Worm-Like Locomotion Systems: An intermediate theoretical Approach 9783486719871, 9783486713046

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Worm-like Locomotion Systems

An intermediate theoretical Approach by

Joachim Steigenberger Carsten Behn

Oldenbourg Verlag München

Joachim Steigenberger is a retired University Professor of Mathematics who spent the first half of his scientific life at the Mechanics Department at Ilmenau University of Technology. Carsten Behn studied Mathematics at Ilmenau University of Technology and obtained his Ph.D. in Mechanical Engineering from the same university at the Department of Mechanics, where he is working presently.

Bibliografische Information der Deutschen Nationalbibliothek Die Deutsche Nationalbibliothek verzeichnet diese Publikation in der Deutschen Nationalbibliografie; detaillierte bibliografische Daten sind im Internet über http://dnb.d-nb.de abrufbar. © 2012 Oldenbourg Wissenschaftsverlag GmbH Rosenheimer Straße 145, D-81671 München Telefon: (089) 45051-0 www.oldenbourg-verlag.de Das Werk einschließlich aller Abbildungen ist urheberrechtlich geschützt. Jede Verwertung außerhalb der Grenzen des Urheberrechtsgesetzes ist ohne Zustimmung des Verlages unzulässig und strafbar. Das gilt insbesondere für Vervielfältigungen, Übersetzungen, Mikroverfilmungen und die Einspeicherung und Bearbeitung in elektronischen Systemen. Lektorat: Angelika Sperlich Herstellung: Constanze Müller Titelbild: Dr. Rüdiger Reinhardt Einbandgestaltung: hauser lacour Gesamtherstellung: Grafik & Druck GmbH, München Dieses Papier ist alterungsbeständig nach DIN/ISO 9706. ISBN 978-3-486-71304-6 eISBN 978-3-486-71987-1

Charles Robert Darwin wrote about worms in 1881: “It may be doubted whether there are many other animals which have played so important a part in the history of the world, as have these lowly organised creatures.”1

1 in Darwin, C. R. (1881): The formation of vegetable mould, through the action of worms, with observations on their habits; John Murray, London, p. 313.

VI

Preface Why? What about? How? Whom for? Why this book? Both authors are with a long-standing experience in teaching mathematics and mechanics for both mathematics and engineering students of various courses. They may not be alone in having made a recurring observation: too many of the students exhibit difficulties in setting up mathematical models for mechanical systems and in handling these models by means of appropriate mathematical tools. This fact is apparently based on a general deficiency in appreciation of mathematical and/or mechanical (physical) thinking and on a regrettable lack of corresponding skills. This book is to be seen as an attempt at contributing to reduce these difficulties. Certainly, there exists a long and ever continuing series of excellent, good, and not so good books - “Mathematics for Engineers”, “Rational and Applied Mechanics” etc. - all aiming at the same end. Roughly, one recognizes two kinds of presentation: - as a textbook that introduces to Mechanics, say, from the very beginning and aims at application of propositions to problems from everywhere; - as a monograph aiming at a comprehensive representation of a more or less closed domain of new results under utilization of the most actual theoretical instruments. The authors do not want just to add something similar to this flood of general treatises. So the question arises: what topics should the book be about? It is worth a try to focus on some strongly restricted class of mechanical objects and, in the course of investigations, to start with a clear description of the objects, to set up a stringent mathematical model, to raise the important questions about the object (now in the context of its mathematical model), and to answer the questions by means of suitable mathematical tools - analytically as far as possible and using computers as well. Principally, any class of objects could do the job. But in order to include some things which hopefully are not yet captured in common literature the investigations will in particular concentrate on items from current research done at Ilmenau University of Technology, Faculty of Mechanical Engineering. There, at the Department of Applied Mechanics, a good deal of research is on legless locomotion systems. Such systems are of significance in diverse domains of application: inspection and cleaning of technical canals, exploration of hazardous, remote, rocky terrain, rescue of earthquake victims, exploration of human intestines and vessels in medicine. Thereof, the present book concentrates on worm-like locomotion systems - in a generality of no strict preference of biological or technical constructs. In practice, artificial worms can accomplish jobs which snake-like wriggling robots (occupying the majority of international theoretical

VIII

Preface

papers on legless locomotion), possibly moving on wheels, are not qualified for: just think of narrow canals or pipeline systems or of minimally invasive surgery (locomotive endoscope). Last not least, the investigation of all these systems has its importance in order to find out and to understand motion patterns of biological paradigms and to utilize them in technology (bionics). How will the investigations be tackled? Obeying what has been said about the aim of the book the authors’ intention is not to use most sophisticated concepts and methods (from differential geometry, algebra, and measure theory, e.g.) but to proceed on some intermediate level. That is, basic knowledge of mechanics (Newton’s laws, energy principles, constraints and reaction forces) and mathematics (sets, linear algebra, calculus, principles of differential equations) is taken as a prerequisite for understanding what is going on. Some items are explained in a footnote or in an Appendix2 . Emphasis is on sharp definitions, thorough modeling, and clear formulations of theorems, whereby frequently topics will be covered which are usually dropped in normal teaching for the sake of saving time. Relevant results are presented in form of graphics achieved by MAPLEr and MATLABr . Concluding, whom is the book written for? In principle, this has already been told at the beginning: it should be studied, first, by mathematicians who wish to gain some useful acquaintance with mechanics and the accompanying applied mathematics and, second, by theoretically interested mechanical engineers who wish to learn something about a special class of objects and their mathematical penetration. Summarizing, the authors recommend the present book to be seen as both a support and a supplement of common mathematics and mechanics teaching and learning at German universities. They hope the engineers to benefit by the mathematical level used throughout and to perceive items for further investigations. They hope the mathematicians to benefit by the way of handling mechanical concepts - and to give absolution for occasional mathematical simplifications.

Organization of the contents of the book How to arrange the various topics - this has almost been dictated by the authors’ intentions sketched at the beginning. So the Introduction addresses mathematical modeling - first on a level adapted to Applied Mechanics in general, secondly, focussing to worm-like systems, worms for short. Worms share some properties with snakes (e.g., slenderness, creeping on ground). They differ by the kind of internal drives: peristaltic (worm) vs. serpentine (wriggling snake). Principally and well planned, no stringent distinction of live and artificial worms is made. The first chapters are confined to establish a theory of worms living in a straight line, straight worms. Emphasis is put on finite DOF systems (systems of mass points) and a discrimination of kinematics and dynamics as far as possible. This exhibits an obvious benefit in Chapter 2: the worm endowed with kinematic drive and spikes allows to 2 The

reader should not misinterpret an Appendix as a crash course in the respective field.

Preface

IX

formulate a simple self-contained kinematical theory whose feasibility can be checked up by dynamics. On this level the construction of gaits (motion patterns) is possible. Generally, these spiked worms show up with a mathematical theory that gives this type of worms priority as an object of investigation and equally for comparison and estimation of theoretical results. In Section 2.2 a continuum model of the straight worm with spikes is introduced and exploited to some extent. In Section 2.3 some steps are gone towards non-straight worms. This regards the spiked worm in a hilly landscape, i.e., moving along a curve in a vertical plane under the action of weight. In Chapter 3 the spikes are replaced by a frictional force interaction worm-environment. This excludes of course a theory that is mainly based on kinematics and restricts the investigations to numerical simulations. Deplorably, friction is very often dealt with in some lax way. In particular, no strict distinction of stiction and sliding friction is made. In the context of worms we wished to give these items some emphasis. And that is why the corresponding chapter is comparably extensive. At least two items demand the theory to become supplemented by control aspects: the practically uncertain knowledge of data and the necessity of realizing certain internal or external motion patterns despite this deficiency. So, after introducing the necessary tools from adaptive control theory, Chapter 4 is dedicated to two tracking problems: (a) how to control the actuator inputs in such a way that, possibly under failures within the actuators, the spiked worm dynamically follows some motion that gets its preference from kinematical theory, and (b) how to control the actuator inputs such that the worm achieves or approaches a preferred motion also under fuzzy frictional interaction with the environment. In particular motions under dry friction, and controlled motions cannot be described without numerical methods realized on a computer. Presented corresponding simulations aim at principal demonstration of kinematical and dynamical macro-phenomena. Therefore, first, they are in general done with no regard to real data of any live or technical systems. And, second, that is why no specially tailored software has been used - calculations are done by means of customary MAPLE and MATLAB routines. Discontinuous functions are unavoidable in our theory (e.g., spikes and Coulomb forces). Usually they cause troubles on the computer (long runtime in best case, rejection in worst case). Appendix D gives some notes about approximations used in simulations in order to overcome these troubles. We add some sketches of program codes to this appendix. Finally, Chapter 5 sketches prototypes of worm systems and some past and current work done in the authors’ environs and some problems to be tackled in future: readers are cordially invited! The authors vigorously hint at the fact that all the contents of the book are to represent an approach to the THEORY of the object class “worms”. It is not the aim to describe or analyze concrete live or artificial worms in detail or even to give instructions for hardware realization. This is why in simulations system data are chosen just for the sake of clear demonstration of possible significant effects. Data are handled as physical quantities throughout, their units of measure are left at arbitrary (and consistent) choice. In representing graphs of time-dependent functions

X

Preface

(in particular, sketching motions) diagrams with horizontal or vertical t-axis are used in like manner. The concluding Appendices are hoped to give the reader some support in retrieving perhaps forgotten concepts from mathematics, mechanics, and control. Thanks are to Helga Sachse for taking care of many figures, to R¨ udiger Reinhardt for being our cartoonist, and, last not least, to Klaus Zimmermann, Head of the Department of Technical Mechanics at TU Ilmenau, for his support. And a great special thank goes to Ursula Steigenberger for her endless patience in seeing her husband do permanent post-retirement work in his ‘playroom’. The authors appreciate any information about the acceptance (or not) of the book, about misprints, (hopefully no) errors and possible improvements. J. Steigenberger & C. Behn (Ilmenau in 2012)

Contents 1

Introduction

1

1.1

Basic concepts: models, bodies, configuration, motion . . . . . . . . . . . . . . . . . .

1

1.2

Models of worm-like locomotion systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

2

The Straight Worm With Propulsive Spikes (“SPIKY”)

13

2.1 2.1.1 2.1.2 2.1.3 2.1.3.1 2.1.3.2 2.1.3.3 2.1.3.4 2.1.3.5 2.1.4 2.1.5 2.1.5.1 2.1.5.2

Straight worm as a system of finite degree of freedom . . . . . . . . . . . . . . . . . . Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kinematic drive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . About spikes of finite strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gait construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A glance back: the simplest guy amongst all worms . . . . . . . . . . . . . . . . . . . . Dynamic drive I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dynamic drive II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Actuator models and realizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The inch-worm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13 14 16 25 25 29 31 36 46 49 51 51 54

2.2 2.2.1 2.2.2 2.2.3

Straight worm as a continuous system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kinematic drive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

58 59 61 63

2.3 2.3.1 2.3.1.1 2.3.1.2

SPIKY in hilly landscape (“HILLY”) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Center-approximation, kinematical drive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HILLY 1 with gait-shift. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HILLY 2 with further slow-down tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71 73 74 78

3

The Straight Worm With Propulsive “Friction”

83

3.1

Pros and cons of frictional propulsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

3.2 3.2.1 3.2.2 3.2.3

Propulsive friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modeling Coulomb forces: stiction and dry sliding friction . . . . . . . . . . . . . . Worm system with ground contact by Coulomb forces (“COULY”) . . . . . Inch-worm with Coulomb forces: a review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.3

Straight worm with viscous friction (“STOKY”) . . . . . . . . . . . . . . . . . . . . . . . 99

84 84 90 92

XII

Contents

3.3.1 3.3.2

General system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 The system with n = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

4

Adaptive Control of Worms

109

4.1 4.1.1 4.1.2 4.1.2.1 4.1.2.2 4.1.3 4.1.3.1 4.1.3.2 4.1.4 4.1.4.1 4.1.4.2

Adaptive control of spiked worms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Our simplest guy: the inch-worm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Adaptive control in case of fixed actuator data . . . . . . . . . . . . . . . . . . . . . . . . . Adaptive control in case of changing actuator data . . . . . . . . . . . . . . . . . . . . . Three-mass-point-worm (n = 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Adaptive control in case of ideal spikes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Adaptive control in case of failing spikes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Adaptive Control of worms with n = 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tracking of different gaits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Change of gaits — gear shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

109 109 112 112 120 121 122 123 125 125 126

4.2

Adaptive control of worms with propulsive friction . . . . . . . . . . . . . . . . . . . . . 128

5

Conclusions

5.1

Worms in literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

5.2

Worms at Ilmenau University of Technology . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

5.3

Current and future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

A

Mathematical Concepts

147

B

Mechanical Concepts

151

C

Control Theory Concepts

159

D

Notes on Simulation Parameters

169

E

Some Program Source Codes

175

135

Bibliography

187

Index

193

1

Introduction

In this chapter we explain some fundamental items which inevitably have to be taken into consideration before starting to investigate any live or technical object. We sketch the way from from the original object to a model which is accessible for mathematical analysis. Thereby, emphasis is put on modeling mechanical systems in general, until, finally, worm-like systems and their models come into view. These few considerations are hoped to lead to a sufficient appreciation of the detailed investigations in later chapters. If this book was a monograph of worm theory then the ouverture should be a comprehensive representation of the state of the art. But, keeping a didactical flair of the book we shifted this item to Chapter 5. Readers who do wish to begin with a glance to worms in the world are kindly invited to read Sections 5.1 and 5.2 as an hors-d’œuvre.

1.1

Basic concepts: models, bodies, configuration, motion

Following the intentions outlined in the Preface let us start with some general topics which - tacitly, as a rule - have to precede every successful ride into a theory. Maybe, somebody could feel this part of the Introduction to be unnecessarily comprehensive. Let us not worry about this. Every theoretical penetration of natural or artificial (technical, virtual) structures and processes is based on a suitable model of the object of interest. Freshmen in science often meet this statement with some disbelief - obviously since they did not yet gain an impression of the actual complexity of the world’s constituents. Concerning the present context just look at the seemingly simple object earthworm: it is segmentally structured, each segment has a skin, i.e., an envelopment by a tissue composite of cells, each segment is equipped with various muscles in diverse arrangements, each a compound of fibers, each segment is filled with liquid, a multitude of biochemical reactions running within. These few words based on observations give only a very crude image of what an earthworm is. In order to clear up this or that detail of this image one must dissect the image and take aside the piece of interest. For example: to investigate what makes a muscle contract there may be no need of simultaneous consideration of how the whole worm is creeping forward; on the other hand, to describe the worm’s motion relative to its environment it seems reasonable not to investigate simultaneously what is happening biochemically within the worm. This illustrates a general principle for doing (theoretical) research in natural or technical

2

1 Introduction

sciences: a natural (or virtual) object can never be submitted to a promising investigation in its totality. Depending on both the goal of the investigation and the skills of the researcher a certain variety of (observed or presumed) properties of the object those which are guessed to be of minor relevance - are dropped. Then what is left over is called a model of the research object. At first it is usually called a physical model and then, after the corresponding physical theories have assigned respective mathematical constructs to the physical objects and their interrelations finally a mathematical model has been established. Application of appropriate mathematical tools (solving equations etc.) is to yield statements about the elements of the model which then undergo a hopefully reasonable interpretation (!) as statements about the real objects. A comparison of these statements with results of experiments may lead to an iterative revision (improvement) of the foregoing models. As a rule, establishing a physical model is always with an eye to the final application of mathematics, thus a strict separation of physical and mathematical model is almost impossible. So at last, a mathematical model is a map from the object of exploration onto a set of mathematical variables (which are supposed to be in some correspondence with physical observables) and their interdependencies (e.g., differential equations). The following very first step in mathematical modeling concerns the fact that mechanics primarily deals with the macroscopic description of how matter in solid or fluid appearance behaves in space and time. Atomic structure and substructures and corresponding phenomena are outside the main stream interests of mechanical science. In Applied Mechanics it is common use to understand (apart from gases) a body as a compact collection of matter that can be observed in R3 , the Euclidean point space. What matter is, what its internal structure is, how its particles are arranged in space - these are, as a rule, no questions in mechanics, there suffices the working hypothesis, that the “particles” somehow are spread over a finite space region and that they can be observed (so that they are individuals, opposite to what is fundamental in quantum theory!). The space region filled up with that matter may - under influence of neighbored matter - change, but it always remains a (3-dimensional) region: a cuboid cannot be turned into a (2-dimensional) patch of a surface. Tearing a body to pieces (i.e., transition of a region into several disjoint regions) as well as the occurrence of holes will not play any role in the present context. Penetration of bodies is excluded anyway, since otherwise particles would coincide and thereby lose their individuality. All these facts can be captured by means of the mathematical concept of a map (for some of connected concepts see Appendix A). The following definition is an adaptation of a definition given in [Noll 57]. Definition 1.1 A 3-dimensional body is a set B of particles together with a set Φ of maps ϕ | B → R3 which have the following properties: (i) every ϕ ∈ Φ is injective, so the inverse map ϕ−1 | ϕ(B) → B exists;

1.1 Basic concepts: models, bodies, configuration, motion

3

(ii) for every ϕ ∈ Φ the image ϕ(B) =: B ⊂ R3 is a 3-dimensional region3 ; (iii) for each pair of maps ϕ, ψ ∈ Φ the composition ψ ◦ ϕ−1 =: h | ϕ(B) → ψ(B) is (the restriction to B of) a diffeomorphism in R3 ; (iv) with any diffeomorphism H ∈ Diff (R3 ) and ϕ ∈ Φ it holds H ◦ ϕ =: ψ ∈ Φ .



Some comments on the foregoing definition are reasonable and supported by Fig. 1.1.

B p j

y

0

j

3

R

x

x B

B’ y j -1

j j -1

B0

0

Fig. 1.1: Set of particles, and configurations.

It is not specified what the set B is, B is kept as an abstract set whose elements are called particles. The maps ϕ ∈ Φ describe embeddings of the particles in the space of observations, R3 : each particle is mapped onto one and only one space point and can be observed individually. These points, then, are called material points. At first, B := ϕ(B) could also be thought of as some disconnected or even dust-like set, but this option is excluded by (ii). The region B = ϕ(B) is called a configuration of the body, the diffeomorphism h := ψ ◦ ϕ−1 | B → B 0 = ψ(B) describes a change of configuration (a homeomorphism could do as well, the smoothness of a diffeomorphism is desirable for to allow the usage of all tools of mathematical analysis). According to (iv), h could be any diffeomorphism acting in R3 , but, regarding some specific context, h (or already the set Φ) is occasionally restricted by additional requirements such as volume-preservation or length-preservation, etc. 3 Here and everywhere in the sequel := or =: means that something, written on the side of the “:”, is to be defined.

4

1 Introduction

As a rule, it is useful to choose one particular map ϕ0 ∈ Φ and to define a reference configuration by B0 := ϕ0 (B). In a particular mechanical problem it is not necessary that the body actually appears in this configuration B0 , but it may be comfortable to describe any (actual) configuration relative to the reference configuration B0 . A spatial reference frame can be introduced by glueing a 3-frame (e1 , e2 , e3 ) to B0 at a point O ∈ B0 . Preferably, (e1 , e2 , e3 ) is chosen as an orthonormal triplet of vectors, then {O, (e1 , e2 , e3 )} is in fact a cartesian coordinate system fixed in R3 (inertial reference frame). If the respective cartesian coordinates in space are denoted by ξ i , i = 1 , 2 , 3, then to each particle p ∈ B the map ϕ0 uniquely assigns a triplet of numbers ξ = (ξ 1 , ξ 2 , ξ 3 )T called body-fixed coordinates which, being used as “names” of the particles, once and for all mark the material points of the body in any configuration. Analogously, every map ϕ ∈ Φ causes an assignment p 7→ x = (x1 , x2 , x3 )T (space coordinates of the particle p in configuration B of the body). Then, the description of configuration B relative to the reference configuration B0 : ξ 7→ p = ϕ−1 0 (ξ) 7→ x = ϕ(p) is given by h = ϕ ◦ ϕ−1 0 : ξ 7→ x = h(ξ) . In view of (iii) in Definition 1.1 h is a diffeomorphism and it describes a coordinate transformation (body-fixed to actual coordinates). A family of maps from Φ, whose members depend on a real parameter t=‘time’, where t is from an interval (t0 , t1 ), is called a motion, i.e., a temporal sequence of configurations (t running from t0 to t1 ), of the body. In coordinates, this means x = h(ξ , t) , if the family is denoted {ϕt | t ∈ (t0 , t1 )} and h := ϕt ◦ ϕ−1 0 . As noted above, at any fixed time h(·, t) is a diffeomorphism, so its inverse, h−1 (·, t) exists and is of the same class of differentiability. By x = h(ξ, t) one is given the position in space at time t of the material points ξ, this is called the Lagrangean representation of a motion. Inversely, the Eulerian representation, ξ = h−1 (x, t) tells which material point ξ coincides with the space point x at time t. In all that follows the Lagrangean representation is preferred (since fluid motions are not considered). As to the dependence on t, a piecewise smoothness h(ξ, ·) ∈ D2 (t0 , t1 ) is required in most cases, i.e., position and velocity (1st derivative with respect to t) continuously depend on t and, with the exception of finitely many instants of time the acceleration (2nd derivative with respect to t) of any material point ξ exists, is continuous, and in the exceptional points the one-sided limits exist (there may be jumps). These facts given by Definition 1.1 still are very comprehensive, they do not regard any physical properties of bodies - those items will be dealt with later on by making B0 support physical fields like mass density, forces, elasticity moduli, etc. It is simple matter to modify Definition 1.1 in such a way as to capture 2-dimensional and 1-dimensional bodies in R3 : just replace in (ii) ‘region’ by ‘surface’ or ‘arc’, respectively. In each case ξ runs through a 1-, 2-, 3-dimensional region, any mechanical theory based on this fact then is a continuum theory (that only under additional rigidity hypotheses may reduce to point mechanics). But finite systems of material

1.2 Models of worm-like locomotion systems

5

points in R3 (0-dimensional bodies, mass points) can be captured by Definition 1.1 as well: suppose B to be a finite set, and drop (ii) and (iii). All such ‘bodies’ serve, adapted to concrete problems, as useful approximations of bodies with some negligible dimensions. A mechanical system or a multibody system is a finite set of 0-, 1-, 2-, 3-dimensional bodies which may both mutually and with an environment interact by forces (see Appendix B). Until now, the diffeomorphisms h which describe changes of configurations may be largely arbitrary: two different configurations B and B 0 = h(B) of a body need not be congruent, i.e., roughly speaking, any two corresponding subsets S ⊆ B and S 0 = h(S) ⊆ B 0 may have different shapes, both total and internal deformations are allowed. For to exclude the occurrence of deformations (practically: to ignore ’small deformations’) the set Φ of maps into space must be restricted. Definition 1.2 A rigid body is a body as described above but undergoing the restriction that for any pair of maps ϕ, ψ ∈ Φ the change of configurations h = ψ ◦ ϕ−1 be a direct congruent transformation.  A congruent transformation is a point transformation of Rn , n = 1, 2 or 3, that is length preserving. Preservation of length means that the distance of any two points keeps unaltered under h. Direct congruence means that additionally orientation is preserved. Preservation of orientation excludes reflections (a 2-dimensional body in the plane R2 will not be turned over by h like a page of a book that needs a third dimension for being moved). For mathematical representation visit Appendix A. Note that the definition claims more than just the preservation of the “outer form”: each part of the body keeps its shape (a body consisting of an undeformable container filled with liquid is something different!). Probably, the present book will not need a thorough theory of (systems of) rigid bodies (opposite to theories of general robots, e.g., [MuSa 93]). The “worm” systems to be considered will modestly appear as finite systems of material points (systems of mass points) or as 1-dimensional deformable bodies in most cases.

1.2

Models of worm-like locomotion systems

In the course of the present treatise the objects addressed in the title will be modeled in various manners emphasizing different aspects of interest. Starting-point is of course a definition of the class of objects to be considered. Definition 1.3 A mechanical system is called an autonomous motion system if it is equipped with internal driving devices which enable the system to change its configuration in

6

1 Introduction absence of driving external forces4 . An autonomous motion system is called a locomotion system if there exists a particular internal drive such that during the corresponding autonomous motion neither the center of mass nor any material point of the system remain fixed in space. A worm-like locomotion system is a mainly terrestrial or subterrestrial, possibly also aquatic locomotion system characterized by one dominant linear dimension with no active (driving) legs or wheels. 

Fig. 1.2: A moving worm system: a) autonomous; b) non-autonomous.

For a detailed representation of items in connection with locomotion see [Stei 11]. This definition of worm-like systems is strong enough as to distinguish the corresponding systems from other locomotion systems regarding the outer form. It is also sufficiently general as to cover both technical and biological systems while not limiting the possible movements in Rn , n = 1 , 2 , 3. Kinematically, this definition is applicable to both the biological paradigms worm and snake in the same manner. A fine distinction is essentially possible by considering the internal structure. Worm: A worm has a hydroskeleton whose (nearly cylindrical) segments are endowed with a system of orbicular and longitudinal muscles. Their activation generates a radial contraction (coupled with axial extension) and axial contraction (coupled with radial expansion), respectively; different contraction of parallel longitudinal muscles may yield a rotation of the segment axis resulting in a bending of the worm’s longitudinal axis. The locomotion is peristaltic. 4 An external force is called a driving one if under its sole action the system could start a motion from rest. So, non-driving forces arise from non-zero velocities and vanish at rest, e.g., friction, Lorentz forces, and workless constraint forces.

1.2 Models of worm-like locomotion systems

7

Snake: A snake has an endoskeleton that is segmented by (rigid) vertebrae which are connected by certain muscles. Their activation may generate a relative rotation of neighbored vertebrae and thereby yield a change of curvature of the skeleton axis, whereas a change of length of the axis is nearly impossible. Disregarding the rectilinear motion of a snake (which follows a different scheme - similar to “walking on hidden legs”) the locomotion is serpentine. This structural distinction is also reflected in dynamics (details see later): the drive of a worm is related to internal longitudinal forces whereas that of a snake is to internal torques. For the following modeling these differences are not important, they only prove to be central when investigating worms in a strong sense. A general guide in modeling is: “make the model as simple as possible (for to enable a thorough analysis) and as comprehensive and complicated as necessary (for to capture all important items)”. Recall the model of a body in 3-space: it is simple since only using the concept of sets and maps, it is comprehensive since it covers solid and fluid matter of what kind ever. (But, maybe, this view will or will not be accepted individually.) Following the above Definition 1.3 of a worm (“thin” object) it seems promising to disregard the 3-dimensional extension and rather base the worm model on a 1-dimensional body (see Definition 1.1 ). This would entail the fortunate fact that, e.g. in describing deformations, only spatially 1-dimensional theories must be set up (confining mathematical tools mainly to ordinary differential equations instead of partial ones). If crosssections (their geometry, areas, inertia moments, etc.) are of importance then they can be attached to the material points of the 1-dimensional model body. Masses, mass densities, external forces, internal stresses, etc., are treated the same way, as it is well-known from the theory of elastic rods, say. Consequently, modeling a worm is based on the following definition. Definition 1.4 The model body of a worm is a 1-dimensional interval that serves as the support of certain fields.  Without loss of generality it may be seen as a unit interval of body-fixed coordinates, 0 ≤ ξ ≤ 1, and to choose B0 := {(ξ, 0, 0) | ξ ∈ [0, 1]} for reference configuration is possible. According to Definition 1.1 and the supplementing remarks any configuration of the worm is then a smooth arc in R3 . A field is a map from (a subset of) [0, 1] into a class of physical objects (or, rather, their respective mathematical models, see below). Thus, every such ξ carries one such object. 5 It may be reasonable to subdivide fields into worm-immanent or intrinsic fields which reflect an internal structure of the worm system and do not depend on the configuration, and into fields which are inseparable of the embedding ϕ ∈ Φ in R3 , extrinsic fields. Some intrinsic fields: 5 Mathematically, one could understand the model body as a measure space and the field as a measure with values in that object class.

8

1 Introduction • Cross-sections (boundary curve, area, inertia moments). • Masses a) in continuous distribution, i.e., on [0, 1] a density function (mass per unit of length) ρ | [0, 1] → R+ : ξ 7−→ ρ(ξ) is defined such that the mass of Rb [a, b] ⊂ [0, 1] is given by a ρ(ξ) dξ;

b) in discrete distribution, i.e., every element of a given subset {ξ1 , ξ2 , . . . , ξn } ⊂ [0, 1] carries a finite mass: ξi 7−→ mi > 0 (or, briefly i 7−→ mi ); thereby, the worm appears as a finite chain of mass points. • Actuators, i.e., discretely or continuously distributed technical elements such as elastic springs, piezo elements, which generate internal forces or internal relative displacements of material points (in a simplified and technologically realizable way representing the complicated system of muscles of a live worm). Some extrinsic fields: • Form of the model body in configuration ϕ(B), described by the three scalar fields arc length, curvature, and torsion, ξ 7−→ (s(ξ), κ(ξ), τ (ξ)). Remind that the worm body in any configuration is a smooth arc. s describes the relative distances of the material points, measured along this arc. The local axis strain relative to the ds − 1. s, κ and τ characterize the shape of reference configuration is then ε(ξ) = dξ the body in these configurations (see Appendix A). • Interactions with environment: a) Weight, corresponds with the mass distribution. b) Friction or stiction in discretely or continuously distributed contact points will show up to be basic for locomotion. In a fluid environment this could be Stokes friction. On a solid ground this could be, dependent on the roughness and texture of the contacting surfaces, an anisotropic (direction-dependent) Coulomb friction. It is composed of sliding friction and stiction (“friction of rest”) which then enter the equation of motion in form of an impressed force (physically given) or reaction force, respectively (“eingepr¨agte Kraft” or “Zwangskraft”, following [Hame 49], see Appendix B). The contacting surface of the worm can be equipped with scales (e.g., snake) or spikes (bristles, thorns, e.g., earthworm) which slide in “forward direction” whereas in counter direction they dig in and prevent any relative displacement. Technologically, the contact can be realized by passive wheels equipped with a ratchet which then allow rolling only in one direction “forward”. In these cases the worm kinematics are governed by a constraint of the form “relative velocity of the contact points is non-negative” that dynamically shows up by a respective constraint force. Certainly, the given list of various fields is not exhausting. For example, a temperature field could be needed if internal biochemical processes or exchange of matter or heat with the environment during the motion are of interest.

1.2 Models of worm-like locomotion systems

9

The subdivision intrinsic-extrinsic is not strict. So the interaction with the environment could determine the spatial-temporal working pattern of the actuators or even the choice of an actuator type (“automatic gears, hybrid drive”). GEOMETRY MASS

{

discrete continuous

ACTUATORS

{

discrete continuous 1-dimensional interval support of certain FIELDS

“GROUND” INTERACTION with environment Stokes friction, weight in contact points (discrete / continuous):

dry friction, stiction, spikes

Fig. 1.3: Relevant fields, schematically.

The activity of an actuator primarily generates a local deformation (change of shape, local contraction, e.g.) which only via some interaction with the environment leads to a global change of position of the worm system. (Clearly, considering the system as a whole, then only external forces - applied to the system by the environment - enter Newton’s law that governs the motion of the system’s center of mass.) As a rule, the actuator activity is periodic in time while the generated global motion is unidirectional (monotonic); examples: piston+cylinder as the actuator of a steam-roller, contracting-expanding segment of an earthworm. In the present context this seems to be the general principle of locomotion. The following definition is adapted from [OBLM 95]. Definition 1.5 Undulatory locomotion means a temporal process during which the actuators receive an activation (exogenous or endogenous signal) and generate a (periodical in time) local deformation (contraction, change of curvature) which via interaction with the environment results in a global change of location, see definition in Section 1.1.  Remark 1.6 Note that, opposite to mechanics, in biology “undulatory” commonly is closely related to “serpentine” movement!  An actuator is, first, a multipole with input activation signal and energy (immanent energy source - e.g., electrical battery, chemical agents - possible as well). Its output

10

1 Introduction

are forces, torques, displacements, twists, respectively, which depend on time and may be connected with the actual state or the state history of the system or with the system’s rheological constitution. Often the internal dynamics of an actuator cannot be modeled, rather the output is connected to the input by means of working hypotheses: the multipole remains a black box. It becomes a (almost) white box if for its internal dynamics a (more or less crude) model is established which yields an output law. ACTIVATION SIGNAL INPUT ENERGY 1 OUTPUT

ENERGY 2

(a) actuator dynamic is modeled (b) actuator dynamic is not modeled working hypothesis

Fig. 1.4: Actuator, schematically.

In the sequel four examples are sketched, where for the sake of clarity the actuator acts to neighbored mass points. 1) Output force: In [Huan 03] the author both theoretically and experimentally investigates a wormlike mobile, where an internal unbalanced rotor is coupled by elastic springs with two mass points having ground contact, see Fig. 1.5. In case (a) the rotor with given angular speed ω is treated as a part of the whole system, there remains the “small” black box “electrical voltage u →angular speed ω”, possibly combined with the working hypothesis “ω proportional u”. The latter could be replaced by modeling the driving electric motor as an electromechanical subsystem. In case (b) the rotor is not modeled at all, there remains a ’big’ black box with input=angular speed ω and output=force F0 sin(ωt) acting on the central mass point (working hypothesis). w a)

b) F = F0 sinwt

Fig. 1.5: Physical models of Huang’s mobile, [Huan 03].

2) Output displacement:

1.2 Models of worm-like locomotion systems

11

In [Stei 04] worm systems are considered in which actuators generate the distance (or the relative velocity) of neighbored mass points as functions of time. The actuators are kept as black boxes (to be implemented, e.g., as piezo elements or screw drive elements, then possibly with a suitable modeling). activation

energy

b)

l.(t) l(t) Fig. 1.6: Actuator, schematically.

3) Mixed case: In [Stei 99] the actuators are spring-damper elements whose original spring lengths are controllable (e.g., by some magnetic flip-flop or by a piezo element). The control element itself is kept not modeled: ‘small’ black box with internal output l0 (t).

Fig. 1.7: Physical actuator models, [Stei 99]; below: relaxed state of actuator.

4) Output torque or rotation: In [Hiro 93] a prototype of a terrestrial snake-like robot is presented: the Active

12

1 Introduction Chord Mechanism (ACM). It consists of a long chain of rigid bodies consecutively coupled by pivots, each body based on two passive but steerable wheels. Actuators are implemented as, say, stepping motors which control the relative angles of the links in the pivots and the rolling-directions of the wheels, see also [OsBu 96].

2

The Straight Worm With Propulsive Spikes (“SPIKY”)

In this chapter we introduce our central (model) object of investigation: the straight worm (not running curves) whose unidirectional mobility relies on internal peristaltic drive in connection with spikes. Kinematics and dynamics, and the corresponding types of drive are discriminated at every stage of the theory. An important item is the construction of gaits, in particular in its application to worms moving “through the mountains”.

The investigations presented in this chapter are concentrated on terrestrial worm systems moving along a rectilinear x−axis and interacting with the environment (ground) via spikes. Spikes restrict the orientation of the velocities (relative to the environment) of the contact points: An inequality of the form x(t) ˙ ≥ 0 for the motion t → x(t) of each material point that contacts the ground can be understood as the mathematical model of the spike. As long as no concrete implementation is dealt with, spike is a synonym for any device which realizes the above inequality: a single thorn, an area of bristles, a passive wheel with ratchet, and whatsoever. The set of contact points may be finite or, in a continuous worm model, infinite.

2.1

Straight worm as a system of finite degree of freedom

In this section worm systems with discrete mass distribution are considered. Let the system consist of n + 1 mass points which are labeled 0 to n, where the point 0 is to be seen as head of the worm. These labels play the role of body-fixed coordinates. All mass points carry equal masses m and are movable along a space-fixed x−axis. Actuators are inserted between (all or some) consecutive mass points and may have variable distances, or relative velocities, or forces as their outputs. Denote the coordinates of the mass points by xi , i ∈ N :={0, 1, . . . , n}, let K be a given nonvoid subset of N, and let the mass points labeled i ∈ K ⊂ N contact the ground via spikes whereas the remaining points are with no contact. The links of the chain of mass points - equipped with actuators or passively viscoelastic - get labels j = 1, . . . , n. Actuators are supposed massless and not to have contact to the ground.

14

2 The Straight Worm With Propulsive Spikes (“SPIKY”)

li xn

l1

xi

x i-1

x1

x0

Si

x

Fig. 2.1: Worm model chain of mass points.

2.1.1

Kinematics

The motions of the worm system, t 7−→ xi (t), i = 0, . . . , n, shall be investigated under the general assumption to be of differentiability class D2 (R), i.e., xi (·) ∈ D2 (R), i = 0, . . . , n, that means xi (·) and x˙ i (·) continuous, x ¨i (·) piecewise continuous (¨ xi ∈ D0 ).

(2.1)

The spikes restrict the velocities of the contact points, x˙ κ ≥ 0, κ ∈ K.

(2.2)

This is a system of constraints in form of differential inequalities which the system’s motions are subject to. Let the distances of consecutive mass points (= actual lengths of the links) be denoted by lj , j = 1, . . . , n, and the actual distance of the mass point i from the head by Si , lj = xj−1 − xj ,

Si = x0 − xi =

i X

lj .

