), Gravity Compensation in Robotics, Mechanisms and Machine Science 115, https://doi.org/10.1007/978-3-030-95750-6_9

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accelerations) and high payload-to-robot-weight ratios; for these reasons, they have been a topic of active research for some decades now [1]. Given that a standard application for such systems is the manipulation of heavy loads, it appears worthwhile to further improve their potential in this regard. Gravity balancing is interesting as a potential approach: it is defined [2] as a method to compensate the effect of gravitational loads acting on the links of a mechanism so that it may remain in static equilibrium at multiple configurations of interest, without any active force or torque applied by the actuators. By a suitable robot design, one can achieve a full balancing, meaning that static equilibrium is obtained for a continuum of poses within the robot range of motion (or even for the entire workspace). Thus, the equilibrium is indifferent (or neutral), namely displacements from an equilibrium configuration do not cause restoring wrenches. When the robot is at static equilibrium, no loads due to gravity are felt by the actuators. This has interesting advantages: static deflections may be smaller, and motor and brake sizes can be reduced. Also, significant energy savings in quasi-static motions can be achieved. For broad reviews and perspectives on balancing of linkages and robots (both serial and parallel), we refer the reader to [2–9]; some textbooks introducing the topic can be found in [10–16]. Gravity balancing can be achieved through a number of methods, with the effects of gravity being usually compensated by passive mechanical elements such as counterweights or springs. An approach used in some works is to use constant-force generators (CFGs) as compensating devices: CFGs are mechanisms that can produce a force F that, though constant in magnitude and direction, can be applied at a point P that can be moved as needed. CFGs have a number of applications in engineering [17], for example delicate manipulation of fragile objects; in particular, they are naturally suited for balancing gravitational loads on the links of a mechanism. The literature on gravity balancing of closed-chain linkages and PKMs in particular is vast, at the time of writing, also due to the fact that the topic is now several decades old [18–21]. In this Chapter, we will only present an overview of gravitybalancing techniques for PKMs taking advantage of CFGs. A special attention will be given to works describing practical techniques and potential pitfalls in the design and applications of CFGs. We believe that this review can be useful for the designer who seeks concrete methods and solutions in this area. The Chapter is organized as follows. In Sect. 2, we introduce some fundamental concepts and briefly summarize previous works on gravity balancing. In Sect. 3 we delve on gravity balancing by way of CFGs, commenting their properties and development, together with practical considerations on their construction. In Sect. 4, we illustrate how the CFG designs examined in Sect. 3 can be combined to define gravitybalanced PKMs. In Sect. 5 we overview the working principles of gravity-balanced PKMs and consider a few particular examples, such as tensegrity and compliant mechanisms. In Sect. 6 we present our conclusive observations, together with ideas and suggestions for future work.

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2 Gravity Balancing: General Observations Gravity balancing has been used since ancient times to move large masses with reduced effort: examples can be seen in counterweights for elevators and cranes [22]. Its advantages are obvious in mechanisms that are to be actuated by human force, such as hand-operated balanced manipulators, used in industrial environments to help users move large loads [23]; it can also significantly improve performances in actuated devices. We begin our review with a brief summary of gravity balancing through the perspective of a designer of PKMs and by introducing some definitions and methods. First, we restrict our discussion to exact or perfect balancing: therefore, we disregard mechanisms that are in static equilibrium only at a number of points within their workspace. Gravity balancing at a discrete set of configurations has received some attention [24] and may prove useful when perfect balancing is not required, since it allows using simpler numerical approaches to be implemented [25]; in this case, static friction in the mechanism joints usually compensates the residual unbalance. While designing an exactly-balanced mechanism [26, 27] is generally a more difficult task, which may also require complex analytical methods, the obtained advantages are generally found to overcome such hurdles.

2.1 Definitions and Balancing Approaches In general, perfect static balancing means that the system has no “preferred” equilibrium configuration (to which the mechanism will move back, if displaced). Thus, the configuration can be changed within a range of possible movements without introducing energy in the system: for this reason, the alternative definition energy-free [13] is also sometimes used for statically-balanced systems. Given the definition of static balance as a continuum of equilibrium configurations, one can derive its mathematical formulation from the conservation of the total potential energy (for a conservative system). Therefore, the general way to achieve static balancing is ensuring that the potential energy of the conservative forces is invariant with respect to the mechanism configuration. Accordingly, neutral equilibrium under the effect of gravity is achieved as soon as the variation of gravitational potential energy is compensated by any other type of potential energy in the mechanism; in practice, it is most common to use either spring forces or the weight of added elements, called counterweights. The total energy Vt due to conservative forces can thus be expressed as (1) Vt = Ve + Vg + Vo where Ve is the elastic potential energy, Vg is the gravitational potential energy, and Vo groups all other sources of potential energy (if present), such as magnetic forces [28–30]. In this chapter, we will assume Vo = 0, unless otherwise specified.

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In order to calculate potential energies, we define a global coordinate frame F = (O, x, y, z) fixed on the ground link of the mechanism (identified by index 0). For the i-th movable link (i = 1, . . . , n l , where n l is the number of moving links), we also define a local coordinate frame Fi = (Oi , xi , yi , z i ) that is attached to the link.1 We will denote ci the position vector of the center of mass (CoM) of the i-th link as seen in the local frame Fi ; its position ci in the global frame is then ci = Ri ci + ti

(2)

where Ri is the rotation matrix between frames F and Fi , and ti = Oi − O is the translation vector between the origins of the frames. Similarly, the generic j-th spring ( j = 1, ..., n s , where n s is the number of springs employed for balancing, if any) can be defined by its stiffness k j , its rest length2 l j,0 , and the locations of its attachment points s j,1 and s j,2 , expressed in the global frame by a linear transformation between coordinate frames, as in Eq. (2). The stretched length of the spring is then (3) l j = s j,1 − s j,2  and the corresponding magnitude of the spring force F j is   F j = F j  = k j l j − l j,0

(4)

Finally, the elastic potential energy of the spring is V j,e =

2 1  k j l j − l j,0 2

(5)

The gravitational potential energy of the i-th mass m i (referring to either a standard link or a counterweight) is (6) Vi,g = m i g(ci · ez ) where g is the (constant) gravitational acceleration and ez = [0, 0, 1]T is the unit vector directed along the positive z axis, pointing upward in the vertical direction. Finally, the total potential energies Ve and Vg in Eq. (1) are obtained as Ve =

ns  j=1

1

V j,e , Vg =

nl 

Vi,g

(7)

i=1

For conciseness, we only consider the most general case of mechanisms having three-dimensional motion; the case of planar motion can be derived as a sub-case, where each frame is defined by only two coordinate axes instead of three. 2 Here, we implicitly assume the springs to be linear. Indeed, some works [25, 31, 32] employ torsional (angular) springs. In this case, the derivation of the potential energy is similar, replacing the stretched and rest lengths by corresponding angles φ j and φ j,0 ; the spring stiffness k j is then defined in angular units.

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If other balancing elements that produce conservative forces are introduced, the corresponding potential energy Vo can be obtained similarly after defining the potential energy of each element. The overall configuration of a mechanism (whether closed- or open-loop) having n Degrees-of-Freedom (DoFs) is, in general, defined by n independent generalized coordinates: (8) q = [q1 , . . . , qn ] The variables qi can be, for example, the joint coordinates of the n actuated joints. Notice that matrix Ri and vector ti in Eq. (2) are dependent on the mechanism configuration, therefore the CoM position ci in the global frame also depends on q (even though the position vector ci in the local frame is constant). It can be seen from Eqs. (6) and (7) that the gravitational potential energy Vg is also configurationdependent. A similar reasoning applies for spring lengths l j and elastic potential energy Ve . In brief, one has Vt = Vt (q, a) (9) where we have expressed the total potential energy Vt as a function of both the mechanism configuration q and the mechanism parameters a. The array contains all dimensional, inertial and design parameters required to define a particular version of a mechanism concept: for example, a includes the components of vectors ci in the local frames Fi , the masses m i and the spring stiffnesses k j . These parameters remain constant once a specific design has been chosen. The conditions for gravity balancing are then derived by setting the partial derivatives of Vt with respect to the generalized coordinates equal to zero, namely ∂ Vt (q, a) = 0, i ∈ {1, . . . , n} ∂qi

(10)

If Eq. (10) holds at a configuration, the mechanism is at equilibrium. If it holds for any value of qi , i ∈ {1, . . . , n} within a range (namely, for a continuous set of configurations defined by qi,min ≤ qi ≤ qi,max ), the mechanism is in neutral equilibrium and the actuators are not exposed to the effects of conservative forces acting on any of the links. Equation (10) yields a set of n equations that depend on the configuration q and the parameters a. Since said conditions must hold independently of the configuration, the approach that is commonly pursued is to separate the terms in q and a in the resulting equations and to set the latter to zero, namely ∂ Vt (q, a) = Q i (q) Ai (a) = 0 ∂qi

⇒

Ai (a) = 0, i ∈ {1, . . . , n}

(11)

which leads to a set of constraints that define a subset within the design space of all possible parameters a. If a particular design is chosen within this subset, Eq. (10) is always verified in the range of interest.

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The partial derivatives of Vt in Eq. (10) define the generalized forces/torques; therefore, an alternative but equivalent approach is to directly write the static equilibrium equations and seek design parameters such that they are identically verified. Indeed, the first works on the topic did not approach the problem from the potential energy viewpoint [26, 33]: this way, however, the derivation of the parameters of interest is usually more convoluted. In the following, unless otherwise specified, the payload is considered to be attached to the EE, for instance by being firmly held by a gripper on the EE; thus, the payload and EE masses can be combined in the analysis. Also, it is assumed that the only masses to be balanced are those of the EE and the links; the latter generally provide the largest terms in the gravitational potential energy Vg , since even for high-performance PKMs the payload-to-robot-weight ratio is rarely greater than 1 for continuous operation—and more commonly in the vicinity of 1/10, at least for common industrial architectures [1, 34, 35]. An important distinction in robot kinematics is made between planar and spatial mechanisms, where the former are those where all points move on parallel planes, whereas the latter have general 6-DoF motions in three-dimensional space. Clearly, when a mechanism moves on a horizontal plane, it does not need to be gravitybalanced [36]: indeed, in this case the weights of the links do not introduce any work in the system nor cause forces or torques to be balanced by the actuators (if friction is disregarded). Having a motion constrained on a horizontal plane, though, is not always feasible. PKMs having general spatial motion, such as for instance the Gough-Stewart platform [37, 38], are more complex to balance against gravity. The main issue making gravity balancing of a linkage a nontrivial problem is the variable transmission ratio between the velocities of different points in the links. Even if the gravity forces are usually constant,3 their effects in terms of wrenches at the actuators are generally nonlinear functions of the mechanism configuration q, due to the terms ∂ Vg /∂qi . In order to compensate nonlinear effects, one needs to use one of the following approaches. (1) Find an acceptable approximate solution. This means that there will be residual unbalanced moments: generally, static friction allows us to compensate for these effects [31, 39]. The approximation can be either global, if one seeks a leastsquare-minimization of the unbalanced moments across the entire workspace, or specifically targeted for a specific task; for instance, it was found in [25] that optimizing the design of linear torsional springs for a particular trajectory can lead to almost perfect balancing over a significant range of motion. Similarly, it was found in [40] that a 2-DoF mechanism can be balanced with counterweights for local balance, thus obtaining a set of poses where the mechanism is balanced, or for unbalance elimination along a specific path.