(2.3)

j=1

Then there holds for the velocities x˙ i = x˙ 0 − S˙ i ,

i = 0, . . . , n ,

and the constraint (2.2) yields x˙ 0 − S˙ κ ≥ 0, i.e., x˙ 0 ≥ S˙ κ for all κ ∈ K. This necessarily entails  x˙ 0 ≥ V0 := max S˙ κ | κ ∈ K . (2.4) Consequently, the head velocity is x˙ 0 = V0 + w ,

w ≥ 0,

(2.5)

and for the others it follows x˙ i = V0 − S˙ i + w ,

i = 0, . . . , n .

(2.6)

2.1 Straight worm as a system of finite degree of freedom

15

Exercise 2.1 Convince yourself that (2.6) indeed ensures x˙ κ ≥ 0, κ ∈ K. What about x˙ i , i ∈ / K? What about sign(V0 )?  By construction and on account of (2.1) the V0 -part of the velocities is piecewise differentiable and the supplementing function w must be of the same class: V0 , w ∈ D1 . Besides its non-negativity and smoothness the function w remains arbitrary at this stage (more will be found out in dynamics). Since w is a common additive term to all velocities x˙ i , it describes a rigid part of the motion of the total system (motion at ‘frozen’ lj ). If 0 ∈ K (head equipped with spike) then because of S0 = 0 the head velocity is non-negative. Often other representations of the velocities prove useful. The coordinate and velocity of the center of mass are obtained by averaging the xi and x˙ i (remind equal masses m for all i), x∗ = x0 − S ,

S :=

v ∗ := x˙ ∗ = W0 + w ,

1 n+1

n P

Si ,

i=0

W0 := V0 − S˙ ,

(2.7)

which entail xi = x∗ + ζi , x˙ i = v ∗ + ζ˙i

ζi := S − Si ,

(2.8)

and in particular x0 = x∗ + ζ0 ,

ζ0 = S ,

W0 = V0 − ζ˙0 .

Now there are three representations of the velocities, x˙ i = x˙ 0 − S˙ i = w + V0 − S˙ i = v ∗ + ζ˙i .

(2.9)

They show that, alternatively, the head velocity x˙ 0 together with S˙ i (mind S˙ 0 = 0), or the rigid velocity part wPtogether with S˙ i , or the velocity of the center of mass n ˙ v ∗ together with ζ˙i (mind 0 ζi = 0) may serve as generalized velocities of the system (degree of freedom = n + 1). It is evident that preferable descriptions of configuration=(position, shape) are       or x∗ (t), ex , ζi (t) | i = 0, .., n . x0 (t), ex , Sj (t) | j = 1, .., n

When considering locomotion under external load it might be of interest to know which and how many of the mass points κ with ground contact, κ ∈ K, are at rest during the motion of the system, these carry the active spikes which transmit the propulsive

16

2 The Straight Worm With Propulsive Spikes (“SPIKY”)

forces from the ground to the system. Now x˙ κ = V0 − S˙ κ + w together with w ≥ 0 and V0 = max{S˙ i | i ∈ K} ≥ S˙ κ imply x˙ κ = 0 ⇐⇒ w = 0 ∧ V0 = S˙ κ ,

κ ∈ K.

(2.10)

If the head is equipped with a spike, 0 ∈ K, then in view of S˙ 0 = 0 and the definition of V0 in (2.4) it follows If 0 ∈ K then x˙ 0 = 0 ⇐⇒ w = 0 ∧ S˙ κ ≤ 0 for all κ ∈ K .

2.1.2

(2.11)

Dynamics

The dynamics of the worm system shall be formulated by means of Newton’s law for each of the mass points. The following forces are applied to mass point i, all acting in x−direction: • gi , the external impressed (physically given) force (e.g., resultant of weight and friction). • µi , the stress resultant (inner force) of the links (let, formally, µ0 = µn+1 := 0). Since the stress appears at a cut of the link, µi will be called a cut force for brevity6 . • λi , i ∈ K, the external reaction force caused by the constraint (2.2), acting on the spiked mass points, they are responsible for a strict non-negativity of the respective velocities.

Ai -m i+1

i

mi

-m i

mi

-m i

i-1 g i-1

gi li

l i-1

m i-1

inner forces external forces

Fig. 2.2: Mass points with forces (Ai : actuator).

Remark 2.2 The classification of the stress resultants as impressed or reaction forces depends on the kind of actuator output, see later.  As the constraint (2.2) describes a one-sided restriction of x˙ κ , velocity and reaction force are connected by a complementary-slackness condition: x˙ κ ≥ 0 , 6 The

λκ ≥ 0 ,

x˙ κ λκ = 0 ,

κ ∈ K.

(2.12)

letter µ is chosen due to the fact that if li is prescribed as a function of t (see Section 2.1.3) then the cut force appears as the corresponding Lagrangian multiplier.

2.1 Straight worm as a system of finite degree of freedom

17

This means that λκ (t) must be zero if at time t the mass point κ is moving forward, i.e., if the velocity inequality is strict, x˙ κ (t) > 0 (“x˙ κ has slack”), whereas λκ (t) may have arbitrary non-negative values as long as x˙ κ (t) = 0 (reaction force at resting spike). Positive λκ (t) implies x˙ κ (t) = 0, simultaneous vanishing, x˙ κ (t) = λκ (t) = 0 is possible (resting, non-active spike). Newton’s laws for the n + 1 mass points mx ¨κ = gκ + µκ − µκ+1 + λκ , mx ¨i = gi + µi − µi+1 ,

κ ∈ K, i ∈ N\ K,

(2.13)

now appear as a system of equations that together with the slack-conditions (2.12) and initial conditions for xi and x˙ i , i = 0, . . . , n, governs the motions of the worm system. On account of the supposition (2.1) the equations are in general only piecewise (on the continuity intervals of x ¨i ) defined. The stress resultants µi shall be specified in the next sections, about the reaction forces already now a general proposition can be formulated. Suppose the spiked mass point κ to be at rest in a time-interval during the motion of the system, x˙ κ (t) = 0 for t ∈ (t0 , t1 ). Then it follows from (2.13) the equilibrium λκ (t) = −fκ , if fκ := gκ − µκ+1 − µκ abbreviates all further forces acting at time t. The slack condition (2.12) demands fκ ≤ 0 (and otherwise fκ would be a force pushing in positive x−direction that could not be compensated by the spike). So the background of the following central lemma will have become clear. Lemma 2.3 7

The complementary-slackness conditions (2.12) are satisfied along any solution of the motion equations (2.13) if and only if
λκ (fκ , x˙ κ ) = 1 − sign(x˙ κ ) fκ− (2.14)

1 = − 1 − sign(x˙ κ ) 1 − sign(fκ ) fκ , κ ∈ K . 2 

Proof. For the sake of brevity the index κ is dropped. (1) Sufficiency a) Owing to (2.14) λ ≥ 0.

b) Owing to (2.14) xλ ˙ = − 21 |x| ˙ − x˙ |f | − f ≤ 0. ◦ If x˙ ≥ 0 then xλ ˙ = 0. ◦ Let x(t) ˙ < 0 at some time t (a contradiction will follow). Due to continuity x˙ is negative on a whole interval around t, so sign(x) ˙ = −1 and λ = |f | − f , and m¨ x = f + λ = |f | ≥ 0 along the solution of (2.13) on that interval. As it has to be, the initial value of x˙ is non-negative, so x˙ has a zero and without loss 7 Concerning a real-valued function remind f + = max{f, 0} ≥ 0 : positive part of f and f − = − min{f, 0} ≥ 0 : negative part of f , such that f = f + − f − , | f |= f + + f − = f · sign(f ).

18

2 The Straight Worm With Propulsive Spikes (“SPIKY”) of generality it can be assumed x(t) ˙ < 0 for t ∈ (t0 , t1 ) and x(t ˙ 0 ) = 0. Then mx ¨(t0 + 0) = |f 0 | ≥ 0 (f 0 denotes the limit value of f at time t = t0 + 0). If f 0 6= 0, then x¨(t0 + 0) > 0, x˙ increases, hence x(t) ˙ > 0 for t ∈ (t0 , t2 ) ⊂ (t0 , t1 ): contradiction! If f 0 = 0 and f = 0 in a right neighborhood of t0 , then x(t) ˙ = 0 for t ∈ (t0 , t2 ) ⊂ (t0 , t1 ): contradiction! On the other hand, if f 6= 0 to the right of t0 , then for these t it holds x ¨ > 0, x˙ goes to positive values: contradiction! Summarizing, x˙ < 0 can never occur, the conditions (2.12) are fulfilled along (2.13), sufficiency of (2.14) has been proven.

(2) Necessity At a time t0 assume x(t ˙ 0 ) = 0. Then (2.12) excludes decrease of the velocity, so mx ¨(t0 + 0) = f 0 + λ(t0 + 0) ≥ 0 ,

i.e.,

λ(t0 + 0) ≥ −f 0

is necessary. If f 0 < 0, then this means λ(t0 + 0) ≥ f − (t0 + 0) > 0, if f 0 ≥ 0, i.e., f − (t0 + 0) = 0, then λ(t0 + 0) ≥ f − (t0 + 0) follows from (2.12). Hence there holds λ(t0 + 0) = (1 + ε)f − (t0 + 0) with some ε ≥ 0, and this entails mx ¨(t0 + 0) = f 0 + (1 + ε) f − (t0 + 0) = f + (t0 + 0) + ε f − (t0 + 0) . Thus, with an ε > 0 the reaction force λ would make a positive contribution to the acceleration: contradiction! Summarizing, there solely remains, in accordance with (2.14), the implication if x˙ = 0 then λ = f − .  On account of this lemma the equations of motion now get the form x˙ i = vi ,

i∈N

m v˙ j = fj , h m v˙ κ = 1 −

1 2

j ∈N\K



i 1 − sign(vκ ) 1 − sign(fκ ) fκ ,

κ∈K

(2.15)

fi := gi + µi − µi+1 .

Of course, in prescribing initial data (xi , vi )(t0 ) one must take care of vκ (t0 ) ≥ 0, κ ∈ K. The D2 −smoothness of the motions assumed in (2.1) demands fi to be piecewise continuous along the motion, the term (1 − sign(vκ )) is piecewise constant thus it implies jumps at most. The physical forces gi follow some law gi = γi (x, x, ˙ t), the assumption γi (·, ·, t) ∈ C 1 , γi (x, x, ˙ ·) ∈ D0 , in the theory of differential equations usually posed for guaranteeing existence and uniqueness of solutions, will do. Statements about the µi must be postponed until actuators come into consideration. This is why the equations of motion in form (2.15) are little suitable to draw general conclusions for the theory, and also for concrete applications the following reformulations will prove useful. In all what follows in this chapter the external physical forces gi are confined to gi := −k0 x˙ i − Γi ,

(2.16)

2.1 Straight worm as a system of finite degree of freedom

19

i.e., to a Stokes friction with a constant coefficient k0 ≥ 0 and a constant force Γi T 0 (gravity component, dry friction, say. Note that Γi < 0 - weight component in forward direction - would be an external propulsive force.). Introducing the velocity of the center of mass by means of (2.8) the system of Newton’s law for the single mass points (2.15) writes (δir : Kronecker’s symbol) m v˙ ∗ + k0 v ∗ + Γi = −m ζ¨i − k0 ζ˙i + µi − µi+1 +

X

δiκ λκ .

(2.17)

κ∈K

Now several first conclusions can be drawn. Conclusion 2.4 Suppose (v ∗ , ζ˙i ) to be known. Then the equations (2.17), i = 0, . . . , n − 1 can be successively solved for the inner forces µj (t, λ), j = 1, . . . , n, where the constraint forces λκ remain as parameters.  Conclusion 2.5 First observe (2.10) and its negation: x˙ κ = 0 ⇐⇒ w = 0 ∧ S˙ κ = V0 x˙ κ > 0 ⇐⇒ w > 0 ∨ S˙ κ < V0

(spike κ active), (spike κ inactive, λκ = 0).

(2.18)

Then, with λ∗ :=

1 X λκ ≥ 0 , n+1

(2.19)

κ∈K

the resultant of all spikes forces averaged upon all mass points (including those without spike), there hold the following implications: (i) w > 0 ⇒ x˙ κ > 0 ∀κ ∈ K ⇒ λκ = 0 ∀κ ∈ K ⇒ λ∗ = 0 , and  (ii) λ∗ > 0 ⇒ ∃κ0 ∈ K : λκ0 > 0 ⇒ x˙ κ0 = 0 ⇒ w=0 . Conclusion 2.6 Summing up the equations (2.17) and observing yields m v˙ ∗ + k0 v ∗ + Γ∗ = λ∗ , where n

Γ∗ :=

1 X Γi n + 1 i=0

Pn

0 (µi

− µi+1 ) = 0,

Pn 0

ζi = 0, (2.20)

20

2 The Straight Worm With Propulsive Spikes (“SPIKY”) is the average of the constant external physical forces. Clearly, equation (2.20) represents the Principle of linear momentum for the total worm system. Note that, apart from Γ∗ (if Γ∗ < 0), the system is propelled forward by the spikes forces! Do you see the analogy with an autocar? 

On account of (2.7) the (total) equation of motion (2.20) is equivalent to m w˙ + k0 w + σ = λ∗ , w ≥ 0 , λ∗ ≥ 0 , w λ∗ = 0 ,

(2.21)

where the accompanying complementary-slackness condition is just the outcome (i) and (ii) of Conclusion 2.6 above, and σ is defined as ˙ 0 + k0 W0 + Γ∗ , σ := m W

W0 = V0 − ζ˙0 .

(2.22)

System (2.21) is called central equations for spiked worms. Now (2.21) gives some information about the rigid part of the velocities during motion of the worm system. Yet this information is incomplete since it suffers from the fact that σ depends on the motion (although not on w). Nevertheless a close investigation of this item will be important and dealt with in the sequel. About the central equation (2.21) Following the emergence of σ (x˙ i → S˙ i = ζ˙0 − ζ˙i → V0 → W0 → σ) it becomes clear ¨ ˙ that during a motion t → xi (t) the function σ is generated as σ(ζ(t), ζ(t)) =: σ e (t), 0 where σ e ∈ D . In the context of the kinematical theory below σ e is even a prescribed function of t! In general, σ e(t) is not known in advance!

Regarding w, the first part of (2.21) appears as an inhomogeneous linear ordinary differential equation for the unknown function w ∈ D1 whereas the unknown function λ∗ ∈ D0 (this D0 -property is necessary for to guarantee w ∈ D2 ) enters the equation algebraically. Equation (2.21) is a heteronomic bimodal differential-algebraic equation (DAE) for w and λ∗ since the complementary-slackness conditions imply two alternative modes of solutions: (a) (w > 0, λ∗ = 0), governed by the differential equation m w˙ + k0 w + σ e(t) = 0 with solution 1 w(t) = − m

Zt

t0

k0

σ e (s) e m (s−t) ds > 0

(2.23)

on t−intervals (t0 , t1 ) where the integral is negative (requires σ e(s) < 0 at least near the lower bound), terminated by t1 such that w(t1 ) = 0, w(t1 + ε) < 0 for ε > 0. (b) (w = 0, λ∗ > 0), described by λ∗ = σ e (t) on t-intervals where σ e(t) > 0.

The alternation of mode (a) and mode (b) is controlled by
+ λ∗ = 1 − sign(w) σ e (t) ,

(2.24)

2.1 Straight worm as a system of finite degree of freedom

21

the proof equals, step by step, that of Lemma 2.3. The central equation (2.21) then formally writes  σ (t), if w > 0 ,  −e e− (t), if w = 0 , m w˙ + k0 w = σ  |e σ (t)|, if w < 0 .

(2.25)

It has to be supplemented by an initial condition w(t0 ) = 0: worm system at rest or possibly terminating mode (a) at t = t0 . Then σ e− (t0 + 0) uniquely decides what happens: σ e− (t0 + 0) > 0, i.e., σ e(t) < 0 on a right neighborhood of t0 , makes w(t) tend to positive values; σ e− (t0 + 0) = 0, i.e., σ e (t) ≥ 0 on a right neighborhood of t0 yields w(t) = 0 (uniqueness of solution of differential equations). Example 2.7 The following figure sketches the behavior of a bimodal system (2.21) with m = k0 = 1 and σ e(t) := − sin(t2 ).

Fig. 2.3: Bimodal system, functions vs. t; fat solid: w with controller (2.24); fat dashed: λ; dashed: w without controller (λ∗ = 0); dotted: σ e. 

Remark 2.8 Note that the controller does not simply cut off the negative-valued part of the uncontrolled w-curve!  Exercise 2.9 Construct further examples (different parameters m, k0 , and functions σ e ) and evaluate them on the computer. (For handling the sign-function in (2.24) see Appendix D.) 

22

2 The Straight Worm With Propulsive Spikes (“SPIKY”)

Knowledge of w is important for the following reason: In mode (a), w > 0, on some t-interval none of the mass points with ground contact is at rest (remind (2.18)), none of the spikes is active, the worm as a whole is skidding forward during these times superimposed by simultaneous internal motion t 7−→ ζi (t). Having in mind that the worm system is to do some fine job (transportation, exploration, repair) then this might be a very unpleasant behavior. In practice those phases of motion in mode (b) are of greater interest. So every information about the possible occurrence of this skidding or about how to prevent it would be a good thing. This item is now investigated in a more detailed way.

Exercise 2.10 What about a worm with only one spike? Can it hubble forward or is it damned to flounder on the spot? Think about the simplest case n = 1, K = {0} first.  Modes of worm dynamics Assume that during the motion of the worm system some of the mass points with ground contact stay at rest on an open time interval (t0 , t1 ) : active spikes. Let A ⊂ K be the set of the indices of all these points, the elements of the complement Ac = K \ A ⊂ K then mark the moving spiked points. Definition 2.11 The dynamics of the worm system are called to be in mode A ⊂ K during the time interval (t0 , t1 ) if x˙ κ (t) = 0 , κ ∈ A ,

x˙ j (t) > 0 , j ∈ Ac = K \ A, t ∈ (t0 , t1 ) ,

accompanied by λκ ≥ 0 , κ ∈ A ,

λj = 0 , j ∈ Ac .

The cardinality of the mode is the number of active spikes during the motion in this mode.  The last part is a consequence of the complementarity condition x˙ κ λκ = 0. Labeling of the set A by the interval (t0 , t1 ) is omitted for brevity. With k := card(K) there are 2k modes (subsets of K), among them mode ∅: no active spike, and mode K: all spikes active, no locomotion of the system. Switching from one mode to another during the motion is governed by the controller (2.14). Mode ∅ coincides with mode (a) of the bimodal system (2.21), every mode A 6=∅ corresponds to mode (b) of (2.21). In order to simplify the foregoing visualization of modes suppose K = N. Then the spikes constraints x˙ i ≥ 0, i ∈ N = {0, . . . , n}, express the geometric fact that the (n + 1)−dimensional generalized velocity vector x˙ = (x˙ 0 , . . . , x˙ n ) - tangent to the trajectory of the representative point of the worm system - is restricted to the closed positive

2.1 Straight worm as a system of finite degree of freedom

23

orthant of Rn+1 (octant in R3 , quadrant in R2 ). During a motion in mode A the coordinates xα , α ∈ A, are kept fixed. Thus the worm system has still n + 1 − a, a = card(A), variable coordinates, it moves like a system of DoF = n + 1 − a. In any mode A 6= ∅ the velocity vector is in a (n + 1 − a)-dimensional hyperplane-part of the boundary of that orthant, whereas in case A = ∅ the velocity is in the interior of the orthant. Figure 2.4 gives a sketch in R3 of what can happen during a motion.

0,2 0 1,2

2 1 0

0 Fig. 2.4: Motion of the representative point (n = 2) under switching mode.

Remark 2.12 Putting all these facts together we find the following observations and conclusions. The worm system can move in this or that mode (2k options). In each mode of cardinality a the dynamics of the system are governed by the (Newtonian) laws of a motion system of DoF = n + 1 − a. At certain instants of time during a motion, switches may occur from one mode to another. The dynamical laws in a fixed mode determine which mode transitions are allowed (and which actually happen). So the modes can be visualized as the nodes of a graph whose edges are the feasible transitions between the modes. Summarizing, the worm system turns out to be a particular kind of what is called a hybrid system, [vdSS 95], [Stei 99]: a family of continuous-time dynamical systems parameterized by the nodes of a transition graph. About the transition from one mode to some other we shall learn something more in Sections 2.1.3.2 and 2.1.3.4. Concluding, we point out that the worm dynamics have two ingredients: - a discrete set-valued part t 7→ A(t) ∈ P( K) (finitely many values, piecewise constant function), and - a continuous Rn+1 -valued part t 7→ x(t) ∈ Rn+1 , with xα (t) = const, α ∈ A(t) (continuous values, piecewise smooth function). 

24

2 The Straight Worm With Propulsive Spikes (“SPIKY”)

Balance of power Recall Newton’s laws for the mass points (2.13) with gi = −k0 x˙ i − Γi , X mx ¨i = −k0 x˙ i − Γi + µi − µi+1 + δiκ λκ , i ∈ N , κ∈K

and xi = x∗ + ζi , x˙ ∗ = v ∗ . For power terms to P become apparent Pn multiply each equation Pn n by x˙ i and add for i = 0, . . . , n. On account of 0 ζ˙i = 0, 0 x˙ 2i = (n + 1) v ∗2 + 0 ζ˙i2 , P P P n n κ ˙ i = 0 (complementarity) and 0 (µi − µi+1 ) = 0, 0 (µi − µi+1 ) ζ˙i = κ∈K δi λκ x P n − 1 µj l˙j this yields    d m ∗2  (n + 1) + k0 v ∗2 + Γ∗ v ∗ v dt 2 +

n n n X X d  m X ˙2  Γi ζ˙i = Pa , (2.26) ζ˙i2 + ζi + k0 dt 2 i=0 i=0 i=0

where Pa := −

n X

µj l˙j

(2.27)

j=1

is the total power of the actuators supplied to the mass points. As can be seen from (2.26) this power supply splits up into parts corresponding to the motion of the center of mass (velocity v ∗ ) and change of shape (velocities ζ˙i relative to center of mass), respectively. Now the motion of the center of mass is governed by (2.20). Multiplication of this equation by v ∗ = W0 + w yields (mind w λ∗ = 0)    d m ∗2  ∗2 ∗ ∗ (n + 1) (2.28) + k0 v + Γ v = (n + 1) λ∗ W0 . v dt 2 Hereby it becomes clear that, Pe := (n + 1) λ∗ W0

(2.29)

is the total power of the external propulsive forces: (n + 1) λ∗ is the resultant of all spikes forces and W0 is the velocity of the center of mass while λ∗ 6= 0. Then the balance can be given the form n n n X X d  m X ˙2  2 ˙ Pe + Γi ζ˙i = Pa . ζi + ζ + k0 dt 2 i=0 i i=0 i=0

(2.30)

If the dynamics are in a mode A 6= ∅: w = 0, λ∗ > 0, then v ∗ = W0 , λ∗ = σ, and the 2 2 ∗ external power is Pe = (n + 1) σ W0 = (n + 1)[ ddt ( m 2 W0 ) + k0 W0 + Γ W0 ]. In mode

2.1 Straight worm as a system of finite degree of freedom

25

A = ∅: w > 0, λ∗ = 0 however, the external P power is Pe =P0, and thePtotal power n n n d m supplied by the actuators then is Pa = dt ( 2 i=0 ζ˙i2 ) + k0 i=0 ζ˙i2 + i=0 Γi ζ˙i , it ∗ has zero influence on v = W0 + w, it just enters lost motion of the system: change of internal kinetic energy and an external dissipation if there is environmental viscosity or dry friction.

2.1.3

Kinematic drive

2.1.3.1

Kinematics

The worm system is called to move under kinematic drive if by means of the actuators all distances lj , j = 1, . . . , n, or, equivalently, the relative velocities l˙j are prescribed as P functions of t. Then the formulas of Section 2.1.1 make S˙ i = ij=1 l˙j and V0 = max{S˙ κ | κ ∈ K} known functions of t. In the velocities (2.6) x˙ i = V0 (t) − S˙ i (t) + w ,

i = 0, . . . , n ,

the rigid part w is now the only free variable. This corresponds to the fact that for the system of n + 1 mass points on the x-axis (DoF = n + 1) the distance relations (2.3), xj−1 − xj − lj (t) = 0 ,

j = 1, . . . , n ,

(2.31)

now represent n independent rheonomic holonomic constraints which shrink the degree of freedom to 1. w is the remaining generalized velocity of the worm system. The differential constraints (2.2) are, as before, satisfied by definition of V0 . The Assumption: The actuator outputs t 7→ lj (t) are piecewise twice continuously differentiable functions, lj (·) ∈ D2 (R) ,

j = 1, . . . , n ,

(i.e., lj and l˙j are continuous on R, l¨j may have isolated jumps) guarantees that the general smoothness assumption (2.1) is met. The known functions l˙j , S˙ i , V0 and W0 are now of class D1 (R), the rigid velocity part w had been classified to be a D1 −function, though arbitrary in kinematics. The representations (2.9) of the velocities x˙ i show that now the head speed x˙ 0 , the rigid velocity part w, or the velocity v ∗ of the center of mass may be alternatively preferred as generalized velocity. Obviously they transform one to each other by rheonomic translations. Once more, the rigid part w of the velocities keeps arbitrary in kinematics. So it seems promising to put it equal to zero, then all velocities of the mass points are known functions of t. Putting w = 0 locks the single degree of freedom, the system has become a compulsive mechanism with ground contact (the latter causing locomotion).

26

2 The Straight Worm With Propulsive Spikes (“SPIKY”)

There remains a nicely simple Kinematical theory: (worm with kinematic drive and w = 0) Prescribe: lj (·) ∈ D2 (R) : t 7→ lj (t) > 0 , j = 1, . . . , n . i  P lj , V0 := max S˙ κ | κ ∈ K ∈ D1 (R) . Determine: Si := j=1

Result: x0 (t) =

Rt

V0 (s) ds ,

xj (t) = x0 (t) − Sj (t) ,

j = 1, . . . , n .

0

(2.32)

Before continuing the theory three examples are considered. Intermediate and final results (function V0 and xi (t)) are found by computer. Example 2.13 Worm with 5 mass points, each with spike. The lengths of the n = 4 links are lj (t) = 1 for t < αj , and lj (t) = 1 + 0.25 sin(2π(t − αj )) for t ≥ αj , αj = 0.2 (j − 1). Starting at t = 0.6 the kinematic drive is periodic in time with period T = 1. The same then holds for the head speed V0 and for the shape of the moving worm. Its average speed per period calculates as v = 1.25. Note that l2 , l3 , l4 are only D1 for t ≤ 0.6, this shows up in the initial discontinuities of V0 ; despite this violation of smoothness assumptions the theory shows reasonable results since accelerations do not play any role in it.

Fig. 2.5: Left: lj vs. t; middle: V0 vs. t; right: worm motion, t−axis upward.

 Example 2.14 Same data as before, but in lj (t) the function sin3 replaces sin, D2 -assumption fulfilled, V0 now is D1 .

2.1 Straight worm as a system of finite degree of freedom

27

Fig. 2.6: Left: lj vs. t; middle: V0 vs. t; right: worm motion, t−axis upward.

 Example 2.15 Same data as in first example above, but spikes at points 1, 3, 4 only! Since now 0, 2 ∈ / K, V0 and x˙ 2 may have negative values (moving backward!). Average speed per period now v = 1.06.

Fig. 2.7: Left: lj vs. t; middle: V0 vs. t; right: worm motion, t−axis upward.

 The bigger average speed in the first case is reasonable since there V0 is generated by maximizing over a bigger set. As a rule: the more spikes the bigger (in general) the speed. In the kinematical theory mass points without spikes do not influence the worm motion at all.

28

2 The Straight Worm With Propulsive Spikes (“SPIKY”)

The map {1, . . . , n}×R → Rn : (j, t) 7−→ lj (t) > 0 ,

j = 1, . . . , n

which characterizes a kinematic drive is called a kinematic gait. In the context of the kinematical theory there appears as an important task: to find a gait that is optimal with regard to a given goal such as maximal speed or minimal/maximal number of simultaneously active spikes, see Section 2.1.3.4. Theorem 1 If the kinematical theory is valid then during locomotion, independently of the gait, not all x˙ κ , κ ∈ K, can simultaneously be positive, i.e., at any time there is at least one active spike.  This theorem excludes mode A = ∅ from the kinematical theory. Proof. In view to (2.18) w(t) = 0 and x˙ κ (t) > 0 for all κ ∈ K means V0 (t) = max{S˙ l (t) | l ∈ K} > S˙ κ (t), κ ∈ K : contradiction! 

Remark 2.16 Note that this simple kinematical theory is a nonlinear theory. This is because of the non-validity of the superposition principle: on account of (2.3) and (2.4) a superposition of gaits, lj (·) = α lj1 (·) + β lj2 (·), α, β ∈ R, in general does not generate V0 (·) = α V01 (·) + β V02 (·). Take, e.g., l1 = l2 , α = 1, β = −2 α = −2. Then, with V0 [l] := max{S˙j }, α S˙ 1 + β S˙ 2 = −S˙ 1 , it follows (let K = N)        V0 α l1 + β l2 = V0 − l1 = max − S˙ j1 = − min S˙ j1 6= − max S˙ j1 in general. This fact holds for the corresponding dynamics as well.



Exercise 2.17 Construct further examples (prescribe different t 7→ lj (t), also for different n) and evaluate them on the computer - just following the simple scheme (2.32) - supplemented by plotting. In this way also a demonstration of Remark 2.16 can be achieved.  Until now it is unclear whether a locomotion described by kinematical theory is dynamically feasible or whether it has to be seen as a limit case of dynamically feasible locomotion or whether it is even generally unfeasible. The dynamics of kinematically driven worm systems have to be considered anyway, since statements about acting forces are of relevance: for the worm system to be able to transport something (in every case that includes its own mass) it needs sufficiently large external propulsive forces, and these are the active spikes forces.

2.1 Straight worm as a system of finite degree of freedom 2.1.3.2

29

Dynamics

The dynamics of worm systems with kinematic drive are governed by all the statements in Section 2.1.2 where now ζ˙i , S˙ i , V0 , W0 are given D1 -functions of time. In the relevant equations (2.21), (2.22), and (2.17) combined with (2.20) these known functions are explicitly denoted f (t): v ∗ = W0 (t) + w , x˙ i = v ∗ + ζ˙i (t) .  P 1 m w˙ + k0 w + σ(t) = λ∗ , λ∗ = n+1 λκ ,    κ∈K ∗ ∗ w ≥ 0, λ ≥ 0, wλ = 0,   ˙ 0 + k0 W0 + Γ∗ , W0 (t) := V0 (t) − ζ˙0 (t) .  σ := m W X µi − µi+1 + δiκ λκ − λ∗ = m ζ¨i (t) + k0 ζ˙i (t) . m v˙ ∗ + k0 v ∗ + Γ∗ = λ∗ ,

(2.33)

(2.34)

(2.35)

κ∈K

The last n + 1 equations (2.35) are linearly dependent (sum=identity). There remain the rigid velocity part w, the n cut forces µj , and the k = card(K) external reaction forces λκ as 1 + n + k unknowns in one bimodal DAE (2.34) and n algebraic equations (2.35). Note that now the cut forces µj are the reaction forces corresponding to the holonomic constraints (2.31). Thus Lemma 2.3 is not of direct usefulness since thereby each λκ 6= 0 depends on the unknown internal reaction forces µj . Recall from Subsection 2.1.3.1 that the central DAE (2.34) can be compressed to
m w˙ + k0 w + σ(t) = 1 − sign(w) σ + (t)

by means of the complementary-slackness condition.

According to Theorem 1 w = 0 implies that at least one spike is active, i.e., λ∗ > 0. Then (2.34) exhibits σ > 0 as a necessary condition for the dynamical feasibility of the kinematical theory. However, σ > 0 is compatible with both (w = 0, λ∗ > 0) and - if w˙ is negative - (w > 0, λ∗ = 0). The more detailed statements of the following theorem need the concept of dynamical modes A. Theorem 2 Consider a worm system with kinematic drive. (1) If σ(t) ≥ 0 for all t, then and only then the system always moves in some mode A 6=∅, i.e., the motion is completely described by the kinematical theory. (2) If the dynamics are not in mode ∅, then the actual mode at time t is A = {κ ∈ K | V0 (t) = S˙ κ (t)}.

(2.36)

In this mode there holds fα = −Γα + µα − µα+1 ≤ 0 for every α ∈ A, the mode is left as soon as one of these fα tends to positive values.

30

2 The Straight Worm With Propulsive Spikes (“SPIKY”) (3) The dynamics change over to mode ∅ at time t0 if and only if σ(t) tends (or jumps) to negative values at that time. Mode ∅ is left again as soon as w, solving (2.34) with λ∗ = 0, is going to negative values. (4) If Γ∗ ≥ 0 (no forward weight component) the system cannot start in mode ∅ from rest to forward locomotion. 

Remark 2.18

(1) In mode ∅ on the maximal time interval (t0 , t1 ), w ∈ D1 is the positive solution of the differential equation m w˙ + k0 w = −σ(t) with initial value w(t0 ) = 0. If σ(t0 + 0) ≥ 0, then w(t ˙ 0 + 0) ≤ 0, so w(t) would decrease to negative values or it remains zero: contradiction. If w(t) = 0 for t ∈ (t0 , t1 ) then (2.34) implies σ = λ∗ ≥ 0 on this interval. This proves part (1) of the theorem. (2) (2.36) is a kinematical characterization of the mode: A is left as soon as at least one of the equations V0 = S˙ α , α ∈ A, is not satisfied anymore. By means of the sign of the fα the termination of this mode is characterized dynamically (recall: λα = −fα in mode A, and λα cannot be negative). This description of shifting from one mode to another (“gear shift”) does not give practical benefit at this stage since the fα depend on unknown reaction forces. (3) If mode A 6= ∅ then σ(t) is proportional to the mean value of the λκ (t). Negative σ(t) would require at least one negative λκ (t), this is impossible in any mode. Formula (2.23) for the solution of the w-differential equation shows that exit from mode ∅ does not occur simultaneously with σ going negative but later (see Fig. 2.3). (4) Rest means w(t) = 0 and l˙j (t) = 0, j = 1, . . . , n for t ≤ 0. Then also v ∗ (t) = 0 for t < 0, and statement (4) follows from the solution 1 v (t) = − m ∗

Zt 0


k0 Γ∗ − λ∗ (s) e m (s−t) ds

of (2.33): in mode ∅ for small t > 0 there holds λ∗ (s) = 0, and Γ∗ ≥ 0 implies a non-forward v ∗ (t) ≤ 0.  The dynamical equations give rise to further theoretically important and practically utilizable conclusions. Unfortunately, the reaction forces λκ could not be determined in detail. But as an essential outcome there appeared the bimodal differential-algebraic system (2.11), now with the known D0 −function t → σ(t). Until now, only statements about change of modes A 6= ∅  A = ∅ have been gained from it. But moreover (2.11) yields some information about the resultant spikes force in the practically important modes A 6= ∅.

2.1 Straight worm as a system of finite degree of freedom 2.1.3.3

31

About spikes of finite strength

In this Subsection we address the problem of failing spikes, i.e., spikes which by accident cannot prevent negative displacements anymore. How to avoid such events in practice?

Consider a worm system with (possibly large) n ≥ k > 1 (recall: n = number of links, k = card(K) = number of mass points with spike), that moves in a mode A 6= ∅. Then w = 0, the motion is governed by the kinematical theory, at every time there is at least one active spike, and (2.21) and (2.22) yield ˙ 0 (t) + k0 W0 (t) + Γ∗ . λ∗ = σ(t) = m W

(2.37)

P W0 = v ∗ is (while w = 0) the velocity of the center of mass, (n + 1) λ∗ = κ∈K λκ is the resultant of all spikes forces (so (2.37) is nothing else but Newton’s law for the total system in mode A 6= ∅). In mode A 6= ∅ it follows from (2.36), (2.7), and (2.8): P P (n + 1) S˙ = i∈N S˙ i = card(A)V0 + i∈N\A S˙ i , and, observing n + 1 − card(A) = card(N) − card(A) = card(N \ A), with W0 = V0 − S˙ finally   P ˙ 1  Si , card(N \ A)V0 −  W0 = n+1 i∈N\A if A 6= ∅ : (2.38) P 1  λα .  λ∗ = n+1 α∈A

Now a thorn can break under overload, a ratchet can fail, and, see Chapter 3, stiction bounds can be exceeded. Therefore it is supposed here that the feasible spikes forces are bounded, b. λκ ≤ λ

Then in mode A 6= ∅ it is necessary that b ˙ 0 (t) + k0 W0 (t) + Γ∗ ≤ card(A) λ 0 < mW n+1

(2.39)

32

2 The Straight Worm With Propulsive Spikes (“SPIKY”)

be satisfied. At time t, card(A) is the number of those α ∈ A, for which S˙ α (t) = V0 (t). ˙ 0 are bounded, and obviously the right inequality W0 is a D1 -function, hence W0 and W could be violated if, e.g., Γ∗ > 0 is too big. Therefore modes with sufficiently many active spikes should be preferred, this is a requirement when searching for a practically suitable gait.

Example 2.19 A worm system with n = 2 and K = N = {0, 1, 2} is considered. A kinematic drive is searched such that at every time a prescribed positive number (1 or 2) of spikes are active. Based on (2.36) two heuristic construction schemes for solving this task will be introduced. Let the drive be T -periodic and put τ := Tt . Assume the alternating modes to be of i duration T /3 each and choose suitable Si0 = dS dτ (schematically, these are constant on the subintervals, their numerical values are arbitrary but must satisfy (2.36); choice is 0 for zero, 1 for positive, and −1 for negative. Furthermore, S00 = 0 and R1 0 0 S dτ = 0 must hold true. Finally, lj0 = Sj0 − Sj−1 follow. 0 i 1st scheme: (cycle of modes: {1} → {0} → {2}) τ (0, 13 ) ( 31 , 23 ) ( 32 , 1)

mode {1} {0} {2}

S00 0 0 0

S10 1 −1 0

S20 0 −1 1

V0 1 0 1

l10 1 −1 0

l20 −1 0 1

2nd scheme: (cycle of modes: {1, 2} → {0, 1} → {2, 0}) τ (0, 13 ) ( 31 , 23 ) ( 32 , 1)

mode {1, 2} {0, 1} {2, 0}

S00 0 0 0

S10 1 0 −1

S20 1 −1 0

V0 1 0 0

l10 1 0 −1

l20 0 −1 1

(Comment: Mode {0, 1} requires S00 (= 0) = S10 = V0 (fat numbers) and S20 < V0 , S20 = −1 forced by 1 above and 0 below.) Using the Heaviside function  1 , if a < τ ≤ b h(a, b, ·) : τ 7−→ h(a, b, τ ) := 0 , else the l0 of the 2nd scheme write

l10 (τ ) = h 0, 31 , τ −h 32 , 1, τ , and l20 (τ ) = −h

1 2 3, 3, τ




+h

2 3 , 1, τ




= l10 τ + 13

(to be continued T -periodically). These piecewise continuous functions should be smoothed out to D1 -functions. This can be done, e.g., by replacing τ 7−→ ±1 by a suitable sin-function τ 7−→ ± π2 sin(3πτ )




2.1 Straight worm as a system of finite degree of freedom

33

having the same integral value. Let l0 be the original length of the links, l0 ε, 0 < ε < 1 the amplitude of its variation in time, let Γ1 = Γ2 = Γ3 =: Γ. Using the abbreviations ω=

2π , T

ρ=

k0 , mω

2ρ , sin(α) = p 9 + 4 ρ2

the time derivatives l˙j = the two gaits.

1 0 T lj

γ=

Γ m l0 ε ω 2

3 cos(α) = p , 9 + 4 ρ2

and further functions can be calculated separately for

1st gait: l˙1 = l˙2 =

3 4 l0 ε ω 3 4 l0 ε ω

V0 (τ ) =


2 sin(3 π τ ) h h 0, 3 , τ ,


sin(3 π τ ) − h 0, 31 , τ + h

h
3 l0 ε ω sin(3 π τ ) h 0, 31 , τ + h 4

2 3 , 1, τ

2 3 , 1, τ

Average speed per period

v1 =

Z1

V0 dτ =


i

,


i

.

1 l0 ε, ω . π

0

h


i ε ω sin(3 π τ ) h 0, 13 , τ − h 13 , 23 , τ + h 23 , 1, τ , h


 ˙ 0 = 3 l0 ε ω 2 cos(3 π τ ) h 0, 1 , τ − h 1 , 2 , τ + h 2 , 1, τ , W 4 3 3 3 3 W0 =

1 2 l0

p 1 2 σ = m l0 ε ω 9 + 4 ρ2 cos(3 πτ − α)· 4 h
· h 0, 13 , τ − h

1 2 3, 3, τ




+h

2 3 , 1, τ


i



+ 4γ .

The results are sketched in the following Fig. 2.8 under arbitrary choice m = 1, k0 = 10, Γ = 2. Left: l1 (solid) and l2 (dashed) vs. τ qualitatively; middle: worm motion (τ upward), center of mass dotted; right: σ vs. τ , qualitatively (τ upward). Note the alternating rest phases of the mass points!

34

2 The Straight Worm With Propulsive Spikes (“SPIKY”)

Fig. 2.8: Worm motion following kinematical theory, first gait.

Discussion: Negative values of σ(t) would entail w > 0, skidding of the worm, too big positive values 0 < σ(t) = λ∗ (t) (= 31 of one spikes force in this gait!) would make the active spike fail. By means of minimum (at τ = 31 ) and maximum (at 3πτ = α) of σ the above formulas give p σ(t) ≥ 0 if 9 + 4 ρ2 ε ω 2 ≤ m4l0 Γ ,   p b+Γ . b if 9 + 4ρ2 ε ω 2 ≤ m4l0 λ σ(t) ≤ 13 λ

So the unsuitable mode ∅ can be avoided if Γ > 0 (gravity acting backward) and ω is not too big. Taking the maximal ω and k0 = 0 there remains the requirement Γ≤

1b λ 6

b and the average speed then is for not to exceed the strength λ, r 2 2 Γ v1 = l0 ε . π 3 m 2nd gait: l˙1 = l˙2 = V0 =

3 4 l0 3 4 l0

h
ε ω sin(3 π τ ) h 0, 13 , τ − h
ε ω sin(3 π τ ) h 31 , 1, τ ,


3 l0 ε ω sin(3 π τ ) h 0, 31 , τ . 4

Average speed per period v2 =

Z1 0

V0 dτ =

1 l0 ε ω . 2π

2 3 , 1, τ


i

,

2.1 Straight worm as a system of finite degree of freedom

35

h


i ε ω sin(3 π τ ) h 0, 13 , τ − h 13 , 23 , τ + h 23 , 1, τ = ε ω sin(3 π τ ) , h


i ˙ 0 = 3 l0 ε ω 2 cos(3 π τ ) h 0, 1 , τ − h 1 , 2 , τ + h 2 , 1, τ W 8 3 3 3 3
= 83 l0 ε ω 2 cos(3 π τ ) sign sin(3 π τ ) , W0 =

1 4 l0 1 4 l0


1 m l0 ε ω 2 3 cos(3 π τ ) + 2 ρ sin(3 π τ ) sign sin(3πτ ) + Γ∗ 8 p
1 = m l0 ε ω 2 9 + 4 ρ2 cos(3πτ − α) sign sin(3 π τ ) + Γ∗ . 8 The results are sketched in Fig. 2.9 in the same way and for same data as above. Again, note the alternating rest phases of the mass points! σ=

Fig. 2.9: Worm motion following kinematical theory, second gait.

By means of minimum (at τ = formulas give σ(t) ≥ 0 : 2b σ(t) ≤ λ : 3

ε ω2

1 3)

and maximum (at 3πτ = α) of σ the above

p 8Γ , 9 + 4 ρ2 ≤ m l0

p εω 9 + 4 ρ2 ≤ 2

8 m l0



2b λ−Γ 3



b But note that in this case for excluding mode ∅ and not exceeding the strength λ. of two simultaneously active spikes this is guaranteed only in the mean, 3λ∗ = P b α∈A λα ≤ 2λ , so one too big and one sufficiently small λ would formally do as well! Taking the maximal ω and k0 = 0 there remains the requirement Γ≤

1b λ, 3

and the average speed per period then is r 2 1 v2 = ε l0 Γ . π 3m

36

2 The Straight Worm With Propulsive Spikes (“SPIKY”) Comparison: The 1st gait admits a larger speed at smaller Γ, whereas the 2nd gait allows for larger Γ although at smaller speed: 1st gait for horizontal motion, 2nd gait uphill. 

Clearly, if on some t-intervals the condition σ(t) ≥ 0 is violated then mode ∅ occurs. In gait 1 this demonstrates the following Fig. 2.10 where k0 = 5 and Γ = 0 are chosen. The left diagram sketches (up to scaling) vs. τ : σ (dotted), λ∗ (dashed), and w (solid line). w adds to every speed of the kinematical theory, this is shown in the right diagram: V0 =headspeed kinematically (dashed), V0 + w =headspeed dynamically (solid line).

Fig. 2.10: w, σ, λ∗ (left), V0 , V0 + w (right) vs. τ .

2.1.3.4

Gait construction

The foregoing Example 2.19 has shown that the applied schematic algorithm for to construct kinematic gaits which guarantee a fixed number card(A) of active spikes indeed works. For low dimension n and with suitable smooth representation of lj (t) it allows to estimate operation data which guarantee not to overload the spikes. But some remarks concerning the heuristic basis of the algorithm are necessary. (a) Consider an n = 2 worm. Formally, there are (n + 1)! = 6 gaits with card(A) = 1 (number of permutations in {0, 1, 2}): {0} {0} {1} {1} {2} {2}

→ → → → → →

{1} {2} {0} {2} {0} {1}

→ → → → → →

{2} {1} {2} {0} {1} {0}

and equally 6 gaits with card(A) = 2 (number of permutations): {0, 1} {0, 1} {0, 2} {0, 2} {1, 2} {1, 2}

→ → → → → →

{0, 2} {1, 2} {0, 1} {1, 2} {0, 1} {0, 2}

→ → → → → →

{1, 2} {0, 2} {1, 2} {0, 1} {0, 2} {0, 1}

2.1 Straight worm as a system of finite degree of freedom

37

In either case each of the last four sequences coincides, disregarding a shift, with one of the first two. This becomes evident if not only one period of the gait but the cycling gait is considered:

{0}

{2}

{0,1}

{1}

{1,2}

{0,2}

Fig. 2.11: Cycling gaits with card(A) = 1 (left) and card(A) = 2 (right).

Moreover it is obvious that the second line gaits are nothing else but the reversed first line ones. Summarizing, the worm with K = N = {0, 1, 2} admits (disregarding the orientation of the cycle) essentially two gaits ensuring a fixed number of active spikes during motion. But, if the cardinality of K considerably exceeds 3 then the number of respective gaits explodes and a systematic way of gait construction becomes unavoidable. (b) Although the gaits in the worked examples had been chosen arbitrarily they now show up as unique ones (except for their functional representation). On the other hand, under an arbitrarily chosen gait a worm with n  1 could show a rather wild behavior, i.e., not every possible pattern of active spikes matches reality or proves reasonable and useful with respect to some objective. Having a glance at the earthworm paradigm one observes certain traveling waves of contraction or extension of the worm’s segments which obviously cause a suitable pattern t → A(t) (also a traveling wave?) of active spikes. Nevertheless, based on Theorem 2(2) it is possible to construct a kinematical gait that realizes any desired motion with arbitrarily given mode function t 7−→ A(t) 6= ∅. The following procedure for to do this at arbitrary n can easily be implemented on a computer. Mind that by definition every A(·) is a piecewise constant (set-valued) function. Suppose K = N for what follows. A gait construction algorithm, see [StBe 11]: (1) Let t be in a maximal time interval (t0 , t1 ) with constant mode A(t) = A0 6= ∅. If 0 ∈ A0 (active head spike) then put V0 := 0 else V0 := 1 (symbolically, means V0 > 0), S˙ 0 := 0. (2) For i = 1, . . . , n do: if i ∈ A0 then S˙ i := V0 else S˙ i := V0 − 1 (symbolically, means S˙ i < V0 ). (3) For j = 1, . . . , n put l˙j := S˙ j − S˙ j−1 . (Symbolically, l˙j ∈ {−1, 0, 1} means link j contracting, invariant, expanding.)

38

2 The Straight Worm With Propulsive Spikes (“SPIKY”) Note that this symbolic gait is unique! Note further that, vice versa, this gait uniquely entails the preceding S˙ i and V0 and hence the mode A0 again.

(4) On [t0 , t1 ] choose a suitable differentiable representation t 7→ L˙ j (t) of β · l˙j , such that (i) β takes care of required data (e.g., amplitudes), (ii) L˙ j (t0 ) = L˙ j (t1 ) = 0 (ensures continuity at t0,1 of the gait whatever in neighboring intervals happens). Determine the corresponding representations of V0 and S˙ j . If the above D1 -gait t 7→ L˙ j (t) yields σ(t) ≥ 0 (this can be ensured by appropriate choice of data, see examples above and investigations below) then it realizes the desired dynamics with modes t 7−→ A(t) 6= ∅.

In the sequel this procedure is sketched in application to a particular case that is, as mentioned above, of some practical relevance. Assumption 2.20 1◦ The mode function A(·) is T -periodic. 2◦ A(·) is of constant positive cardinality, card(A(t)) = a = const ≤ n (same number of active spikes at every time). 3◦ All time intervals with constant value A(t) have the same length, i.e., T /N , since one cycle needs N := n + 1 steps. It is not necessary but proves useful to introduce the supplementing item 4◦ Each A(t) consists of (cyclically on [0, 1, . . . , n]) consecutive spikes labels. 

2.1 Straight worm as a system of finite degree of freedom

39

Example 2.21 n = 6, a = 4. Claimed mode sequence: {0, 1, 2, 3} → {1, 2, 3, 4} → {2, 3, 4, 5} → .... → {6, 0, 1, 2} → {0, 1, 2, 3} → . . . . The underlined part characterizes one cycle of N = n + 1 steps. The algorithm given above yields the following result: step k

mode

S˙ n . . . S˙ 0

l˙n . . . l˙1

V0

1

{0, 1, 2, 3} [-1,-1,-1, 0, 0, 0, 0] 0 [ 0, 0,-1, 0, 0, 0]

2

{1, 2, 3, 4} [ 0, 0, 1, 1, 1, 1, 0] 1 [ 0,-1, 0, 0, 0, 1]

3

{2, 3, 4, 5} [ 0, 1, 1, 1, 1, 0, 0] 1 [-1, 0, 0, 0, 1, 0]

4

{3, 4, 5, 6} [ 1, 1, 1, 1, 0, 0, 0] 1 [ 0, 0, 0, 1, 0, 0]

5

{4, 5, 6, 0} [ 0, 0, 0,-1,-1,-1, 0] 0 [ 0, 0, 1, 0, 0,-1]

6

{5, 6, 0, 1} [ 0, 0,-1,-1,-1, 0, 0] 0 [ 0, 1, 0, 0,-1, 0]

N = 7 {6, 0, 1, 2} [ 0,-1,-1,-1, 0, 0, 0] 0 [ 1, 0, 0,-1, 0, 0]

W0 3 7 3 7 3 7 3 7 3 7 3 7 3 7

(2.40)

 Inspecting this schedule closely, it is easy to draw the following conclusions which hold for any n and a. (a) In each step there are exactly a indices α such that S˙ α = V0 and therefore N − a indices such that S˙ α = V0 − 1. This implies, in this symbolic representation, n

W0 = V0 −

1 X ˙ N −a Si = n+1 0 N

(2.41)

as the expected velocity v ∗ of the center of mass. (b) Picking l˙a out of the last column we recognize the temporal sequence (−1, 0, 0, 0, 1, 0, 0)T that is characterized by the a − 1 zeros between the contraction −1 and the extension 1. The neighboring columns are obtained by a time shift. This structure uniquely marks the property 4◦ formulated above: cyclically, the sets A(t) are without gaps. It is a fortunate feature for the implementation of the gait l˙j . Finally, the result can be seen as a contraction wave running from head to rear through the links. (c) In this symbolic representation the time rates l˙j can be written as functions of t l˙j (t) =

N X

k=1

rj (k) · h



k−1 k t , , N N T



,

(2.42)

40

2 The Straight Worm With Propulsive Spikes (“SPIKY”) where rj (k) ∈ {−1, 0, 1} is the constant value on the respecting k th time interval to be taken out of the schedule above. So the l˙j are piecewise constant functions with jumps at the interval boundaries. Of the same kind then are the S˙ i , and also W0 is in fact not constant throughout rather it is discontinuous and zero valued at the interval boundaries. Recalling the definitions of S˙ i and V0 by means of the l˙j we obtain

V0 (t) =

N X

k=1

V0k · h



k−1 k t , , N N T



, where V0k

  i X rj (k) . = max  i

j=1

(2.43)

(d) Both theory and technical implementation (l˙j as actuator outputs) demand at least continuity and piecewise differentiability of every velocity. That is why on each of the above time intervals the Heaviside function h has to be replaced by a differentiable function e h that vanishes at the boundary points and has the same integral value as h. Let the links be of original length l0 and suppose εl0 rj (k) to be the length increment. Then we obtain the sinusoidal representations    N  N k−1 k t N π X (1) ˙ , (2.44) Lj (t) = ε l0 ω sin t , , rj (k) · h 4 T N N T k=1

and

N (2) L˙ j (t) = ε l0 ω sin2 π



  N  Nπ X k−1 k t t , , rj (k) · h T N N T

(2.45)

k=1

which are of class D1 and D2 , respectively. As usual, ω = 2π T . Clearly, the functions have to be continued T -periodically before any virtual or practical application. For this end just replace Tt in the Heaviside function by its fractional part f rac( Tt ). It is indeed a pleasant fact that this entails the nice single-term (and already representations   N − a N (1) ω sin ω t , W0 (t) = ε l0 4 2 (2)

W0 (t) = ε l0

N −a ω sin2 π



N ωt 2



T N −periodic)

(2.46)

(2.47)

which allow a consideration of σ(t) as easily as it has been done in the worked examples above. This is done in the sequel using the D2 -representation and, to keep formulas simple, confining to the case k0 = 0. To do the same for the D1 -representation and to compare the results with Example 2.19 is left as an exercise.

2.1 Straight worm as a system of finite degree of freedom

41

Then ˙ 0 (t) + Γ∗ = m ε l0 ω 2 N (N − a) sin(N ω t) + Γ∗ . σ(t) := m W 2π

(2.48)

In order to guarantee validity of the kinematical theory (w = 0: no skidding) σ must be non-negative for every t, its minimum at N ωt = 32 π yields m ε l0 ω 2

N (N − a) ≤ Γ∗ . 2π

(2.49)

b means Not to exceed the total strength of all active spikes, N λ∗ = N σ(t) ≤ a λ, m ε l0 ω 2

N (N − a) a b λ − Γ∗ . ≤ 2π N

(2.50)

In combination there results the Proposition 2.22

Every motion with card(A(t)) = a = const in the above D2 sinusoidal representation is governed by the kinematical theory, i.e., dynamically feasible, and ensures not to exceed the strength of the active spikes if and only if 0


N Γ∗ . b λ

So any gait with 2

be realized (possibly with small εω ) whereas spikes as being undersized. 8 Note

that the factor

1 2π

N Γ∗ b λ

N Γ∗ b λ

< a ≤ N − 1 can

≥ N − 1 qualifies the

may change if another smooth representation is used.

42

2 The Straight Worm With Propulsive Spikes (“SPIKY”) - Eventually Γ∗ could vary during the motion owing to change of inclination in hilly landscape or change of roughness if Γ∗ has a Coulomb friction part. This may cause σ(t) to tend to either negative or too big values. Then, watching σ(t) online, two options are available to prevent this tendency in due time: a) switch to some other gait with bigger a (“gear shift”), b) diminish εω 2 by means of amplitude ε or by means of frequency ω (“throttle down”); F at given εω 2 search for Γ∗ to match: - Obviously, Γ∗ must be large enough but satisfying the inequality (2.52); - A big value of Γ∗ could occur by a big gravity component backwards, Γ∗g = m g sin(β) (inclination β), this is bounded above by m g and, possibly, not big enough; - In a gait with constant a always the same number, N − a, of mass points are sliding in positive x-direction along the ground. If a dry friction applies to these points (see Subsection 3.2.1), then Γ∗f = NN−a F + (F + magnitude of sliding friction in positive direction) is the respective constant Coulomb sliding friction acting at the center of mass. Finally, Γ∗ = Γ∗g +Γ∗f may ensure σ(t) > 0, then mode ∅ does not occur and, thus, the constancy of a is also dynamically justified; - Γ∗f may depend on the gravitational pressure between mass point and ground. If the spikes are realized by wheels with ratchet the friction could be internally generated as a controllable braking torque!

The average speed per cycle, v ∗ = v∗ =

1 T

1 ε l0 ω (N − a) . 2π

RT 0

W0 (t)dt calculates as (2.53)

The maximal acceleration with no skidding is (disregarding bounded λ∗ ) l0 ε ω 2 = 2 π

Γ∗ 1 . m N (N − a)

This entails the average speed (maximal without skidding) r r 1 ε l 0 Γ∗ a ∗ v = 1− . 2π m N

(2.54)

(2.55)

The maximal acceleration (2.54) is feasible also for bounded spikes force if the stronger inequality Γ∗ ≤ is satisfied.

a b λ 2N

(2.56)

Concluding remark: The algorithm for gait construction remains the same also if the subintervals of one period are not of equal lengths, moreover the algorithm also works

2.1 Straight worm as a system of finite degree of freedom

43

for non-periodic modes A(·). In the functional representations of the generated gait the k Heaviside functions h( k−1 N , N , ·) must be replaced by appropriate h(tk−1 , tk , ·) together with suitable smooth equivalents. We repeat the foregoing Example 2.21 in a pure visual version.

Example 2.23 We draw the desired successive configurations of the worm. The vertical dotted lines in Figure 2.12 mark those mass points that are to be at rest during the respective time interval. Doing this it is easy to note the symbolic value 0, 1 or −1 of l˙j (t) at the corresponding links. l6’(t)

l5’(t) 0

l4’(t)

0

l3’(t)

l2’(t)

l1’(t)

-1

0

0

0

0

1

0

-1

0

0

-1

0

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

t

0

0

1

1

1

1

x

-1

0

-1

0

-1

0

0

Fig. 2.12: Worm motion.

The reader should do this construction using a different a and then check the total displacement. Note that this procedure relies on the minimal number of lengths to be changed per step! (The first step kept its feasibility if we would choose lj0 (t) = −1 for j = 4, 5, 6.) Following the cyclic mode sequence {0, 1, 2, 3} → {1, 2, 3, 4} → . . . we arrive at the tableau of Figure 2.13.

44

2 The Straight Worm With Propulsive Spikes (“SPIKY”)

l6’(t) 0 0 -1 0 0 0 1 0 0 -1 0 0 0 1

l5’(t) 0 -1 0 0 0 1 0 0 -1 0 0 0 1 0

l4’(t) -1 0 0 0 1 0 0 -1 0 0 0 1 0 0

l3’(t) 0 0 0 1 0 0 -1 0 0 0 1 0 0 -1

l2’(t) 0 0 1 0 0 -1 0 0 0 1 0 0 -1 0

l1’(t) 0 1 transient 0 behavior 0 -1 0 0 T=7t 0 1 0 0 -1 0 0

}

Fig. 2.13: Link tableau.

The following Fig. 2.14 shows the corresponding continuous worm motion:

Fig. 2.14: Worm motion xi (solid) and center of mass (dashed) vs. representation (2.45).

t T

, using sin2 -

Figures 2.13 and 2.14 clearly show a traveling contraction wave running from head to rear (known from the earthworm) entering link 1 after four transient steps. Every T step is of duration τ = N = T 71 and contributes 17 to the stride length L = 1 (measured in any unit of length, not specified here). 

2.1 Straight worm as a system of finite degree of freedom

45

The examples above have shown that the condition σ > 0 for exclusion of mode ∅ (skidding worm) cannot be satisfied under arbitrary system data. Though general statements are seemingly not possible a closer consideration may be in order. Exercise 2.24 Repeat all steps above for (e.g.) the following motions of a worm with n = 6. a) Gait {0, 1, 2, 3} → {6, 0, 1, 2} → ... (gait of Example 2.21, reversed). b) Gait {0, 1, 3, 4} → {1, 2, 4, 5} → ... (analogue to Example 2.21 but with gap!). Observe possible variations of results.  For the following let K = N: each mass point of the system contacts the ground via spike. Then there holds Lemma 2.25 In case K = N the D1 -function W0 has the following properties: (1) W0 (t) ≥ 0 for all t, (2) W0 (t) = 0 if and only if l˙j (t) = 0, j = 1, . . . , n.



P Proof. By definition of W0 and V0 it holds (n + 1) W0 (t) = (n + 1) V0 (t) − n0 S˙ i (t) = Pn ˙ 0 [V0 (t) − Si (t)] geq0, since each summand is non-negative. Hence W0 (t) = 0 ⇐⇒ ˙ V0 (t) − Si (t) = 0, i = 0, . . . , n. Taking i = 0 this means V0 (t) = 0, therefore also S˙ j (t) = 0, and this entails vanishing of all l˙j (t).  In view to statement (2) W0 (t) = 0 means that at time t the lengths of all links are ‘frozen’, the system is either at rest at this instant or it performs a rigid motion with speed w(t) > 0. If K 6= N then W0 need not exhibit these properties (1) and (2). For instance, in P P ˙ ˙ (n + 1) W0 = κ∈K (V0 − Sκ ) + i∈K / (V0 − Si ) now only the first sum is certainly non-negative (see definition of V0 ) whereas the second could cause a negative value of W0 . Exercise 2.26 Run the gait construction algorithm for an example of your choice - but with K 6= N. Discuss the results.  The minimal supposition lj ∈ D2 (R) entails W0 = V0 − S˙ ∈ D1 , because of the maxprocess in forming V0 any stronger supposition would not lead to a better smoothness

46

2 The Straight Worm With Propulsive Spikes (“SPIKY”)

˙ 0 ∈ D0 : piecewise continuous, i.e., continuous up to isolated jumps. of W0 . Therefore W ˙ 0 may not be defined at all, but the one-sided limits exist and are At a t0 with jump W ˙ 0 is in particular bounded on every bounded t-interval. finite. So W In regard to undulatory locomotion let lj be T -periodic. Then also W0 is T −periodic and, following Lemma 2.25, always positive if the l˙j do not have common zeros, hence, since W0 is continuous, it cannot be monotonic (if not constant) on a periodicity interval. ˙ 0 , which is T -periodic, too, has alternating positive and negative Thus the derivative W values on subintervals. This observation leads to some statements about the sign of ˙ 0 (t) + k0 W0 (t) + Γ∗ . σ(t) = m W First, it is obvious that σ has alternating sign in case k0 = 0, Γ∗ = 0: enduring or temporal occurrence of mode ∅ is unavoidable (i.e., skidding forward is an inertia phe˙ 0 could be compensated nomenon). Negative values of W

a) by means of large values of k0 (possibly creeping on ground with a layer of strong viscosity); b) by means of large values of Γ∗ . Several options are possible: they are outlined in the sequel of Proposition 2.22 above.

There is no need to worry about negative σ(t) if we put m := 0, i.e., cancel any inertia effects. This hypothetical scenario comes close to what happens during slow motion (big ˙ 0 = O( 12 )), in a strongly viscous environment (big k0 ), see T in undulation, since W T [TINK 11], where a theory is set up under such premises. In a general context it remains unclear whether at a given gait the condition σ ≥ 0 (possibly accompanied by additional conditions for bounded spikes strength) can be satisfied by means of practically suitable values of k0 , Γ∗ and ω.

2.1.3.5

A glance back: the simplest guy amongst all worms

The foregoing description of the dynamics of spiked worms might imply the feeling that these systems exhibit a behavior that is, compared with system behavior discussed in customary mechanics textbooks, a bit strange. Although this behavior has been well captured by the theoretical means developed above it should be appreciated as a relaxing repetition to have another look at the simplest worm system with only two mass points under kinematic drive and see what happens in detail.

2.1 Straight worm as a system of finite degree of freedom

47

So let n = 1, take the T -periodic kinematic drive ˙ = ε l0 ω sin(ω t) , l(t)

ω = 2 π/T ,

ε ∈ (0, 1) , t ≥ 0 ,

and suppose first that there are no impressed external forces (k0 = 0, Γ = 0). Figure 2.2 sketches the dynamical scenario: i = 1, µ0 = µ2 = 0. The internal forces ±µ1 , applied by the actuator (of what kind ever) to the neighbored mass points, are the reactions to the holonomic constraint x0 − x1 − l(t) = 0, while the external forces λ0,1 are the non-negative reactions to the differential constraints x˙ 0,1 ≥ 0, transmitted from the ground to the mass points via the spikes. Let the worm be at rest until t = 0 : x˙ 0 (0) = x˙ 1 (0) = 0, l(0) = l0 . At t = 0 the link ˙ begins to expand: l(t) > 0 for t ∈ (0, T2 ). As the spike prevents x1 from decreasing, mass point 0 gets the (positive) acceleration ¨ l(t) = εl0 ω 2 cos(ωt). Then x˙ 0 (t) > 0 and consequently λ0 = 0. Therefore it holds m ¨l(t) = −µ1 (t), and the internal force µ1 (t) applied to the mass point 1 remains negative and compensated by the spike force λ1 for t ∈ (0, T4 ): mass point 1 stays at rest during this time interval and the dynamics are in the mode A(t) = {1}. Then what happens at time t = T4 ? Starting at t = T4 the expansion rate slows down, i.e., the actuator pulls inward, thus it applies a positive force µ1 to mass point 1. This force cannot be compensated by the spike, so λ1 (t) = 0, and m x ¨1 (t) = µ1 (t) > 0 implies x˙ 1 (t) > 0 for t > T4 . Since still x˙ 0 (t) > 0 either mass point is moving: the dynamics are in mode A(t) = ∅ during some time interval ( T4 , t1 ). The system moves now under no external forces, so the center of mass has constant speed x˙ ∗ (t) = x˙ ∗ ( T4 ) = 12 x˙ 0 ( T4 ) = 21 εl0 ω. How long? The velocities of the mass points, ˙ and x˙ 1 = x˙ ∗ − l/2 ˙ remain positive up to isolated instants of time and this x˙ 0 = x˙ ∗ + l/2, means A(t) = ∅ for all times t > T4 . The movement of the worm is a uniform skidding with speed 21 εl0 ω superimposed by an unprofitable harmonic “pumping” l(t). However, if there was then an external viscous damping force (k0 > 0) then the differential equation m x ¨∗ = −k0 x˙ ∗ , x˙ ∗ ( T4 ) = 21 εl0 ω would make the center of mass exponentially slow down. Then one of the x˙ 0,1 would tend to negative values (which one?) and a switch to some mode A(t) 6= ∅ occurs.

48

2 The Straight Worm With Propulsive Spikes (“SPIKY”)

Let us check the above results by means of the formal tools. Differential equations: mx ¨0 = −k0 x˙ 0 − Γ − µ1 + λ0 mx ¨1 = −k0 x˙ 1 − Γ + µ1 + λ1 mx ¨∗ = −k0 x˙ ∗ − Γ∗ + λ∗

x0 = x∗ + ζ0 x1 = x∗ + ζ1 λ∗ = 12 (λ0 + λ1 ) .

ζ0 = 21 l(t) , ζ1 = − 21 l(t) ,

Speeds: ˙ = l˙+ , w ≥ 0 , x˙ 0 = V0 + w , V0 = max{0, l} 1 x˙ ∗ = W0 + w , W0 = | l˙ | . 2 Central equation: m w˙ + k0 w + σ(t) = λ∗ ,

˙ 0 + k0 W0 + Γ∗ . σ = mW

˙ = ε l0 ω sin(ωt) simple calculations yield for the first period9 With l(t)   1 ε l0 ω sin(ωt) h(0, π, ωt) − h(π, 2π, ωt) , 2    1 σ(t) = ε l0 ω m ω cos(ωt) + k0 sin(ωt) h(0, π, ωt) − h(π, 2π, ωt) + Γ∗ 2   p m = 1 + ρ2 cos(ωt − α) h(0, π, ωt) − h(π, 2π, ωt) + 2 γ . ε l0 ω 2 2 k0 ρ 1 Γ∗ ρ= . , sin(α) = p , cos(α) = p , γ= mω m ε l0 ω 2 1 + ρ2 1 + ρ2

W0 (t) =

It is simple matter to guarantee the validity of the kinematical theory (σ(t) ≥ 0 for all t) by choosing γ sufficiently big. But to come close to the foregoing discussions let now γ = 0. Then σ(t) is sketched in the following Figure 2.15 (m ε l0 ω 2 = 9.87 chosen).

Fig. 2.15: σ vs. t/T , solid: ρ = 0, dashed: ρ > 0. 9 It is useful to write | l(t) ˙ |= ε l0 ω sin(ωt) [h(0, π, ωt)−h(π, 2π, ωt)]. Differentiation (piecewise) then acts simply on the sin factor.

2.1 Straight worm as a system of finite degree of freedom

49

Fig. 2.16: Left: head speeds V0 (thin line) and V0 + w (fat line); right: corresponding worm motions.

Owing to Theorem 2(3) the time intervals with negative σ(t) are definitely occupied by mode A = ∅, whereas modes A 6= ∅ are possible on the σ ≥ 0 - intervals. Clearly, the σ < 0 - intervals shrink with increasing ρ, but the occurrence of the skidding mode A = ∅ is unavoidable. Switching among the modes is governed by the central equation above and Theorem 2. It is evident that a Γ∗ < 0 that is large enough to ensure σ < 0 causes a permanent downhill skidding, see next figure, where Γ∗ = −4 is used.

Fig. 2.17: Left: head speeds V0 (thin line) and V0 + w (fat line) for ρ > 0 and Γ∗ < 0; right: corresponding worm motions.

2.1.4

Dynamic drive I

In this subsection a worm model is considered whose actuator outputs are forces which are prescribed functions of time and are immediately applied to the adjacent mass points. The question of how to realize such actuators shall not be answered here, it is just to be observed how the worm might behave under this kind of drive. Realizable and already realized principles of dynamic drives are considered in the next subsection.

50

2 The Straight Worm With Propulsive Spikes (“SPIKY”)

As usual the actuators are supposed massless. Assuming that every link is endowed with an actuator then the equations of motion are given by (2.13), where now the cut forces µi are given functions of time (and, thus, impressed forces): µi (·) ∈ D0 ensures the required smoothness of the motion. The worm motions are governed by mx ¨i = −k0 x˙ i − Γi + µi (t) − µi+1 (t) +

X

δiκ λκ ,

i = 0, . . . , n .

(2.57)

κ∈K

Putting µi (t) − µi+1 (t) =: Fi (t), mass point i appears Pn under the action of one timedependent actuator force, by definition there holds i=0 Fi = 0, and this reflects the fact of Fi to be internal forces. The complementary-slackness conditions (2.12) belonging to (2.57) can be satisfied using Lemma 2.3. Because of fκ = −k0 x˙ κ −Γκ +Fκ (t) the ‘control law’ (2.14) does not contain unknown reaction forces anymore, so it can successfully be used.

Example 2.27 Worm system n = 2, harmonic actuator forces. Using two different sets of data the following two figures show the system behavior: coordinates xi , x∗ and velocity v ∗ vs. t. Data 1: m = 1, k0 = 0.5, Γi = 0.1, µ1 (t) = 50 sin(Ωt), µ2 (t) = −25 sin(Ωt), Ω = 10 π, T = 0.2.

Fig. 2.18: x0,1,2 , x∗ vs. t/T (left), v ∗ vs. t/T (right).

Data 2: m = 1, k0 = 0, Γi = 0, µ1 (t) = 20 sin(Ωt), µ2 (t) = −60 sin(Ωt), Ω = 4 π.

2.1 Straight worm as a system of finite degree of freedom

Fig. 2.19: x0,1,2 , x∗ vs. t/T (left), v ∗ vs. t/T (right).

51



These examples show that a suitable choice of pulsating actuator forces µi (t) leads to a worm-like locomotion. However the obvious running into each other/running away from each other of neighboring mass points cannot be tolerated. The dubiousness of this way of modeling is evident: disregarding the questionable realizability of this type of force-generating actuators the mass points of the system form a very loose ensemble. Particularly in the representation with Fi (t) := µi (t) − µi+1 (t) the Newton laws (2.57) point to the fact that the mass points Pn essentially move independently of each other, the only weak coupling is given by i=0 Fi = 0.

2.1.5

Dynamic drive II

2.1.5.1

Actuator models and realizations

The modeling of the internal drive can be improved by assigning, besides the pure force output, also certain simple rheological properties to the actuators. In the following, not a real actuator or its physical model shall be the starting-point, rather the investigations are based on some mathematical ansatz the possible realizations of which to be discussed after that. Let the cut forces µi be qualified as impressed forces obeying the following law:
(2.58) µi (t, x, x) ˙ := ci xi−1 − xi − li0 + k00i (x˙ i−1 − x˙ i ) + ui (t) .

This mathematical relation describes the parallel arrangement of a linear-elastic spring with a constant stiffness ci and original length li0 , a Stokes damping element with constant coefficient k00i , and a time-dependent force ui (t).

Fig. 2.20: Actuator, general physical model.

52

2 The Straight Worm With Propulsive Spikes (“SPIKY”)

The Figure 2.20 shows the corresponding physical model of this actuator, where now the small circular box represents a non-modeled device generating the force output ui (t), see Fig. 2.2. Four possibilities for realizing this actuator are sketched in the following. 1st realization: Formally writing ui (t) = −ci li0 εi (t) turns the elastic term into

ci xi−1 − xi − L0i (t) with L0i (t) := li0 1 + εi (t) .

This describes a linear-elastic spring with time-dependent original length L0i (t), and εi (t) has to be seen as to-be-prescribed strain. The physical model of the actuator is shown in the next figure, where now the small circular box represents a device generating L0i (t) as output (see Fig. 1.7). In practice it might be implemented as a piezo element, a stroke magnet, or pneumatic element. For worm systems with this kind of actuators see [Stei 99].

Fig. 2.21: Actuator, physical model 1.

2nd realization: In [Weis 02] and [Huan 03] a worm-like system with n = 1 was described, designed, and realized as a hardware prototype. Ground contact is achieved via (areas of) spikes, driving device is a piezo element P with compliant amplitude magnifiers (see Fig. 2.22 left and Section 5.2).

Fig. 2.22: Actuators 2 and 3.