3

Notice that the gravity force acting on the EE may vary after the payload is changed; some designs that allow the mechanism to adapt to changes of the EE total weight while keeping the global gravitational balance of the mechanism are reported in Subsect. 5.2.

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(2) Use balancing elements with nonlinear force-displacement characteristics; one may use specifically-designed nonlinear springs [41–43] or energy-storage elements based on different principles, such as magnetism [28–30]. (3) Employ nonlinear transmission mechanisms that connect the balancing devices with the actuators. The (variable) transmission ratios τi j (between the i-th actuator and the j-th balancing element) ought to be chosen by taking into account both the wrenches to compensate and the forces/torques provided by the balancing elements. In general, one obtains the required ratios τi j = τi j (q) which can be achieved by using cam mechanisms [27, 44–48] or linkages. Among the possibilities described above, option (1) may introduce significant errors in balancing and is beyond the scope of this review, as defined at the beginning of this Section. Nonlinear spring systems (option (2)) are quite complex to design, manufacture and apply, especially if compact designs are sought [49]; the same applies for other nonlinear energy-storage systems, where often the balancing forces cannot be derived analytically and are only known from empirical results. Therefore, option (3) is the most frequently pursued for its practical advantages: this will be our main focus in the following.

2.2 Force and Moment Balancing In Subsect. 2.1, we have shown how to achieve balance by requiring the total potential energy to be constant at any configuration. In this case, no resultant force or torque due to the gravitational acceleration on the links is felt at the actuators under static or quasi-static conditions. In some cases, however, gravity balancing is not enough and it is desirable to balance the mechanism not only when it is at rest, but also for any motion with non-negligible velocities and accelerations. This approach is called dynamic balancing of a mechanism. It is known from the general principles of Newtonian mechanics that the resultant of all external forces acting on a system is given by the time derivative of the linear nl m i is the total mass of the mechanism and c is momentum M c˙ , where M = i=1 the position vector of the global CoM, namely c=

nl 1  m i ci M i=1

(12)

Since the total mass M is constant, the aforementioned time derivative becomes 

Fe =

d (M c˙ ) = M c¨ dt

(13)

 For a mechanism, Fe is also the total force that is applied on the ground link and it is usually called the shaking force. Similarly, one obtains the resultant of

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the external moments Me at a point O by differentiating the total angular moment  of the mechanism with respect to time: then, Me is called the shaking moment applied at the base.4 The shaking force and moment will be transmitted to the larger structure on which the mechanism operates, generally causing vibrations and noise [50]: therefore, it is clearly desirable to reduce them (or completely cancel them, when feasible). From Eqs. (2) and (12) it can be seen that the position c = c(q) of the overall ˙ q) ¨ CoM is dependent on the configuration. Therefore, its acceleration c¨ = c¨ (q, q, ¨ From Eq. (13), the shaking also depends on the joint velocities q˙ and accelerations q. force is zero if and only if the acceleration c¨ of the CoM is null. This could be achieved by having the CoM move with constant velocity c˙ ; however, it can be proven that, if it is desired that Eq. (13) hold for any possible motion law q = q(t) of the actuators, the only possibility is to have a constant c = c0 , that is, a CoM fixed at an (arbitrary) constant position [50, 51]. It can be shown from Eqs. (6) and (7) that, if the total CoM of the mechanism is fixed, the gravitational potential energy Vg is constant: this is sufficient to obtain gravity balancing, without using elastic elements (Ve = 0). In fact, it would be sufficient to have c · ez constant, that is, the mechanism is gravity balanced if the CoM has constant z-axis component and thus does not move in the direction of gravity (even if the x- and y-axis components vary). In theory, it is not difficult to conceive a mechanism satisfying this more general condition; however, in practice, almost all mechanisms found in this bibliographic research that are gravity balanced by keeping Vg constant (that is, by only using counterweights and/or mass redistribution) have a fixed CoM; an exception is the 3-DoF mechanism from [52] discussed in Subsect. 3.3. The theory of dynamic balancing is a classical topic in mechanics of machines due to its practical applications, for instance in the design of piston-crank assemblies in internal combustion engines; in this sense, such a theory also applies to gravity balancing. It is worth to point out that the balancing of the shaking moment is not required for gravity balancing, since balancing the shaking force is sufficient for this purpose. Notice that dynamic balancing is a very broad concept, which also includes topics such as partial dynamic balancing, for instance by minimizing the amplitudes of the shaking force/moment at some frequencies of interest [53, 54]; these issues are beyond the scope of the present work and are not considered here. For a standard reference on dynamic balancing (with some applications to gravity balancing), we refer the reader to [15]. For a fully-balanced mechanism, the force acting on its base will be constant, equal to the total weight Mg of the system, no matter the motion being performed. This total force must be supported by the structure to which the robot is attached. For robots operating on mobile vehicles (such as robot arms operating on space stations),

4

Notice that some authors [50] define instead the shaking force as the total force applied by the fixed ground to the mechanism; a similar notation applies then to the shaking moment.

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this structure will be the vehicle frame: in this case, dynamic balancing becomes essential, to avoid changing the momentum of the vehicle uncontrollably [50]. In this chapter, we will also mention relevant works on dynamically-balanced linkages (including some where the shaking moment is also balanced), when it is useful for a broader understanding of gravity balancing. Generally, full dynamic balancing requires introducing not only counterweights (to balance the shaking forces) but also counter-rotating inertias, to balance the shaking moment [55–59]. However, it was found for some simpler planar mechanisms that counter-rotating elements are not strictly necessary: it was found in [60] that, with a careful choice of both the inertial and the kinematic parameters of a four-bar linkage, one could obtain full dynamic balance without additional elements. These reactionless four-bar linkages were later used as “building blocks” for more complex 3-DoF [50, 61] and 6-DoF [62] robots, both planar and spatial; these systems are also gravity balanced. More recently, a generalization of the concept of dynamically-balanced linkages with a minimal usage of counter-mechanisms was explored in [63]. Force balancing of linkages can be done by carefully designing the links in order to obtain a favorable mass distribution, while keeping the desired kinematic performance of the linkage; generally the kinematic parameters (such as the link lengths) are considered as a given input in this phase and therefore cannot be modified. This way, gravity balancing can be obtained without adding inertia nor other components to the mechanism. However, designing the links so that they respect both kinematic and dynamic constraints (together with constraints on their resistance to static stresses and fatigue) is not always feasible. Another approach, then, is adding counterweights on the links, in which case the total mass of the i-th link becomes m i∗ = m i + m c,i , where m c,i is the mass of the added counterweight. This way, the total mass of the mechanism increases (quite often significantly), but the desired balancing can be reached with a relatively simple design. Gravity balancing by way of force balancing may introduce some constraints on the mechanism design. In many works [64] on force balancing for planar linkages, it is assumed that, for each link, there is at least one RR...R kinematic chain connecting said link to the fixed frame.5 This condition is generalized for a spatial linkage by requiring each link to have a connection with only R or S joints to the ground link. Indeed, for some time it was generally believed [36, 65, 66] that the force balancing of a linkage strictly required this condition. For mechanisms where this condition is not respected, one should then add kinematic chains where needed [64]; these are generally called idler loops, in the sense that they are passive chains which are not used for actuating the mechanism. Adding idler loops clearly increases the complexity and increases the risk of interference between the members. In later works [52] it was shown that this constraint can in fact be relaxed by using movable counterweights; the motion of the counterweights may be controlled by the same actuators that move the robot, by using simple transmissions with constant ratio. This way, it was possible to fully balance a 6-DoF spatial PKM (a Gough-Stewart 5

Here and in the following, R denotes a rotary joint, P a prismatic joint, U a universal joint and S a spherical joint; a line above each symbol (such as P) denotes that a joint is actuated.

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platform, which has an EE supported by six UPS legs) against gravity without adding other kinematic chains. In some works [51, 58, 59], dynamic balancing is achieved by employing duplicate mechanisms, namely by adding linkages whose links have the same dimensions and masses of the original linkage to be balanced. A transmission (having constant ratio) is generally used between the “original” mechanism and the copies, so that the total number of DoFs does not change: the added mechanisms thus perform the same motion of the original, but with a phase angle difference and/or moving in the opposite direction. In particular, adding a mechanism which is symmetrical to the original with respect to a horizontal axis allows to fully balance the gravity forces. Proposed concepts use either two [67] or four [51] duplicates, depending on whether full or partial dynamic balancing is sought. A closely-connected concept is achieving dynamic balance for mechanisms having mirror symmetry with respect to a moving plane [63, 68]: by properly taking advantage of the internal symmetries, no additional linkages are to be added, but this approach is not applicable to generic linkages. Duplicate mechanisms can offer full static and dynamic balancing, but at the expense of a significant increase in cost and complexity of the complete system; on the other hand, the design of the balancing mechanism itself is immediate once the original mechanism has been defined. Notably, as observed in [51], duplicate mechanisms can outperform simpler devices for full dynamic balancing (such as counterweights and counter-rotating inertias) in terms of the total inertia that is added to the original (unbalanced) mechanism. Adding duplicate mechanisms can thus be useful for simpler linkages where several “copies” of the same working principle must operate in parallel, such as in presses, combustion engines or automated machines; conversely, it is less commonly used in complex robotic systems.