3rd realization: For theoretical reasons and presumably also for simpler technology it seems reasonable to substitute the two chains of joints by an elastic circular lune, the latter admits a fairly simple mathematical modeling using common procedures from

2.1 Straight worm as a system of finite degree of freedom

53

(linear) elasticity theory. The outcome is as follows. Let opposite vertical forces ±F2 act at the corners and opposite horizontal forces ±F1 at the vertices (see Fig. 2.22 right). Then the resulting changes of the horizontal and vertical diameters, d1 and d2 , show up as functions of F1,2 with coefficients aij depending on the geometry and elasticity of the lune, d1 = a11 F1 + a12 F2 ,

d2 = a21 F1 + a22 F2 .

If the piezo element is characterized by a linear relation of longitudinal force F , dilation d, and electrical voltage U , AF + B d + C U = 0, then there results a linear elastic device with static characteristics F1 = c d1 + u , where u is proportional to the voltage U . Positioning the actuator between the mass points i and i − 1 then just the velocity-independent term in (2.58) is met. 4th realization: A formal rewriting of the force, ui (t) = −ki a˙ i (t) turns the viscous term into
k00i x˙ i−1 − x˙ i − a˙ i (t) ,

and this could be seen as the description of a viscous element consisting of two consecutive damping elements whose pistons have a distance ai that is a prescribed (smooth) function of t. The physical model of the actuator is shown in the next figure, where now the small circular box represents a device generating ai (t) as its output.

Fig. 2.23: Actuator, physical model 4.

Under the assumption that all actuators have the same data (c, l0 , k00 ) the equations of motion follow from (2.57) in the actual form mx ¨0 = −c (x0 − x1 − l0 ) − k00 (x˙ 0 − x˙ 1 ) − k0 x˙ 0 − u1 (t) − Γ0 + λ0 , mx ¨j = −c (2 xj − xj+1 − xj−1 ) − k00 (2 x˙ j − x˙ j+1 − x˙ j−1 )+ −k0 x˙ j + uj (t) − uj+1 (t) − Γj + λj , mx ¨n = c (xn−1 − xn − l0 ) + k00 (x˙ n−1 − x˙ n ) − k0 x˙ n + un (t) − Γn + λn , (2.59)

54

2 The Straight Worm With Propulsive Spikes (“SPIKY”)

which holds for each of the above realizations (including dynamical drive I if c = k00 = 0). The accompanying complementary-slackness conditions are satisfied through expressing the λi by meansPof the “controller” (2.14). If there are mass points without spike, K 6= N, then λi = κ∈K δiκ λκ . At this stage, the ui are to be seen as prescribed functions of t, off-line controls, later on they will also be handled as depending on the state (x, x), ˙ too: feedback, on-line controls. Alternative forms of the equations of motion correspond to (2.33), and (2.34), of course, with ζi , W0 and σ not being known functions of t. Clearly, the elastic and viscous parts of the actuator could equally be characterized by nonlinear laws. 2.1.5.2

The inch-worm

In the sequel the simplest worm system, n = 1, both mass points with spikes, K = {0, 1}, dynamically driven, is considered in detail, some examples undergo simulation. This system models the above mentioned prototypes after [Weis 02] and [Huan 03], and the classical inch-worm as well (see Fig. 2.24 - actually, the friendly worm in the middle moves like a caterpillar, nevertheless it is captured by the worm concept: the actuator works by bending the material link, thus shortening and re-stretching the distance l of the mass points).

Fig. 2.24: Inch-worm: natural and modeled - the circle box indicates an actuator as in Fig. 2.20.

Supposing Γ0 = Γ1 = Γ the equations of motion are given by (2.59): mx ¨0 = −c (x0 − x1 − l0 ) − k00 (x˙ 0 − x˙ 1 ) − k0 x˙ 0 − Γ − u1 + λ0 , mx ¨1 = c (x0 − x1 − l0 ) + k00 (x˙ 0 − x˙ 1 ) − k0 x˙ 1 − Γ + u1 + λ1 , x˙ i ≥ 0 , λi ≥ 0 , x˙ i λi = 0 , i = 1, 2. (2.60) Let as usual x0 = x∗ + ζ, x1 = x∗ − ζ, where x∗ is the coordinate of the center of mass, v ∗ = x˙ ∗ and let u := u1 − c l0 (in order to get rid of the l0 -terms). With 2 ζ = l = actual length of the link and exploiting kinematics from Subsection 2.1.1, i.e., x˙ 0 = V0 + w ,

˙ = l˙+ x˙ 1 = V0 + w − l˙ where V0 = max{0, l}

2.1 Straight worm as a system of finite degree of freedom

55

and W0 = 21 l˙ ,

v ∗ = 12 l˙ + w ,

w ≥ 0,

˙ ˙ (m ¨l + k0 l) σ = Γ + 12 sign(l)

the equations of motion can be written as  m ¨l + (k0 + 2 k00 ) l˙ + 2 c l = −2 u + (λ0 − λ1 ) ,     1 ∗  m w˙ + k w + σ = (λ + λ ) = λ , 0

σ =Γ+ w ≥ 0,

2

1 2

0

1

˙ , ˙ (m ¨l + k0 l) sign(l)

λ∗ ≥ 0 ,

w λ∗ = 0 .

    

(2.61)

First, analyzing these relations the various modes of the motion are described in detail. Mode A = ∅: λ0 = λ1 = 0, m ¨l + (k0 + 2 k00 ) l˙ + 2 c l = −2 u , m w˙ + k0 w = −σ , w ≥ 0 , ˙ (u + k00 l˙ + c l) . σ = Γ − sign(l)

(2.62)

The new form of σ without a second derivative followed by means of the first equation. This mode will be entered as soon as σ becomes negative, return to some mode A 6= ∅ occurs when w is going to get negative values. Modes A 6= ∅: w = 0, ˙ , ˙ (m ¨l + k0 l) λ0 + λ1 = 2 Γ + sign(l) λ0 − λ1 = 2 u + m ¨l + (k0 + 2 k00 ) l˙ + 2 c l .

(2.63)

A = {0}: x˙ 0 = 0, x˙ 1 = −l˙ > 0, hence l˙ < 0, λ1 = 0. (2.63) is equivalent to m ¨l + (k0 + k00 ) l˙ + c l = Γ − u , l˙ < 0 , λ0 = Γ + u + k00 l˙ + c l = 2 Γ − m ¨l − k0 l˙ > 0 .

(2.64)

A = {1}: x˙ 0 > 0, x˙ 1 = x˙ 0 − l˙ = 0, hence l˙ > 0, λ0 = 0. (2.63) is equivalent to m ¨l + (k0 + k00 ) l˙ + c l = −Γ − u , l˙ > 0 , λ1 = Γ − u − k00 l˙ − c l = 2 Γ + m ¨l + k0 l˙ > 0 .

(2.65)

In both cases the first equation determines l(·) (damped oscillation with excitation u). After that the signs of l˙ and σ fix the feasible interval of this solution. Different modes ˙ smoothly follow each other (continuous l and l). If t 7→ l(t) was a prescribed function (kinematic drive) then the first equation would yield the corresponding t 7→ u(t): realization of gait l(·) via dynamic drive. This item will be taken up towards the end of this section.

56

2 The Straight Worm With Propulsive Spikes (“SPIKY”)

A = {0, 1}: x˙ 0 = x˙ 1 = 0, l˙ = 0, l(t) = l1 . (2.63) is equivalent to λ0 = Γ + u + c l1 ≥ 0 , λ1 = Γ − u − c l1 ≥ 0 .

(2.66)

This entails c (l1 − l0 ) − Γ ≤ u1 ≤ Γ + c (l1 − l0 ) as a necessary condition for the state of rest (requires Γ ≥ 0, else skidding forward; no need of constant u1 ). Finally, it is interesting to see how the equations of motion for the modes {0} and {1} combine and then govern the motion provided σ > 0. Simply write the Γ-term on the ˙ then it follows right hand sides in (2.64) and (2.65) in the form −Γsign(l), ˙ = −u(t) . m ¨l + (k0 + k00 ) l˙ + c l + Γ sign(l)

(2.67)

This is nothing else but the differential equation of a damped linear oscillator with Coulomb sliding friction (last term left hand side) and excitation −u. The validity of (2.67) is ensured if n
o ˙ = 1 Γ − sign(l) ˙ ¨l + k0 l) ˙ k00 l˙ + c l + u(t) > 0 . σ = Γ + 21 sign(l)(m 2

Since the terms within the parentheses are bounded functions of time this can be fulfilled by sufficiently large Γ.

Exercise 2.28 Same investigations with K = {0} or K = {1}.



Now the Γ-term in (2.67) formally reminds of a Coulomb sliding friction: opposite to velocity, constant magnitude. Therefore, a discussion supported by the differential equation (2.67) of the inchworm motions will be postponed. This item is taken up again after dry friction models have been introduced in Section 3.2.1. Another question demands attention. If the inchworm is to move in a prescribed kinematic gait t → l(t) then the formulas (2.64) and (2.65) allow to find that dynamic drive t → u(t) which realizes just this wanted motion (the inequality σ ≥ 0 guarantees the validity of the kinematical theory). In the following simulations the prescribed gait is   
 0 , l0 = 1 , a = 0.25 , Ω = π . t 7→ l(t) = l 1 − a 1 − cos Ωt The external forces are chosen with k0 = 1 and Γ = 1.2336, where the latter yields
min σ(t) = 0 which justifies the kinematical theory.

2.1 Straight worm as a system of finite degree of freedom

57

Fig. 2.25: Inchworm with gait l(·) given above: (a) x0 , x∗ , x1 vs. t; (b) v0 , v ∗ , v1 vs. t; (c) λ0 , λ1 vs. t.

This motion is achieved by a dynamic drive t 7→ u(t) given in (2.67). In Figs. p2.26 and 2.27 these drives are sketched vs. t for some actuators of different data ω0 = c/m and k00 = 2 m ω0 k.

Fig. 2.26: Dynamic gaits u vs. t for inchworm of Fig. 2.25 with different actuator data: (left) ω0 = Ω (resonance), k = 1 (aperiodic limit); (right) ω0 = Ω (resonance), k = 0.1 (weak damping).

58

2 The Straight Worm With Propulsive Spikes (“SPIKY”)

Fig. 2.27: Dynamic gaits u vs. t for inchworm of Fig. 2.25 with different actuator data: (left) ω0 = 1.5 Ω (sub-resonance), k = 0.1 (weak damping); (right) ω0 = 0.75 Ω (supra-resonance), k = 0.1 (weak damping).

Unfortunately, there appears a serious drawback for application: The procedure sketched above requires exact knowledge of all the system parameters m, k0 , k00 , c, Γ. As a rule, this is not the case, even more, some parameters - k0 , Γ for example - may change during motion. Then it becomes necessary to use appropriate tools from control theory to overcome these insufficiencies. Questions like this will be tackled in later sections. Finally, note that the problem ‘find dynamic drives to realize kinematic gaits’ - accompanied by parameter uncertainties - arises for systems with n > 1 as well, and, regrettably, for large n there is a severe lack of simple formulas as above.

2.2

Straight worm as a continuous system

Let us think of a worm system with large n, i.e., consisting of a large number of mass points, or recall the paradigm earthworm - with many consecutive segments with no indication of any mass point. One might feel then a continuum model to be a useful or

2.2 Straight worm as a continuous system

59

even a better means for a locomotion theory. That is why in this section worm systems are modeled as continua of material points located in a rectilinear x-axis. Let the reference configuration B0 be the interval [0, l0 ], where l0 is the length of the relaxed worm. The body-fixed coordinate ξ ∈ [0, l0 ] now replaces the label i ∈ {0, . . . , n} in the discrete model of Section 2.1, ξ = 0 marks the head, ξ = l0 the end of the worm. The field of actuators on [0, l0 ] might be pictured as a discrete one (e.g., piezo elements) or as continuous (relying on electro-magnetic or biochemical properties of the worm’s matter). Howsoever it serves for local change of shape (local strain of worm axis, ε(ξ, t), see below). The continuum theory to come exhibits the same structure as the finite DoF theory, encloses the latter while showing new options of describing motions.

x

x=0

x(x, t)

x(0, t)

x = l0

x Fig. 2.28: Continuous worm with coordinates (see Fig. 2.1).

2.2.1

Kinematics

A motion is now a family of maps x(ξ, ·) | R+ → R : t 7−→ x(ξ, t),

ξ ∈ [0, l0 ]

with the following properties: (i) for every t let x(·, t) be a diffeomorphism [0, l0 ] → R (so the actual configuration Bt = x([0, l0 ], t) is a closed interval); (ii) x,t exists on [0, l0 ] × R+ and is continuous, x,t (·, t) is a smooth function for every t; (iii) x,tt (·, ·) exists piecewise and it is piecewise continuous. Let K be a non-void subset of [0, l0 ], the material points ξ ∈ K and no points else contact the ‘ground’ via spikes. K may be a finite set, a finite union of closed intervals, or the complete interval [0, l0 ]. Therefore every motion will be restricted by the differential constraints ∀ξ∈K

∀ t ∈ R+

x,t (ξ, t) ≥ 0 .

(2.68)

60

2 The Straight Worm With Propulsive Spikes (“SPIKY”)

Let x0 (t) := x(0, t) denote the actual coordinate of the worm’s head and s(ξ, t) the actual distance of the material point ξ from the head at time t, then x(ξ, t) = x0 (t) − s(ξ, t) .

(2.69)

Differentiation yields the velocity of the material point ξ, v(ξ, t) := x,t (ξ, t) = x˙ 0 (t) − s,t (ξ, t) ≥ 0 ∀ ξ ∈ K , and this entails

 x˙ 0 (t) ≥ max s,t (ξ, t) | ξ ∈ K =: V0 (t) .

(2.70)

(The maximum exists since K is introduced as a closed set, else it had to be replaced by supremum). Therefore it holds w(t) ≥ 0 ,

x˙ 0 (t) = V0 (t) + w(t) ,

(2.71)

v(ξ, t) = V0 (t) − s,t (ξ, t) + w(t) .

Again, w is a common additive term to all velocities v(ξ, ·) - a rigid part of the motion, it remains arbitrary in kinematics. Now suppose a constant mass density along the relaxed worm (reference configuration), i.e., mass per original unit of length ξ 7−→ ρ0 = const > 0. Hence the worm’s total mass is M = ρ0 l0 . (Of course, in general the density in the actual configuration is not constant anymore but ρ(ξ, t) = ρ0 /(1 + ε(ξ, t)), where ε(ξ, t) denotes local strain.) So the center of mass has the actual coordinate ∗

x (t) = x0 (t) − S(t) ,

1 S(t) := l0

Zl0

s(ξ, t) dξ ,

(2.72)

0

and its velocity is v ∗ (t) = x˙ ∗ (t) = W0 (t) + w(t) ,

˙ . W0 (t) := V0 (t) − S(t)

1

Position in space R can be described by means of the frame there holds x(ξ, t) = x∗ (t) + ζ(ξ, t) ,

Ft∗

(2.73) ∗

= {x (t), ex }, then

ζ(ξ, t) = S(t) − s(ξ, t) ,

∗

v(ξ, t) = v (t) + ζ,t (ξ, t) ,

(2.74)

and in particular x(0, t) = x∗ (t) + ζ(0, t) ,

ζ(0, t) = S(t) ,

v(0, t) = V0 (t) − ζ,t (0, t) .

(2.75)

Clearly, shape at time t is described by {ζ(ξ, t) | ξ ∈ [0, l0 ]} or {s(ξ, t) | ξ ∈ [0, l0 ]} when using the frames Ft∗ = {x∗ (t), ex } or Ft0 = {x0 (t), ex }, respectively, for position. In either case change of shape is characterized by local strain (relative change of length at ξ), ε(ξ, t) := lim

dξ→+0

x(ξ, t) − x(ξ + dξ, t) − dξ = −(1 + x,ξ ) . dξ

2.2 Straight worm as a continuous system

61

Remark 2.29 The somewhat unusual minus sign in the definition of the local strain is due to the opposite orientations of x- and ξ-axis. Therefore ε(ξ, t) = −(1 + ζ,ξ (ξ, t)). 

For an active spike ξ ∈ K there holds v(ξ, t) = 0 ⇐⇒ ζ,t (ξ, t) = −v ∗ (t) ⇐⇒ w(t) = 0 ∧ s,t (ξ, t) = V0 (t) . (2.76) The reader should compare the above formulas with the respective ones in Subsection 2.1.1.

2.2.2

Dynamics

As in Subsection 2.1.2 theory the external impressed forces per original unit of length are confined to g(ξ, t) := −k0 v(ξ, t) − Γ , i.e., to a Stokes friction with constant coefficient k0 ≥ 0, and a constant force Γ R 0 (weight component, dry friction, e.g., constancy here in contrast to Γi used in Subsection 2.1.2). Then (see Fig. 2.2) the equations of motion write ρ0 v,t (ξ, t) = −k0 v(ξ, t) − Γ − N,ξ (ξ, t) + λ(ξ, t) ,

ξ ∈ [0, l0 ],

(2.77)

where N denotes the x-component of the cut force at ξ. λ is the constraint force (per original unit of length) corresponding to (2.68). Of course λ(ξ, t) = 0 for ξ ∈ / K, and the complementary slackness condition v(ξ, t) ≥ 0 ,

λ(ξ, t) ≥ 0 ,

v(ξ, t) λ(ξ, t) = 0 ,

ξ∈K

(2.78)

reflects the unilateral action of the spikes. Rl Averaging the equations of motion, l10 0 0 . . . dξ, yields the equation of motion for x∗ (principle of linear momentum for the total worm system): x˙ ∗ = v ∗ , ρ0 v˙ ∗ = −k0 v ∗ − Γ + λ∗ , ∗

v = W0 + w , w ≥ 0,

W0 = V0 −

λ∗ ≥ 0 ,

λ∗ := 1 l0

Rl0 0

w λ∗ = 0 .

1 l0

Rl0

λ(ξ, t) dξ ,

0

s,t (ξ, t) dξ ,

(2.79)

62

2 The Straight Worm With Propulsive Spikes (“SPIKY”)

Remark 2.30

(a) Observing that the configuration at time t of the continuous worm system can be described as {position, shape} = {{x∗ (t), ex }, {s(ξ, t) | ξ ∈ [0, l0 ]}} then (2.79) puts into evidence how locomotion t 7−→ x∗ (t) depends on both change of shape (s,t ) and interaction with the environment (λ). Of course, at this stage the equations do not serve for any immediate evaluation since nothing is known about shape s(ξ, t) yet. (b) In calculating λ∗ integration is actually about K (a sum if K finite). (c) If additional external forces F0 , F1 were applied to head and rear end of the worm, respectively, then N (0, t) = F0 , N (l0 , t) = −F1 , and the v ∗ differential equation must be supplemented by the term (F0 + F1 )/l0 (e.g., payload).  The core of (2.79) is again the well known bimodal system “central equation” ρ0 w˙ + k0 w + σ(t) = λ∗ , λ∗ ≥ 0 ,

w ≥ 0,

w λ∗ = 0 ,

(2.80)

˙ 0 + k0 W0 + Γ ∈ D1 σ := µ0 W with modes [w > 0 ∧ λ∗ = 0] : no active spike, and [w = 0 ∧ λ∗ ≥ 0] : at least one active spike. Formally, the slackness conditions can be satisfied by a “controller” (2.24). Clearly, from (2.77) a balance of power can be deduced. Multiplication by v(ξ, t) and Rl integration 0 0 . . . dξ yield, observing (2.74), 

M d ∗2 v + l0 k0 v ∗2 + l0 Γ v ∗ 2 dt





ρ0 d + 2 dt

Zl0

ζ,2t dξ + k0

0

Zl0 0



ζ,2t dξ  = Pa .

(2.81)

Using the actual local strain (local change of shape!) ε(ξ, t), the total power of actuators (actuators are somehow distributed along the worm) is

Pa := −

Zl0

N (ξ, t)ε,t (ξ, t) dξ .

0

Introducing the total power of the external propulsive forces Pe := l0 W0 λ∗

2.2 Straight worm as a continuous system

63

(resultant of propulsive forces l0 λ∗ , velocity of center of mass v ∗ = W0 + w) the balance writes (see balance (2.30)) 

ρ0 d Pe +  2 dt

2.2.3

Zl0

ζ,2t dξ + k0

0

Zl0 0



ζ,2t dξ  = Pa .

(2.82)

Kinematic drive

In the context of the continuous worm kinematic drive means to prescribe the longitudinal local strain ε(ξ, t). Any such function (ξ, t) 7−→ ε(ξ, t) is called a kinematic gait. It entails the additional holonomic constraints

(2.83) x,ξ (ξ, t) = − 1 + ε(ξ, t) , ε(ξ, ·) ∈ D2 R+ , (−1, 1) given. for the system. These replace the former constraints (2.2) and, in integrated form

x(ξ, t) = x(0, t) − s(ξ, t) ,

s(ξ, t) = ξ +

Zξ

ε(η, t) dη ,

0

now represent the distance relations (2.69) with known s(ξ, t). Corresponding constraint forces per unit of original length are the cut forces N (ξ, t). Consequently, W0 and σ in any foregoing equation are now to be treated as known functions of time. Merely w remains as free kinematic variable: the DoF of the continuous system reduced to 1, w is its generalized velocity. Putting w := 0, a kinematical theory arises, whose dynamical feasibility is governed by (2.80), the corresponding Theorem 2 applies almost literally: Theorem 3

(1) If σ(t) ≥ 0 for every t then w = 0, and the motion is completely governed by the kinematical theory. (2) If w(t) = 0 then the set of actually active spikes is n

A(t) = ξ ∈ K | V0 (t) =

Zξ 0

o ε,t (ζ, t) dζ .

(3) Mode [w > 0, λ∗ = 0] begins as soon as σ(t) becomes negative, the mode is left as soon as w (solution of the differential equation (2.80)) goes to negative values. (4) With Γ ≥ 0 the system cannot start a motion from rest in mode [w > 0, λ∗ = 0]. 

64

2 The Straight Worm With Propulsive Spikes (“SPIKY”)

It should be emphasized again that in mode [w > 0, λ∗ = 0] the worm system with spikes does not perform a proper locomotion, rather the worm is skidding on ground (while, internally driven, varying its shape but keeping any propulsive interaction with ground “switched-off”). So in fact the kinematical theory is the right one to rule locomotion under kinematic drive. All investigations to come in the sequel are restricted to kinematic drives. If we preferred dynamic drive in this continuous model then the internal forces N (ξ, t) had to be seen as outputs of continuously distributed actuators. To this end a special rheology for the material the worm is made of must be introduced (piezo-electric, shapememory, ferrofluid, etc.). Such things would break the scope of this book (but see [Zimm 07] and, on the other hand, the remarks in Section 5.1 about the paper [TINK 11] where the drive is mainly a contraction wave, i.e., a kinematic one). Finite set K of spikes Let K = {ξ0 , ξ1 , . . . , ξn } ⊂ [0, l0 ], ξi+1 > ξi ; head without spike, i.e., ξ0 > 0, may be allowed. All the basic kinematic relations from the beginning of this section remain unaltered. V0 (t) = max{s,t (ξi , t) | i = 0, . . . , n} might be negative if ξ0 > 0, and, since maximizing is over a finite set, it will in general be smaller than in case K = [0, l0 ]. Using Dirac’s “δ-function” the equations of motion (2.77) get the form ρ0 v,t (ξ, t) = −k0 v(ξ, t) − Γ − N,ξ (ξ, t) +

n X

λi (t) δ(ξ − ξi ) ,

ξ ∈ [0, l0 ] .

i=0

Integration from ξi − ε to ξi + ε with some ε > 0 followed by ε → +0 yields the obvious jump condition for the longitudinal cut force N (ξi + 0, t) − N (ξi − 0, t) = λi (t) . The open intervals (0, ξ0 ), (ξ0 , ξ1 ), . . . , (ξn , l0 ) are free of constraint forces λ, so a pieceRξ Rξ Rl wise integration ( 0 0 + ξ01 + . . . + ξn0 ) . . . dξ yields the well-known equation of motion for x∗ ρ0 v˙ ∗ = −k0 v ∗ − Γ + λ∗ ,

λ∗ =

n 1 X λi . l0 i=0

In combination with the basic kinematic relations and the representation vi (t) := x,t (ξi , t) = v ∗ (t) + ζ,t (ξi , t) of the velocities of the material points with spikes again a theory of worms with finite DoF has formally been attained. It describes the motion t 7→ x(ξi , t) of the material points ξi . This is in exact correspondence with the motion t 7→ xi (t) of the mass points in the former discrete model, where now, of course, the total mass M = ρ0 l0 (= (n+1)m formerly) is spread over the interval [0, l0 ].

2.2 Straight worm as a continuous system

65

In this discrete picture of the continuous worm

li (t) =

Zξi

ξi−1


1 + ε(ξ, t) dξ ,

i = 1, . . . , n ,

are the actual lengths of the links (distances of neighbored spikes). A (biological) interpretation closer to reality is met when choosing ξi := l0

2i−1 , 2n

i = 1, . . . , n .

l0 These n equidistant points imply a subdivision of [0, l0 ] into n subintervals [ξi − 2n , ξi + l0 l0 M ] each of length . Each of these intervals carries the mass , an actuator with 2n n R ξi +ln0 /2n output li (t) = ξi −l0 /2n (1 + ε(ξ, t))dξ and a spike. This is a quite nice model of the good old paradigmatic earthworm, see Fig. 2.2910 . Simulation results are presented later.

Fig. 2.29: Picture of an earthworm with bristles (left); zoom (right), [Courtesy by Ralf Weber and by the worm himself ].

Fig. 2.30: Model of an earthworm. 10 Arranged

by Danja Voges, TU Ilmenau

66

2 The Straight Worm With Propulsive Spikes (“SPIKY”)

Drive by traveling strain wave The continuous model admits the consideration of motions which are driven by strain waves running along the worm axis (as one can recognize when observing live earthworms)11 . This is independent of the set K of spikes. The investigations will focus on wave trains of finite length u0 , called solitons for short. Let p | R → R be a smooth function and let, using the Heaviside cut function h, Π : u 7−→ Π(u) := p(u) h(−u0 , 0, u) ,

u0 > 0

describe the smooth wave profile (note that p and its derivative must vanish at u = 0 and u = −u0 ). With positive κ and ω ε+ (ξ, t) := Π(κ ξ − ωt) = p(κ ξ − ωt) h(ω t − u0 , ωt, κ ξ)

(2.84)

and

ε− (ξ, t) := Π κ(l0 −ξ)−ω t = p κ (l0 −ξ)−ω t h(κ l0 −ω t, u0 +κ l0 −ω t, κ ξ) (2.85) are ε-solitons of wavelength l = u0 /κ and phase velocity c = ω/κ, traveling in positive and negative ξ-direction, respectively, entering the interval [0, l0 ] at time t = 0. p(u) < 0 means contraction, p(u) > 0 extension. ε is called short soliton if its wavelength is smaller than l0 , i.e., 0 < u 0 < κ l0 . Coming investigations are done within kinematical theory only and focussing on short solitons. In order to find the most important kinematic function V0 some not very exciting stepby-step calculations have to be done. For soliton ε+ : Rξ

Rξ

κ ξ−ωt R Π(κ ζ − ωt) dζ = ξ + κ1 Π(u) du , −ωt 0  0   s,t (ξ, t) = − ωκ Π(κ ξ − ωt) − Π(−ωt) = − ωκ ε+ (ξ, t) − ε+ (0, t) .

s(ξ, t) = ξ +

ε+ (ζ, t) dζ = ξ +

For soliton ε− likewise

 ω s,t (ξ, t) = + ε− (ξ, t) − ε− (0, t) . κ  Now V0 (t) = max s,t (ξ, t) | ξ ∈ K can be calculated: h  i V0 (t) := c ε+ (0, t) + max − ε+ (ξ, t) | ξ ∈ K ← − h  i = c Π(−ωt) − min Π(κ ξ − ωt) | ξ ∈ K h  i = c Π(−ωt) − min Π(u) | u ∈ −ωt + κK ,

11 Remind the examples in Section 2.1.3.4 where gait constructions for discrete worms resulted in traveling contraction or extension waves.

2.2 Straight worm as a continuous system

67

h  i V0 (t) := −c ε− (0, t) + max − ε− (ξ, t) | ξ ∈ K − → h  i = −c Π(κl0 − ωt) − min Π(u) | u ∈ κl0 − ωt + κK ,

where −ωt + κK := {−ωt + κξ | ξ ∈ K} has been used for short.

The soliton needs time T = (κl0 + u0 )/ω for one run through [0, l0 ]. In the kinematical theory V0 is the velocity of the worms head. Therefore, the average speed during one run is 1 v := T

ZT

V0 (t) dt .

0

This entails Theorem 4 Contraction waves and extension waves of profiles which are equal in magnitude but opposite in traveling direction generate equal average speeds v.  Proof. V0 (t) and V0 (t) are formally with profile ±Π, respectively. In both cases the − → ← − minima are the same at times t and t − c l0 , so the equality of the v follows through the RT RT R0 equality of the integrals 0 Π(−ωt) dt = . . . = 0 Π(κl0 −ωt) dt = . . . = ω1 −u0 g(u) du.  In the following one particular short soliton Π(u) := εb sin2 (u) h(−π, 0, u) ,

κl0 > π ,

is used for driving different worms.

εb ∈ R

1st worm: Set of spikes K = [0, l0 ]. To minimize εb sin2 (u) h(−π, 0, u) with u ∈ [−ωt, −ωt+ κl0 ] is the same as to minimize εb sin2 (u) with u ∈ [−π, 0] ∩ [−ωt, −ωt + κl0 ]. The next figure (Fig. 2.31) sketches the intervals relevant in this example (u0 = π). 1) Soliton ←−, εb > 0 (extension): Fig 2.31 shows min(. . .) = 0, hence V0 (t) = εb c sin2 (ωt) h(0, π, ωt) . ← −

2) Soliton ←−, εb < 0 (contraction): Fig 2.31 shows min(. . .) = εb sin2 (ωt) in (a), = εb in (b), and = εb sin2 (−ωt + κl0 ) in (c). Hence n ε c cos2 (ωt) h( π2 , π, ωt) + h(π, κ l0 + π2 , ωt) V0 (t) = −b ← − o + sin2 (κ l0 − ωt) h(κ l0 + π2 , κ l0 + π, ωt) .

68

2 The Straight Worm With Propulsive Spikes (“SPIKY”) The averaged speed follows as 1 v := ← − T

ZT

1 V0 (t) dt = |b ε| c ← − 2

0

(

π κ l0 +π 2κl0 κ l0 +π

, ,

εb > 0, εb < 0.

Therefore a (short, head to rear) extension wave yields v < under contraction wave.

1 4

εb c, whereas v >| εb | c

Fig. 2.31: Intervals for minimization, cases (a),(b),(c) top to bottom.

3) Soliton −→, εb > 0 (extension): As above n V0 (t) = εb c sin2 (ωt) h(0, π2 , ωt) + h( π2 , κ l0 , ωt) − → o + cos2 (κ l0 − ωt) h(κ l0 , κl0 + π2 , ωt) . 4) Soliton −→, εb < 0 (contraction):

ε c sin2 (−ωt + κ l0 ) h(κ l0 , κ l0 + π, ωt) . V0 (t) = −b − →

The average speeds − v are the same as the ← v above but with εb > 0 and εb < 0 → − interchanged: the extension soliton −→ and the contraction soliton ←− generate same speeds! Summarizing this yields the Observation: In this case of drive by means of a short soliton (π < κl0 ) a contraction strain wave running in positive ξ-direction (←) generates a larger speed

2.2 Straight worm as a continuous system

69

v than an extensional strain wave of the same direction but the same speed as an extension wave in opposite direction (→). Conjecture: The same holds true for arbitrary wave profiles Π. 2nd worm: K 6= [0, l0 ]. To calculate the required minima is a bit more involved and will be done only numerically. The following simulation results demonstrate the motion of worms with length l0 = 1 and various sets K of spikes. Drive is by short strain solitons of data Π(u) := εb sin2 (u) h(−u0 , 0, u) , | εb |= 0.25 , u0 = π , κ = 1.5π , ω = π .

Run time (ξ : 0 → l0 or l0 → 0) is T = 2.5, wavelength l = 0.67, phase velocity c = 0.67. Supposing appropriate dynamical data the motions rely on the kinematical theory. In the figures the actual positions of the soliton are sketched. Example 2.31 K = [0, l0 ], one run of contraction strain wave head to rear:

Fig. 2.32: Worm at various time levels, t-axis upward. Solid lines show positions of chosen material points. Average speed v = 0.1. 

Example 2.32 8 equidistant spikes (segmented earthworm), same drive as before:

70

2 The Straight Worm With Propulsive Spikes (“SPIKY”)

Fig. 2.33: Worm at various time levels, t-axis upward. Solid lines show positions of segment borders, dashed lines positions of spikes. Average speed v = 0.09.

Fig. 2.34: Lengths of segments li vs. t.



2.3 SPIKY in hilly landscape (“HILLY”)

71

Example 2.33 Only 3 spikes at ξ = 0, 0.25, 0.75. Contraction strain wave entering at head and reflected at rear end with phase jump (extension wave back), total run time T = 4.

Fig. 2.35: Worm at various time levels, t−axis upward. Solid lines show positions of chosen material points, dashed lines positions of spikes. Note the ‘useless wriggling’ between spikes. Av erage speed v = 0.07.

2.3

SPIKY in hilly landscape (“HILLY”)

72

2 The Straight Worm With Propulsive Spikes (“SPIKY”)

Until now we considered nearly exclusively spiked worms moving along a straight line (x-axis). A few passages pointed to the possible variability of the external forces Γi during motion (see remarks following Proposition 2.22). In the present section we will address worm motions under state-dependent forces Γi . Emphasis is on Γi depending on position, Γi (xi ). The most important scenario with this type of forces is of course a worm motion under influence of gravity (Γi gravity component backward) in a hilly landscape or within a curved tube. In both cases the non-constancy of Γi is determined by the local inclination β, i.e., Γi = G sin(β(xi )). Another scenario is the worm under influence of a Coulomb sliding friction caused by an environment with x-dependent surface roughness. The problem is sketched in the following figure (compare Fig. 2.2).

Fig. 2.36: (a) Worm on hill; (b) weight component; (c) worm along s-axis.

Let the hill profile be given as y = h(x) or, in parameter representation, as x = x(s), y = y(s) := h(x(s)), using arc length s. In either case the worm kinematics is (supposing K = N: every mass point with spike) t 7→ si (t) ,

s˙ i (t) ≥ 0 ,

i = 0,1,...,n,

(2.86)

together with all the conclusions drawn in Section 2.1.1 (there, just replace xi by si ). Equally, the dynamics are comprised by the differential equations (see Section 2.1.2 and Appendix B) and complementary-slackness conditions m s¨i = −k0 s˙ i − Γi + µi − µi+1 + λi , s˙ i ≥ 0 , λi ≥ 0 , s˙ i λi = 0 ,

i = 0,1,...,n,

(2.87)

with Γi = Γ(si ). Again, µi are the actuator generated internal forces, and the complementary-slackness conditions can be satisfied by means of a corresponding rule like (2.14).

2.3 SPIKY in hilly landscape (“HILLY”)

2.3.1

73

Center-approximation, kinematical drive

Given a kinematic gait t 7→ lj (t), j = 1,P . . . , n, we obtain (see Section P 2.1.3.2), using the n n 1 1 ∗ ∗ ∗ center of the s-coordinates, s∗ := n+1 s , and s ˙ = v , λ = 0 i 0 λi n+1

s˙ i = v ∗ + ζ˙i (t) , v ∗ = W0 (t) + w , W0 (t) = V0 (t) − ζ˙0 (t) , ∗ ∗ m w˙ + k0 w + σ = λ , w ≥ 0 , λ ≥ 0 , w λ∗ = 0 , (2.88) ∗ ˙ σ = m W0 + k0 W0 + Γ . Pn 1 ∗ Here Γ∗ = n+1 0 Γ(s + ζi (t)) and the center-approximation relies on the

Assumption 2.34

The distances from s∗ , |ζi |, are small w.r.t. some characteristic length of the hill, so that Γ (s∗ + ζi (t)) ≈ Γ (s∗ ) .

(2.89) 

This means that the inclination β does not vary too much along the worm length. The theory now reduces to the following task: Find t 7→ s∗ (t) solving the equations s˙ ∗ = W0 (t) + w , ˙ 0 (t) + k0 W0 (t) + Γ∗ (s∗ ) , σ(s∗ , t) = m W ∗

∗

m w˙ + k0 w + σ(s , t) = λ ,

w ≥ 0,

(2.90) ∗

λ ≥ 0,

∗

wλ = 0.

As formerly, indicated arguments mean that the respective function is known. The new feature is now that σ, whose sign rules the occurrence of skidding forward, w > 0, depends on the position s∗ . If we had a means to guarantee positive σ then w(t) = 0 would follow reducing the last equation to an expression for λ∗ , σ(s∗ , t) = λ∗ .

(2.91)

(σ = 0 would do as well - but on the hills active spikes, i.e., σ > 0, are well desirable.) b then this gives another requirement concerning If the spikes are of bounded strength λ σ, so that, finally, b 0 < σ(s∗ , t) ≤ λ

(2.92)

has to be ensured. The left inequality is satisfied if and only if the dynamics are in any mode A 6= ∅, while the right one could depend on A. Now it seems promising to generate the gait lj (t) by a gait construction procedure that relies on card(A) > 0 (as presented in Section 2.1.3.4) and to find out options for on-line control of σ and on-line change of the mode. To a certain extent this is the aim of the following section.

74

2 The Straight Worm With Propulsive Spikes (“SPIKY”)

2.3.1.1

HILLY 1 with gait-shift

The following investigations are certainly only a first step towards a controlled change of gait during motion. The investigations are restricted to those gaits which are based on the class of modes emphasized in Section 2.1.3.4, particularly in Example 2.23: each mode A is a subset of N = {0, 1, . . . , n} that is cyclically running through N, has (cyclically) no gaps and fixed cardinality a, 0 < a ≤ n. In the following let N = n + 1. Changing the gait then means changing the cardinality, t 7→ a(t) ∈ {1, . . . , n}. Naturally, a(·) is a piecewise constant function. Two questions must be answered: (i) how to decide about the value a(t) during a constancy interval (certainly in connection with the inequality above), (ii) how to choose the switching times where a(·) is to have its jumps? As to the second question, there is a quick possible answer: since every cycle (following the formal construction) ends with a relaxed configuration of the worm it seems promising to perform any a(·)-switch optionally at the end of a cycle. Taking this as a basic convention the decision mentioned in (i) has to be done at the end of each cycle. What do we postulate to be the aim of an a(·)-switch (“gear shift”)? As a guide we shall use the following three items: • The modes A(t) 6= ∅ of the constructed gait have to be dynamically feasible, i.e., σ(s∗ (t), t) ≥ 0 : no skidding forward; b i.e., N σ(s∗ (t), t) ≤ • The spikes forces must not exceed the given spikes strength λ, b a λ during a mode with a active spikes; • Obeying the foregoing demands, a(t) should be minimal (means maximal speed). (Equally, one could decide for a mean value between minimal and maximal.)

(The reader should try a comparison with the aim of shifting gears in driving a car.) Now remind the essentials of Section 2.1.3.4: Every gait is characterized by the following items: (i) it is T -periodic in time, (ii) its modes t 7→ A(t) run cyclically through N (N steps per period) and have fixed cardinality a (number of active spikes), and (iii) a smooth representation of t 7→ l˙j (t), j = 1, . . . , n. Using the sinusoidal D2 -representation (2.45) and k0 = 0 then we are given the center velocity and σ by (2.47) and (2.48), respectively, and the rules for choosing a follow from Proposition 2.22:

2.3 SPIKY in hilly landscape (“HILLY”) Let

1 2π

75

b − N Γ∗ (s∗ )} =: α(s∗ , a). m εl0 ω 2 N 2 =: F and min{N Γ∗ (s∗ ) , a λ

Then during any cycle of modes with cardinality a 0 < (N − a) F ≤ α(s∗ (t), a)

(2.93)

has to be satisfied for all t. (The left inequality is trivial, it just demands a < N .) Now it has become clear that, before entering a new cycle, the feasible values of a to come have to be determined. This requires to forecast the coming period. Howsoever this might be managed, knowledge of the occurring values of Γ∗ (s∗ (t)) is needed possibly leading to practical exploration problems in unknown landscape. Fortunately, we do know the hill profile completely, so that every required value Γ∗ (s∗ ) and thereby α(s∗ , a) can be calculated. Since the inequality (2.93) is based on extremal values of σ (see deduction of Proposition 2.22) one value per step does the job. Mind that we are given s˙ ∗ = v ∗ (t) as the closed analytical expression (2.47), so s∗ (t) = Rt T ]. In the s∗0 + τ0 v ∗ (τ ) dτ , where the preceding step ended at τ0 , s∗0 , and t ∈ (τ0 , τ0 + N T following simulations we use the final step time t = τ0 + N for that single point needed per step. We sketch the algorithm to be done for each cycle. For each step k = 1, . . . , N of the cycle l to come check the feasibility (inequality satisfied) of each a ∈ {1, . . . , N − 1}. Mark those a which are feasible in every step, and choose the minimal one of them (gait of maximal speed). This then serves for calculating the motion during this cycle l just scanned. Mind that no feasible a might exist, e.g., if Γ∗ (s∗ ) tends to negative values (motion downwards) or α is negative (failing spikes) even for a = N − 1. In the present context the calculation then terminates. Help could come via manipulation of F or Γ∗ (in driving a car one applies brake and throttle besides gear! See next section). The following figures sketch Maple12-simulations using the above algorithm.

Example 2.35 Arbitrarily chosen data: - hill profile: piecewise linear; b = (3, 1, 1, 0.35, π , 6, 8); - worm data: (n, m, l0 , ε, ω, Γmax, λ) 4

Resulting a-sequence: 111 2 33333 1 222 333333, terminates at top of hill (next cycle: Γ∗ < 0).

76

2 The Straight Worm With Propulsive Spikes (“SPIKY”)

Fig. 2.37: Motion along profile, small circles: s∗ at step ends, big circles: begin of cycles, crosses: gait shift occurred.

Fig. 2.38: Same with n = 5 and resulting a-sequence: 444 33 44 3 44444 55.

Fig. 2.39: Same with n = 8 and resulting a-sequence: 888888 5 666666 7.



2.3 SPIKY in hilly landscape (“HILLY”)

77

Example 2.36 - hill profile: y = x + 0.95 · sin(x); b = (6, 1, 1, 0.35, π , 6, 8) - worm data: (n, m, l0 , ε, ω, Γmax, λ) 4

Resulting a-sequence: 6666 5555 666 4 55 66666 . . . (repeating).

Fig. 2.40: Motion along profile, small circles: s∗ at step ends, big circles: begin of cycles, crosses: gait shift occurred. 

Why is the presented procedure only a first step towards a controlled change of gait? The first point is that it relies only on observing λ∗ and not on the λi : it may happen that b if some of the spikes forces exceed the bound while the majority λ∗ is smaller than λ of spikes forces keep small. It is not exactly known what happens after one or more spikes failed temporarily or broke forever (imagine what may happen while running on ice). The second point concerns the center-approximation which might be not feasible for motions in a landscape of short-scale variation of the inclination. So what has been introduced is just a measure of precaution.

78 2.3.1.2

2 The Straight Worm With Propulsive Spikes (“SPIKY”) HILLY 2 with further slow-down tools

In this section we want to see what to do if there is no more gait-shift to continue the motion (see Examples 2.35 and 2.36). To this end we take up a close inspection of the ruling inequality (2.93), b − N Γ} , (N − a) F ≤ α(s, a) = min{N Γ, aλ

a ∈ {1, . . . , N − 1} ,

(2.94)

1 where F = 2π m ε l0 (N ω)2 , and, for short, s := s∗ (t), Γ := Γ∗ (s∗ (t)). The left-hand term (N − a) F is the dynamic part of σ, see (2.48).

The investigation must envisage the ultimate a = N − 1 which, in context of HILLY 1, may have been reached by gait-shift (remind: increasing a entails decreasing left hand side and increasing right hand side in (2.94)). Then the motion stops if F > α(s, N − 1). This may happen in two main cases: b −N Γ ≤ 0, Case 1 : α(s, N − 1) ≤ 0 ⇔ Γ ≤ 0 ∨ (N − 1)λ b −N Γ > 0. Case 2 : α(s, N − 1) > 0 ⇔ Γ > 0 ∧ (N − 1)λ We start with b − N Γ > 0.) Case 1: (Note that Γ ≤ 0 implies (N − 1)λ Γ ≤ 0 means downhill or horizontal motion, skidding forward seems unavoidable. The only (and evident) way out is to “brake”, i.e., to substitute Γ by e := Γ + B > 0. Thereby we must take care of keeping the second some Γ member in α positive (B not so big as to overload spikes). Altogether, this demands to choose B such that b − N Γ. −N Γ < N B < (N − 1) λ

We shall be satisfied with the medium value choice B :=

1 N − 1b λ − Γ. 2 N

This entails the final requirement F ≤

1 b, (N − 1)λ 2

which can be fulfilled by decreasing F : “throttle down” by means of reducing the link elongation ε or the frequency ω. b − N Γ ≤ 0 there is no help besides increasing λ b howsoever this If (N − 1)λ might be realized (maybe by additional thorns at each mass point). Next consider Case 2:

2.3 SPIKY in hilly landscape (“HILLY”)

79

Since now α(s, N − 1) is positive there is no need to manipulate α by braking (though it might be done if there are special reasons). As in Case 1 above, the inequality F ≤ α(s, N − 1) with fixed and positive right hand side can be satisfied by throttle down: decrease F to a sufficiently small value. The results given below rely on the following algorithm: - forecast every cycle to come and determine the minimal values Γ, α of Γ and α; - if Γ ≤ 0 then introduce B as given above with Γ = Γ; reset B to zero as soon as during one of the coming cycles the hill inclination gets positive; - if Γ > 0 and F > α > 0 choose a new ε such that F = 0.8 · α, reset ε to its original value as soon as, at the end of one of the coming cycles, F < 0.75 · α(x, N − 1). Example 2.37 Arbitrarily chosen data: - hill profile: parabola with superimposed cosine - worm data: (n, m, l0 , ε, ω, G, b λ) = (3, 1, 1, 0.35, π/2, 6, 12)

Resulting a-sequence: 2 33 22 33333 2 33 22 3 222 3 222 33 1 3 2 11 Mean speed: v ∗ = 0.147

Fig. 2.41: Motion along profile. Circles: cycle ends (steps are dropped in figure), crosses: gaitshift; dotted lines, downward: braking, upward: throttle-down. Upper curve: HILLY 2 (L = 30 cycles prescribed); lower curve: HILLY 1 (stops after 12 cycles, up to this stop coincidence with HILLY 2).

80

2 The Straight Worm With Propulsive Spikes (“SPIKY”)

Fig. 2.42: Velocity v ∗ vs. t.



Remark 2.38 During cycle 29 the last (in cycle 27) generated brake power B = 6.126 is still active e = 1.671.  (no further one), it fits with the local values G sin(β) = −4.455 and Γ

Example 2.39

Same profile, same worm data but n = 9. Resulting a-sequence: 999999 777 99 77 888 7 9 88 77777 88 Mean speed: v ∗ = 0.182

Fig. 2.43: Motion along profile (HILLY 2, n = 9), same data as above.



Maybe, all this shows up to have a purely theoretical and queer flair. A physical realization of braking could be to unfold additional bristles at the mass points or to eject some substrate that generates sliding friction (remind that our gait used here guarantees a constant number of non-active, thus sliding, spikes12 ). It is even simpler if the spikes 12 One

might note some relationship with a theoretical tool used in [TINK 11], see end of Section 5.1.

2.3 SPIKY in hilly landscape (“HILLY”)

81

are realized by passive wheels with ratchet: just generate a classical braking torque. To realize throttle down is nothing else but to re-adjust actuator data. The very aim has been to point to options for intervention into a movement. Refinements could be achieved (i) by executing the investigations with some fixed a < N − 1, and (ii) by thinking about a throttle-up after a preceding slow-down as above. b should be big enough as not to let the spikes become overloaded. Finally, note that λ A reasonable scale to be envisaged is b> λ

N mg, N −1

(2.95)

which means that N − 1 spikes together may hold the non-moving worm (of total weight N m g) at a vertical cliff. A concluding remark As already done in the former simulations we have accepted arbitrary data, chosen just for to generate clear demonstration of effects. The presented HILLY-scenario as a whole must however be recognized as an unrealistic one: the total length of the considered part of the hill profile is about 25 (in which length unit ever) whereas the relaxed length of the worm is (N − 1)l0 . The displacement per cycle is d∗ = v ∗ T = ε l0 (N − a), so that after L cycles with some gait shift we have ε l0 L ≤ d∗ ≤ ε l0 L (N − 1) . So l0 = 1 leads to a giant worm that does not fit with the demands for utilizing the center-approximation. To be more realistic one had to stretch the profile by, say. a factor 103 , but this would entail a long computation time and certainly very boring resulting plots. A reasonable interpretation of the figures could be to observe a mass point with mass mN and state (x∗ , v ∗ ) that is endowed with a fictitious switching device making it move in the represented way.

3

The Straight Worm With Propulsive “Friction”

In this chapter we tentatively replace the ground contact via spikes by “friction”, which many people understand as a better model. Since friction is often used in a loose way, we are comparatively cautious in introducing this model. Various scenarios are exemplified with the inch-worm. A simple and rather thorough theory is given for an aquatic worm (where, anyway, spikes are unsuitable).

3.1

Pros and cons of frictional propulsion

The investigations in the foregoing Chapter 2 have proven the model of the ideal spike a very useful tool in the theory of locomotion systems. Basically, it consists of the differential inequality-constraints (2.3). For systems with kinematic drive it permits to set up a purely kinematical theory the dynamical validity of which can be checked up by means of the constraint forces λκ which undergo the complementary-slackness conditions (2.13). In the context of dynamics the slackness conditions could be satisfied by means of the ’control law’ (2.15) for λκ . The constraint forces λκ , however, keep undetermined if the internal forces of the system themselves depend on constraint forces (as it is the case in kinematic drive), solely their average λ∗ can be found from the principle of linear momentum for the total system (central equation (2.21)). Systems with dynamic drive naturally do not permit a covering kinematical theory, rather they require right from the beginning a description by dynamical equations of motion which contain the λκ in form of (2.15). Generally, it may not be possible to judge how excellent “spike” as a model for the environmental contact of a real system in fact is. In Section 2.1 some considerations b had been done: How to drive about systems with spikes of bounded strength (λκ < λ) the system such that the spikes keep working (do not ’break’) and, thus, suppress any backward displacement? This question, too, could not be answered in detail, but only by means of the average λ∗ of the spikes forces. For giving up this theoretically oriented kinematic model “ideal spike” of the contact with the environment obviously the only way will be to model the interaction systemenvironment not by a constraint (with corresponding reaction force) but rather a priori by a force-interaction. Howsoever this might be done, it excludes an exhausting kinematical theory, and it demands the use of dynamics from the very beginning. Modeling the forces of interaction requires certain physical assumptions and ingredients which reflect properties of the contacting media: lower/upper bounds for the forces, a

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3 The Straight Worm With Propulsive “Friction”

physical law of force with measurable parameters (in case of impressed force). Hereby a certain amount of arbitrariness, doubt, and uncertainty enters the theory. Again, the choice of a model will essentially be determined through both context and aim and subjective skills. Within this framework also backward displacements are feasible. As long as these remain small this fact could also fit with real spikes, since every thorn, bristle, wheel with ratchet owns some compliance which the abstract ideal spike strictly disregards.

3.2

Propulsive friction

Two scenarios are taken into consideration for a theory of worm systems with no ideal spikes: (1) Straight worm on solid ground without lubrication. The corresponding coupling forces are described, using rules dating back to Coulomb (1736 - 1806), by anisotropic friction of rest - influence of viscous lubricants between the contacting surfaces may enter the theory through extra impressed forces. Friction of rest is a force that acts between contact points which are at relative rest, i.e., the response to the constraint “relative speed zero”. Note therefore that friction of rest, for short also stiction, is a reaction force, thus lacking of any general physical law! Its peculiarity is the existence of a lower and an upper bound such that a relative motion of the contact points will start as soon as either bound had to be exceeded for maintaining the constraint. The magnitudes of the bounds are determined by physical and structural properties of the contacting surfaces. If these values depend on the orientation within the contacting surfaces (anisotropy) then they can be responsible for motion in preferred direction. Practically, stiction is accompanied by (anisotropic as well) dry sliding friction, this force then enters the theory as an extra, impressed force undergoing Coulomb’s law (or some refined law, see [ArDC 94], [AwOl 05] or [COAL 95]). (2) Straight worm in viscous hydrosphere. During motion in a fluid environment of strong viscosity there are interactions in form of viscous friction forces depending on the velocity (impressed forces). If appropriate devices cause anisotropy then locomotion in preferred direction will occur. In what follows only the simplest laws for the contact forces are used, since micro-events in the contact layer are apparently not essential for the macroscopic worm motion. However the modeling is open for every refinement.

3.2.1

Modeling Coulomb forces: stiction and dry sliding friction

To set up a mathematical model of the respective forces talked about above it suffices to consider a DOF = 1 mechanical system x˙ = v ,

v˙ = f (x , v , t) + F .

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85

Here F is to represent a Coulomb type force occurring if the system contacts some ground, and f is the resultant of all further external forces. Recalling something from above, F compensates f if and only if the velocity v is zero and f does not exceed certain limit values. If v 6= 0 then F takes constant values. These values and the above limit values as well are characteristic for the contacting surfaces, they may depend on the orientation within the contact area (determined by direction of v 6= 0: anisotropic sliding friction, or determined by direction of f at v = 0: anisotropic stiction, respectively). Often they are supposed to be proportional to the normal force N acting between the surfaces (a constraint force in general) with friction coefficient µ. Following Hamel’s classification of forces [Hame 49], this description qualifies F to be an impressed force during motion v 6= 0 (sliding friction), whereas F appears as a constraint force with given bounds as long as v = 0 (friction of rest, stiction). Consequently, the Coulomb force F depends on both velocity v and the further force f ,    − + 0 0   −f , v = 0 ∧ f ∈ − F0 , F0 ,
(3.1) F (f , v) = F − , v < 0 ∨ v = 0 ∧ f 0 < −F0− , 
 + + 0 −F , v > 0 ∨ v = 0 ∧ f > F0 . Here f 0 := f (x , 0 , t), F ± ≥ 0 are the values of sliding friction, whereas F0± ≥ F ± are the bounds of friction of rest. Anisotropy in sliding and at rest means F + 6= F − and F0+ 6= F0− , respectively. The dynamics of the DOF = 1 system then are
x˙ = v , v˙ = f (x , v , t) + F f (x , v , t) , v . (3.2)

As a first check of the correctness of (3.1) mind what happens in the last case above: v = 0 ∧ f 0 > F0+ at some time t0 implies mv(t ˙ 0 + 0) = f 0 − F + > F0+ − F + ≥ 0 so that + v(t) turns to positive values and −F begins to act; if, however, 0 < f 0 < F0+ then the first line yields m v(t ˙ 0 + 0) = f 0 − f 0 = 0 so that v(t) remains zero. Again, despite the seeming law given by the first line in (3.1) stiction F |v=0 is not an impressed force since it does not depend on any physically measurable parameter via some general and unique law! (3.1) simply gives a connection with the context “motion system with external force f ”. Let F = F (0) + F (1) be the formal partition of the Coulomb force into reaction and impressed term. Then stiction is given by (   −f (x , v , t) , v = 0 ∧ f (x , 0 , t) ∈ − F0− , F0+ , (0) f (x , v , t) , v) = F 0, else . (3.3) Being a bit lax with regard to the F (1) -values at v = 0 it is common use to drop the dependence on f and to accept for the sliding friction the simple force law  v = 0,  0, (1) − F (v) = F , v < 0 , (3.4)   + −F , v > 0 .

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3 The Straight Worm With Propulsive “Friction”

Again, the values F ± can be found by experiments and, thus, they qualify F (1) as an impressed force. However the F0± , though physically measurable as well, do not determine the actual values of F (0) which, via f (·, ·, ·), depend on the context. The following Fig. 3.1 sketches the graph of F . The projection of that graph to a (v , F )−plane shows a diagram F vs. v that is very often preferred in literature; there, F appears as a set-valued function, and so the corresponding dynamics are sometimes tackled by means of fitting theories, [BaSL 02].

F

F v

-

f

F0 F

-

-F + v

-F0+

Fig. 3.1: Graph of Coulomb force.

In practice, the friction values are only roughly known because of considerable measuring uncertainty and a lack of insight into what really happens in the thin layer between the contacting surfaces at different relative velocities. This caused several authors, based on physical model considerations to present refinements of the Coulomb rules and others to introduce certain relaxations. In the following we prefer the second way - not so much in view to physical modeling as rather in order to achieve a fairly good handling on the computer. At first the relaxation consists of replacing v = 0 by −ε < v < ε with some

3.2 Propulsive friction

87

small ε > 0, see [Karn 85]13 . Using the Heaviside function h ( 1, a ≤ x < b, h(a, b, x) := 0 , else , and ignoring some inconsistencies at ±ε, F can be given the form  F (f, v) = −f h(−ε , ε , v) h(−F0− , F0+ , f )    − − (3.5) +F h(−∞ , −ε , v) + h(−ε , 0 , v) h(−∞ , −F0 , f )    + + −F h(ε , +∞ , v) + h(0 , ε , v) h(F0 , +∞ , f ) .  − − To have another check mind what happens if f < −F : F (f , v) = 0+F h(−∞ , −ε , v)+ 0 + − + h(−ε , 0 , v) − F h(ε , +∞ , v) = F h(−∞ , 0 , v) − F h(ε , +∞ , v). Then this yields: if v < −ε: m v˙ = f + F − < −F0− + F − ≤ 0, v(t) decreases; if v ∈ (−ε , ε): m v˙ = f + F − h(−∞ , 0 , v) < 0, v(t) decreases towards zero or to negative values; if v > ε: m v˙ = f − F + < −F0− − F + < 0, v(t) decreases. The following Fig. 3.2 shows the graph of F after this ε−relaxation.

Fig. 3.2: Graph of F with v = 0 relaxed to ε−strip (with ε = 0.5, F ± = 2, F0+ = 6 and F0− = 5).

In order to avoid difficulties in computing caused by the discontinuities of F the hfunction is often replaced by a smooth approximation, see Appendix D. (Physically this may also be justified by the above mentioned fuzziness of the boundary layer microevents. And do jumps actually occur in real processes?) 13 At

first sight ε ≈ 10−12 might be seen as to model a computer accuracy, which is present anyway.

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3 The Straight Worm With Propulsive “Friction”

So the second step of relaxation can be based on a tanh-approximation of the signfunction, sign(x) ≈ tanh(Ax) with some A  1 and h(a , b , x) :=

1 sign(x − a) + sign(b − x) , 2

(3.6)

apart from some inconsistencies at x = a and x = b, then h(a , b , x) ≈ H(a , b , x) :=



o 1n tanh A (x − a) + tanh A (b − x) . (3.7) 2

The smooth (C ∞ ) mathematical model of Coulomb forces now is  F (f , v) = −f H(−ε , ε , v) H(−F0− , F0+ , f )    − − +F H(−∞ , −ε , v) + H(−ε , 0 , v) H(−∞ , −F0 , f )    + −F H(ε , +∞ , v) + H(0 , ε , v) H(F0+ , +∞ , f ) .

(3.8)

Its graph is sketched in the following Fig. 3.3:

Fig. 3.3: Graph of C ∞ −model of Coulomb force F (f , v) with ε = 0.5, A = 10, F ± = 2, F0+ = 6 and F0− = 5.

The rather big ε and small A used in the figures aim at clearness. In the following simulations A = 104 and A ε = 1 is used.

3.2 Propulsive friction

89

Example 3.1 Consider v(t) ˙ = f (t) + F (f (t) , v(t)), v(0) = 0, with f (t) = 2 (e−2 t − 1). Let (F0− , F0+ , F − , F + ) = (1 , 1.5 , 0.5 , 0.5). How does the system move? For t ∈ [0, 0.35) the Coulomb force F compensates f (t) = 2 (e−2 t − 1) < 0, hence v(t) = 0. For t > 0.35 a jump of F from 1 to 0.5 occurred, and (t) + 0.5,  v is governed by v˙ = f
 v(0.35) = 0. The corresponding space curve t 7→ v(t) , f (t) , F f (t) , v(t) lies on the graph of the friction function, see Fig. 3.4.

Fig. 3.4: The space curve (thick) on the friction graph.

Fig. 3.5: Left: (F0− , F0+ , F − , F + ) = (1 , 1.5 , 0.5 , 0.5), f , f + F , v vs. t (bottom to top); right: (F0− , F0+ , F − , F + ) = (1 , 1.5 , 0 , 0) f , v vs. t (bottom to top).

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3 The Straight Worm With Propulsive “Friction”

Fig. 3.6: Left: (F0− , F0+ , F − , F + ) = (0 , 0 , 0.5 , 0.5), f , f + F , v vs. t (bottom to top); right: v vs. t (zoomed).

The Figures 3.5 and 3.6 present the solution v(t) of the differential equation together with the actual forces. They clearly show that the influence on the motion of sliding friction alone is rather obscure (stop phase at the beginning?, see Fig. 3.6), whereas a sole stiction produces a clear resting phase from the beginning, see Fig. 3.5 (right). For further numerical details the reader should consult Appendix D.  Exercise 3.2 In the last example play with the parameters (F0− , F0+ , F − , F + ) and with the function f , evaluate on the computer with some integration routine and observe the results.  Exercise 3.3 Consider the standard problem: Determine the motion of a mass point under the influence of Coulomb forces, drawn along the x−axis by an elastic spring the free end of which gets a constant positive speed v0 > 0 (e.g., by a motor). 

3.2.2

Worm system with ground contact by Coulomb forces (“COULY”)

The foregoing examples raise the following observation: If ideal spikes with their constraints x˙ κ ≥ 0 and corresponding reaction forces λκ ≥ 0 are to be replaced by a force interaction wormground then, in the Coulomb setting, it is stiction that deserves primary attention. Firstly, this gets some evidence by noting that sliding friction as an impressed force could have already been treated within the framework of Chapter 2 simply as a term of the external force gκ . Further insight can be gained by reconsidering the spikes forces λ = (1 − sign(x))f ˙ − (for short the index − κ in (2.14) has been dropped). The negative part f of the force f can be represented as f − = −f h(−∞ , 0 , f ). That means that λ in the equation of motion, m x ¨ = f + λ, compensates any force f with values in (−∞ , 0). If this ability of compensation is to be

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91

bounded below by some −F0− < 0 and by some F0+ > 0 above, then the f -dependence of b = F −, λ at v = 0 should be −f h(−F0− , F0+ , f ). This points to a breakable spike with λ 0 + see (2.39), that, additionally, causes some resistance (maximal force F0 > 0) against starting a motion in positive direction. Disregarding what happens at the upper limit F0+ this describes exactly the job of stiction F (0) . Let us now consider a discrete worm as in Chapter 2 whose mass points labeled κ ∈ K ⊂ N interact with the ground via Coulomb forces. Supposing the same stiction and friction parameters for all these interactions the equations of motion of the worm, owing to (2.13), are (0)

mx ¨κ = fκ + Fκ

mx ¨i = fi ,


fκ , x˙ κ ,

κ ∈ K,

i ∈ N\ K.

The stiction term can be written (

−fκ h − F0− , F0+ , fκ if x˙ κ = 0 , (0) Fκ fκ , x˙ κ = 0 if x˙ κ 6= 0 ,

(3.9)

(3.10)

or in the form

Fκ(0) fκ , x˙ κ = −fκ h − F0− , F0+ , fκ h(−ε , ε , x˙ κ )

(3.11)

if relaxing x˙ κ = 0 to x˙ κ ∈ [−ε , ε).14 The forces fi , i ∈ N, are fi = gi + µi − µi+1 ,

where µi is an internal cut force and gi is the resultant of all external impressed forces acting at the mass point i. For i ∈ K the latter may contain a dry friction term F (1) (x˙ i ). One should keep an eye on the fact that in case of kinematic drive the µi are unknown reaction forces (corresponding to the holonomic constraints xi−1 − xi − li (t) = 0) and, thus, cause troubles in dealing with stiction! The way towards a central equation as in Subsection 2.1.2 in order to get rid of these troubling µi at least partly, cannot be followed here, since nothing like complementary-slack is in sight. For the sake of brevity we use in the sequel the following - non-official! - nomenclature: SPIKY: Worm with ground contact by spikes; COULY: Worm with ground contact by both stiction and dry sliding friction; STICKY: Worm with ground contact by stiction only; SLIDY: Worm with ground contact by dry sliding friction only. 14 Principally,

this realizes the exact form

F (0) (f , x) ˙ = −f h − F0− , F0+ , f






1 − sign(x) ˙ 1 + sign(x) ˙ ,

whereby ε clearly exhibits itself as a computing accuracy.

92

3.2.3

3 The Straight Worm With Propulsive “Friction”

Inch-worm with Coulomb forces: a review

In order to see what may happen under the influence of Coulomb forces instead of spikes we consider first the simple inchworm equipped with a viscoelastic actuator (Section 2.1.5.1). Its equations of motion are (see equation (2.59)) (1)

− u 1 + Λ0 ,

(1) F1

+ u 1 + Λ1 ,

mx ¨0 = −c (x0 − x1 − l0 ) − k00 (x˙ 0 − x˙ 1 ) − k0 x˙ 0 − Γ + F0 0

mx ¨1 = c (x0 − x1 − l ) + k00 (x˙ 0 − x˙ 1 ) − k0 x˙ 1 − Γ +

(0)

(1)

where in the following Λ0,1 represent either a spikes force λ0,1 or a stiction F0,1 ; F0,1 are sliding friction forces. The total force acting upon mass point 0 (mass point 1 analogously) is the sum of actuator force: external impressed force: external reaction force:

−c (x0 − x1 − l0 ) − k00 (x˙ 0 − x˙ 1 ) − u1 , (1) −k0 x˙ 0 − Γ + F0 , Λ0 .

In the sequel we examine this inchworm under various data. But once for all the dynamic drive u1 is chosen according to (2.67), h i ˙ . u1 = u + c l0 , u := − m ¨l + c l + (k0 + k00 ) l˙ + Γ sign(l) (3.12) That is, u is to generate the gait t 7→ l(t) for the classical inchworm with spikes (kinematical SPIKY), provided the dynamics are in alternating modes {0} and {1} and there is no sliding friction. So u takes care of external forces k0 x˙ and Γ, additional external forces like Coulomb sliding friction or replacing spikes by stiction forces F0− , possibly accompanied by stiction F0+ , cannot be overcome by u and thus imply certain effects. The aim of the following examples is to show such effects. The inchworm does this job at lowest expense, and moreover it admits the gait-mimicking actuator force u as a common basis for comparisons. For the gait we take l(t) := l0 [1 − a (1 − cos(Ω t))] . Common data for all the following examples are taken over from Section 2.1.5.2: l0 = 1 , a =p 0.25 , Ω = π (T = 2 π/Ω = 2) , m = 1, c/m = ω0 = Ω (resonance), actuator: k00 = 2 m ω0 k , k = 1 (aperiodic limit), external: k0 , Γ at choice. gait:

For the sake of comparison we start with the kinematical SPIKY. In later graphics the corresponding gray-filled diagram of this ideal inchworm motion emphasizes respective effects. Signum and Heaviside functions are generally used in a smooth version (see Appendix D).

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93

Example 3.4 Kinematical SPIKY (see Fig. 2.25). Remind that the internal actuator data and any external impressed forces are of no influence over the motion which is described by kinematical theory. But according to the central equation (see Section 2.1.3.5) the latter determine λ0 and λ1 . We choose k0 := 0. Average speed is v 0 = 0.25, max(v0,1 ) = 0.7854. σ varies in the interval [−1.2336, 1.2336], let 1.2336 =: Γ0 .  Remind that the above motion (from kinematic theory) is dynamically feasible if and only if σ ≥ 0. This is realized by choosing Γ ≥ Γ0 . Now we use Γ := Γ0 together with k0 := 0. Then the motion is the same as the latter one, now dynamically justified, and the central equation yields λ∗ ≡ 12 (λ0 + λ1 ) = σ ∈ [0, 2Γ0 ]. Since modes {0} and {1} alternate it follows max(λ0 ) = max(λ1 ) = 2 max(σ) = 4 Γ0 = 4.9344 . Example 3.5 Dynamical SPIKY with Γ := Γ0 . The motion visually coincides with the foregoing one, average speed is v 0 = 0.24994, the maximal value of λ0 is 4.9613 (transiently at start). So the dynamical SPIKY model driven by the gait-mimicking u and using the above mentioned continuous signum approximations has proven reliable.  Example 3.6 Dynamical SPIKY with Γ := Γ0 /4 (smaller than Γ0 , the worm must skid forward; describable also by the central equation giving w 6= 0). The next figures clearly show this effect.

Fig. 3.7: left: x0 , x1 , x∗ vs. t, compared to motion by kinematic theory; right: velocities, spikes forces (solid: v0 , λ0 ; dashed: v1 , λ1 ), gait l and actual length x0 − x1 , all vs. t (top to bottom).

The kinematical gait l(t) is not exactly generated (because of the actual mode sequence {0} → ∅ → {1}), the average speed is v = 0.2955, and the maximal λ-value is λ0 (+0) = 3.085. 

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3 The Straight Worm With Propulsive “Friction”

Example 3.7 Dynamical SPIKY with Γ := Γ0 , k0 := 2.5. Now the motion essentially coincides with the one in Example 3.5 (k0 -influence suppressed by u), average speed is v = 0.2499, maximum of the λ is now 5.6307.  Example 3.8 Dynamical SPIKY with Γ := Γ0 , k0 := 0, and an additional sliding friction F + := 2.5. (Mind that any Stokes friction k0 is compensated by the mimic u, whereas Coulomb sliding friction F + is not; Coulomb sliding friction F − is of no relevance since negative velocities are prevented by the spikes). As expected the motion is slowed down: average speed is v = 0.09105.

Fig. 3.8: left: x0 , x1 , x∗ vs. t, compared to motion by kinematic theory; right: velocities, spikes forces (solid: v0 , λ0 ; dashed: v1 , λ1 ), gait l and actual length x0 − x1 , all vs. t (top to bottom). 

Interestingly, there are now rest phases, the mode sequence is {0} → {0, 1} → {1}, and max(λ0 ) = 7.7341, max(λ1 ) = 7.4276. Considering a motion of one of the types above one expects that nothing changes if the spikes are replaced by stiction F0− not smaller than the maximal spikes force. This proves true, e.g., in the following example by modifying Example 3.5. Example 3.9 STICKY with (F0− , F0+ ) := (4.962 , 0), Γ := Γ0 , k0 := 0. In short we write as from now (F0− , F0+ , F − , F + , Γ , k0 ) = (4.962 , 0 , 0 , 0 , Γ0 , 0) . Reproduces exactly Example 3.5.



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95

Example 3.10 STICKY as in Example 3.9 and additional Stokes friction, (F0− , F0+ , F − , F + , Γ , k0 ) = (4.962 , 0 , 0 , 0 , Γ0 , 0.5) . (We know from Examples 3.5 and 3.7 that, via u, the latter has no influence on the motion, but it raises the spikes forces.) Now the chosen F0− is too small, and a catastrophe happens: worm running backward.

Fig. 3.9: x0 , x1 , x∗ vs. t, compared to motion by kinematic theory.

Taking k0 := 2.5 as in Example 3.7 we have to choose larger stiction values than λmax = 5.6307. Hence the worm is not running backward (see Fig. 3.10, left) with the following set (F0− , F0+ , F − , F + , Γ , k0 ) = (5.631 , 0 , 0 , 0 , Γ0 , 2.5) . Observing Example 3.8 with λmax = 7.7341 then the following set avoids backward motion, see Fig. 3.10 (right): (F0− , F0+ , F − , F + , Γ , k0 ) = (7.735 , 0 , 0 , 2.5 , Γ0 , 0) .

Fig. 3.10: x0 , x1 , x∗ vs. t for two parameter sets, compared to motion by kinematic theory.



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3 The Straight Worm With Propulsive “Friction”

Example 3.11 STICKY with (F0− , F0+ , F − , F + , Γ , k0 ) = (9.962 , 5 , 0 , 0 , Γ0 , 0) . The motion has slowed down to an average speed v = 0.2196 due to rest phases caused by F0+ (compare with Example 3.9).

Fig. 3.11: left: x0 , x1 , x∗ vs. t, compared to motion by kinematic theory; right: velocities, impressed forces (solid: v0 , f0 ; dashed: v1 , f1 ), gait l and actual length x0 − x1 , all vs. t (top to bottom). 

Example 3.12 The following is a comparison of STICKY and SLIDY with data structure (at same Γ = Γ0 , k0 = 0) (F0− , F0+ , F − , F + ) = (a , b , 0 , 0) ↔ (F0 − , F0+ , F − , F + ) = (0 , 0 , a , b) . We obtain the following results:

Fig. 3.12: Worm motion: (F0− , F0+ , F − , F + ) = (5.962 , 1 , 0 , 0) (left); (F0− , F0+ , F − , F + ) = (0 , 0 , 5.962 , 1) (right).

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97

Fig. 3.13: Worm motion: (F0− , F0+ , F − , F + ) = (9.962 , 5 , 0 , 0) (left); (F0− , F0+ , F − , F + ) = (0 , 0 , 9.962 , 5) (right).

 The figures point to characteristic differences. STICKY moves still well after introducing a (not too big) F0+ simultaneously adding this F0+ to F0− , with some slow-down due to resting phases. On the other side, using the same values for F ± , the SLIDY may still move (but strongly slows down due to damping during non-zero velocity phases) whereas sufficiently big F ± -values can make the worm unmoveable. Example 3.13 Until now most of the investigated motions took place at Γ = Γ0 > 0, i.e., the worm was to move uphill. Let us finally go to the horizontal plane, choosing Γ = 0 (makes SPIKY skidding forward). We compare the motions of all four types of worms.

Fig. 3.14: Worm motion: dynamical SPIKY (left); STICKY with (F0− , F0+ , F − , F + ) = (2 Γ0 , Γ0 , 0 , 0) (right).

98

3 The Straight Worm With Propulsive “Friction”

Fig. 3.15: Worm motion: SLIDY with (F0− , F0+ , F − , F + ) = (0 , 0 , 2 Γ0 , Γ0 ) (left); COULY with (F0− , F0+ , F − , F + ) = (2 Γ0 + 1 , Γ0 + 1 , 2 Γ0 , Γ0 ) (right).