2.3 Gravity Balancing: Goals, Advantages, Limitations The first obvious advantage of gravity balancing is obtaining a robot which can move, under quasi-static conditions, having very low forces/torques at the actuators: indeed, the motors only have to compensate the friction in the joints. Therefore, the energy consumption is very small. Reducing the motor torques under static conditions is especially important for direct-drive robots [69]: in these designs, no speed reducers (such as gearboxes) are used and friction is much lower. Thus, the motors have to directly provide large torques for long times when the robot is at rest: size and overheating can then become issues, which may be mitigated through gravity balancing. Also, serial or parallel robots that are programmed by play-back (namely, the operator “trains” the robot by manually driving the system through a number of desired poses) require gravity balancing [70]. Even in low-dynamics conditions, gravity-balanced mechanisms tend to perform better than unbalanced ones; on the other hand, balancing can worsen the performance in high-dynamics conditions, as the weight of the added elements can increase inertial effects. This is especially significant when counterweights are used, as they tend to

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add large masses. For these reasons, springs are usually preferable for balancing mechanisms moving at high speeds [71], as they have no effect on the shaking force and usually a negligible impact on the inertia (however, they do have an effect on the reaction forces in the joints). In the following, both spring- and mass-based balancing systems are considered, as they have their own merits and applications. Other known advantages of balancing [50] are in general a reduction of vibrations and noise of the mechanisms; the joints are less subject to wear and fatigue and the stresses in the links are usually reduced. These advantages should however be weighted against the added complexity of the system and the risk of worsening the dynamic performances: therefore, a case-by-case analysis of the advantages and disadvantages of balancing is recommended for original robot designs. For serial robots, the advantages of gravity balancing are generally found to be prevalent: with a serial architecture, each actuator has to withstand the weight of all subsequent elements up to the EE (including the other actuators). Therefore, gravity balancing of serial architectures is a standard design step [44, 72]. Frequently, the motor themselves can be used as counterweights for partial balancing, given that their masses are quite significant; this, for example, is the design of the KUKA R360 and PUMA 200 robot arms [2]. Even if only the first joint of the robot is balanced, such as in the ABB IRB 4400 or Fanuc M400/M900 series, significant reductions in actuator torques can be achieved with a modest increase in complexity [32]. On the other hand, since PKMs are appreciated for their capability to work even at very high accelerations, the application of gravity balancing to these robots is somewhat controversial. Nonetheless, gravity balancing has shown to be effective in several applications of PKMs. The methods for gravity balancing of robots can have the goal to either present original robot architectures or provide compensators that can be used to retrofit an existing system [30, 36, 73, 74]. Generally, it is desirable to develop gravitybalancing devices such that they may be added to any given robot within a specified class: this way, the design stages for the robot and for the balancing devices are separated. Therefore, the kinematic and design parameters of the robotic platform can be optimized independently, and the parameters of the gravity-balancing system can be derived at a later stage. This approach is especially useful for industrial robots: these are expensive devices, which are expected to have a service life of several years. In these cases, adding a gravity-compensator to an existing robotic system (with only limited changes to the robot structure) is a cost-effective method to reduce energy consumption [36]. In general, however, there is no guarantee that a given mechanism can be balanced while keeping the same general architecture [75]. Even if such a solution can be found, the balanced mechanism might not respect size constraints: if these are especially stringent, it is recommended to integrate the design phases of both robot and gravity compensators [5] for a global optimization of the entire system. This generally leads to conceiving new robot architectures. An issue closely connected to the previous distinction is the derivation of balancing conditions Ai (a) = 0 from Eq. (11) that are both necessary and sufficient. Indeed, in most previous works the conditions derived are sufficient, but not strictly necessary: while acceptable for a practical design in most cases, this means that

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potentially advantageous designs may be discarded. However, deriving a minimal set of conditions for balancing (either static or dynamic) is for most mechanisms a non-trivial problem [7, 50, 76]: up until very recently, the problem has been fully solved only for the four-bar linkage [60]. A general approach that can derive necessary and sufficient conditions for full dynamic balance to characterize all possible design instances of a given mechanism has been proposed in [63]; here, however, spring balancing was not considered as a possible approach. By comparing the applications of spring- and mass-balancing, we note that, broadly speaking, using springs generally introduces more unknown design parameters in the mechanism definition, as for each spring two position vectors sj,1 and sj,2 , a spring stiffness k j and a free length l j,0 need to be determined, whereas each  and a mass m c,i . Thus, spring counterweight is defined by only a position vector cc,i balancing can provide more freedom in design and the subset of the design space containing all possible mechanisms respecting the constraints for balancing (see Eq. (11) in Subsect. 2.1) will be high-dimensional.

3 Constant-Force Generators In this Section, we introduce some basic concepts about CFGs by analyzing their most simple instances. Then, we explore some other CFGs proposed in the literature. Finally, we offer some practical observations for the designer.

3.1 1-DoF CFG In this Section, we present the analysis of the simplest CFG known in the literature [19, 21, 22, 26, 33, 77–80]: this is a 1-DoF, planar mechanism with one link connected to the fixed frame by an R joint with axis passing through point O in the plane of motion. A single spring (having stiffness k and rest length l0 ) is attached to the fixed frame 0 and to the mobile link 1, respectively in points A and B. Link 1 has a mass m and its CoM is in point G; a counterweight of mass m c is attached on point G c and moves with link 1. The mechanism is to be gravity balanced and has to apply a constant force F at point P of link 1. We thus consider the most general case where both springs and counterweights are employed; see Fig. 1. With the symbols presented in Subsect. 2.1, we define the local frame F  = (O, x  , y  ), having the x  -axis directed from O to G, and the fixed global frame F = (O, x, y), having the same origin as F  but constant orientation. The position vector of the CoM of link 1 in the local frame is thus c = (G − O) = [c, 0]T . The counterweight added on the link has position vector cc = (G c − O) in the local frame, at a distance cc = G c − O from the origin. Similarly, we define the position vectors of the spring attachment points s1 = (A − O) and s2 = (B − O), respectively on links 0 and 1. Having defined s1 in F and s2 in F  , these vectors have

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Fig. 1 Basic 1-DoF, planar CFG employing both spring and mass balancing

constant components; their lengths are s1 = A − O and s2 = B − O. The constant force F, having magnitude F = F, is applied in point P. The latter position on link 1 is defined by the local vector p = (P − O) having length p = P − O. Finally, θ is the angle between vector c and the x-axis, φ is the angle between vector (A − O) and the y-axis, γ is the angle between force F and the x-axis, ψ =   O P. Angles are positive along the counterclockwise G O B, α = G OG c , and β = G direction. Angle θ varies as the link moves and can be chosen as the joint coordinate; thus, in this case vector q in this case is simply a scalar. The transformation between local and global frames in Eq. (2) simplifies as   cos(θ ) − sin(θ ) v = Rv , R = sin(θ ) cos(θ ) 

(14)

where v and v correspond to a generic vector whose components are expressed, respectively, in F and F  . On the other hand, angles φ, ψ, α and β are constant. The array of (constant) mechanism parameters is thus a = k, m, m c , c, cc , s1 , s2 , p, F, φ, ψ, α, β, γ . Force F is constant in the sense that its components Fx = F cos(γ ) and Fy = F sin(γ ) in the global frame F do not change; its application point and line of action, however, change as P rotates around O. Therefore, the potential energy term V f due to F is simply given by the force itself dot-multiplied by the displacement of P from a reference position, which for simplicity we take in the origin O, namely Vf = F · p

(15)

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Combining Eqs. (6) and (7) the gravitational potential energy is Vg = mg(c · e y ) + m c g(cc · e y )

(16)

where e y = [0, 1]T . Finally, the elastic potential energy Ve in Eq. (5) becomes Ve =

1 k (A − B − l0 )2 2

(17)

Applying Eq. (14) and simplifying through trigonometrical identities, we can rewrite the potential energies as functions of the input variable θ . Equations (15) and (16) become, respectively,  V f = p Fx cos(θ + β) + Fy sin(θ + β) = p F cos (θ − (γ − β))

(18)

Vg = g [mc sin(θ ) + m c cc sin(θ + α)]

(19)

The length of the spring can be derived by observing from Fig. 1 that π  AO B = − θ + (φ − ψ) 2

(20)

One then obtains, from the law of cosines applied to triangle AO B, the stretched spring length as A − B =



s12 + s22 − 2s1 s2 sin (θ − (φ − ψ))

(21)

Substituting Eq. (21) into Eq. (17), we finally obtain Ve as a function of θ . Comparing the resulting expression with Eqs. (18) and (19), we immediately notice that the latter are linear combinations of functions sin(θ ) and cos(θ ); on the other hand, Ve is a more complex expression containing a square root that cannot be simplified in terms of trigonometric functions of θ . Therefore, it is not possible in general to achieve a constant total potential energy Vt = V f + Vg + Ve , as this requires that the terms in θ cancel each other. However, if l0 = 0, Eq. (17) simplifies as Ve =

1  2 k s1 + s22 − 2s1 s2 sin (θ − (φ − ψ)) 2

(22)

Obtaining a zero-free-length spring (ZFLSs, see [81]) is indeed possible from a design point of view and in fact is the main assumption from most previous works [19, 26, 33, 77–79]; we will provide further details in Subsect. 3.5 on the design and application of ZFLSs. For now, we observe that Ve can now be written as a linear combination of sin(θ ) and cos(θ ). We can finally write the total potential energy Vt as (23) Vt (θ ) = V f (θ ) + Vg (θ ) + Ve (θ ) = Vt,0 + Vt,c cos(θ ) + Vt,s sin(θ )

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where  1  2 k s1 + s22 2 Vt,c = p F cos() + m c cc g sin(α) + ks1 s2 sin(δ) Vt,s = p F sin() + m c cc g cos(α) + mcg − ks1 s2 cos(δ) δ =φ−ψ

Vt,0 =

 =γ −β

(24) (25) (26) (27) (28)

Differentiating Eq. (23) with respect to variable θ and setting it equal to zero yields d Vt = −Vt,c sin(θ ) + Vt,s cos(θ ) dθ

(29)

Since sin(θ ) and cos(θ ) are functions of θ which are non-linearly dependent, having constant d Vt /dθ = 0 implies a condition on Vt,c and Vt,s , namely Vt,c = Vt,s = 0

(30)

whereas the constant term Vt,0 can have any value. As long as the conditions in Eq. (30) are satisfied, the mechanism will be in neutral equilibrium in all configurations. If an actuator is used to control angle θ (for instance, by attaching a motor at O between links 0 and 1), no torque will be required for any quasi-static motion. The conditions for gravity balancing can in fact be satisfied by infinitely many choices of the parameter array a; the solution space is 12-dimensional. Such, for example, is the CFG employed in [77, 78] to balance a four-bar linkage. Some special cases of interest can be distinguished and are discussed below.6 (I) F = 0 and k = 0 (no spring): this corresponds to a gravity-balanced mechanism through a counterweight. Here, we consider the non-trivial case in which m > 0, c = 0, since the trivial cases do not require any counterbalancing system. From Eqs. (25) and (26) we obtain α = 0, m c cc = −mc

(31)

The first condition implies that the counterweight on point G c is aligned with the line through OG. Since the countermass m c is also positive, the latter condition implies that cc is negative, thus G c is on the opposite side of G with respect to O. These conditions define the simplest counterweight design, that can be commonly seen in designs of cranes and gates since antiquity [22]. In this Chapter, we always implicitly assume g = 0; indeed, for mass-balanced systems it can be readily seen that the formulas hold for any value of g. Therefore, a mechanism mass-balanced on Earth (g = 9.80665 ms−2 ) retains its properties even in different gravity environments, such as in space or on the surface of other planets; this can be useful for designs targeted for space exploration.

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(II) F = 0 and m c = 0 (no counterweight): this is a purely-spring-balanced mechanism. Here, again, we assume m > 0 and c = 0; thus, we disregard trivial cases where either k, s1 or s2 are zero, in which case the elastic potential energy Ve from Eq. (22) is constantly nought. From Eq. (25) we find sin(δ) = 0, which implies that7 φ − ψ is either 0 or π . Setting Vt,s = 0 then provides mcg = ±K s1 s2

(32)

with the right-hand side being positive if δ = 0 and negative otherwise. In this latter case, since both m, g and K are necessarily positive, at least one of the lenghts c, s1 or s2 is negative; this simply means that points G, A or B (respectively) must be on the other side of O with respect to the case illustrated in Fig. 1. This defines the simplest spring-balanced mechanism known from the earliest works on gravity balancing [19, 21, 82, 83]. (III) F = 0, m = m c = 0 (spring-based CFG): this corresponds to the design of a CFG by using a spring and a link of negligible mass. The force F can be, for example, the reaction from a link attached to link 1 in P. From Eqs. (25) and (26) and known trigonometric formulas, one derives the following conditions: p F = ks1 s2 cos( − δ) = 0

⇒

(33) π =δ± 2

(34)

Excluding trivial cases, Eq. (33) sets a condition on the spring, while Eq. (34) is a geometrical condition on the attachment points; this condition can be verified by considering the position at which both the spring and the force F have a null moment around point O. Similar results can be obtained from Eqs. (25) and (26) when only a counterweight is used instead of a spring. The conditions derived from these special cases can also be combined, thanks to linear superposition: for example, combining the conditions from cases (I) and (III) one obtains a CFG with non-negligible link mass m (which is gravity balanced by the counterweight m c ), where the spring balances the constant force F. Notice, however, that the linear superposition of cases (I), (II) and (III) provides sufficient, but not necessary, conditions for balancing; the most general case is defined by Eqs. (25), (26) and (30). In the discussion above, we considered a mechanism with only one spring, for simplicity. Mechanisms that are gravity balanced through several springs, for example acting in parallel between links 0 and 1, have been studied in previous works [80]; it was found that this allows us to relax the condition of having l0 = 0 (see Subsect. 3.5). In any case, these generalizations are conceptually very similar to the simplest CFG here discussed. In fact, it is known [84] that several springs of stiffnesses k j , acting in parallel between a frame 0 and a rotating link 1, can be substituted by a single 7

Without loss of generality, we assume all angles φ, ψ, α, β and θ to be in the range [0, 2π [.

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 spring, having stiffness K = k j ; the attachment points of the statically-equivalent spring are found by considering the equilibrium equations. To the best of our knowledge, the first CFG to be studied in academic works is the 1-DoF, spring balanced mechanism considered here (case (II)): for example, this is the base design presented in [33]. Indeed, this simple mechanism has been rediscovered several times in history, in more or less general versions. The first analytical study of this mechanism was proposed by K. Hain in the 1950s [82, 83]; in [13], however, the concept was recognized to be older, as the first technical descriptions were in patents from the early 1900s by W. Lynen [21] and G. Carwardine [18]. The earliest publication on the topic was presented by LaCoste [19]; here, the author assumed s1 = s2 (see Fig. 1), which is not a necessary condition. Notably, mass-balanced mechanisms can also be seen as a form of CFGs, as observed in case (III). Indeed, a mechanism with counterweights can be easily designed to provide a constant (vertical) force F = Fez at a given point P on its i-th link (under static conditions). Notice that the sum of the link weight m i gez , applied on point G, and of force F is a constant force Ft , applied at a point Pt , whose position pt = Pt − O is constant with respect to the link itself. In other words Ft = (m i g + F) ez m i gci + Fp pt = Pt − O = mi g + F

(35) (36)

Therefore, one can proceed to balance the mechanism with either counterweights or mass redistribution with the approach outlined in Subsect. 2.2, considering for the i-th link a fictitious equivalent mass m i,t = (m i g + F) /g applied on point Pt . The mechanism thus obtained will be in neutral equilibrium against a force F and can thus be seen as a CFG.

3.2 2-DoF CFGs In this Subsection, we will show how the results of Subsect. 3.1 can be expanded to obtain 2-DoF CFGs that can follow generic planar displacements of a point P, where a constant force F is applied. The generalization to spatial CFGs (with point P having full 3-DoF motions) will be presented in Subsect. 3.3. A first possibility to achieve a 2-DoF motion is to attach a link 2 on the CFG shown in Fig. 1 at point P. Link 2 can be mass-balanced by either applying a counterweight or redistributing the link mass, such that the conditions in Eq. (30) are fulfilled, with k = 0 in Eqs. (25) and (26), while a constant force F is applied on a generic point of link 2, as discussed in case (III). It is easy to see that the force applied by link 2 on link 1 at P is then constant, therefore link 1 can again be balanced using Eqs. (25), (26) and (30). Clearly, as each one of the links is balanced, the resulting 2-DoF serial linkage works as a CFG under static conditions.

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Fig. 2 Schematic of a planar, two-DoF CFG. Points O1 , P1 , P2 and O2 form a parallelogram

Another option, which will be analyzed in detail in the following, is to use spring balancing. It is convenient to devise a mechanism such that each link can be balanced independently with a spring, for instance by applying the results from case (III). However, since link 2 in the kinematic chain is not directly attached to the frame, it is not easy to directly use springs to balance it, as Eqs. (25), (26) and (30) no longer apply. Recently, a method to balance similar CFGs by only adding ZFLSs was proposed in [85]; a simpler result can however be obtained by adding auxiliary links. It is well known that a parallelogram linkage has purely-translational motion at the coupler link; we can then use the latter to attach the “fixed” end of the spring for link 2 and the spring will behave as if it were attached to the frame. With reference to Fig. 2, link 3 is designed to translate, so that d = (O2 − O1 ) = (P2 − P1 )

(37)

In Fig. 2, points P1 (on link 1) and O3 (on link 3) are coincident, in the center of the corresponding R joint; the same condition applies to points P3 and O4 . The global frame F  has origin in O1 . For each link i, we define (see also Fig. 1) the following angles, which are not shown in Fig. 2 for simplicity: θi between vector (G i − Oi ) and the x-axis, φi between (Ai − Oi ) and the y-axis, ψi = G i Oi Bi , and     βi = G i Oi Pi . On each link, a mobile frame Fi = (Oi , xi , yi ) is defined having xi oriented along (G i − Oi ); moreover, each link has associated vectors ci = (G i −   = (Ai − Oi ) and si,2 = (Bi − Oi ), whose lengths are ci , Oi ), pi = (Pi − Oi ), si,1 pi , si,1 and si,2 , respectively (for i = 1, 2, 3, 4). A constant force F, having magnitude F and at an angle γ with respect to the x-axis, is applied at the distal link of the chain (here, P ≡ P4 ). The spring attached

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on the i-th link has stiffness ki (where k2 = k3 = 0); we assume that both springs have zero free length.8 For simplicity, no counterweights are shown; they can be introduced in the design, if desired, by defining the CoM of each link G i as the combined CoM of the original link and of the added counterweight. The position vector of point P is given by p = p (θ1 , θ4 ) = (P4 − O) = p1 + p3 + p4 = p1 (θ1 ) + p3 + p4 (θ4 )

(38)

where p3 is a constant vector, since link 3 translates and θ3 is constant. Here, vectors pi are given by pi = Ri pi , Ri =

    cos(β) cos(θi ) − sin(θi ) , pi = pi sin(θi ) cos(θi ) sin(β)

(39)

The position vectors of the CoMs are given by Eq. (2), where ci = [ci , 0]T (from the choice of the local frames) and t1 = 0, t1 = d, t3 = p1 and t3 = p1 + p3 . The position vectors of the attachment points for the i-th spring are given by si,1 si,2

  − sin(φi ) = (Ai − Oi ) = si,1 cos(φi )   cos(ψi ) = (Bi − Oi ) = Ri si,2 sin(ψi )

(40) (41)

Notice that vectors s1,1 and s4,1 are constant: A1 is on the fixed frame, while A4 is on a link that does not rotate. Finally, we can write the total potential energy Vt from Eq. (1): the definitions in Subsect. 2.1 directly apply, with l j,0 = 0 in Eq. (5) and replacing ez with e y in Eq. (6). The potential energy Vo = V f due to F is again given by Eq. (15). As noted, θ3 is constant, while θ2 can be derived from the condition that O2 − P2 is parallel to O4 − P4 : this implies θ1 + β1 = θ2 + β2 . Accordingly, Vt is a function of two independent variables (θ1 and θ4 ), as expected, and q = [θ1 , θ4 ]T . Differentiating Vt with respect to each angle, we obtain two trigonometric functions of θ1 and θ4 with expressions similar to those in Eq. (23). The coefficients of each variable term, in sin(θi ) and cos(θi ), must then be set to zero, thus yielding the conditions p1 F cos(1 ) + m 2 c2 g sin(α) + k1 s1,1 s1,2 sin(δ1 ) + p1 (m 3 + m 4 )g sin(β1 ) = 0 (42) p1 F sin(1 ) + m 2 c2 g cos(α) + m 1 c1 g − k1 s1,1 s1,2 cos(δ1 ) + p1 (m 3 + m 4 )g cos(β1 ) = 0 (43) 8

Only two springs are employed, for simplicity, but more springs could be introduced; this provides greater freedom in design, but does not significantly change the results shown here. For instance, a spring between links 2 and 0 could be introduced; since link 2 and 1 move at the same angular speed, however, it can be shown that its effect is equivalent to that of a spring between links 1 and 0.