 Example 3.14 This example must be seen in comparison with Example 3.9. There, with Γ = Γ0 , stiction F0− = 4.962 showed up sufficient for a SPIKY-like motion. The present SLIDY still shows a globally forward motion although its sliding friction (which substitutes stiction) is F − = 3.5 < 4.962. This behavior is obviously due to the fact that, when the velocity is tending to negative values, stiction just compensates the actual (possibly small) impressed force whereas sliding friction slams with the additional force F − . Note the intervals of negative and zero velocities

Fig. 3.16: SLIDY with (F0− , F0+ , F − , F − , Γ, k0 ) = (0, 0, 3.5, 0, Γ0 , 0), worm motion (left) and velocities (right) vs. t.



3.3 Straight worm with viscous friction (“STOKY”)

99

Summarizing, the simulations presented in this section show that the effects, caused by Coulomb forces, in motion are manifold. And again they point to the fact that one has to be very cautious in interchanging stiction and sliding friction when doing simulations. There are two further items the reader’s attention should be focussed on. First, it has turned out that the friction model is rather sensitive with respect to the values F0± , F ± and in particular to the approximation data A, ε. So one should not blindly rely on a single simulation result (if it is not based on a very special and sophisticated software). Hence the above results are to be seen just as hints to what could happen in the world of non-spiky worms. Second, any practical application suffers from ignorance of exact F0± and F ± values. Therefore it is unavoidable to ensure a desired dynamical behavior of the worm (gained within SPIKY theory) by applying appropriate control measures (to overcome the data uncertainty). This item is addressed in Chapter 4.

3.3

Straight worm with viscous friction (“STOKY”)

3.3.1

General system

This section is dedicated to the mathematical model of a fictitious undulatory locomotion system that interacts with its strongly viscous environment via Stokes’ friction. To our best knowledge there is no biological or technical prototype yet. But at least the corresponding framework gives some convincing insight into what is undulatory locomotion. Starting-point is a system of mass points as in Fig. 2.1, where now the spikes are omitted and the constraint forces λi are replaced by friction forces. They are impressed forces, subject to a law of the form ri = −ki vi with positive coefficients ki . The latter may depend on the velocity vi or, in particular, on its sign (direction) thereby causing anisotropy of the friction. The simplest dependence of this kind is  +  k , 1 − 1 − + + k(v) = (k + k ) − (k − k ) sign(v) = 12 (k − + k + ) ,  2 2  − k ,

v > 0, v = 0, v < 0, (3.13)

with constant k ± equal for all mass points. For a desired direction v > 0 of locomotion the respective anisotropy is given by 0 ≤ k+ < k− .

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3 The Straight Worm With Propulsive “Friction”

Fig. 3.17: Friction coefficient k vs. v, left; friction force r vs. v, right; (solid: after (3.13), dashed: smoothed by some tanh(A v)).

Possible realizations could be achieved by appropriate shaping (rigid caps) or by folding flaps, see the next figure.

l i(t)

i

i-1

i vi

l i(t)

i-1 vi-1

Fig. 3.18: Mass points with rigid or foldable flaps.

It is evident that all this is certainly a rather crude modeling, since the surrounding variable flow field has been totally disregarded. The smoothing of k(v) as often used for numerical reasons could then be seen as an approximate consideration of the transient behavior of the flow caused by a change of sign of v. Now the dynamics of the worm system are described by ) mx ¨κ = gκ + µκ − µκ+1 + rκ , κ ∈ K , mx ¨i = gi + µi − µi+1 ,

i ∈ N \ K,

(3.14)

with friction forces of the form rκ = −k(x˙ κ ) x˙ κ = −k0 x˙ κ + k1 |x˙ κ | , k0 :=

1 2

(k − + k + ) ,

k1 :=

1 2

(k − − k + ) .

(3.15)

rκ shows up as a superposition of a viscous damping (first term) and (if k − 6= k + ) a propulsive force of fixed direction. The mass points i ∈ N \ K do not interact with the environment. The external forces g shall be dropped here, since in particular its viscous part could be captured by modifying k0 .

3.3 Straight worm with viscous friction (“STOKY”)

101

Again, adding up the equations (3.14) the result is the linear momentum law of the total system (the internal forces µ mutually cancel). This single equation governs the motion in case of kinematic drive (DOF = 1). So the following considerations are confined to the worm with kinematic drive using the notations from Subsection 2.1.3, providing K = N: xj−1 − xj = lj (t) , ∗

xi = x + ζi (t) ,

x˙ i = v ∗ + ζ˙i (t) .

Pn Adding up the equations (3.14) and observing (3.15) and 0 ζi = 0, the resulting equation is a nonlinear heteronomic differential equation for the velocity v ∗ of the center of mass, m v˙ ∗ = −k0 v ∗ +

n X ∗ 1 v + ζ˙i (t) . k1 (n + 1) i=0

(3.16)

The last term on the right hand side shows very clearly the mechanism of undulatory locomotion: the internally generated deformation rates ζ˙i (t) are transformed (depending on properties of the surfaces with environmental contact - reflected by the coefficient k1 ) to an external propulsive force of fixed direction. In the following some examples of worms with n = 1 and n = 2 are considered. The results are achieved by numerical integration of (3.16). Friction coefficients are k − = 2, k + = 0.2. The figures show the velocity of the center of mass v ∗ , the coordinates xi , x∗ , and the propulsive force (last term right in the differential equation). Example 3.15 n = 1, kinematic gait h
i l1 (t) = l0 1 − ε 1 − cos(Ω t) ,

l0 = 1 , ε = 0.25 , Ω = 2 π .

0.5

1.5 0.4

1 0.3 0.5 0.2 0 0.1 -0.5 0

2

4

6

8

10

0

1

2

3

4

5

∗

Fig. 3.19: left: v vs. t; middle: x0 , x1 and x∗ (dashed) vs. t; right: propulsive force vs. t. ∗ The velocity for large t is v ∗ = v∞ ≈ 0.492, smoothing |x| ˙ = x˙ sign(x) ˙ ≈ x˙ tanh(A x) ˙ ∗ ∗ yields of course smaller values: v∞ ≈ 0.469 with A = 5, v∞ ≈ 0.392 with A = 2. As expected, the limit velocity decreases monotonically for k − − k + → 0:

102

3 The Straight Worm With Propulsive “Friction”

∗ Fig. 3.20: v∞ vs. k+ /k− (with above data).



Example 3.16

h
i n = 2, gait: l1 (t) = l2 (t) = l0 1 − ε 1 − cos(Ω t) ,

l0 = 1 , ε = 0.25 , Ω = 2 π .

Fig. 3.21: left: v ∗ vs. t; middle: x0 , x2 and x∗ (= x1 ) vs. t; right: propulsive force vs. t; limit ∗ velocity is v∞ ≈ 0.867. 

Example 3.17


  n = 2, gait: li (t) = l0 1 − εi 1 − cos(Ωi t) , i = 1, 2

with l0 = 1, ε1 = 0.3, Ω1 = 2 π, ε2 = 0.15, Ω2 = 4 π.

Fig. 3.22: left: v ∗ vs. t; middle: x0 , x1 , x2 and x∗ vs. t; right: propulsive force vs. t; limit ∗ velocity is v∞ ≈ 0.816. 

3.3 Straight worm with viscous friction (“STOKY”)

103

Power balance Let n ≥ 1, arbitrary. Multiplying the differential equations mx ¨i = µi − µi+1 − k0 x˙ i + k1 |x˙ i | ,

i = 0,...,n,

by x˙ i and summing over i there results the total power balance (n + 1)

 m 2

v

∗2

·

+ k0 v

∗2



+

 X n m 2

ζ˙i2

i=0

= Pa + k1

·

+ k0

n X i=0

n X i=0

ζ˙i2



∗
v + ζ˙i v ∗ + ζ˙i , (3.17)

Pn where again Pa := − j=1 µj (ζ˙j−1 − ζ˙j ) is the power output of the actuators that Pn is injected into the mass point system, and k1 i=0 |v ∗ + ζ˙i | (v ∗ + ζ˙i ) =: Pe is an externally (through the anisotropic friction) generated power entering the system from the environment (note that it vanishes if k − = k + , and that its terms may have different signs). Multiplication of (3.16) by v ∗ yields an external power balance n X ∗ d  m ∗2  k1 v + ζ˙i − k0 v ∗2 , = v v∗ dt 2 n+1 i=0

(3.18)

where the first term on the right hand side represents the driving power (first part of the external power) whereas the second, negative, term describes an external dissipation. Then (3.17) turns into an internal power balance   n n n X X d m X ˙2 |v ∗ + ζ˙i | ζ˙i . ζ˙i2 = Pa + k1 ζi + k0 dt 2 i=0 i=0 i=0

(3.19)

It shows how the actuator power and a second part of the external power are used up for to change the internal kinetic energy and for internal dissipation. This is an unavoidable effect which should desirably be small: Pa is wanted to be converted to ∗2 locomotion energy m and not to be wasted for “internal worm gymnastics”. 2 v The total power of the Stokes forces ri , n X i=0

  n n X X ∗ v + ζ˙i (v ∗ + ζ˙i ) , ζ˙i2 + k1 ri x˙ i = k0 (n + 1) v ∗2 + i=0

i=0

enters both balances, this clearly reflects the fact that the forces ri influence both the locomotion (v ∗ ) and the deformation (ζi ) of the system.

104

3.3.2

3 The Straight Worm With Propulsive “Friction”

The system with n = 1

In the sequel the behavior of this system with two mass points is considered in some more detail. The equation of motion is rather simple although it is not a differential equation found in every picture book. Using the kinematic drive h
i l(t) = l0 1 + ε 1 − cos(Ω t) ,

ζ0 =

1 1 l , ζ1 = − l , 2 2

the equation of motion (3.16) is m v˙ ∗ = −k0 v ∗ + where a0 :=

1 2 l0

o 1 n ∗ k1 v + a0 sin(Ω t) + v ∗ − a0 sin(Ω t) 2

ε Ω > 0. Putting

Ω t =: s ,

v ∗ (t) =: a0 w(s) ,

M :=

mΩ , k0

k :=

the equation of motion gets the clear form n o M w0 = −w + k w + sin(s) + w − sin(s) .

1 k1 , 2 k0

(3.20)

At first sight this is a nonlinear heteronomic differential equation of the form w0 = f (w , s) with π-periodic (!) right-hand side f (w , ·) that is (for |w| < 1) only piecewise smooth. If |w| ≥ 1 then the differential equation is simply M w0 = −(1 − 2 k) w. It describes an exponential decrease or constancy of the solutions since by definition the viscosity parameter k is no greater than 12 . For velocities |w| < 1 the periodicity property suggests some oscillatory behavior of the solutions (see the following phase portrait).

Fig. 3.23: Direction field and some solutions of the differential equation (3.20) with a0 = 0.79, M = 2.5, k = 0.30.

The sketched solutions in this figure indicate the behavior of a system with constant viscous damping (exponential) superposed with a π-periodic oscillation. These solutions can be smoothed out by replacing them with the mean velocity per period, 1 s → w(s) := π

s+π Z

w(σ) dσ .

s

3.3 Straight worm with viscous friction (“STOKY”)

105

Fig. 3.24: Mean velocities corresponding to Fig 3.23.

The most interesting thing here is the limit of the velocity, w∞ := lims→∞ w(s), which gives the asymptotic speed of the worm disregarding any pulsation. Both foregoing figures suggest the existence of a π−periodic particular solution w e that attracts any other solution for increasing s: lims→∞ | w(s) − w(s) e |= 0. Let us assume the existence of w e and let us try to compute the asymptotic speed w(∞) e = w∞ , which is also the mean value per period of w. e To this end we consider the R s+π differential equation (3.20) which is an identity for w = w. e We apply π1 s . . . ds to (3.20) obtaining
M e w(s e + π) − w(s) e = −w(s) π s+π Z   k w(σ) + e + sin(σ) + w(σ) e − sin(σ) dσ . π s

Now the left hand side is zero due to periodicity, and writing w(s) e = w ∞ + pe(s), where pe is π-periodic with mean value zero, we get πw ∞ = k

s+π Z s

  w ∞ + pe(σ) + sin(σ) + w∞ + pe(σ) − sin(σ) dσ .

The integrand has period π, so s = 0 will do. Certainly 0 ≤ w(s) e = w ∞ + pe(s) ≤ 1 for positive k (remind exponential decrease of solutions if w ≥ 1), so the first term of the integrand is simply w ∞ + pe(σ) + sin(σ) for σ ∈ [0, π], and its integral is πw ∞ + 2. If we knew the intervals of fixed sign of w ∞ + pe(σ) − sin(σ) then the integral could easily be calculated as a sum of integrals over these intervals. But nothing is known about pe. To proceed at least approximately, we drop pe at this place. w ∞ ≤ 1 implies

106

3 The Straight Worm With Propulsive “Friction”

a := arcsin(w ∞ ) ∈ [0, π2 ]. Now the integral splits Zπ 0

w∞ − sin(σ) dσ =

Za


w∞ − sin(σ) dσ

0

−

π−a Z a


w ∞ − sin(σ) dσ +

Zπ

π−a


w ∞ − sin(σ) dσ

and, calculating this, we obtain finally q h i 2 =0 w ∞ π − 4 k arcsin(w ∞ ) − 4 k 1 − w ∞

(3.21)

as an equation for the approximate determination of the asymptotic average speed w ∞ . − + Note that w ∞ only depends on k = 21 kk− −k +k+ . The graph of the function k 7→ w∞ (k) is shown in the next figure.

Fig. 3.25: Asymptotic average speed w∞ vs. k.

Balance of power The following is just an adaptation of equations (3.17), (3.18), and (3.19) to (3.20). Normalizing Pa = 2 k0 a20 pa ,

Pe = 2 k0 a20 pe ,

(3.19) gives the actuator power in the form pa =

    M cos(s) + sin(s) − k w + sin(s) − w − sin(s) sin(s) .

Its average value per period is — again approximating w ≈ w ∞ for large s — pa =

1 −k = 2

−1  k− . 1+ + k

3.3 Straight worm with viscous friction (“STOKY”)

107

d The power of the external forces, Pe := dt (m v ∗2 ) = 2 m a20 Ω w w0 (total mass= 2 m) gets the representation   pe = −w2 + k w + sin(s) + w − sin(s) w ,

and its asymptotic average value per period is pe = (2 k − 1) w ∞ 2 .

4

Adaptive Control of Worms

This chapter concerns adaptive control of worm systems. In particular in case of unknown actuator data or failing actuators, or varying external forces (failing spikes, changing friction or gravity) control has to keep the worm’s locomotion close to an optimal pattern (gained in kinematical theory). Various controllers are introduced and applied to worm motions.

4.1

Adaptive control of spiked worms

4.1.1

Introduction

In this chapter we take up the problems arising in connection with uncertain data of a worm system. Uncertainty means that there are internal (actuator) or environmental (contact) data which are not exactly known or unknown or vary in an unknown way during motion. In Section 2.1.5.2 the investigations were focussed on a worm with dynamic drive through viscoelastic actuators of the following form

Fig. 4.1: Viscoelastic actuator.

(see Fig. 2.20). The corresponding cut forces applied upon the neighbored mass points are µi (t, x, x) ˙ | {z }

actuator force output

= ci (xi−1 − xi − li0 ) + k00i (x˙ i−1 − x˙ i ) +

ui (t) | {z }

actuator force input

(4.1)

(see (2.58)). The general task is to find an actuator input ui (t) that ensures a desired motion pattern (which might have been constructed in the kinematical theory). In Section 2.1.5.2 we presented a formula solution to this task for the inch-worm (n = 1) to move in a prescribed gait t 7→ l(t). The corresponding gait-mimicking actuator input

110

4 Adaptive Control of Worms

had been found as
˙ − c l(t) − Γ sign l(t) ˙ u(t) = −m ¨l(t) − (k0 + k00 ) l(t)

(4.2)

(−k0 l˙ − Γ: external forces), provided the dynamics are always in mode A = {0} or A = {1}, i.e., equivalently, σ(t) ≥ 0 for all t. So at this stage, u is a given function of t, to be seen as an open-loop or off-line control15 . Moreover it depends on both external (k0 , Γ) and internal (m, k00 , c) data which are, possibly, uncertain. Furthermore, in real applications emphasis is on worm systems with n > 1 which apparently do not present that simple control laws. The way out of this dilemma is adaptive control, i.e., the search for a closed-loop controller that despite of not or not completely known system parameters achieves tracking of the desired motion. It is not the aim to identify the uncertain data as a basis for to build a control force u(t) but, rather, to design a learning controller (an adaptive high-gain output feedback controller) that successively approximates the necessary forces ui (t) on its own. Thereby we do not focus on exact tracking, instead we rely on the approximate λ-tracking control objective, i.e., we tolerate a prespecified tracking error of size λ > 0 - the price for the design of fairly simple closed-loop controllers. Let us consider the worm system equations (2.59). For to track a given gait t 7→ lj (t), j = 1, . . . , n, we take the actual lengths of the links as the system outputs: yj (t) := xj−1 (t) − xj (t) ,

j = 1, . . . , n .

Then we search for a control law y = (y1 , . . . , yn ) 7→ u = (u1 , . . . , un ) which, when applied to the equations of motion (2.59) - thus making them a closed-loop system -, realizes approximate tracking of the reference signal

yref (t) = yref1 (t), . . . , yrefn(t) := l1 (t), . . . , ln (t) . In detail, λ-tracking means that

(i) every solution of the closed-loop system is defined and bounded for all t ≥ 0, and (ii) the output y(·) tracks yref (·) with asymptotic accuracy, quantified by λ > 0, in the sense that n o

max 0, y(t) − yref (t) − λ −→ 0 as t → ∞ .

Setting up the controller then goes along the following steps.

15 Readers not familiar with control theory are kindly recommended to inspect Appendix C where the respective fundamentals are sketched.

4.1 Adaptive control of spiked worms

111

a) Calculate the tracking error, i.e., the deviation of the actual output y(t) from the desired reference output yref (t), e(t) := y(t) − yref (t) .

(4.3)

b) Determine an adaptation law that rules a high-gain parameter k as a function of t in accordance with the actual tracking error: a big error is to generate increase of k; the “classical adaptor”, see Appendix C, to be used first is  n o2

˙ k(t) = γ max 0, e(t) − λ ,

k(0) ∈ R , γ ≥ 1 .

(4.4)

Clearly, the adaptor causes k to increase monotonically as long as the error norm is bigger than λ else it keeps k constant (dead zone behavior). Sometimes the increase of k is too slow, then a parameter γ  1 may serve for acceleration. c) Design a feedback of the information obtained from a) and b):   u(t) := −k(t) e(t) + κ e(t) ˙ , κ ∈ R.

(4.5)

The foregoing considerations are valid for arbitrary n ∈ N.

But, at first, the following investigations are confined to n = 1, just in order not to overburden the simulations and to obtain a better insight into what happens during the action of an adaptive controller. Though the first simulations are done with examples containing known (but arbitrarily chosen) data the adaptive nature of the controller is untouched (since “the controller doesn’t know these data”), and later on the certainty of data will be seriously relaxed. Exercising the inch-worm we proceed by the following strategy. We start by using the above mentioned “classical adaptor” (4.4) and demonstrate its mode of action and its insufficiencies as well. Guided by the latter we successively modify this adaptor and the last one in this series of adaptors is finally used in all control tasks to be tackled later.

112

4 Adaptive Control of Worms

In any case it seems natural to expect that the adaptive control u approaches umimic for t → +∞. We refer to the fact that the classical adaptor is justified by theory (see [Behn 05]) whereas its modifications, tested in various problems, proved reliable but still live with an experimental flair.

4.1.2

Our simplest guy: the inch-worm

For all simulations in this section we choose the following data; additional data needed in simulations are given on the spot. • worm system: m0 = m1 = 1. • environment: k0 = 0, Γ = 1.2336 (ensures kinematical theory to be dynamically feasible). h
i • reference gait: l(t) = l0 1 − ε 1 − cos(Ω t) , l0 = 1, ε = 0.25, Ω = π.

• actuator: c0 = m ω02 = m Ω2 = m π 2 , k00 = 2 π. • controller: k(0) = 0, λ = 0.02, κ = 1.

In spikes approximation we use sign1 : x 7→ tanh(Ax), with A = 104 . Exercise 4.1 Evaluate and plot t 7→ umimic(t) for the given data. 4.1.2.1



Adaptive control in case of fixed actuator data

The equations of motion are, see (2.59), mx ¨0 = −c0 (x0 − x1 ) − k00 (x˙ 0 − x˙ 1 ) − Γ − u + λ0 , mx ¨1 = c0 (x0 − x1 ) + k00 (x˙ 0 − x˙ 1 ) − Γ + u + λ1 ,

(4.6)

output y := x0 − x1 , with spikes forces λ0,1 according to (2.14) and u as an adaptive control
u(t) = k(t) e(t) + κ e(t) ˙ , feedback
˙ k(t) = A k(t), e(t), λ, parameters , adaptor e(t) = y(t) − l(t) , error. ADAPTOR 1 (the “classical” one): We choose  o2

 ˙ k(t) = γ max e(t) − λ, 0 .

(4.7)

(4.8)

4.1 Adaptive control of spiked worms

113

Let us start with γ := 100. The following figures sketch what happens in a time interval 0 . . . 10: the worm motion is far from that of the ideal SPIKY (shadowed grey in figures), the error is far from the λ-strip, the gain k is growing, and the control u does not approach the gait-mimicking umimic. For smaller γ everything is considerably worse.

Fig. 4.2: Worm motion and error vs. t with γ = 100.

Fig. 4.3: u and umimic vs. t, gain k vs. t with γ = 100.

Using γ := 1000 we observe things going to the better, see the following figures.

Fig. 4.4: Worm motion, error vs. t with γ = 1000.

114

4 Adaptive Control of Worms

Fig. 4.5: u and umimic vs. t, gain k vs. t with γ = 1000.

The error seems to be attracted by the λ-strip, the gain k shows a slower increase after a transient phase at the beginning, and u exhibits a clear trend towards umimic , see Figs. 4.4 and 4.5. The reason of this still unsatisfactory behavior is evident: the growth of k when the error ˙ is leaving the λ-strip is too slow (caused by the quadratic dependence ke(t)k − λ → k(t) with zero slope at start), this low growth rate can be increased by use of a sufficiently big γ, but at the price of possibly undesired big limit values of k. In the following we try to gain a faster growth of k by modifying the adaptor. ADAPTOR 2: We replace the (max{. . .})2 by a square root (having infinite slope at zero): ˙ k(t) =γ



o1/2

 max e(t) − λ, 0 .

(4.9)

Now with γ = 100 the results are comparable with those from Figs. 4.4 and 4.5 though k takes bigger values:

Fig. 4.6: Worm motion, error vs. t with γ = 100.

4.1 Adaptive control of spiked worms

115

Fig. 4.7: u and umimic vs. t, gain k vs. t with γ = 100.

Let us choose γ = 500:

Fig. 4.8: Worm motion, error vs. t with γ = 500.

Fig. 4.9: u and umimic vs. t, gain k vs. t with γ = 500.

Finally, γ = 500 has made the error stay in the λ-strip, this is reflected by constant (though big) k(t) and u nearly identical with umimic .

116

4 Adaptive Control of Worms

Obviously the nearby-attraction of the λ-strip has become stronger but for ke(t)k > λ+1 ˙ the quadratic law of Adaptor 1 shows up to be the better one due to larger values of k:

Fig. 4.10: Sketch of increase of k˙ vs. kek via Adaptor 1 and 2.

ADAPTOR 3:   2

e(t) − λ ,     1/2

˙ k(t) =γ

e(t) − λ ,    0,



e(t) ≥ λ + 1 ,

λ ≤ e(t) < λ + 1 ,

e(t) < λ .

  2

e(t) − λ ,  γ    1/2  

˙k(t) = γ e(t) − λ ,     0, −δ · k(t) ,



e(t) ≥ λ + 1 ,

λ ≤ e(t) < λ + 1 ,



e(t) < λ ∧ t − te ≤ td ,

e(t) < λ ∧ t − te > td .

(4.10)

This is a combining of both foregoing adaptors. Possibly a power (. . .)α , 0 < α < 21 , in the second line of Adaptor 3 could do the job even better. In our present context the error norm is always smaller than 1, therefore we drop the visualizations which are the same as under Adaptor 2. Instead we address another item. Figure 4.9 exhibits a constant value of k while ke(t)k stays within the λ-strip. Possibly this value of k is unnecessarily big and could be diminished within another modification of the adaptor. ADAPTOR 4:

(4.11)

Here te means the current entry time to the λ-strip, after te the gain k(t) is kept constant for a duration td whereas k(t) undergoes an exponential decrease with a decay rate δ afterwards (this holds until the error leaves the strip whereby then k is caused to increase again). The positive parameters δ and td have to be chosen suitably. A positive td is reasonable since it could happen that e very rapidly traverses the λ-strip. Then it would be inadequate to immediately decrease k after e entered the strip, rather the

4.1 Adaptive control of spiked worms

117

decay begins only after the duration td has been exceeded. Neither δ nor td should be too big, this is demonstrated by the following two figures (both with γ = 500).

Fig. 4.11: Error e (left) and gain k (right) vs. t for δ = 0.2, td = 5.

Obviously, the interval of big, constant k is unnecessarily long: choose td = 1.

Fig. 4.12: Error e (left) and gain k (right) vs. t for δ = 1, td = 1.

The error quickly leaves the strip, k has a very steep decay. Further on, choose δ = 0.2.

Fig. 4.13: Worm motion, error vs. t for δ = 0.2, td = 1.

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Fig. 4.14: u and umimic vs. t, gain k vs. t for δ = 0.2, td = 1.

The observed effects from Fig. 4.12 could now be suppressed. The Figures 4.13 and 4.14 clearly show that k diminishes so far as to let e leave the λ-strip. This behavior could be damped by letting δ depend on the distance of e from the strip boundary. This leads to the next modification. ADAPTOR 5:   2

  γ e(t) − λ ,   1/2  

˙k(t) = γ e(t) − λ ,   0, 


 −δ 1 − λ1 e(t) k(t) ,



e(t) ≥ λ + 1 ,

λ ≤ e(t) < λ + 1 ,

e(t) < λ ∧ t − te ≤ td ,

e(t) < λ ∧ t − te > td .

(4.12)

Now the decay rate equals δ if and only if e(t) = 0 and it tends to zero as ke(t)k % λ. This entails some smoothing of t 7→ k(t) and a weakened tendency for leaving the λ-strip, see next figure (and compare with Figs. 4.13 and 4.14).

Fig. 4.15: Error e (left) and gain k (right) vs. t for δ = 0.2, td = 1.

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119

Remark 4.2 If (repeated) modest violation of the λ-restriction cannot be tolerated then a simple parameter variation could help: in the adaptor, replace λ by some suitable ε < λ which then asymptotically forces e to stay within the desired λ-strip (certainly at the price of larger k(t)-values), see the following figures.

Fig. 4.16: Worm motion, error (with λ- and ε-tubes) vs. t for δ = 0.2, td = 1 and ε = 0.015 < λ = 0.02.

Fig. 4.17: u and umimic vs. t, gain k vs. t for δ = 0.2, td = 1 and ε = 0.015 < λ = 0.02. 

Summary. Reviewing the foregoing figures we observe that Adaptor 5 works best. The common overshooting of the error e, accompanied by a steep increase of the gain k is caused by the (arbitrary) initial value k(0) = 0. If we had chosen a very big k(0) then k(t) would decrease until e stays in the vicinity of the strip. Use of Adaptor 1, 2, or 3 with a sufficiently big k(0) yields k(t) ≈ const (high-gain stabilizability). Unfortunately, one never knows the least possible value of such k(0) in advance. It must be estimated by simulations as, equally, useful controller data must be found. The effect of taking for k(0) the guessed value of k(+∞) (supposed it exists) from the foregoing figures is demonstrated in Fig. 4.18.

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Fig. 4.18: Error e (left) and gain k (right) vs. t for Adaptor 5, using ε = 0.015, as in Figs. 4.16 and 4.17 with estimated suitable k(0).

4.1.2.2

Adaptive control in case of changing actuator data

We use the same parameters as presented at the beginning of this chapter besides the actuator data. We let the spring stiffness and the viscous damping factor vary in time: t 7→ c(t) =



c0 , 0,

t ∈ [0, 10) , t ∈ [10, 50]

  k00 , t 7→ k00 (t) = k00 −  0,

k00 10

(t − 20) ,

t ∈ [0, 20) t ∈ [20, 30) t ∈ [30, 50]

(with c0 and k00 as before) we simulate a failing actuator: breaking spring at t = 10, and loss of viscosity of damper (by heating) starting at t = 20. Using Adaptor 5 (and tracking ε = 0.015 < λ) with estimated suitable k(0) we obtain the following result:

Fig. 4.19: Error e vs. t (left) and gain k vs. t (right).

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121

Fig. 4.20: Control input u vs. t.

It is obvious that u tends to umimic , so the adaptor makes the actuator robust against failures. Exercise 4.3 Evaluate and plot t 7→ umimic (t) for t ≥ 30 and compare with u in Fig. 4.20.

4.1.3



Three-mass-point-worm (n = 2)

In this section we consider a worm system consisting of three mass points, see Fig. 4.21

Fig. 4.21: Worm with n = 2 and viscoelastic actuators in link 1 and 2.

The equations of motion are, see (2.59), m0 x ¨0 = −c1 (x0 − x1 ) − k001 (x˙ 0 − x˙ 1 ) − Γ − u1 + λ0 , m1 x ¨1 = c1 (x0 − x1 ) − c2 (x1 − x2 ) +k001 (x˙ 0 − x˙ 1 ) − k002 (x˙ 1 − x˙ 2 ) − Γ + u1 − u2 + λ1 , m2 x ¨2 = c2 (x1 − x2 ) + k002 (x˙ 1 − x˙ 2 ) − Γ + u2 + λ2 ,     y1 x0 − x1 with output y := = , y2 x1 − x2 (4.13)

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with spikes forces λ0,1,2 according to (2.14). The adaptive control u = (u1 , u2 )T is
u(t) = k(t) e(t) + κ e(t) ˙ ,   (4.14) y1 − y1ref e(t) = , y2 − y2ref where we choose Adaptor 5 and the following reference gait (periodically repeated) for simulations:    1 + ε (1 − cos(π t))  t ∈ [0, 1) ;   1 − ε (1 − cos(π t))       1 + ε (1 − cos(π t)) (4.15) t ∈ [1, 2) ; yref = l0 1−2ε        1    1 − ε (1 + cos(π t)) t ∈ [2, 3) . For all simulations in this section we choose the following fixed data: • worm system: m0 = m1 = m2 = 1; • environment: k0 = 0, Γ = 2.7; • reference gait: l0 = 1, ε = 0.25; • actuator: cj = m ω02 = m π 2 , k00j = 2 π. • controller: k(0) = 10, λ = 0.05, δ = 0.2, td = 1, γ = 500, κ = 1. In spikes approximation we use sign1 : x 7→ tanh(Ax), with A = 104 . 4.1.3.1

Adaptive control in case of ideal spikes

The following figures sketch the usual most important state functions vs. time t. (Mind the shortened time interval in the y-diagrams for a better visuality.) It is obvious that the Adaptor 5 acts in a very good way again.

Fig. 4.22: Worm motion (left) and outputs y with λ-strip (right) vs. t for Adaptor 5.

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123

Fig. 4.23: Control inputs (left) and gain k (right) vs. t for Adaptor 5.

The foregoing figures (Figs. 4.22 and 4.23) serve as a basis for to recognize what happens under system failures in the following.

4.1.3.2

Adaptive control in case of failing spikes

We confine the investigation of failure effects to the case of failing (breaking) spikes, while the actuator data are kept fixed. 1st accident: one failing spike All three spikes are working well until t = 15, then the middle spike 1 breaks (is removed, λ1 = 0) and does not act until t = 25. For t > 25 all three spikes are working well again. The effect is easily seen by the temporarily occurring negative velocity v1 (Fig. 4.25), producing “useless” oscillations of the middle mass point, and entailing decrease of the worm’s mean speed (longer rest phases of mass points 0 and 2). After “spike repair” the system behavior returns to the beginning one.

Fig. 4.24: Worm motion (left) and gain k (right) vs. t for Adaptor 5.

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Fig. 4.25: Velocities (left) and control inputs u (right) vs. t for Adaptor 5.

2nd accident: two failing spikes Now we have the foregoing scenario until t = 25. Then spike 1 keeps failing and, additionally, also spike 2 breaks.

Fig. 4.26: Worm motion (left) and gain k (right) vs. t for Adaptor 5.

Fig. 4.27: Velocities (left) and control inputs u (right) vs. t for Adaptor 5.

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125

So, for t > 25, two of the mass points temporarily undergo negative velocities, while the worm’s head suddenly comes to rest (this fact could possibly change under different data, e.g., Γ as small as to allow forward skidding of the system).

4.1.4

Adaptive Control of worms with n = 3

In this section we consider a dynamical SPIKY with n = 3 links. First, we want to track three kinematical gaits obtained by the gait construction algorithm presented in Section 2.1.3.4 (with sin2 -smoothing). Second, we present two simulations of a worm with gear-shift (switching between these kinematical gaits) to climb in a hilly landscape16 . We use the following data for all simulations. Additional data is given on the spot: • worm system: m0 = m1 = m2 = m3 = 1; • environment: k0 = 0, Γ = 4.14; • actuator: cj = m ω02 = m π 2 , k00j = 2 π. • controller: Adaptor 5 — k(0) = 10, λ = 0.05, δ = 0.2, td = 2, γ = 500, κ = 1. In spikes approximation we use sign1 : x 7→ tanh(Ax), with A = 104 . 4.1.4.1

Tracking of different gaits

Exercise 4.4 Verify the worm motions in the following simulations by means of the gait construction algorithm given in Appendix E.  Gait 1: a = 1, A[1] = {0}, head to rear. (One active spike at every time, mode sequence: {0} → {1} → {2} → {3}.)

Fig. 4.28: Worm motion (left) and gain parameter k (right) vs. t for Adaptor 5. 16 These

simulations are done by the bachelor student Silvan Schwebke, using MATLAB.

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4 Adaptive Control of Worms

Gait 2: a = 2, A[1] = {0, 2}, head to rear. Mind the gap in the modes!

Fig. 4.29: Worm motion (left) and gain parameter k (right) vs. t for Adaptor 5.

Gait 3: a = 3, A[1] = {0, 1, 2}, head to rear.

Fig. 4.30: Worm motion (left) and gain parameter k (right) vs. t for Adaptor 5.

Exercise 4.5 In the left diagrams, check the decrease of the mean speed for increasing a. Mind the not yet sufficient tracking until t ≈ 2 in Fig. 4.30, left. 

4.1.4.2

Change of gaits — gear shift

Gait shift 1: Gait 1 → Gait 2 → Gait 3, shift times t = 10 , 20, using a constant Γ = 4.14.

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127

Fig. 4.31: Worm motion (left) and gain parameter k (right) vs. t for Adaptor 5.

Gait shift 2: Same as above. At the same shift times there is a change of Γ   4.14 , t ∈ [0, 10) Γ(t) := 6.31 , t ∈ [10, 20)  8.04 , t ∈ [20, 30]

Fig. 4.32: Profile of the hilly landscape.

induced, e.g., by a change of hill inclination 25◦ → 40◦ → 55◦ . Note further that Fig. 4.32 does not correspond to the center approximation used for the HILLY in Section 2.3.1.

Fig. 4.33: Worm motion (left) and gain parameter k (right) vs. t for Adaptor 5.

Note the same mean speed as under gait shift 1. This is achieved by bigger values of the gain k in the 2nd and 3rd time interval.

128

4.2

4 Adaptive Control of Worms

Adaptive control of worms with propulsive friction

In this subsection we replace the worm-ground interaction via spikes by stiction combined with Coulomb sliding friction. This might be seen as a more realistic description of the interaction or as a model of practical failing of the spikes (being of finite strength). We use the mathematical model (3.8) in Subsection (3.2.1). Again, adaptive control has to be used when considering uncertain or randomly changing friction data (rough terrain). Successful application is shown by the following simulation results: 0. spikes (for comparison), 1. and 2. stiction only, 3. sliding friction only, and 4., 5. and 6. both. We do not present a closed mathematical theory of such systems. The following results have to be seen as a first demonstration. Data: We use the same data as in Subsection 4.1.3 and we track the reference gait (4.15) (constructed due to the gait generation algorithm in Subsection 2.1.3.4 with n = 2 and a = 1) using controller (4.14) with Adaptor 5. Example 4.6 Adaptively controlled dynamical SPIKY for comparison.

Fig. 4.34: Worm motion (left) and outputs y with λ-strip (right) vs. t.

Fig. 4.35: Velocities (left) and gain k (right) vs. t.



4.2 Adaptive control of worms with propulsive friction

129

Example 4.7 Here, we consider only stiction, i.e., (F0− , F0+ , F − , F + ) = (14, 0, 0, 0). The value F0− corresponds to the maximum of the spikes forces due to (2.48) and (2.49). We get:

Fig. 4.36: STICKY: (F0− , F0+ , F − , F + ) = (14, 0, 0, 0) — worm motion (left) and outputs y with λ-strip (right) vs. t.

Fig. 4.37: STICKY: (F0− , F0+ , F − , F + ) = (14, 0, 0, 0) — velocities (left) and gain k (right) vs. t.

There are some short backward motions at the beginning, afterwards the motion coincides with that of Example 4.6 (SPIKY). A sufficiently large F0− suppresses this negative velocity in the beginning. 