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and p4 F cos(4 ) + k2 s4,1 s4,2 sin(δ4 ) = 0 p4 F sin(4 ) + m 4 c4 g − k2 s4,1 s4,2 cos(δ4 ) = 0

(44) (45)

α = β1 − β2 δi = φi − ψi , i = 1, 4

(46) (47)

i = γ − βi ,

(48)

where

i = 1, 4

Comparing these constraints with Eqs. (25) through (28) and (30), some similarities appear. Equations (44) and (45) are simply those of a 1-DoF CFG as described in Subsect. 3.1, without the term due to the counterweight (since its mass has been lumped with that of the fourth link at G 4 ). Equations (42) and (43) are similar to Eqs. (25) and (26), too: they can be obtained after the equations for a 1-DoF, springbalanced CFG with the following elements: • a counterweight of mass m 2 : indeed, the potential energy term due to link 2 can be grouped with that of link 1, since these links have constant transmission ratio; their angular positions (with respect to the R joints in O1 and O2 ), however, differ by a constant amount, equal to angle α; • a constant force F at an angle γ with respect to the x-axis, applied in point P: the force applied on link 4 is transmitted to link 1; • another constant force g(m 3 + m 4 )e y at an angle γ with respect to the x-axis, applied again in P: this is the force (directed upwards) that link 1 exerts to balance the weight of links 3 and 4. The constraints in Eqs. (42) to (45) are decoupled, meaning that the first two only apply to link 1 and the remaining two to link 4; also, all terms in Vt (θ1 , θ4 ) only depend on one of the two variable angles. This is due to the geometrical constraint in Eq. (37), which greatly simplifies the analysis and allows us to find exact solutions. Indeed, the parallelogram architecture is frequently employed in these designs, as discussed in Sect. 5. Often, the links are designed to be slender and compact; therefore, in many cases links have collinear points Oi , G i and Pi . We remark that the design described in this Section is by no means the only possibility to achieve a 2-DoF CFG; for example, a recent concept of a two-DoF system was presented in [86].

3.3 3-DoF CFGs In this Subsection, we will present two possible examples of CFGs that are designed for general spatial motion of the force application point P. A first example can be

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Fig. 3 A 3-DoF CFG with RRR architecture. This design is derived from a special case of Fig. 2, where O1 , P1 , O2 and P2 form a parallelogram with two vertical sides O1 − O2 and P1 − P2 ; the linkage rotates around the z-axis through O1 − O2

easily derived from the design in Fig. 2: since the mechanism is planar, with the gravitational acceleration vector g contained in the plane of motion, its static equilibrium does not change if the linkage (including frame 0) is allowed to rotate around a vertical axis in the plane. An advantageous design of this concept is presented in Fig. 3: if points A1 , O1 and O2 on frame 0 are collinear on the same vertical axis, frame 0 can be kept fixed. Then, a single extra R joint has to be added (with respect to the design in Fig. 2) to achieve spatial motion; the joint in A1 becomes an S joint, while the rest of the mechanism remains identical. It is easy to see that, if the constraints from Subsect. 3.2 hold, the total potential energy Vt (q) remains constant as the con T figuration q = θ1 , θ4 , θz changes: the mechanism is thus balanced and applies a constant force at P. Designs conceptually similar to the one in Fig. 3 can be found in the literature. For instance, in [74] the authors propose leg architectures for gravity-balanced 6-DoF spatial PKMs, where, in one of such designs, each leg is a parallelogram linkage similar to the one considered here, but the vertical axis for out-of-plane rotation is introduced between links 3 and 4. A number of linkages such as the one in Fig. 3 can be arranged in series, to obtain a serial mechanism as the one presented in [49].

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Fig. 4 The 3-DoF CFG presented in [52]. This CFG was designed to be employed as a leg for a gravity-balanced 6-DoF PKM. The leg can rotate with respect to the fixed frame around a U joint in point O, and is supposed to be attached to the moving platform by a spherical joint centered in P. A number of ZFLSs are attached between the frame and the leg (only one spring is shown, for simplicity)

As an alternative design, we also discuss a spatial CFG first introduced in [52]. This is a 3-DoF device which is gravity balanced throughout its motion; the allowed movements are rotations around the two axes of a U joint (intersecting at point O in Fig. 4) and a translation of link 2 with respect to link 1, along the axis of a P joint, parallel to P − O. We wish to define a UP spatial linkage capable of applying a constant force F at point P on the distal link 2. Since the mechanism includes a P joint, it is not possible to balance it by simply redistributing the link masses or attaching fixed counterweights [36, 64–66]; accordingly, the original element of this design is adding a movable counterweight 3, which is driven by the same actuator that controls the distance from O to P. To this end, a transmission with constant ratio τ is employed; therefore, if link 2 moves with respect to link 1 at a speed ρ, ˙ the counterweight 3 will move at speed τ ρ. ˙ Each of the three links has a mass m i and a CoM G i ; n s ZFLSs are connected between link 1 and the fixed frame. Again, we disregard trivial cases in which the j-th spring has stiffness k j = 0 or its attachment points A j and B j are coincident with O. Let F = (O, x, y, z) denote the fixed global frame, while F  = (O, x  , y  , z  ) is the frame attached to link 1; since the two frames share the same origin, the translation vector in Eq. (2) is zero, while the rotation matrix R can be defined by the two angles φ1 , φ2 that describe the rotations around the axes of the U-joint. We also

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set z  directed along P − O. In the following, we employ the following subscripts for conciseness: • 0: denotes a quantity evaluated at reference position ρ = 0 (namely, when the slider is completely retracted); • ⊥: denotes the component of a vector orthogonal to the z  axis; • z: denotes the component of a generic vector v along the z axis of the global frame F (respectively, along the z  axis for a vector v expressed in frame F  ). The position vectors of points G i can then be directly expressed (in frame F  ) as functions of the input variable ρ: c1 =

  ez c1,z + c1,⊥     c2 = ez c2,z,0 + ρ + c2,⊥     c3 = −ez c3,z,0 + τρ + c3,⊥

(49)

From Eqs. (49) and (12), one finds the position vector of the global CoM for the mechanism in the mobile frame, namely 3    c = ez Mcz,0 + (m 2 − m 3 τ )ρ + m j cj,⊥

(50)

j=1

where M = m 1 + m 2 + m 3 is the total leg mass and  = cz,0

   + m 2 c2,z,0 − m 3 c3,z,0 m 1 c1,z M

(51)

is a constant. From the previous definitions, we can write the components of the total potential energy Vt in Eq. (1) (where Vo = V f is the energy due to the constant force F) as Ve =

ns ns ns    1 1 k j s j,1 − Rsj,2 2 = k j s j,1 2 + sj,2 2 − k j sTj,1 Rsj,2 2 j=1 2 j=1 j=1   =V˜ (constant)

      Vg = M Rc · g = Mcz,0 + (m 2 − m 3 τ )ρ Rez + MRc⊥ ·g      V f = F · Rp = p0 + ρ F · (Rez )

(52) (53) (54)

where p = (P − O) in frame F  . Differentiating Vt with respect to ρ and setting it to be constant for all values of φ1 and φ2 , it can be shown that it must hold F = (m 3 τ − m 2 ) g = m f g

(55)

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and thus the constant force generated by the leg is necessarily vertical. Furthermore, it can be seen that, if F is positive, namely pointing upward (which is the case of most practical interest, if the CFG has to be used to balance a body coupled to it) one needs a counterweight of mass m 3 = m 2 + m f /τ > 0. In [52], it was also shown that the total potential energy Vt is equivalent to that of a single link (with fixed CoM C) connected to the frame by a U joint and to n s ZFLSs, namely Vt = V˜ −

ns 

   k j sTj,1 Rsj,2 + M + m f RcC · g

(56)

j=1

where cC =

Mc + m f p  = cC,0 M + m f

(57)

meaning that the position of C does not change as ρ varies; its position is the global CoM of the leg, considering F = m f g as the force due to a point mass m f applied in P. Expressing the potential energy as in Eq. (56) allows us to advantageously apply previous results. Indeed, the gravity balancing of a link with fixed CoM connected to the frame with a U joint was already studied in [87] (for joints where the fixed axis is horizontal) and [88] (where the fixed axis is vertical). These results were later expanded and generalized in [52] for arbitrary joint and link geometry. The 3-DoF CFG could also be balanced by having the global CoM C in point O; in this case, the conditions for complete force balancing are fulfilled (the CoM does not move). Gravity balancing (without inertia-force balancing) could also be achieved by having C on the moving axis of the U joint and g aligned with the fixed axis; in this case, the CoM moves, but has zero displacement in the direction of gravity, a possibility noted in Subsect. 2.2. In both cases, no springs are required for balancing; however, as noted in [52], this may lead to onerous constraints on the size of the counterweight, which have to be installed beneath the base (thus making their installation more complex).

3.4 Other CFG Mechanisms A number of other CFG designs have been proposed beside the ones discussed in Subsects. 3.1 to 3.3. In [17], a review of previous works is presented, with a toplevel classification of designs based on whether the CFG is based on a conventional or on a compliant mechanism. Compliant mechanisms are designed taking the link deflections into account and often have a monolithic design, meaning that they are composed of a single link that deforms thanks to its own internal elasticity: the conventional joints are thus substituted by flexural joints, which are zones of the

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mechanism designed to have controlled deflections. As observed in [17], common limits to the performance of CFGs are friction and backlash, which limit the accuracy in the force generation: compliant mechanisms are then an interesting design alternative, as they can almost entirely eliminate these issues. Some designs reviewed in [17] also avoid the requirement of using ZFLSs. A few CFG devices have been patented for hand-operated mechanisms, but could also be applied to actuated linkages and robots [89–91]. Common applications of CFGs include manipulation of small, delicate objects (for instance in the biomedical sector), as they intrinsically provide protection against the risk of overload being manipulated, without using force-control feedback: in fact, the mechanism used to grip on the objects always applies the same force on it. For this same reason, CFGs can also offer a friendlier interaction when a robot must safely operate in close contact with human operators. Finally, CFGs are also applied as vibration isolators [28, 29, 92, 93]: when used to suspend a vibrating load, they transmit a constant force to the base. For a global overview on vibration isolators, we refer the reader to [94]: several designs presented therein can be recognized to be generators of a force that is either constant or quasi-constant over a large range of motion. The stiffness of a system capable of deforming in a given direction nˆ is defined as the ratio k = F/Δ between the magnitude F of the force F nˆ along said direcˆ by tion and the magnitude Δ of the corresponding (elastic) deflection Δnˆ along n; combining elastic elements with links and joints or redefining spring designs [42], several mechanisms have been proposed that offer a non-constant stiffness k, which can even become negative for some range of motion [95]. In this perspective, CFGs can be considered as zero-stiffness devices [17, 28, 29, 93, 94, 96], since F does not change as Δ increases. This is in fact what allows us to use CFGs as vibration isolators: since the resonant frequency ωn is proportional to (the square root of) k, such mechanisms offer extremely low values of ωn and can thus always work well beyond the resonant frequency. For example, the mechanism proposed by LaCoste in [19] is designed to realize a seismograph having null resonant frequency. Special designs of zero-stiffness mechanisms can be obtained by taking advantage of the post-buckling behavior of flexible beams. Indeed, a system in buckled mode undergoes large displacements under a constant load; while this is generally seen as a risk of failure for structures, buckling was successfully employed for the design of vibration isolators [92, 94, 95] and zero-stiffness sytems [96, 97]. As observed in [98], the theory of buckling can provide a new strategy to achieve gravity-balanced mechanisms. More recently [99], post-buckling analysis was applied to a five-bar (RRRRP) mechanism with springs in each joint: the linkage was shown to be gravity balanced and the spring stiffnesses compensated each other. Finally, we mention yet another class of CFGs, where an elastic element is designed such that only a part (having constant volume) of the element undergoes elastic deformation: this way, the elastic force does not vary as the deflection increases, but remains constant instead. This is for example the case of “Neg’ator” (or constant-force) springs [42, 43, 96]. The tape springs commonly used in carpenter tapes are another example: these are thin elastic strips with circular cross-section

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that display a form of localized buckling [100, 101]. Other systems with a constant amount of material deformation were also proposed in [97] (inspired by lotus receptacles) and [102] (a patented linear bearing with an elastic element, but no preferred equilibrium position). Such designs can be applied in gravity balancing, by attaching a constant-force spring to the mass that is to be balanced against gravity [42].

3.5 Design Observations As observed in Sect. 3, the gravity compensators shown in Figs. 1, 2 and 4 require the use of a ZFLS in order to achieve perfect balancing. In fact, most spring-based gravity-balanced mechanisms proposed so far introduce springs with l0 = 0: this way, the deflection Δ = l − l0 equals the spring length and the force characteristic of the spring is a simple linear function F = kΔ. Clearly, a compression spring does not naturally have a zero free length. Extension springs, on the other hand, can be manufactured to be ZFLSs [19]: this implies that, while the spring is unloaded (at rest length l p ), there are internal stresses (due to, for example, the manufacturing process) such that the spring is preloaded with a force F p and its coils are pressed against each other: a nonzero force F p = kl p is then required to separate the coils and deform the spring. However, this introduces significant stresses in the spring and may reduce its fatigue life; moreover, a high preload force F p may lead to loss of stability in the spring [13]. Several designs were therefore proposed to obtain ZFLSs by combining normal springs with other mechanical elements. For example, a common method [13] is to introduce cables and guides (such as pulleys) to transmit the force from the springs to the links: the length of the cable is chosen such that, when the spring is at rest length l0 > 0, the cable attachment point B on the link coincides with the cable exit point A on the frame. This way, the spring-cable system collectively acts as a ZFLS. Another approach is to combine elastic elements with negative and positive free length in series, such that the global equilibrium point of the system occurs when the system is at zero length [2]. We also remark that some works [103] showed the possibility of achieving exact gravity balancing while using conventional springs with l0 > 0: these simplify the design and manufacturing, as they can be bought as off-the-shelf elements. It was however showed [80] that, for the 1-DoF system in Fig. 1, this requires using more than a single spring. In general, the advantage of using normal springs should be balanced against the greater complexity of the resulting design [103–105]. It is worth noting that, once gravity balancing is achieved for a given mechanism, it is not influenced by the choice of the actuated joints: since motion occurs without energy transfer under quasi-static conditions, the actuators do not have to apply any force or torque to the linkage. This is important in the design of PKMs, which have many more joints than DoFs to be controlled: for a given mechanism, the designer can choose which joints ought to be actuated and which should be passive. This choice can be made independently of the design considerations regarding balancing.

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Mass-balancing a mechanism is achieved by adding counterweights: their masses are found depending on the problem at hand, through conditions such as Eq. (31). Afterwards, it is an easy design problem to realize a counterweight with the required mass, which can have any real value within a continuous range (compatibly with manufacturing limitations). Using springs, on the other hand, usually constrains the designer to choose from a discrete catalog by suppliers: therefore, there is a finite set of stiffnesses k to choose from. The balancing equations generally involve k: see for instance Eq. (32). Therefore, it is advisable to choose a convenient value of k from a catalog; then, the other design parameters, such as s1 and s2 in Eq. (32), can be freely chosen as long as the constraint equations are respected. More generally, all spring-balanced mechanisms found in this review allow one to arbitrarily choose the spring parameters and then accordingly vary the link dimensions. In order to reduce the reactions transmitted by the balancing elements to the links, such as side loads and torques, said elements should be as close as possible to the vertical plane containing the gravity forces to be balanced. For instance, in the mechanism shown in Fig. 1, one could move the spring attachment points A and B along the normal to the view plane: it can then be shown [80] that the balancing conditions remain the same, since a translation of the attachment points along an axis normal to the plane of motion does not influence the balancing moment, as long as ZFLSs are used [84]. This way, one must use S joints to connect the spring ends to the links (instead of R joints); the designs are otherwise unchanged. However, using off-plane springs clearly introduces lateral loads on the pins in A and B and also increases the energy Ve stored in the spring [106]. Nevertheless, several works [80, 88, 107] propose using off-plane springs, as this allows greater design flexibility and reduces the risk of spring-link interference for planar systems. An issue for most gravity-balanced systems is lack of compactness. Adding counterweights of significant mass without significantly increasing the inertia of the links (which is undesirable for dynamics applications) is complex and generally requires using large counterweights; for planar mechanisms, these also protrude significantly from the plane of motion. Springs attached to the frame reduce the accessibility if a user has to interact with the system; it is preferable in this case to reroute the spring force transmission system through mechanisms (such as those used in household appliances, like oven or dishwasher doors, which are usually balanced against their own weight) that are not directly accessible, at the expense of an increase in complexity. Also, springs clearly increase the risk of interference with other links or with the environment; while in theory there is a significant leeway in choosing which links the springs should be attached to, it is recommended to avoid solutions where the springs cover long “spans” across links that are not directly connected to each other by joints. Some concepts for more compact spring-balanced joints were proposed, for instance, in [47, 105, 108, 109]. As seen in Subsect. 3.4, CFGs such as those used for gravity balancing have intrinsically zero or quasi-zero stiffness, therefore the resonant frequency is close to zero. Therefore, gravity-balanced systems are less commonly employed in applications which alternate high- and low-dynamic phases, as there is a risk of inducing vibrations [110] when the mechanism motion starts or stops and low-frequency excitations

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are introduced. Furthermore, gravity-balancing elements tend to add inertia to the links, thus worsening dynamic performance in some applications, and in general the advantage of balancing gravity forces is less relevant when inertia forces become predominant. For the reasons outlined above, in almost all works reviewed in this Chapter, links are considered to be perfectly rigid and dynamic effects negligible. Since weight forces are frequently the most significant external actions on a robot, it is safe to assume that the stresses on the links and the corresponding deflections due to the internal link elasticity can be disregarded. Thus, the elasticity is thought to be concentrated in the springs or in the compliant elements (where present). Instead, dynamically-balanced linkages (see Subsect. 2.2) frequently operate at high speeds and accelerations; therefore, the effect of link elasticity is often non-negligible and kinetoelastodynamic effects ought to be taken into account [111, 112]. A general requirement for all robots, especially those that work in contact with users (such as those for medical or rehabilitation robotics) or in industrial environments, is that they must be fail-safe: in practice, this implies that, should control signals or power no longer be supplied to the robots, they cannot have uncontrolled motions. For example, a serial-arm robot would abruptly “fall down” during a power failure under the effect of its own weight. To avoid this risk, passive brakes are usually installed on the actuators: these automatically activate when the robot is turned off, but require energy to be deactivated when the robot is in use [39]. Gravity-balanced robots are thus intrinsically safer and have reduced energy consumption. If a change occurs in either the payload on the EE or the link weight of a gravitybalanced robot, in general it will no longer be in neutral equilibrium and will instead start moving until it reaches the closest (stable) equilibrium configuration qeq . Massbalanced robots will reach the configuration with the lowest potential energy Vg : the new equilibrium configuration (after payload variation) may however be far away from the current one (even if the payload change is small). On the other hand, a springbalanced mechanism will have an elastic displacement under a different load to reach equilibrium, but the displacement from the start configuration will be proportional to the payload change [39]. From a safety viewpoint, this latter situation appears preferable, as it is less likely that the robot will move uncontrollably. On the other hand, installing and maintaining high-energy-storage springs (such as those used for balancing large masses) introduces safety issues, as they may unload violently if not properly handled [113]. The performance of a theoretically-balanced mechanism will degrade due to various error sources, such that in practice balance will not be exact. Some of such errors are detailed as follows. • Manufacturing tolerances will cause the actual mechanism parameters to be different from the theoretical ones; in practice, no mechanism can be at exactly neutral equilibrium [97], since this is a “border” condition between stability and instability. • The nonlinearity of the spring force-displacement characteristic can have an influence: conventional springs are designed to have a linear relationship as shown

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•

•

•

•

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in Eq. (4), but in practice there is always a small deviation from an ideal linear behavior [44]. It is frequently necessary to add auxiliary members to a given linkage so that it can be balanced. For example, up until recently, all designs for spring-balanced mechanisms having more than three R joints used auxiliary bodies [85]. These members are optimized to be as lightweight as possible in order to minimize the added mass; thus, they are often disregarded in the computation of the total weight [114], which introduces a small balancing error. The effect of spring masses, which are usually disregarded, may in fact alter the global equilibrium. The spring masses will in general be small in relative terms, but can be quite substantial in absolute terms: some works [113] on gravity balancing report spring masses up to 40 kg, for instance. Moreover, as derived in [115] from considerations on the maximum allowable energy storage, for very large mechanisms the mass of the springs can even exceed the mass of the mechanism itself. A few of the works cited here [13, 66, 116] take into account the mass of each spring, usually by introducing two point masses (one at each spring terminal) each having half of the total mass; these masses can be added to the masses of the links to which the spring is attached. Where possible, it is preferable to have one end of each spring fixed to the frame and the other to a moving link, so that half of the spring mass can be considered as fixed and does not influence balancing [116]. Even better, in this sense, is to employ force-transmission systems from the springs to the links, such as cables and pulleys; this way, the springs can keep the same orientation [104], which will reduce inertial effects, especially if the springs are in a horizontal position [115]. The effect of spring mass can thus be completely removed from gravity balancing (note, however, that it will still influence the total inertia under dynamic conditions). Where cables are introduced in the design to transmit spring forces, they are generally employed together with pulleys [13, 104], cams [27, 44, 46, 48, 117], fusees [117] or other cable-guide systems, to guide the cables or alter the forcetransmission properties. Usually, these elements are taken into account in the mechanical model with a number of simplifying approximations; for instance, it is common to assume that the point of contact between a cable and a pulley is a fixed exit point, while in reality it changes depending on the cable orientation. This error may be compensated by correcting the contact profiles [44], but this complicates the design equations significantly. Alternatively, some ingenious mechanisms have been proposed where the cable loops on several pulleys and the pulley wrapping angles compensate each other [104]. The effect of friction cannot be included in the total potential energy in Eq. (9), since it is a non-conservative force. Yet, friction will in practice be present in all joints. Its effects are detrimental when the robot has to move, as the system will no longer be energy-free: even in quasi-static conditions the actuators will have to compensate the energy dissipated by friction, thus the energy requirements will increase, especially when the robot has to move frequently. At the same time, friction helps to compensate for small unbalances due to the error sources just

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listed, as observed in point (1) of Subsect. 2.1. Friction effects are especially useful (in small amounts) for hand-operated systems [23]. It is worth noting that the aforementioned errors are not negligible, as errors up to 20% of the moment to be balanced were reported [2].