Example 4.8 Now we change the anisotropy for stiction to (F0− , F0+ , F − , F + ) = (14, 3, 0, 0). We get mainly the same results as in Example 4.7 except of a diminished mean speed caused by F0+ > 0. 

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4 Adaptive Control of Worms

Example 4.9 Now replace stiction by sliding friction. We put (F0− , F0+ , F − , F + ) = (0, 0, 14, 3), this yields:

Fig. 4.38: SLIDY: (F0− , F0+ , F − , F + ) = (0, 0, 14, 3) — worm motion (left) and outputs y with λ-strip (right) vs. t.

Fig. 4.39: SLIDY: (F0− , F0+ , F − , F + ) = (0, 0, 14, 3) — velocities (left) and gain k (right) vs. t.

Though there is again a good tracking of the desired gait from kinematical theory, we observe an unsatisfactory external behavior of the worm (recurring negative velocities, no locomotion anymore), obviously owing to the cancellation of stiction. 

Example 4.10 So, we switch back to the previous stiction values and add a small positive sliding friction of magnitude 1, i.e., (F0− , F0+ , F − , F + ) = (14, 3, 0, 1). This yields:

4.2 Adaptive control of worms with propulsive friction

131

Fig. 4.40: COULY: (F0− , F0+ , F − , F + ) = (14, 3, 0, 1) — worm motion (left) and outputs y with λ-strip (right) vs. t.

Fig. 4.41: COULY: (F0− , F0+ , F − , F + ) = (14, 3, 0, 1) — velocities (left) and gain k (right) vs. t.

Now worm runs backwards.



Example 4.11 Therefore we need a stiction value F0− = 14 + 2 to overcome the forward sliding friction F + = 1 (simultaneously acting upon two mass points). We have (F0− , F0+ , F − , F + ) = (16, 3, 0, 1). This is essentially the same effect as it would be caused by an increase of Γ from 2.7 to 4.7 since two mass points are sliding at every moment during the motion. So F + leads to an additional backward force of magnitude 2 that has to be compensated by stiction. We get the same results as in Example 4.7.  Example 4.12 Now, applying (F0− , F0+ , F − , F + ) = (16, 3, 5, 1) (nearly the same anisotropy of stiction and sliding friction), we obtain:

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Fig. 4.42: COULY: (F0− , F0+ , F − , F + ) = (16, 3, 5, 1) — worm motion (left) and outputs y with λ-strip (right) vs. t.

Fig. 4.43: COULY: (F0− , F0+ , F − , F + ) = (16, 3, 5, 1) — velocities (left) and gain k (right) vs. t.

Good behavior, comparable with the worm motion in Example 4.6.



Example 4.13 Here we investigate what happens under a sudden change of the external friction. Having in mind Example 4.12 we choose now

(F0− , F0+ , F − , F + ) :=

and obtain:

  (16, 3, 5, 1) , 

t ∈ [0, 15) ∪ [30, 50] ,

(8, 1.5, 2.5, 0.5) , t ∈ [15, 30) ,

4.2 Adaptive control of worms with propulsive friction

133

Fig. 4.44: COULY: worm motion (left) and gain k (right) vs. t.

Fig. 4.45: COULY: velocities (left) and control inputs (right) vs. t.

We observe that, after a temporary slithering backwards, the adaptive control forces the worm to return to its original locomotion.  R´esum´e: In the foregoing examples the (adaptive) control has been directed to ensure a prescribed gait (i.e., a temporal pattern of shape - something internal!). It is intelligible that a changing environment or changing type of interaction influences the global movement (and the driving forces ui , not shown here) despite a good tracking of the gait. A comparison of the simulations points at stiction as the essential part of Coulomb interaction with the ground and gives a warning of a careless reducing of the interaction to pure sliding friction. The simulations show a good tracking of the gait, but the global behavior may be bad. We have to track a prescribed global movement of the worm. A first step could be to track a reference head speed (a pure tracking of the prescribed gait is not sufficient to do this), possibly coupled with appropriate change of gaits (“gear shift”, see Subsection 4.1.4.2).

5

Conclusions

Here we give a glance to international and home worm systems. Some items are sketched which had to be done now and later on.

5.1

Worms in literature

We do not try nor do we intend to give an exhaustive review of papers about worms. It is just to be shown that the contents of this book are part of international work that aims at both theoretical insight and practical application of (artificial) worms. Apparently the majority of papers on slender legless locomotion systems deal with snake-like systems. Here we only quote a few of them: [Gans 70], [ChBu 95], [OsBu 96], [Cher 05], more can be found in the lists of references therein. It is understandable that the early investigations of the motion of worms (and of snakes as well) were done by biologists or appeared with a certain biological flair. We mention the 1968 standard book Animal locomotion [Gray 68] by Gray and the 1983 paper Crawling of worms [KeFa 83] by Keller and Falkovitz. The latter authors present an equation of motion that is similar to our equation (2.77). Included are gravity and Coulomb friction as external forces and internal pressure and tension which are “controlled by the worm”. Focus is put on the construction of pulse and traveling wave solutions. There are no stringent and convincing mechanics and mathematics. We picked up the idea of using spikes to ensure unidirectional locomotion first from the 1988 paper The motion dynamics of snakes and worms by Miller [Mill 88]. Without using mechanical details the paper tells one that a worm is modeled as a chain of cu-

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5 Conclusions

bic segments. Each segment is built up by mass points in the corners and viscoelastic actuators with controllable springs (see Fig. 2.21) in every edge and diagonal. The investigations (not presented by the author) are to serve for realistic animation of animals in computer graphics. A practical application of spikes is shown in the 2000 paper [VaCQ 2000] by Vaidyanathan et al. There, the authors present a segmented artificial worm (up to 5 segments). Each segment consists of a fluid-filled bladder between two rigid discs which are equipped with spikes. The discs are connected by at least two springs made of shape memory alloy wires. The chain of segments forms a hydroskeleton, bladders and springs represent antagonistic muscles. By controlled heating and relaxing of some or all springs of a segment the discs change their relative position which in turn results in a (not necessarily rectilinear) motion of the whole system. No theory but experimental results are given in the paper. Strictly oriented to application in medical endoscopy is the 1995 paper The development of a robotic endoscope [SlBG 95] by Slatkin et al. The authors describe segmented worm-like robots constructed by alternating gripper and extensor elements. These elements, respectively, expand radially and longitudinally through variation of the internal pressure by means of electronically controlled valves. Due to a passive lateral compliance the worm is qualified for locomotion within, e.g., the human intestine. Various earthworm-like gaits are considered, propulsive forces (caused by pressing grippers against the surrounding wall) are not. Quite similar are the worms considered in [CYHG 99] by Chen et al. The investigations are theoretical and based on kinematics, they aim at algorithms for gait construction. In the 2006 paper [NKIM 06] by Nakamura et al. the authors describe a laboratory crawling robot consisting of four segments. The latter are built up by two rigid discs which are connected by several belts made of flexible plates. In the interior there is a servo motor combined with a crank mechanism that serves for changing the length of the segment. Variation of length is coupled with a radial bending of the belts which then press against the ground or a surrounding tube thereby generating friction (or stiction) as a propulsive force. Results of laboratory experiments are presented and a mathematical model to come is announced. The 2010 paper [BPZZ 11] by Bolotnik et al. presents purely theoretical investigations concerning the correlation of periodic kinematic drive and environmental resistance during worm motion. Various properties of the friction force F (v) are considered: linear/nonlinear, continuous/discontinuous, isotropic/anisotropic. Supposing small friction forces the results are based on an averaging method applied to the heteronomic differential equation for the center velocity. One outcome is, that under certain circumstances locomotion may take place also under isotropic propulsion. At this place it is worth having a comparing glance to [TINK 11] by Tanaka et al. There, the authors present a biomechanical discussion of “peristalsis-like locomotion driven by contraction waves propagating along the body axis” (observed, e.g., in earthworms and snails). They set up a continuous model for slow motion in a “wet” environment, i.e., the investigations focus on quasi-static processes (neglecting all inertial effects) under propulsion by strong isotropic viscous forces (no Coulomb forces nor even spikes

5.2 Worms at Ilmenau University of Technology

137

are introduced). The tool for ensuring locomotion despite the isotropy of the external influence is a spatiotemporal control of the friction coefficient (k0 (ξ, t) in our notation, this mimics a suitable activation of the earthworm’s bristles or secretion of mucus by a snail) in correlation with the wave propagation. The contraction is seen in connection with time-dependent original lengths of internal springs (ε(ξ, t) in our notation). The outcome is a linear diffusion equation for the displacement u(ξ, t) = x(ξ, t) + ξ (mind x(ξ, 0) = −ξ : axes-orientation) with (ξ, t)-dependent diffusion coefficient and source term, u,t = D(ξ, t) (u,ξξ −e(ξ, t))

(5.1)

where D = C/k0 , C some internal elasticity coefficient. Appropriate choice of the functions D and e admits wave-like solutions of this diffusion equation. In our context this equation appears out of equation (2.77): put ρ0 := 0 (no inertia), drop Γ and λ (not present in above theory). Introduce Hooke’s law N (ξ, t) = C(ε(ξ, t) − e(ξ, t)), where ε = −u,ξ is the actual strain (see Remark 2.29) whereas the strain e(ξ, t) is endogenously generated (by muscles).

5.2

Worms at Ilmenau University of Technology

This section gives a chronological overview of worm-like locomotion systems developed at the Department of Technical Mechanics at Ilmenau University of Technology. Since 1995, starting with an Innovation College “Motion Systems” (founded by the German Science Research Foundation), one research focus is on modeling, analysis and control of worm-like locomotion systems.

Fig. 5.1: First prototype “ALWIN”.

The prototype “ALWIN” (ArtificiaL Worm IlmeNau) was developed for demonstration purposes only. Prototypes were developed very quickly: the first followed the peristaltic principle of earthworm locomotion with a pneumatic drive system.

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5 Conclusions

Fig. 5.2: Prototypes with pneumatic drives: designed by Lysenko (DE 10027447A1) (left) / designed by Preuß and Stubenrauch (DE 10231057A1).

In 1998 a micro-robot was developed by Riemer (DE 19853324A116) with focus on some applications in minimally invasive surgery. The robot actuators are a combination of SMA elements and elastic springs. For details see [Huan 03]. Following the trend towards miniaturization a first prototype using piezo elements was developed in 2002 by Weise [Weis 02]. This peristaltic robotic system consists of two thin plates which are connected by a compliant mechanism that amplifies t he amplitudes of the piezo element (arrangement in Fig. 5.3, left).

Fig. 5.3: Micro-robot: assembly from top view (left) and side view (right).

Global movement, driven by piezo vibrations of some 100 Hz frequency, is achieved by equipping the plates with spikes made by bundles of glass fibers (see Fig. 5.4), which realize an anisotropic friction with µ− > µ+ .

Fig. 5.4: Micro-robot: spikes structure.

In 2003 prototypes with a drive by unbalanced rotors are developed by [Huan 03]. The first prototype, shown in Fig. 5.5 left, has one unbalanced rotor. Experiments have 16 The

codes in parentheses are German Patent Codes.

5.2 Worms at Ilmenau University of Technology

139

proven the functionality (achievement of locomotion of the system). But the single unbalanced rotor gives rise to disturbing oscillations of the ground friction (through variation of normal force) besides a gravity-induced non-uniformity of the rotation.

Fig. 5.5: Two prototypes with unbalanced rotors.

Hence, the second prototype (Fig. 5.5 right) has two exciters in counter-rotation. The new arrangement leads to the fact that the sum of the vertical components of the centrifugal forces vanishes. Both systems are equipped with the same special bristle structure as shown in Fig. 5.6. Some modification is presented by Zimmermann et al. in [ZZBP 09]. The system, sliding along a straight line, consists of two plates connected by a spring, each plate equipped with one unbalanced rotor. The advantage of this construct is the controllability of speed and direction of motion via angular velocity and phase shift of the rotors. In [SBZA 05] the authors introduce a prototype, called “TM-Robot I”, which was to justify the kinematical theory (Subsection 2.1.1). It consists of two stepping motors and a dummy, so it forms a three mass point worm with n = 2 links. Each mass point is equipped with a bristle-structure to prevent slipping backwards, see Figure 5.6. This structure models the ground contact via spikes. Each stepping motor can travel separately along a threaded rod in both directions with controllable velocities to generate the l˙j . The l˙j produce the change of shape of the worm.

Fig. 5.6: One ‘mass point’ with bristle-structure.

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5 Conclusions

During experiments with the TM-Robot I short-term negative velocities could be observed [Abaz 06]. So the differential constraints x˙ i ≥ 0 were not ideally realized. The spikes could be replaced by wheels with ratchets (supposing the wheels are rolling without sliding), Fig. 5.7.

Fig. 5.7: One “mass point” with passive wheels.

Both design and hardware realization of the above prototypes were essentially supervised by Peter Walkling. Thanks for all! In 2007 we split in three main directions in designing worm-like robotic locomotion systems: 1. vibro-driven motion systems using piezo elements as actuators, see [ZiZB 09], [LZCB 11] and [Beck 10],

Fig. 5.8: Vibration-driven locomotion system “MINCH Robot”.

2. ferrofluid-based locomotion systems (based on the deformation of fluid magnetic material in a traveling magnetic field), see in [Zimm 07], and 3. mathematical framework, analytical methods, development of optimal kinematic gaits, and adaptive control of worm systems. Both authors belong to the group working in the latter direction. A new locomotion system developed under their guidance is the “TM-Robot II”. It is designed to have a better dynamical behavior than the TM-Robot I, the drive is by motors, pulleys and springs. Adaptive control algorithms are prepared to be implemented in order to track reference motions.

5.3 Current and future work

141

Fig. 5.9: CAD model of the TM-Robot II (left); prototype (right).

Finally, we mention a locomotion system “CREST”, developed in the Department of Biomechatronics at TU Ilmenau. Actually, it is mainly moving in a gait similar to that of a gripper-extensor system. So it could, disregarding structural details, be captured by SPIKY-theory.

5.3

Current and future work

Actually, our investigations did explicitly not aim at a model of the live earthworm. Nevertheless, one cannot flee from the idea of realizing the peristaltic movement of an artificial worm through a segmental structure. In the first instance this had been done by modeling the worm as a chain of mass points each or some equipped with a spike. (It is a bit vague how to define segment within this model). Guided by the paradigm earthworm one could prefer the spikes to be positioned in the middle of the links, i.e., at ξj = 12 (xj−1 + xj ), j = 1, . . . , n. The kinematic constraint then is ξ˙j = 12 (vj−1 + vj ) ≥ 0, and the theory develops in the well-known way. Some few calculations using this model did not show significant new results. But furthermore it could possibly be useful to consider worms with “dynamic spikes”, i.e., worms whose set of spikes, K ⊂ N, undergoes a control. K may be changed on-line or off-line as required (remind the HILLY-investigations in Chapter 2.3) or, if the spikes are at the ξj mentioned above, K may change in dependence on the actual link lengths lj (e.g., j ∈ K if lj contracted) or their time rates l˙j (e.g., j ∈ K if l˙j < 0). There are no respective investigations done yet, but see [TINK 11] for continuous models and corresponding remarks at the end of Section 5.1. All the worms treated in this book exhibit a kind of homogeneity: equal mass points, equal original lengths of the links, and equal actuators in the discrete model, a constant mass density in the continuous model. Dropping this feature would certainly overwhelm the theory with details which are unnecessary in a first introduction. Nevertheless, some things have to be changed in particular in view to applications of artificial worm systems: in considering payload transportation different masses mi are needed, and an additional force acting at the head is required for instance to describe the influence of a drilling tool or of some surgical instrument.

142

5 Conclusions

An alternative way to capture a segmental structure is to model the worm as a chain of true balloon-like segments, each segment contacting the ground by a spike. The segments, each made up by two rigid circular discs and a cylindrical skin, are fluid-filled and radially/longitudinally deformable either by variation of the internal pressure or by means of suitable orbicular and longitudinal muscle-like contractors. Some hints to respective prototypes and laboratory experiment are given in Chapter 5.1. In [Stei 03], [StAb 08] and [StAb 09] static problems of isolated segments are considered: isochoric free deformation of fluid-filled segments under axial forces and isobaric deformations of such segments within rigid or compliant tubes, respectively. Expansion, contraction, and pressure against tube are analyzed. First steps have been done to extend the theory by dropping symmetry suppositions.

Fig. 5.10: Segment filled up to a (big) working pressure, followed by isobaric deformation via axial force, qualitative (longitudinal cut, upper half, dashed: at zero pressure, bold: at working pressure).

Fig. 5.11: Segment within rigid constricted tube under different internal pressures (longitudinal cut, dashed: at zero pressure).

5.3 Current and future work

143

In connection with worm-like locomotion a future challenge is to investigate chains of such segments and their dynamics - free or within cylindrical or constricted tubes. This contains plenty of detailed questions: incompressibility (of each segment or whole ensemble of segments) or constant internal pressure, distribution of the actuators, electrorheological properties of the deformable hull, fields of application (e.g., pipe crawler, autonomous endoscope in surgery), [Meie 04], [SlBG 95], [NKIM 06]. In any case it is of primary importance to find a design that needs a minimum number of feed lines. Close to such segmental worms is the presented continuous model of spiked worms, though it has not been explored in all directions, in Section 2.2. At least Example 2.32 indicated some features of a segmental worm. The dynamics of continuous worms and dynamic drives have not yet been considered. Quite a different current approach to continuous worms (maybe, better snakes) is based on ferrofluids which can be driven by external magnetic fields - although the latter feature makes them lose the property of autonomy, [Zimm 07].

Fig. 5.12: Magnetizable elastic body (worm) in a traveling magnetic field, [ZZBP 09].

Currently, there is some gap between worm-like locomotion systems with slow drive as considered here and (medium to high frequency) vibro-driven systems which are not yet supported by a thorough theory. Some effort should be made to bring them together in future.

144

5 Conclusions

In fact, our investigations are confined to straight worms, worms which essentially live in a straight line. Even our friend HILLY, moving along the mountains (but in a fixed vertical plane!) has an orthogonal projection into a horizontal plane that moves like a straight worm. Live worms are not restricted this way. So what about worm motion along curved paths in a horizontal plane? Remind that a worm motion is understood as peristaltic, i.e., driven by internal longitudinal forces and not by internal torques bending the worm’s axis. The majority of literature on plane non-straight motion deals with snake-like systems, [Hiro 93], [OsBu 96] and [OBLM 95]. A typical such system consists of a chain of consecutively joined rigid links, each link is settled on a pair of passive wheels rolling on ground whose axes directions are controllable (steer) whereas each joint connecting two links is equipped with a controllable rotatory actuator (drive). A corresponding worm system (of finite DoF) had to be made up by finitely many mass points with spikes connected in pairs by more than one massless links of controllable lengths which realize both drive and steer (paradigm given in [VaCQ 2000], see Chapter 5.1). There is no attached theory yet. Generally, one should think over the priority of peristaltic slender motion systems which are able to follow serpentine locomotion patterns in a plane. Peristaltic worm systems certainly occupy the first place in narrow canals and animal vessels (see end of Appendix B), no doubt. This determined the orientation of all the foregoing theory. Most systems presented here are based on anisotropic friction, but they can only move in one direction because of the fixed pitch angles of the bristles. Prototypes should have mechanisms which realize a selective anisotropic friction by switching the orientation of the bristles to achieve a bidirectional locomotion system. In [Hart 11], a first technical realization is based on a special technical principle (principle of Category D in [Schu 11]): there are several support systems with bristles of opposite orientation realizing a selective anisotropic friction due to the contact with the ground. Then, once a support system is having contact (while the others are free), we get a forward motion. Otherwise the backward motion is achieved. This switching mechanism is a rotatory drive in combination with a screw gear and a transmission by a wedge gear. The lifted and lowered support systems of one segment are shown in Fig. 5.13.

Fig. 5.13: top: Uplifted support system; bottom: Lowered support system, [Hart 11]

5.3 Current and future work

145

The prototype of the artificial worm with three segments (n = 2) is presented in Fig. 5.14.

Fig. 5.14: Prototype of the artificial worm, [Hart 11]

We emphasize that some sub-functions are summarized, i.e., the periodic drive and the alignment of the bristles need only one central drive. Realizing the periodic change of position of the elements the drive is built up by a stepping motor and a cylindrical cam gearing, which realizes the kinematic gait {1, 2} → {0, 2} → {0, 1} of Fig. 2.11 (right). The rotational direction is an additional information and can be used to switch the alignment of the bristles changing the direction of locomotion. A self-toggling clutch with limited travel range (operating by different rotational directions) is the basis for the functional integration.

A

Mathematical Concepts

A map f | A → B from a set A into a set B is called injective, if every element b of the image set f (A) ⊂ B has only one original a ∈ A such that f (a) = b. Then the inverse map f −1 | f (A) → A exists, f is a one-to-one map. f is called surjective, if f maps onto (f (A) = B), bijective means injective and surjective. Let f | A → B and g | B → C, then their composition g ◦ f | A → C maps a 7→ (g ◦ f )(a) := g(f (a)). The composition f −1 ◦ f := id is the identity map on A. R3 will always be endowed with the Euclidean metric (distance): Euclidean 3-space. An open set O is characterized by the fact that with each point x ∈ O there is a (small) ball around x that is a subset of O, in a connected set C each pair of points x, y ∈ C can be linked by an arc that is a subset of C. A set is called bounded if it is a subset of a ball (‘cage’ of finite diameter!). The boundary ∂A of a set A ∈ R3 is the set of all points a (belonging to A or not) such that every ball with center a intersects both A and its complement R3 \A. A domain D ∈R3 is an open and connected set, a region B is the closure of a domain, B = D ∪ ∂D, where the boundary set ∂D will always be supposed to be a piecewise smooth surface. Analogously, this holds true for R2 , ∂D is then a piecewise smooth arc; in R1 , domain means an open interval (x, y), region means closed interval [x, y], the boundary is the set {x, y} of the two end points of the interval. Given two open sets A and B from Rn , n = 1, 2 or 3, and a map f | A → B. The map f (acting in Rn !) is called to be of class C 0 (A, B) (for short: C 0 ), if f is continuous. f is a homeomorphism, if f is injective and both f and f −1 are continuous. f is of class C k (A, B), k ∈ N, if f is continuous and has continuous partial derivatives up to order k (f is a smooth map of order k). f is a diffeomorphism of class C k , if f is injective and both f and f −1 are of class C k , k ≥ 1. f is of class C ∞ (A, B), if f has k continuous partial derivatives of any order, C ∞ = ∩∞ k=0 C . A map f from Rn into Rn , x 7→ y = f (x) := y0 + Ax, with y0 ∈ Rn and a matrix A ∈ Rn×n is called an affine map (inhomogeneous linear map). Its image f (Rn ) is all of Rn if and only if A is a non-singular matrix if and only if f is a C ∞ -diffeomorphism of Rn with constant partial derivatives. In this case f can be seen as a (affine) point transformation (point x one-to-one mapped to point y) or as a coordinate transformation (point with coordinates x gets new coordinates y). Convention in notation: (a) It is just for the sake of convenience (to make coordinate and matrix notation compatible) that any n-tuple of objects carrying lower or upper indices will be written as a line or column matrix, respectively. As a rule, elements of a vector n-tuple will be marked by lower indices, the respective coordinates by upper ones.

148

Mathematical Concepts

(b) There holds a summation convention for indices of different position: ai bi := a1 b1 + . . . + an bn = (a1 , . . . , an ) · (b1 , . . . , bn )T . (c) Partial derivatives are denoted by comma:

∂f ∂xi

=: f,xi =: f,i .

A homeomorphism h | R3 → R3 is a congruent transformation if and only if it is an affine transformation, h(x) = h0 + Ax, h0 ∈ R3 , A ∈ R3×3 , that takes any orthonormal vector triplet (e1 , e2 , e3 ) to the triplet (E1 , E2 , E3 ) := (e1 , e2 , e3 ) · A (matrix product), which is orthonormal as well. This is true if and only if A is an orthonormal matrix: A−1 = AT . The transformation h is a direct congruent transformation if and only if det(A) = +1. The orthonormality of (E1 , Ep 2 , E3 ) is expressed by the scalar products (Ei | Ei ) = 1, (Euclidean norm kEi k := (Ei |Ei ) = 1), and (Ei | Ej ) = 0, i 6= j, i, j = 1, 2, 3 (orthogonality). These are in fact six equations for the nine elements of A. Hence an orthonormal matrix depends on 3 parameters (for instance Euler’s angles). Under a direct congruent transformation h the cartesian coordinate system (frame at origin) F0 = {o; (e1 , e2 , e3 )} is mapped to h(F0 ) =: F ={h0 ; (E1 , E2 , E3 )} which is cartesian as well (o null vector, h0 = h(o) = hi0 ei ). During a motion t 7→ Ft = {h0 (t); (E1 (t), E2 (t), E3 (t))} of an orthonormal frame there exists a vector ω(t) = ω i (t)Ei (t) such that d Ei = ω × Ei , i = 1, 2, 3. dt In explicit form this reads d dt E1 d dt E2 d dt E3

= −ω 3 E2 +ω 2 E3 , = ω 3 E1 −ω 1 E3 , 2 1 = −ω E1 +ω E2 .

d i So the rate of change of Ft is captured by the vector pair (v0 , ω), where v0 = ( dt h0 )ei is the translational velocity and ω the angular velocity.

Curves in R2 A smooth map of a real interval into R2 is called a regular oriented plane curve. Using the position vector x = xex + yey in a cartesian coordinate system of R2 the curve is represented in the form t 7→ x(t) = x(t) ex + y(t) ey ,

x(·) ∈ C k , k ≥ 2 .

Here t is any real parameter, running through an interval [t0 , t1 ], say. The image c = x([t0 , t1 ]) is called the arc of the curve. The introduction of the arc length s :=

Mathematical Concepts

149

Rt

d x(t)kdt is a feasible parameter transformation, s 7→ x(s) is the normal k dt representation of the curve. If t means time, then t 7→ s(t) describes a motion along the curve and c might be called the path of the moving point. t0

e1 (s) :=



d x(s) = x0 (s) ex + y 0 (s) ey = cos ϕ(s) ex + sin ϕ(s) ey ds

is the unit tangent vector,



e2 (s) := −y 0 (s) ex + x0 (s) ey = − sin ϕ(s) ex + cos ϕ(s) ey ,

emerging from e1 by a π2 −rotation, is the unit normal vector. Clearly, ϕ(s) is the angle from ex to the tangent vector e1 (s). The pair (e1 , e2 ) is called a moving frame d ϕ is called curvature, if κ(s) > 0 or or Frenet’s frame. The rate of change κ := ds κ < 0 then the curve is left-handed (tends to the left-hand-side of the tangent) or right-handed at s, respectively, a point s where κ(s) = 0 is a flat point of the curve. The moving frame solves Frenet’s equations e01 = κ e2 , e02 = −κ e1 . Given s 7→ κ(s), these differential equations together with x0 = e1 determine the curve uniquely up to congruent transformation, i.e., up to arbitrary x0 , e10 at some s0 . Curves in R3 A regular oriented curve in R3 can be given in normal representation s 7→ x(s) = x(s) ex + y(s) ey + z(s) ez ,

x(·) ∈ C k (R; R3 ) , k ≥ 2 .

Again, the unit tangent vector is defined as e1 := x0 = x0 ex + y 0 ey + z 0 ez . Its unit norm entails (e1 | e1 ) = 1 → (e1 | e01 ) = 0 → e01 ⊥ e1 . If e01 6= 0, i.e., if the arc is not a straight line, the principal normal is defined as the unit vector e2 := e01 ke01 k−1 , and further the binormal as e3 := e1 × e2 . Now the orthonormal triplet (e1 , e2 , e3 ) represents the moving frame of the curve, which is governed by Frenet’s equations e01 = κ e2 , e02 = −κ e1 +τ e3 , e03 = −τ e2 . Here the curvature is κ := ke01 k (note κ ≥ 0 for curves in 3−space!). τ is called the torsion of the curve. If κ 6= 0 it calculates as τ = (e1 | e01 × e001 )/κ 2 . Given κ and τ as smooth functions of s then the curve is fixed by Frenet’s equations together with x0 = e1 up to congruent transformation. τ = 0 characterizes a plane curve, κ = 0 a straight d line. A comparison with the motion equations dt Ei = ω × Ei , i = 1, 2, 3, of a general 2 orthonormal vector triplet shows ω = 0 in Frenet’s case, hence Frenet’s equations can be given the short form e0i = Ω × ei using Darboux’s vector Ω := τ e1 + κe3 .

B

Mechanical Concepts

Inertia objects Inertial properties of a body are described by its mass - a non-negative time-independent physical quantity attached to every part of the body.16 Consider a body (of dimension 3, 2, or 1) in its reference configuration B0 . Then there exists a density function ξ 7−→ ρ(ξ) ≥ 0 (a scalar field on B0 ) such that the mass of an infinitesimal element of B0 around ξ is dm(ξ) := ρ(ξ)dξ - where, for short, dξ stands for dξ 1 dξ 2 dξ 3 , dξ 1 dξ 2 , or dξ 1 , respectively. The total mass of the body in any configuration B is Z Z dm(ξ) = ρ(ξ) dξ . M= B0

B0

In case of a mass point at ξ ∈ R3 let simply dm(ξ) = m > 0 and M = m. SK For a system of bodies B0 = k=1 B0k - possibly of different dimensions 3 , 2 , 1, 0 - the total mass is K K Z X X dm(ξ) =: B0 dm(ξ) , M= Mk =

S

k=1

k=1B 0k

where the boldface integral sign abbreviates the respective sum of integrals of different dimensions. In this sense the symbol is used in all that follows.

S

The center of mass is the point with coordinates ξ ∗ := x∗ :=

1 M 1 M

S S

B0 ξ

dm(ξ) in reference configuration B0 ,

B0 h(ξ)

dm(ξ) in configuration B = h(B0 ) .

Mind that possibly ξ ∗ ∈ / B0 , x∗ ∈ / B (think of a donut- or horseshoe-shaped body). During a motion ξ 7−→ h(ξ, t) the position vectors in a space-fixed coordinate system {O ; (e1 , e2 , e3 )} attached to B0 (see following figure) are ξ := ξ i ei , x(ξ, t) = h(ξ, t) ,

h(ξ, t) := hi (ξ, t)ei , x∗ (t) =

1 M

S

B0 x(ξ, t)

dm(ξ) .

16 In [Noll 57] mass is introduced by means of measure theory. In order to avoid this setting in the following the more familiar concept of a physical field shall be used.

152

Mechanical Concepts

Since ei , B0 , dm(ξ) do not depend on t, the corresponding velocities are v(ξ, t) = x,t (ξ, t) = hi ,t (ξ, t)ei ,

v∗ (t) = x˙ ∗ (t) =

1 M

S

B0 v(ξ, t)

dm(ξ) .

Either of the following two vectors represents a compound of inertia properties and motion of a body system. The linear momentum of a body system is given by P(t) :=

S

B0 v(ξ, t)

dm(ξ) = M v∗ (t) ,

and the angular momentum about a space-fixed point O is LO (t) :=

S

B0 x(ξ, t)

× v(ξ, t) dm(ξ) .

Letting x = x∗ (t) + z(ξ, t), where z is the position vector relative to the center of mass and v = x,t = v∗ + w, where w(ξ, t) = z,t (ξ, t) is the velocity relative to the center of mass there follows the useful representation LO (t) = x∗ (t) × P(t) + L∗ (t) .

S

L∗ (t) := B0 z(ξ, t) × w(ξ, t) dm(ξ), the angular momentum about the center of mass, is called the spin of the body system. Principles of motion All the considerations in this paper are done in a Newtonian setting: (i) space and time are absolute in the sense of forming a “stage” where physical events take place, (ii) motions of a body B1 start or change only if the body is influenced by a force exerted by another body B2 (principle of inertia), (iii) B1 then exerts a force onto B2 that is equal in magnitude and of opposite direction (principle of reaction)17 . Forces are classified under two points of view [Hame 49]. 1) If B1 and B2 belong to the same system of bodies the the mutually exerted forces are called internal forces (relative to the system). On the other hand, if B1 is part of the system while B2 belongs to the system’s environment then the force exerted to B1 is an external force. 17 If moreover the two forces act along a common straight line then they are said to obey the “full principle of reaction” [Hame 49].

Mechanical Concepts

153

2) A force law (e.g., Newton’s gravitational law, Stokes’ viscous friction law) connects the force vector with the state of the system (position, shape, and their rates of change) and certain physical parameters (which can be measured and characterize the physical circumstances responsible for influencing the motion). If a force does depend on any physically measurable quantity (e.g., a relative velocity, an elasticity modulus, a surface roughness) then it is called a physically given or impressed force18 . If, on the other hand, the occurrence of a force solely originates in the presence of a constraint (geometric or kinematic restriction of the motion, like rolling without sliding on a rigid surface) then the force is called a constraint force or reaction to the constraint. (There is nothing like a force law given in advance, the force solely serves for to ensure the non-violation of the constraint during motion.) Summarizing, there result four types of forces: internal impressed force, internal constraint force, external impressed force, external constraint force. Remark B.1 Internal forces come into consideration by dissecting a system or a single body. The formerly hidden forces then appear as opposite forces distributed on both sides of the cut manifold (cut forces). They might be impressed forces (as in an elastic body) or reactions (as responding to “rigidity”, or “fixed distance”, or “no sliding”).  Now let B0 be the reference configuration of a body, a system of bodies, or of any dissected part of it. In the actual configuration Bt during a motion ξ 7→ x = h(ξ, t) let - symbolically, and dropping time and any physical ingredients for short - dF(ξ) be the force actually exerted to an infinitesimal neighborhood of ξ ∈ B0 (dF might be represented by means of a vector-valued force density as it was analogously done above with mass). The resultant force applied to the body in configuration Bt is F :=

S

B0 dF(ξ) ,

and the resultant moment about a point O ∈ R3 is defined as MO :=

S

B0 x(ξ, t)

× dF(ξ) .

Using again x = x∗ + z the moment splits into two terms MO := x∗ × F + M∗ , the moment of the resultant force F (to be seen as acting at the center of mass) and M∗ as the resultant moment about the center of mass. 18 “Eingepr¨ agte

Kraft” in [Hame 49].

154

Mechanical Concepts

The Newton-Euler setting of classical mechanics is based on the following Axiom B.2 Any motion of a system of bodies is subject to (A1) Principle of linear momentum: For every part B0 of the system the resultant force is equal to the rate of change of the linear momentum, d P = F, dt (A2) Principle of angular momentum: Let O be a fixed point of R3 , then for every part B0 of the system the resultant moment about O is equal to the rate of change of the angular momentum about O, d LO = MO . dt  Corollary B.3 (a) (A2) remains valid if the resting point O is replaced by the (moving) center of d ∗ mass of the part B0 : dt L = M∗ .

Mechanical Concepts

155

(b) Of course, (A1) and (A2) hold true for the system as a whole. (c) According to the principle of reaction the internal forces mutually compensate in forming F, so F is in fact the resultant external (with respect to B0 !) force. If the internal forces satisfy the full principle of reaction then MO is the resultant moment of the external forces. (d) If the external forces have zero resultant force and moment (special case: no external forces acting at all) the (c) entails conservation of linear momentum and spin: P(t) = const, L∗ (t) = const. 

The rigid body Consider a single rigid body (a) in its reference configuration B0 with attached bodyfixed coordinate system {O ; (e1 , e2 , e3 )} and (b) in configuration Bt = h(B0 , t) during motion with actual body-fixed coordinate system {Ot ; (E1 , E2 , E3 )}, see Figure B.1.

e2 0 x

0t

x0 e1 x

B0

E1

E2 r

x

Bt

Fig. B.1: Rigid body, coordinate systems.

Then x(ξ, t) = x0 (t) + r(ξ, t) , x0 (t) = hi (0, t)ei , r(ξ, t) = ξ i Ei (t) , and hence ˙ i (t) , v0 (t) = hi ,t (0, t)ei . v(ξ, t) = x,t (ξ, t) = v0 (t) + ξ i E ˙ i = ω × Ei with the angular velocity ω = ω j Ej (see Appendix A) yields Euler’s E velocity formula v(ξ, t) = v0 (t) + ω(t)×r(ξ, t) ,

156

Mechanical Concepts

a splitting of the velocity vectors in translational (independent of ξ) and rotational part. In particular the velocity of the center of mass is v∗ (t) = v0 (t) + ω(t)×r(ξ ∗ , t) . Euler’s formula yields a representation of the spin by means of the angular velocity in the following way: L∗ (t) =

S

B0 z(ξ, t)

× w(ξ, t) dm(ξ)

with z(ξ, t) = ζ i (ξ)Ei (t) , ζ i = ξ i − ξ ∗i , w(ξ, t) = ω(t)×z(ξ, t) . So L∗ becomes a linear function of ω. Mass points A rigid body motion is a pure translation if and only if ω = 0. Then L∗ = 0, and the Principle (A2) entails 0 = M∗ as a necessary condition: zero resultant moment about x∗ , i.e., resultant force acting at x∗ . Vice versa, if this condition is fulfilled and ω(t0 ) = 0 at some instant t0 , then (A2) yields ω(t) = 0 for all t. The motion is then governed by (A1), M v˙ ∗ = F , i.e., the body moves as if all its mass was concentrated at the point x∗ with F acting upon it. The principle of angular momentum with respect to any fixed point O appears then only as a consequence. So the concept of a mass point is born out of continuum mechanics. Strictly speaking, this concept is justified only within this scenario (no rotation). Nevertheless, it is commonly in preferred use if the rotatory inertia of a rigid body is guessed to be of negligible effect in a given context. Mass point on a fixed space-curve Let s 7→ x(s) = x(s) ex + y(s) ey + z(s) ez be a given curve in R3 as described in Appendix A. Consider a mass point moving along this curve under the action of an impressed force F. Then its actual position is given by some D2 −function t 7→ s(t), and the force law is of the general form F(t, s, s). ˙ The geometric constraint “mass point on fixed curve” causes the occurrence of a reaction force R⊥ orthogonal to the curve. If moreover a kinematical constraint is present (like stiction or influence of spikes) then another, tangential, reaction rq has to be considered. The motion is governed by m x ¨ = F + rq + R⊥ . The moving frame of the curve, {x(s); e1 (s), e2 (s), e3 (s)}, serves for representing vectors at x(s), e.g., F = F i ei . Using

Mechanical Concepts

157

the Frenet equations and noting

d dt x

=

d ds x

· s˙ = e1 (s) · s˙ the equations of motion are

m s¨ = F 1 (t, s, s) ˙ + r1 , m κ(s) s˙ 2 = F 2 (t, s, s) ˙ + R2 , 0 = F 3 (t, s, s) ˙ + R3 . In simple cases the first equation governs the motion completely, whereas the others determine the reaction R⊥ . Chain of mass points on a fixed space-curve The following is to provide tools for the investigation of, e.g., worms within a curved tube. Consider n + 1 mass points numbered 0, . . . , n arranged as a chain along a fixed space-curve, see Figure B.2.

m i+1

-m i mi

y -m i+1 e1

s

e2

x z

e3 Fig. B.2: Mass points on space-curve.