4 Statically-Balanced PKMs by Way of CFGs In this Section, we will show how CFGs can be combined to provide gravity-balanced PKMs. For most PKMs, the weight of the links is small in comparison with the total weight of the EE (including the payload); this holds especially for large, heavyduty manipulators such as flight simulators. Disregarding the weight of the links, a gravity-balanced PKM can be seen as a CFG, which has to apply at the CoM of the EE a constant force equal in magnitude and opposite in direction to the platform weight. The problem of providing a constant force at a point of the EE moving with multiple DoFs has been approached with two main techniques: (a) some authors introduce an auxiliary kinematic chain, designed as a CFG (for example, such as those described in Sects. 3.2 and 3.3), from the base to the EE and attached to its CoM (for example, by way of an S joint). (b) other authors design the legs of the PKM such that each leg exerts a constant force, which is a fraction of the total EE weight; this way, each leg is a CFG and no auxiliary kinematic chains need to be introduced. In this Section, we will take the work from [52] as an example for discussion; this work takes the approach outlined in case (b). Indeed, the method in case (a) obviously leads to a balanced design and requires no further considerations; on the other hand, for case (b) it remains to be shown that the EE can be in static equilibrium through the application of several CFGs (one for each leg). Some other works will also be cited later as a perspective on CFGs applied to gravity balancing of PKMs.

4.1 Motion of a Rigid Body in Space Assume that the EE is coupled to n F legs, each one of them working as a CFG (for example of the RRR o UP type, as described in Sects. 3.2 and 3.3, or a parallelogramtype CFG of the types discussed in [116]) connected to the EE in point P by an S joint. Therefore, n F constant forces Fi act on the EE as it moves, and their attachment points P1 , . . . , P6 have fixed positions with respect to the EE itself. Two coordinate frames F = (O, x, y, z) and F  = (O  , x  , y  , z  ) are defined, respectively on the fixed frame and on the mobile body; see Fig. 5. A constant force F, such as the

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Fig. 5 Schematic of a rigid body upon which n F constant forces Fi are acting (at points Pi ), together with n τ constant torques τ i , while it moves in space; the system of forces balances a single force F applied in O  , which is the origin of the link-fixed coordinate frame

weight, acts on the body in point O  . From the principle of virtual work, the body is in equilibrium if and only if F · δ O +

nF 

Fi · δ Pi = 0

(58)

i=1

where δ O  and δ Pi are the virtual displacements of the corresponding points on the body. In this case, Pi − O  = Rpi , where pi = (Pi − O  ) is the position vector of the i-th force application point (in the local frame), and the rotation matrix R = R jk can be expressed as a function of three rotation angles Φ = [φ1 , φ2 , φ3 ]T (according to a preferred convention). One thus has δ Pi = δ O  + δ(Rpi ) = δ O  +

3  ∂R  p δφ j ∂φ j i j=1

(59)

Equation (58) can be rewritten as  F+

nF 

 Fi

· δ O  + (∇W) · δΦ = 0



nF 

(60)

i=1

where



L = L jk =

 T Fi pi

i=1

W = W (φ1 , φ2 , φ3 ) =

3 3   j=1 k=1

(61) R jk L jk

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Since Eq. (60) must hold for arbitrary displacements δ O  and rotations δΦ, it necessarily follows F+

nF 

Fi = 0

(62)

∇W = 0

(63)

i=1

where it can be shown that Eq. (63) implies L=

nF 

 T Fi pi = 0

(64)

i=1

While Eq. (62) simply represents the force equilibrium equation for the body, Eq. (64) sets a geometrical condition: as shown in [52], it implies that O  lies in the affine space of points Pi . This condition sets a constraint on the design: if all Pi ’s are coplanar on Π , then O  should be on Π , too. As O  coincides with the platform CoM, if F is the weight to be balanced, this constraint may be difficult to realize in practice; for a flight simulator, for instance, the weight is usually on one side of the plane passing through the leg attachment points [76]. In these cases, an approach is to introduce a CFG leg whose attachment point Pi is not coplanar with the others and applying a negative force, meaning that it pushes the platform downwards. While theoretically feasible, this solution proportionally increases the total load to be balanced by the other legs, thus increasing stresses and reducing fatigue life.

4.2 Gravity-Balanced PKMs To the best of our knowledge, the earliest application of a CFG for balancing a closed-loop linkage was in [22], where a pantograph linkage (with a counterweight) was attached to the CoM of a four-bar linkage; physically, the CoM was located by auxiliary devices. This design is an example of case (a) as previously defined. In [66], the author proposed a 6-DoF spatial parallel robot where each leg is statically balanced: the static analysis was simplified by assuming that the total mass of the EE could be replaced by three equivalent point masses located on the leg attachment points. General results for planar linkages (including closed-loop ones) were offered in [116]: it was shown that a generic planar mechanism can be balanced against gravity, by adding a CFG to each link in its CoM. Although theoretically feasible, this clearly increases the complexity; therefore, a comparison of two planar CFG designs in terms of the number of auxiliary elements was proposed. Later, a general approach to device spring-balanced mechanisms (both serial and parallel) that could

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balance both their own weight and external constant loads, but without any auxiliary links, was presented in [85]. In [118], the 1-DoF CFG studied by Nathan in [33, 91] and extended to multi-DoF, spatial serial systems in [49] was extended to parallel systems, by devising staticallybalanced legs; the authors thus designed a 3-DoF platform (with one translation and two rotations). A class of PKMs with CFG legs was presented in [76]: each leg had a parallelogram design, inspired by previous works on planar mechanisms [36, 116], and balancing was achieved through ZFLSs. As in [52], it was found that the CoM of the EE ought to belong to the affine space defined by the attachment points; the possibility of using legs applying a negative force on the EE was also considered. In [119], a 6-DoF parallel platform with PUS legs was proposed: the EE is balanced by attaching a pantograph-like mechanism (with a counterweight) to its CoM. Counterweights are also applied at the distal end of each leg, so that the CoM of the entire linkage coincides with the CoM of the EE. Some practical remarks were also offered on the optimal design in order to reduce inertias. A similar concept was proposed in [3], where a Delta robot was balanced by a pantograph mechanism, again applying a constant force on the CoM. The (constant) balancing force on the proximal side of the pantograph was provided by an actuator and is thus a form of “active” balancing, but the mechanism proposed is general and can be employed in combination with passive CFGs or for other robot platforms. An auxiliary actuator is however needed to move the balancing system out of singular configurations, which occur in the central axis of the robot’s workspace. A prototype was also presented, which showed that balancing greatly reduces pose errors due to elastic deformations. More recently, an Orthoglide (a 5-DoF parallel manipulator with a “Linear-Delta” architecture) was considered in [120] for static balancing; in this robot, the EE has purely-translational motion, while the two rotational DoFs are controlled by a spherical wrist on the EE. The authors advanced the idea of balancing the platform through springs that could be attached either directly to the platform or through cable-routing systems; the latter option is preferable in terms of design, as in this way the springs are mounted to the base and their inertial effects are negligible, as observed in Subsect. 3.5. Each balancing device is designed as a CFG. Since the platform has translational motion, Eqs. 62 and 64 can be considerably simplified; it was found that, due to the special architecture, it was sufficient to attach a CFG on the vertical leg of the robot. Later, in [121], a 5-DoF, fully parallel robot with UPS was proposed and shown to be statically balanced by transforming each leg into a CFG through the use of springs.

5 Applications of Gravity-Balanced PKMs As noted in Subsect. 2.3, gravity balancing is commonly employed in serial robots. On the other hand, gravity balancing is somewhat overlooked in PKMs: these robots

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usually have high stiffness, thus making the loss of accuracy due to static deflections a smaller concern. Moreover, closed-loop linkages complicate the design of balancing systems, especially for spatial robots. Finally, PKMs offer high dynamic performances and thus are frequently applied to tasks where they move at high speeds and accelerations: since gravity-balancing devices usually add mass and inertia to the links, their usefulness is less obvious. Gravity balancing is therefore applied mostly for PKMs where the EE exhibits to carry large loads, for instance due to its own weight when the moving platform is large; also, balancing becomes more useful when the robot has low or moderate dynamics, with long periods being at a rest configuration. Examples of PKMs more suited for gravity balancing are thus those used for machining tasks [120, 121], loading and unloading heavy objects on pallets [113, 122], medical devices [32], robotic-assisted surgery [3], and flight simulators [76]. The latter application appears especially suited for balancing: hydraulic actuators are frequently employed, but if they have to withstand large loads, fluid losses will reduce efficiency [73]. Given the complexity of directly balancing a given closed-loop mechanism, several works [3, 49, 51, 64, 114, 123] propose adding auxiliary idler loops or linkages on a principal linkage that has been previously designed; these auxiliary elements can be designed as CFGs, acting on the CoM of each link [116]. Conversely, a general approach to achieve exact balancing of planar linkages by adding ZFLSs (but no auxiliary elements) was presented in [85]. Adding auxiliary linkages provides the designer a greater flexibility, since they yield less “constrained” solutions (namely, the space of parameter arrays a satisfying the gravity balancing conditions is higher-dimensional). Since auxiliary links are generally subject to only moderate loads (as the total weight of the robot is shared among the principal and the auxiliary links) and are usually designed to work in favorable load conditions (pure axial loads), these links can be quite slender; thus, the increase in mass and inertia is usually acceptable. The main issues of this approach are the increases in friction dissipation due to the added joints, in the risk of interference (between the principal and the auxiliary links), and in the overall mechanism complexity. Commonly, auxiliary linkages are designed as articulated parallelograms [18, 20, 33, 36, 39, 49, 66, 72, 74, 76, 91, 108, 123–125], which can be arranged to define pantographs [3, 22, 58, 90, 114, 116, 119, 126]. These linkages have been successfully applied both for spring and mass balancing. In particular, pantographs “rescale” the motion of a point P at another point Q (with respect to the origin O) by a constant factor f , thus (P − O) = f (Q − O); they allow the balancing linkage to follow large displacements of an attachment point P (on the link to be balanced) with smaller displacements of the balancing element. This way, the robot can have a large workspace while maintaining a compact gravity compensator [3, 119]. Also, pantographs can be used to physically locate the CoM of a mechanism [22, 114], which then coincides with a joint of the balancing linkage; a CFG can then be applied at the joint [123].

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5.1 Mass Balancing As observed in Subsect. 2.3, mass balancing is generally found to be more effective for PKMs moving at low speeds [71]; otherwise, the inertia forces due to counterweights may overcome the advantages brought by balancing. The total mass added in order to constrain the CoM position to remain constant can indeed be significant: for a four-bar linkage taken as a reference example [77, 112, 127], the total mass of the counterweights is about 50% greater than the total mass of the original (unbalanced) mechanism. Preferably, masses should be added on the proximal links, namely, those connected to the fixed frame; this way, the counterweights have smaller movements during robot operation and inertial effects are proportionally smaller [115]. With reference to the 1-DoF CFG in Fig. 1, notice that Eqs. (26), (25) and (30) only set a constraint on the value of the total product m c cc , but do not define the mass m c of the counterweight. It is therefore desirable to choose the shape of the counterweight that minimizes the total moment of inertia Jtot around point O, which will create an inertia moment during dynamic motions. Clearly, the best results can be obtained by using high-density materials and concentrating the inertia in the direction normal to the plane of the figure; however, the choice of materials that can be practically employed in a robot design is limited, as is the maximum allowable dimension of the counterweight in the out-of-plane direction. For a given density and thickness of the counterweight, it is known [15, 54] that the lowest Jtot is obtained by designing the counterweight planar section as a circle tangent to point O; the circle radius is then found from the constraint on m c cc . These results generalize to other mass-balanced, multi-DoF mechanisms. As for reducing the total number of counterweights, it was shown in [65] that, for a planar linkage having n l links, the minimum number of counterweights sufficient for full force balance is n l /2. Finally, we remark that, if size constraints are especially tight, one could use remote counterweights that are not directly connected to the manipulator; the gravity compensation force can then be transmitted to the robot through compact transmission devices, such as hydraulic systems [128]. A notable advantage of mass balancing for PKMs, first observed in [118], is that it is independent of the direction of the gravity vector g, if the CoM position c is constant. In other words, if the orientation of the fixed frame is changed, the mechanism remains balanced, whereas spring-balanced robots are in neutral equilibrium only for a specific orientation. This feature makes mass balancing attractive for portable manipulators (for instance on vehicles).

5.2 Elastic Balancing Compared to mass balancing, elastic balancing offers greater freedom in the design phase and allows us to achieve gravity balance with a limited increase in the total inertia of the mechanism; on the other hand, a spring-balanced mechanism will

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no longer operate as desired if the base orientation is changed. For these reasons, springs are preferable in mechanisms operating at high dynamics; some designs with a combination of springs and counterweights were also presented [77, 78, 113, 122]. Notably, springs cannot in general balance inertia forces. While it is theoretically possible to use spring compensators for this purpose [15, 129], this generally requires the use of cam-follower mechanisms; most importantly, the actuator torques will be balanced only for a specific motion q = q(t). If a mechanism has to perform different motions that can be programmed by the user, as is almost always the case with robots, dynamic balance is required, which cannot be achieved through springs. An advantage of balancing through springs is that they allow us to achieve forms of adaptive balancing, meaning that the gravity-compensator mechanisms can be regulated by the user: this is essential to have gravity balancing after a change in the payload weight on the EE. Considering again the 1-DoF CFG in Fig. 1, case (II), one can see from Eq. (32) that, if the weight m to be balanced changes, the mechanism can still be in equilibrium, for instance, by simply changing the distance s1 from point A on the fixed frame to the R joint in O. Designing an attachment point on the base that can be moved along a slot is technically quite easy, while changing the position of a counterweight on a moving link is more complex. However, changing the position of A (for fixed orientation angle θ of the CFG) in this design is not an energy-free operation, since the spring potential energy varies; one needs either an actuator or the force provided by an user to regulate the mechanism, which may require considerable effort. Thus, in general, adapting a compensator for a different payload requires an independent actuator [47, 109]; nonetheless, some spring-based designs were presented [104, 130, 131] that allow the adaptation of gravity balancing in an energy-free manner. This is usually achieved by introducing a spring-to-spring compensator, where the “main” balancing spring is balanced against a second spring that compensates the potential energy variation for adapting the mechanism to the payload change. A special class of PKMs are tensegrity mechanisms: these are defined [132] by having all links either in tension or in compression (where the elements in compression are not directly connected). Having the links under purely axial loads is advantageous for design and allows using slender, light-weight links. Generally, these mechanisms are actuated by controlling the lengths of some of their links. Notably, tensegrity mechanisms are usually kinematically indeterminate [133], as some DoFs are not controlled: the final configuration of the mechanism thus depends on the applied loads and the static and kinematic problems must be solved simultaneously. The tensile members in a tensegrity mechanism can be springs [132], in which case it is natural to ask whether the resulting mechanism can be statically-balanced: indeed, in [133] it was shown that a tensegrity system can be designed to have zero stiffness as defined in Subsect. 3.4. The first gravity-balanced tensegrity mechanisms were presented in [134] and had 3-DoFs; more recently, a 6-DoF, gravity-balanced mechanism was presented in [135]. For these systems, it was found that balancing makes the static analysis much easier and allows analytical solutions to be found.

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265

Compliant Mechanisms

Compliant mechanisms, as first defined in Subsect. 3.4, are defined by having flexible links and elastic joints that are generally monolithic: for instance, the pin in a conventional R joint is replaced by a flexure, namely a zone within the mechanism of controlled stiffness that allows large rotational deformation and negligible translational displacements. Given the elasticity intrinsic in compliant-mechanism design, we consider gravity-balanced compliant systems as a sub-case of mechanisms that are gravity balanced through elastic elements. An issue of compliant mechanisms is that, due to the elastic deflections in the flexures, their motion is in general not energy-free, even in quasi-static conditions and ignoring the effects of gravity. The elastic potential energy Ve due to the flexure deformations is in general a function of the configuration q: therefore, compliant mechanisms have preferred equilibrium positions. It then appears natural to apply the results already known from the theory of static spring-to-spring balancing [84, 133] to the design of compliant mechanisms [9, 98, 99]; this way, the gravitational loads may also be compensated, if required. Basic design criteria for static balancing of compliant mechanisms were presented in [136]. A compliant device, specifically designed for gravity balancing (to compensate the weight of the leg in a humanoid robot), was presented in [137]; the gravity compensator was designed as a CFG and attached in parallel to the leg. Interestingly, this gravity compensator does not require prestressing the compliant elements, a common approach which may reduce accuracy due to stress relaxation. More recently, compliant, monolithic elements were used for gravity-balancing a 1-DoF rotational [138] motion; later, similar compliant balancers were designed for more general planar motion, using either flexible beams [139], shells [140] or planar springs [86], again achieving constant-force generation over a large range of motion. It was found in [5] that compliant mechanisms generally offer the best results in terms of the energy that can be stored (and, consequently, of the force that can be compensated). Compliant mechanisms were also applied to CFG design, for instance by developing a constant-torque device with potential applications in gravity balancing [141]. A review on CFGs, with some examples of compliant systems, is presented in [17].

6 Conclusions In this Chapter, we presented a perspective on parallel kinematic manipulators that are balanced against gravity. In particular, we focused on the application of constantforce generators, a class of mechanisms capable of applying a force with constant direction and magnitude but movable application point. Relying on the available literature, we showed that constant-force generators offer a suitable design paradigm for gravity balancing, especially for parallel robots: since these use several kinematic chains (“legs”) to support the movable end-effector, each leg can be advantageously

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designed as a constant-force generator, carrying a fraction of the total load, while applying a constant force on the end-effector. Our focus was on passive devices for gravity balancing, that require no extra actuators besides those used to control the robot; we also mostly considered devices that achieve exact balancing, as opposed to those which approximately balance the forces and torques due to gravity at the actuators. Throughout this review, we stressed practical considerations that may be useful in guiding the designer during the project of such gravity-compensated robots. For example, several works employ devices using springs with zero free length: thus, we described some options to achieve elastic forces with this characteristic. Massbalanced systems were also considered, for completeness; these can achieve not only balancing against gravity under static conditions, but also against inertia forces (dynamic balancing). We offered a number of example applications of balanced parallel robots, which can offer superior performances (with respect to conventional, non-balanced alternatives) especially where large loads have to be supported and the robot moves at relatively low speeds. A few directions can be suggested for future work. First, we notice that the vast majority of works on gravity balancing employ either counterweights or elastic elements; conversely, relatively little was written on other elements that exert conservative forces (such as magnets). Even restricting the focus on elastic elements, non-conventional spring designs (such as air springs) could provide effective alternatives. Almost all reviewed works found sufficient, but not strictly necessary conditions for balancing. It is in fact known that finding minimal conditions for static balancing is a nontrivial problem for all but the simplest planar linkages. General approaches that avoid imposing unnecessarily restrictive conditions would lead to greater freedom in design and could pave the road to optimized architectures. Most works on gravity-balanced robots consider all links to be ideally rigid; even the works on gravity-balanced compliant mechanisms mostly assume the elastic deflections to be concentrated in localized flexures. On the other hand, the gravity balancing of devices with large-scale, distributed deflections (such as parallel continuum manipulators) seems to have received little attention. Finally, relatively few works offered practical design observations or results from tests on prototypes; some effects such as joint clearance, frictional effects, nonlinearities in the springs (where used) and manufacturing tolerances with respect to the exact design are often disregarded in theoretical analyses, but could significantly affect the actual performance. This Chapter shows that machine balancing, while being an old topic, is still an active area of research and can greatly improve the performance of robots and mechanisms. It is our belief that future work in this topic can offer new insights on machine design and novel devices with improved performance in a number of applications.

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