Each mass point achieves a motion t 7→ si (t), i = 0, . . . , n, under the action of impressed forces Fi and reactions ri , Ri as above while consecutive mass points undergo a coupling that is described by forces (impressed or reactive cut forces) µi . Based on the assumption that the coupling forces acting upon a mass point are tangential to the curve (thereby excluding, e.g., coupling by taut strings or rigid rods) the vector equations of motion mi x ¨i = Fi + µi − µi+1 + rqi + R⊥ i ,

i = 0, . . . , n ,

yield (simply taking scalar products with the vectors of the corresponding Frenet frames

158

Mechanical Concepts

(e1 , e2 , e3 )(si )) mi s¨i = Fi1 (t, s, s) ˙ + µi − µi+1 + ri1 , mi κ(si ) s˙ 2i = Fi2 (t, s, s) ˙ + Ri2 , 0=

Fi3 (t, s, s) ˙

+

Ri3

i = 0, . . . , n .

,

(Here s stands for s0 , . . . , sn since F might depend on all si .) The above considerations refer to mass points. So, if one thinks of an artificial worm whose components are of considerable dimensions then it might be inevitable to take rotatory inertia into account. The principle of angular momentum contributes to the theory.

C

Control Theory Concepts

In this appendix we present only some facts from control theory and system theory which may be necessary for the reader to understand the control strategies in Chapter 4. There are numerous books on this subject, for example [NvdS 90], [Zabc 92], [Isid 95] and [Khal 02]. Control theory deals with processes which are exposed to influences (in particular external ones, partly disturbing, partly prescribed in order to gain desired effects). The following definition gives a sufficiently general mathematical model of a control system19 . Definition C.1 A control system is a pair (f , g) of sufficiently smooth functions f |R1+n+m → Rn and g|R1+n → Rm which describe a process t 7→ x(t) (t: time, x(t): state) through the following equations:  x˙ = f (t, x, u) , x(0) = x0 ∈ Rn , Σ : (C.1) y = g(t, x) . The dynamics of the process are governed by the ordinary differential equation (C.1.a) that is entered by the m-dimensional input variable u. The m−dimensional output y comprises (physically measurable) variables of particular interest for analyzing the process. In most cases the only information one can get about the process is carried by t 7→ y(t) = g(t, x(t)). (There are theories admitting t 7→ y(t) = g(t, x(t), u(t)).)  The input u describes effects or influences the system is exposed to. Principally, there are two different types of input variables: completely known (and externally prescribed as a rule) and not completely known (both exogenous and endogenous) variables (disturbances). In contrast to [KnKw 86] we shall model either type of influence on its own, i.e., the input u is understood as a completely known intervention to the system and does not contain any disturbance components. If there is need for to deal with disturbances then these shall be considered as an extra compound of influences. In particular, control theory deals with the synthesis of control inputs for dynamical systems. The main point is, to what extent a system can be affected in such a way that its behavior is (or comes close to) a desired one. So the task is to design appropriate control 19 We confine the definition to systems with continuous, real-valued variables. Discrete systems are out of the scope of our investigations.

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Control Theory Concepts

strategies such that the output (of the system with input following this strategy) has favored properties. There are two alternative controls (or control strategies): open-loop control and closed-loop control. • An open-loop control is an a priori given function u(·) such that the initial value problem (C.1.a) has a well-defined solution with desired properties of the output y(·). In our context of worm systems a simple open-loop control is a given kinematic gait function t → u(t) := (l1 (t) , . . . , ln (t)). This example puts into evidence that an open-loop control realizes an open action flow (off-line control), it does not admit any possibility to react on sudden disturbances like changes in the system’s environment.

Fig. C.1: Open-loop system.

• A closed-loop control is defined by a map h|(t, x) → Rm as u(t) := h(t, x(t)) such that the initial value problem (C.1.a) has a well-defined solution and the system output exhibits desired properties. The dependence on x means that the actual state x(t) is led back to the controller (a device that “produces” the system input u(t)): h is called state feedback.

Fig. C.2: Closed-loop system, state feedback.

Practically, only physically measurable quantities can be fed back to the controller. Thus, output feedback h|(t, y) → Rm as u(t) := h(t, y(t)) is the most important and powerful control strategy. Since now the controller gets knowledge of the actual output (or even of the full actual state) it produces in fact an on-line control and, thus, may react to sudden events.

Control Theory Concepts

161

Fig. C.3: Closed-loop system, output feedback.

Example C.2 A plausible control objective in our worm context is to adjust (or to approximate) a gait (either kinematic or dynamic) such that the worm system exhibits a desired motion pattern; this could be realized off-line (a priori gait construction, see Subsections 2.1.3.4 and 2.1.5.2) or online (“automatic gear shift”, see Subsection 2.3.1.1, or compensation of actuator failures).  This control objective leads us to the notion of tracking control: given a favored reference output yref (·) find a control u(·) that keeps the error e(t) := y(t) − yref (t) as small as possible. Several types of tracking are distinguished: • asymptotic tracking: lim e(t) = 0; t→∞

• λ-tracking (or approximate tracking): for any prescribed λ > 0 : lim sup ke(t)k < λ ;

(C.2)

t→∞

this control objective tolerates a tracking error of size λ but allows the design of simple control strategies; • tracking with a pre-specified transient behavior or objective achievement in finite time. Note that, considering the actual problem type an appropriate norm can be chosen. Let e(t) = (e1 (t), . . . , en (t))T ∈ Rn , then preferred norms are

ke(t)k2 :=

n X

2

e(t)

! 21

: Euclidean norm,

(C.3)

ke(t)k∞ := max{|ei (t)| , i = 1, . . . , n} : maximum norm.

(C.4)

i=1

The λ-tracking control objective is to determine an online control strategy that achieves tracking of a given reference signal in the following sense:

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Control Theory Concepts

(i) every solution of the closed-loop system is defined and bounded for t ≥ 0, and (ii) the output y(·) tracks yref (·) with asymptotic accuracy λ > 0 in the sense that t→∞

max{0, ky(t) − yref (t)k − λ} → 0 .

(C.5)

Visually, this means that the output y(t) tends to a tube of radius λ around yref (t), see Fig. C.4.

Fig. C.4: Reference signal and λ-tube.

In principle, the construction of a suitable feedback law has to obey the following scheme: 1: Let e = y − yref be the error, i.e., the deviation of the output y from yref . 2: Comparison: (a) if ke(t)k < λ then there is no need of intervention but, (b) if ke(t)k ≥ λ then the dynamics are to be changed such that ke(s)k decreases for s ≥ t. 3: Design a feedback law t 7→ u(t) := Ft [e]

(C.6)

that depends on the actual time t and the actual error e(t) (or on the error history {e(s), s ∈ [0, t]}). (Mind that a notation Ft [·] means a functional dependence on Rt e, for example Ft [e] = a0 e(t) + a1 e(t) ˙ + a2 0 e(s)ds, whereas Ft (e) would indicate a pure function like Ft (e) = a0 e(t). To find out a working (i.e., achieving λ-tracking) and handy Ft [·] a tricky intuition or a background theory is needed. We prepare this set of problems by some examples and explanations of related basic concepts. The examples are restricted to linear systems and asymptotic tracking because of their better transparency.

Control Theory Concepts

163

Example C.3 (1) Consider the 1−dimensional control system x˙ = x + u ,

x(0) = x0 ,

y = x.

The aim is to asymptotically track the reference signal yref = 0 by means of a feedback linear in y (P-feedback): u(t) := −k e(t) = −k x(t). The closed-loop system then is x˙ = x − k x = −(k − 1) x ,

y = x,

with solution x(t) = x0 e−(k−1) t tending to zero for any initial value x0 , i.e., approximating yref asymptotically, if and only if, k > 1. Common formulation: The equilibrium point x = 0 of the uncontrolled system has become globally asymptotically stabilized by means of this feedback control. (2) Remark: Since the controller in the foregoing example achieves asymptotic tracking it also realizes λ-tracking: there is some t∗ < ∞ such that kx(t)k < λ for all t > t∗ . (3) Consider the second order control system x ¨ = a x˙ + b x + u ,

y=x

with initial data x(0) = x0 ∈ R, x(0) ˙ = v0 ∈ R and fixed arbitrary real a and b. We choose a PD-feedback (linear with respect to e and e) ˙ using yref = 0, u = −k0 e − k1 e˙ with k0 > a , k1 > b . The closed-loop system then is x ¨ = −(k1 − a) x˙ − (k0 − b) x having the characteristic equation (via ansatz x(t) = eµ t ) µ2 + (k1 − a) µ + (k0 − b) = 0 with solution µ1,2

k1 − a =− ± 2

r

(

k1 − a 2 ) − (k0 − b) . 2

k0 > a and k1 > b yield Re(µ1,2 ) < 0, hence with every initial data the solution x(t) tends to zero, i.e., yref is asymptotically tracked. If moreover ( k12−a )2 > k0 − b, then the tracking is even (monotonically) exponential. (4) Remark: One may check that the above feedback law with k1 = 0 (pure Pfeedback) will not do the job for arbitrary coefficients a, b. 

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Control Theory Concepts

It might be evident that the following questions are of some significance for a control design. (i) How strong is the influence of u on the output y? (ii) Could the control objective be achieved if the feedback is “big enough”? (iii) Has the system some hidden uncontrollable malignant subdynamics that could destroy any control effect? In a sense the first item is described by a natural number called the relative degree of the control system. This important concept is defined in the following constructive way. Take the control system Σ and inspect its output y = g(t, x). This output does not directly depend on the input u, rather y is influenced by u only via x and the system dynamics. We wonder about the intensity of this indirect influence. So we look how the rate of change of y depends on u. Now y˙ = g,t (t, x)+g,x (t, x)x˙ = g,t (t, x)+g,x (t, x)f (t, x, u) may indeed explicitly depend on u, then the relative degree is defined to be equal to 1. Else (if (g,x f ),u ≡ 0) we differentiate again, define the relative degree to be 2 if and only if y¨ explicitly depends on u, and, by iteration, we come to the following rule: Differentiate the output with respect to time until you see the control. If u shows up in the r-th derivative for the first time then r is called the relative degree of the control system. %

d So relative degree r > 0 means that r is the lowest order of output derivatives dt % y, % ∈ N, which u directly acts upon whereas all lower derivatives are influenced by u only indirectly. Roughly speaking, the higher the degree r the weaker the influence of u on y and the more involved the feedback design will be.

Remark C.4 If the control system models the dynamics of a mechanical device then the differential equations are of the principal structure (2n-dimensional if n is the degree of freedom) x˙ = v ,

v˙ = f0 (t, x, v) + f1 (t, x, v)u ,

i.e., f1 u models a force that, naturally, enters the principle of linear momentum. Thus, any such system has relative degree 2 if the output does not depend on the velocity v, otherwise the degree is 1.  Example C.5 The control system (1) of Example C.3 shows y˙ = x˙ = x + u, therefore its relative degree is r = 1. Write the control system (3) of Example C.3 as a system of first order equations, putting x =: x1 , x˙ =: x2 , x˙ 1 = x2 , x˙ 2 = b x1 + a x2 + u , y = x1 .

(C.7)

Control Theory Concepts

165

Now y˙ = x2 , y¨ = x˙ 2 = b x1 + a x2 + u, relative degree is r = 2. Clearly, if the system was changed with a new output y = x1 + x2 , say, then it would get relative degree r = 1.  Supplement: Consider a linear control system with x(t) ∈ Rn , u(t) , y(t) ∈ Rm ,  x˙ = A x + B u , Σ: (C.8) y = Cx where A, B, C are matrices of appropriate dimensions. Then it is simple matter to see Σ has relative degree r ⇔ CAi B = 0, i = 0, . . . , r − 2,

CAr−1 B 6= 0 .

If the last matrix is even non-singular, det(CAr−1 B) 6= 0, then r is called strict relative degree. In this case the equation y (r) = . . . + CAr−1 Bu can be solved for u, so u appears as a linear function of x and y (r) . Check that the relative degrees in Example C.3 are strict. Unfortunately, there is no general concept to find the right feedback law to make a linear system track the zero output yref = 0. But the following laws have been approved in the literature: • strict relative degree 1: u(t) = Ft [e] = Ft (e) = k e(t), • strict relative degree 2: u(t) = Ft [e] = Ft (e, e) ˙ = k0 e(t) + k1 e(t). ˙ The real k 0 s in the above feedback laws are called gain parameters. In Example C.3 they had to be chosen positive and big enough in order to solve the tracking task successfully. Remark C.6 Whether the gain parameters have to be positive or negative depends on the spectrum of the matrix CAr−1 B.  This hints to the second item: under certain conditions a linear system is stabilizable (yref = 0 asymptotically tracked) by the feedback u = −ky for any k > k ∗ ≥ 0. This property is called high-gain stabilizability. Therefore the feedback design is simple as soon as k ∗ is known; in general it is not. Example C.7 Consider the linear control system Σ in the above supplement. If (i) every eigenvalue of CB has positive real part (implies CB 6= 0: relative degree r = 1), and

166

Control Theory Concepts 

sIn − A B C 0 minimum-phase condition)

(ii) D(A, B, C) := det



6= 0 for every s with Re(s) ≥ 0 (the so-called

then Σ has the high-gain property.



The next example sheds a light also on the third question above. Example C.8 Consider the system x˙ 1 = a x1 + x2 + u , x˙ 2 = α x2 , y = x1

(C.9)

    1 a1 , , B = where a ∈ R is fixed and α = 1 or α = −1. Here A = 0 0 α
C = 1 0 , and it is easy to check CB = 1, D(A, B, C) = (s − a)(s − α), so the high-gain property is ensured if and only if a < 0 and α = −1. Check: Write the system as y˙ = a y + x2 + u ,

x˙ 2 = α x2 .

The second differential equation represents an uncontrollable subdynamics. We try to track yref = 0 by means of u = −k y. Then the closed-loop dynamics are y˙ = (a − k) y + x2 ,

x˙ 2 = α x2 .

The subdynamics gives x2 (t) = x20 eα t , x2 is bounded for t → ∞ and tending to zero if and only if α = −1 (α < 0 would do as well), and only then y tends to zero if we choose k > a (solve the differential equations by standard methods). So, with negative a, we have k ∗ := 0. (Besides we see that the condition in Example C.7 is only a sufficient one since any k > a := k ∗ would do the job for non-negative a.)  In the foregoing simple examples lower bounds or even a lower limit k ∗ for working gain parameters could be found - based on complete knowledge of the system parameters. This may be difficult for systems of large dimension (although a theory guarantees the existence of k ∗ ), it is impossible in case of incompletely known or randomly changing system parameters (uncertain systems). Then adaptive control shows a way out of this awkward situation: let the gain parameter k (a constant until now) increase in time until it takes a sufficiently big value (> k ∗ ). Adaptive means that this increase takes place simultaneously with an observation of the error while the latter then determines the time rate of the increasing k(t). For λ-tracking control the adaptation law has to be designed in such a way that • k(t) increases as long as ke(t)k ≥ λ and

Control Theory Concepts

167

• k(t) stays constant as long as ke(t)k < λ. The design of an adaptor is independent of the system’s relative degree. Literature presents various adaptation laws among them the classical one ˙ k(t) = (max{k e(t) k −λ, 0})2 .

(C.10)

˙ As this entails k(t) ≥ 0 the gain parameter shows monotonic (possibly unbounded) increase that might be not desirable in practice. Modifications avoiding this drawback are introduced in Section 4.1.2.

D

Notes on Simulation Parameters

Recall a general observation from the preceding chapters. As soon as we are dealing with the dynamics of worms - maybe as the central equation or the single mass point equations of motion of a spiked worm or a worm with Coulomb forces, with or without adaptive control - we encounter a situation as follows. The differential equations to be investigated are of the principal form

x˙ = v , v˙ = f (x, v, t) + λ ,

1 λ = − 1 − sign(v) 1 − sign(f ) f . 2 So we have inevitably to cope with the signum-function particularly in computer simulations. Now signum is a handy object in theory but it is well known that the discontinuity of signum eventually makes it very unpleasant on the computer. Most DE-solvers in MAPLE react unfriendly to jumps produced by signum in the right hand side of a DE, in particular if terms of the form signum(x(t)), x : dependent variable, appear. Essentially the same situation appears if λ represents Coulomb forces. Then λ is made up by Heaviside functions h(a, b, ·) with argument v or
f , where h is composed of signum-functions, h(a, b, x) = 21 sign(x − a) + sign(b − x) .

In order to tame respective DEs in computing one had to work either with specially tailored software or with hopefully suitable approximations of the signum function. Considering the aim of this book we decided for the second way. Two approximations showed up preferable:

sign1 : x 7→ tanh(Ax) ,  1 , x ≤ −A  −1 , 1 sign2 : x 7→ A x , − A ≤ x ≤  1 1, A ≤ x with some big positive A, say, A = 103 . . . 106 .

1 A

, ,

170

Notes on Simulation Parameters

Fig. D.1: Sketch of signum and its approximations with A = 5 (left) and A = 10 (right).

Now the choice sign1 or sign2 is like choosing pest or cholera: (a) sign1 is a C ∞ -function, thus well-liked by any routine (if A is not too big), but the values ±1 do not belong to its range (sign1(x) = 0.99 for x ≈ 2.64665); (b) on the other hand, sign2 has values ±1 on the side intervals, it is a D1 -function (continuous and piecewise continuously differentiable), but by definition it appears in MAPLE as a piecewise-function, not always well accepted by dsolve procedures. Both functions have slope A at the origin. These pros and cons of approximations equally show up in the context of Heaviside functions:

Fig. D.2: Sketch of h(−1, 1, x) and approximations with A = 5 and A = 10: by means of sign1 (left), sign2 (right).

This could be of significant influence on motions under Coulomb forces which are described by means of approximate h-functions. The stiction term −f h(−F0− , F0+ , f ) h(−ε, ε, v) has lost its abruptness of activity at the respective bounds and - possibly falsifying its proper effect - it acts also for v 6= 0. (In this context mainly A = 104 , A ε = 10 . . . 100 were used.) To gain some impression of various phenomena we consider the following simple DE that describes a motion under the influence of the force f (t) = t and an isotropic Coulomb

Notes on Simulation Parameters

171

sliding friction with F ± = 1, v˙ = t − sign(v) ,

v(0) = v0 ,

(D.1)

where sign stands for signum or for one of its approximations. These DEs are simple enough as to find their solutions analytically. We drop these formulas here and present only the resulting phase portraits (solution graphs for v0 close to 0). (i) sign := signum: The corresponding figure (Fig. D.3) shows that the solutions for v0 ∈ (−6, 2) terminate at the t-axis whereas v0 > 2 admits parabolas, and v0 < −6 yields parabolas which can be continued to the upper half plane with a break (jump of derivative); moreover, note that the line elements at points (t, 0) have slope t (not drawn).

Fig. D.3: Phase portrait (D.1) with sign = signum.

By the figure it may become evident that in simulation with v0 = 0 the computer “does not know what to do” and yields a solution graph which only macroscopically looks like v(t) = 0 on a first t-interval but after zooming shows a chaotic jittering of small amplitude (Fig. D.4).

Fig. D.4: Solution for v0 = 0 (by MAPLE dsolve), right: zoomed.

172

Notes on Simulation Parameters

(ii) sign := sign1: The following figures (Fig. D.5) sketch the phase portrait using A = 10 (for demonstration), left, and A = 104 , right. The latter is close to that from Fig. D.3.

Fig. D.5: Phase portrait (D.1) with sign = sign1, A = 10 (left), A = 104 (right).

The next figures show the corresponding solution graph with A = 104 for v0 = 0:

Fig. D.6: Solution for v0 = 0 (by MAPLE dsolve), right: zoomed.

(iii) sign := sign2: The following figures (Fig. D.7) sketch the phase portrait using A = 10 (for demonstration), left, and A = 104 , right. Again, the latter is close to that from Fig. D.3.

Notes on Simulation Parameters

173

Fig. D.7: Phase portrait (D.1) with sign = sign2, A = 10 (left), A = 104 (right).

The next figures show the corresponding solution graph with A = 104 for v0 = 0:

Fig. D.8: Solution for v0 = 0 (by MAPLE dsolve), right: zoomed.

Comparing the above simulation results the sliding friction roughly seems to mimic a stiction effect. But in fact it does not, rather it admits non-zero velocities, in its exact version on the computer and also theoretically in either of the approximations. Similarly do the approximate h-functions, part of the stiction law in form h(−ε, ε, v).

E

Some Program Source Codes

In this section we show three program source codes concerning worm motion. They are of a restricted size in order just to give some insight into simulations the results of which are presented in foregoing chapters. Mind that these program codes are written by non-professionals in MAPLE and MATLAB. They should serve as a first approach in handling worm systems numerically. Obviously, these programs offer a personal touch of the authors and do not represent codes of high-end. Any proposals for improvements are welcome! The first program has Sections 2.1.3.1 and 2.1.3.2 as its context. It runs in MAPLEr 12 and analyzes the motion of a spiked inch-worm, i.e., SPIKY with n = 1. First, the motion comes out of the kinematical theory and it is, secondly, dynamically checked by means of the central equation and eventually superimposed by a skidding-forwardmotion. One may experiment by using different data ε, k0 , Γ for various gaits l : t 7→ l(t). The second program is for MAPLEr 15 and concerns the gait construction algorithm in Section 2.1.3.4. It is confined to the principal non-smooth motion by doing without sine-representations. One may experiment by displaying the dimensions n and a and various mode sequences t 7→ A(t). The third program is written in MATLABr R2011a and it reproduces the simulations done in Section 4.1.2. Experiments are possible by choosing various spring stiffnesses, damping coefficients, adaptors, etc. Mind that the softwares eventually are not willing to accept the discontinuous signumand Heaviside-functions. To overcome such unfriendly gestures some smooth approximations of these functions have been used (see Appendix D).

Kinematical SPIKY, n = 1 > restart:with(plots):

## auxiliaries: > _Envsignum0:=0: A:=1000: # choice sgn:=z->tanh(A*z): # smooth signum pos:=z->(z+abs(z))/2:

176

Some Program Source Codes

## choose a gait: > l0:=1:eps:=.4:OM:=2*Pi: # choices T:=2*Pi/OM: # period l:=unapply(l0*(1-eps*sin(OM*t)),t): # choice dl:=D(l):ddl:=D(dl):

## kinematics: > v:=pos@dl:V[0]:=v:V[1]:=v-dl: kin:=dsolve({diff(x(t),t)=v(t),x(0)=0},{x(t)}, numeric,method=rosenbrock, output=listprocedure): X[0]:=subs(kin,x(t)):X[1]:=X[0]-l: speed:=(X[0](2*T)-X[0](0))/(2*T): plX:=plot([X[0],X[1]],0..2*T,color=black, linestyle=[SOLID,DASH],thickness=3, axes=box,title="SPIKY: x0, x1 vs. t, "):%; 'meanspeed'=evalf(speed,3); plV:=plot([V[0],V[1]],0..2*T,color=black, linestyle=[SOLID,DASH],thickness=3, axes=box,title="SPIKY: v0, v1 vs. t, "):%;

## dynamics: > W0:=V[0]-dl/2:dW0:=D(W0): #### caution: eventually, confine following to t in (0,T) #### because of non-differentiability of W0 ! > m:=1:k0:=5:Gam:=0: # a choice sigma:=m*dW0+k0*W0+Gam: sigmacheck:=plot(sigma,0..T): %; # no skidding iff sigma>=0 for i from 1 to 201 do s[i]:=sigma((i-1)*T/201) od: -min(seq(s[i],i=1..201)): Gam0:=evalf(%); # no skidding for Gam>=Gam0 ! > Gam:=Gam0/3; # choose new Gam sigma:=m*dW0+k0*W0+Gam: lam:=unapply((1-sgn(w))*pos(s),(w,s)): cent:=dsolve({diff(z(t),t)=w(t), diff(w(t),t)=-k0*w(t)-sigma(t)+lam(w(t),sigma(t)), z(0)=0,w(0)=0},{z(t),w(t)},numeric, method=rosenbrock,output=listprocedure):#central equation Z:=subs(cent,z(t)):W:=subs(cent,w(t)): plz:=plot(Z,0..T):plw:=plot(W,0..T,title="skidding velocity w vs. t"): plw; > plVw:=plot([V[0]+W,V[1]+W],0..2*T,color=black,thickness=1): plXw:=plot([X[0]+Z,X[1]+Z],0..2*T,color=black,thickness=1): display(plX,plXw);display(plV,plVw); print(`bold: kinematical SPIKY, normal: (m,k0,Gam)=`,(m,k0,Gam));

Some Program Source Codes

177

### spikes forces: > lam0:=t->(1-sgn(V[0](t)+W(t)))*lam(W(t),sigma(t)): pllam0:=plot([lam0,V[0]+W],0..2*T,thickness=[2,1],linestyle=SOLID, color=black,title="lam0 (bold) and v0+w vs. t"): lam1:=t->(1-sgn(V[1](t)+W(t)))*lam(W(t),sigma(t)): pllam1:=plot([lam1,V[1]+W],0..2*T,thickness=[2,1],linestyle=DASH, color=black,title="lam1 (bold) and v1+w vs. t"):

### plot all: > display(array([[plX],[plV],[pllam0],[pllam1]])); print(`bold: kinematical SPIKY, normal: (m,k0,Gam)=`,(m,k0,Gam));

Gait construction algorithm > restart:with(plots): ## auxiliaries: pi:=evalf(Pi): h:=(a,b,x)->piecewise(a(sign1(x-a)+sign1(b-x))/2: # smooth heaviside perT:=t->T*frac(t/T): # T-periodification > ## Choose n: n:=5; N:=n+1: # N=steps per cycle K:=[seq(k-1,k=1..N)]: # set of spikes cyc_hr:=z->N*frac((z+1)/N): # cycling head to rear cyc_rh:=z->N*frac((z+n)/N): # cycling rear to head > ## Choose a,startmode A[1],mode direction dir: a:=3: # a < N ! A[1]:=[0,1,3]: # card(A)=a ! dir:=rh: # hr:head to rear; # rh:rear to head if dir=rh then cyc:=cyc_rh else cyc:=cyc_hr fi: for k from 2 to N do for j from 1 to a do AA[j]:=cyc(A[k-1][j]) od: A[k]:=[seq(AA[jj],jj=1..a)] # mode in step k od: print('n'=n,'a'=a, 'dir'=dir\n, 'mode_sequence_per_cycle'\n, seq(A[k],k=1..N));

178

Some Program Source Codes

> ## Formal calculation of S',V0, and l': for k from 1 to N do B:=convert(A[k],set):if ({0} intersect B)={0} then v0[k]:=0 else v0[k]:=1 fi: ds[k,0]:=0: for j from 1 to n do if ({j} intersect B)={j} then ds[k,j]:=v0[k] else ds[k,j]:=v0[k]-1 fi: dl[k,j]:=ds[k,j]-ds[k,j-1] od: print('k'=k,'v0'=v0[k],'dl'=[seq(dl[k,j],j=1..n)], 'ds'=[seq(ds[k,j],j=1..n)]); od: #end k > ## Formal motion T:=1: # period for j from 1 to n do v[0]:=unapply(add(v0[k]*h((k-1)/N,k/N,t/T),k=1..N),t): ds[j]:=unapply(add(ds[k,j]*h((k-1)/N,k/N,t/T),k=1..N),t): v[j]:=v[0]-ds[j]: od: ds[0]:=0: > Tp:=3.5*T: # choice plot endtime for i from 0 to n do y:=v[i]@perT:V[i]:=%:plV[i]:=plot(V[i],0..Tp): # T-periodic sx:=dsolve({diff(x(t),t)=y(t),x(0)=-i},{x(t)}, numeric,method=rosenbrock, output=listprocedure): X[i]:=subs(sx,x(t)): plX[i]:=plot(X[i],0..Tp, color= black,thickness=2,axes=box): od: display(seq(plX[i],i=0..n)); print(seq(A[k],k=1..N));

Adaptively controlled inch-worm The following program does not consist of one source file, rather it is built by various MATLAB-function files which are listed in the following.

Some Program Source Codes % main file for exexution % all other files are only function files clear all; close all; global m omega0 D k0 l0 a Omega A e spikes_flag global adapt_switch lambda kappa gamma delta td % data m=1; omega0=pi; D=1; k0=0; x00=0; x01=-1; x10=0; x11=0; %initial conditions kbegin=0; ini=[x00;x01;x10;x11;kbegin]; % Gait l0=1; a=0.25; Omega=omega0; % controller gamma=500; lambda=2/100; kappa=1; delta=0.2; td=1; % smoothness A=10^4; e=10/A; % Flags spikes_flag=1; adapt_switch=5; % time horizon t_0=0; t_1=50; % solution options=odeset('RelTol',1e-5,'AbsTol',1e-8'); [T Z]=ode15s('dgl',[t_0 t_1],ini,options); --------------------------------------------------------------------% function file to build the ODEs function dzdt=dgl(t,z) global m k0 l0 gamma kappa global old vec_error dot_vec_error gain_k % adapt u input vec_error=z(1)-z(2)-ref(t); gain_k=z(5); dot_vec_error=z(3)-z(4)-refdot(t);

179

180

Some Program Source Codes

u=z(5)*vec_error+kappa*z(5)*dot_vec_error; % total impressed forces fa0=-c(t)*(z(1)-z(2)-l0)-k0*z(3)-k00(t)*(z(3)-z(4))-GAMMA(t)-u; fa1=-c(t)*(z(2)-z(1)+l0)-k0*z(4)-k00(t)*(z(4)-z(3))-GAMMA(t)+u; % ODEs dzdt=[z(3); z(4); 1/m*( fa0+F0(t,z(3),fa0) ); 1/m*( fa1+F1(t,z(4),fa1) ); k_dot(t)*gamma]; old=vec_error; % save actual error vector for adaptation law --------------------------------------------------------------------% function file for various adaptation laws function result_ctrl = k_dot(t) global lambda gamma t_e old vec_error dot_vec_error global delta td gain_k adapt_switch result_ctrl = 0; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% if (adapt_switch == 1) temp1 = norm(vec_error,2)-lambda; if temp1 >= 0 result_ctrl = temp1^2; end end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% if (adapt_switch == 2) temp1 = norm(vec_error,2)-lambda; if temp1 >= 0 result_ctrl = temp1^(0.5); end end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% if (adapt_switch == 3) temp1 = norm(vec_error,2)-lambda; if (temp1 >= 1) result_ctrl = temp1^2; end; if (temp1 < 1) && (temp1 >= 0) result_ctrl = temp1^(1/2); else result_ctrl = 0; end end

Some Program Source Codes

if (adapt_switch == 4) if ((norm(vec_error)-lambda-1) >= 0) if (norm(old)-lambda)*(norm(vec_error)-lambda)= 0) if (norm(old)-lambda)*(norm(vec_error)-lambda) is activated if (t >= 0) && (t= 10) regout(1)=0; end --------------------------------------------------------------------% function file for actuator damping function regout=k00(t) global D omega0 m k000=2*D*omega0*m; regout=zeros(1,1); % constant damping coefficient -> please activate % regout(1)=k000+t*0; % variable damping coefficient -> is activated if (t >= 0) && (t= 20) && (t= 30) regout(1)=0; end --------------------------------------------------------------------% function file for GAMMA function regout=GAMMA(t) GAMMA0=1.2336; regout=zeros(1,1); % activate GAMMA on your own regout(1)=GAMMA0; % if (t >= 0) & (t= 20) regout(1)=2*GAMMA0; end

183

184

Some Program Source Codes

--------------------------------------------------------------------% function file for reference gait function regout=ref(t) global l0 a Omega regout=zeros(1,1); regout(1)=l0*(1-a*(1-cos(Omega*t))); --------------------------------------------------------------------% function file for 1st derivative of reference gait function regout=refdot(t) global l0 a Omega regout=zeros(1,1); regout(1)=-l0*a*Omega*sin(Omega*t); --------------------------------------------------------------------% executable file for plots global k0 m [length_T, width_T]=size(T); for i=1:length_T REF(i)=ref(T(i)); REFDOT(i)=refdot(T(i)); LAMBDA(i)=lambda; EPS(i)=save; GAM(i)=GAMMA(T(i)); Umim(i)=-GAM(i)*SGN(dotl(T(i)))-m*ddotl(T(i)) -(k0+k00(T(i)))*dotl(T(i))-c(T(i))*l(T(i))+c(T(i))*l0; end % worm motion figure(1); plot(T,Z(:,1),'color','k','LineWidth',2); hold on; plot(T,Z(:,2),'--','color','k','LineWidth',2); axis([0 50 -1.5 12]); set(gca,'FontSize',12); % Error figure(2); plot(T,Z(:,1)-Z(:,2)-REF(:),'color','k','LineWidth',2); hold on;

Some Program Source Codes

plot(T,LAMBDA(:),'--','color','k','LineWidth',2); plot(T,-LAMBDA(:),'--','color','k','LineWidth',2); plot(T,EPS(:),'-.','color','k','LineWidth',2); plot(T,-EPS(:),'-.','color','k','LineWidth',2); axis([0 50 -0.02 0.14]); set(gca,'FontSize',12); % Velocities figure(3); plot(T,Z(:,3),'color','k','LineWidth',2); hold on; plot(T,Z(:,4),'--','color','k','LineWidth',2); set(gca,'FontSize',12); % Gain parameter figure(4); plot(T,Z(:,5),'color','k','LineWidth',2); axis([0 50 0 350]); set(gca,'FontSize',12); % Control input figure(5); plot(T,Z(:,5).*(Z(:,1)-Z(:,2)-REF(:)) +kappa*Z(:,5).*(Z(:,3)-Z(:,4)-REFDOT(:)),'color','k','LineWidth',2); hold on; plot(T,Umim(:),'--','color','k','LineWidth',2); set(gca,'FontSize',12);

185

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The book ends. The worm has overcome. Hope the reader does not feel as disappointed as the bird. Best regards! The authors

Index acceleration, 4 actuator, 8, 9, 16 force input, 109 force output, 109 gait mimicking input, 109 model, 9–11, 51, 109 output, 25 realization, 51–53 adaptation law, 111, 166 adaptor, 167 Adaptor 1, 112 Adaptor 2, 114 Adaptor 3, 116 Adaptor 4, 116 Adaptor 5, 118, 119 autonomous motion system, 5 bimodal, 20, 21, 29, 62 body, 2 1-dimensional, 4 2-dimensional, 4 3-dimensional, 2 rigid, 5, 155 center of mass, 15 central equations, 20, 21 complementary-slackness condition, 16, 17 configuration, 3, 15, 62 reference, 4 constraint, 14 rheonomic holonomic, 25 spike, 22 control adaptive, 109, 110, 112, 166 closed-loop, 160 input, 159 off-line, 54, 110, 160 on-line, 54, 160 open-loop, 110, 160

strategy, 160 system, 159 control system linear, 165 coordinate body-fixed, 59 coordinates body-fixed, 4 COULY, 90, 91, 98 curve, 148, 149 drive dynamic I, 49 dynamic II, 51 kinematic, 25, 29, 47, 63 feedback law, 162 output, 110, 111, 160 state, 160 field extrinsic, 7, 8 intrinsic, 7 force constraint, 19, 153 Coulomb, 84, 85 mathematical model, 85, 87 smooth mathematical model, 88 cut, 16, 51, 153 external, 9, 152 external constraint, 153 external impressed, 16, 18, 153 graph of Coulomb, 86–88 impressed, 8, 153 inner, 19 internal, 152 internal constraint, 153 internal impressed, 153 internal reaction, 16 law, 153

194 propulsive, 19 reaction, 8 resultant, 153 spike, 19 friction, 8, 83 anisotopic, 99 anisotropic, 138 Coulomb, 8, 84 dry sliding, 84, 85 mathematical model, 85 of rest, 8, 85 propulsive, 84 selective anisotropic, 144 Stokes, 8, 19, 61 viscous, 84, 99 gain parameter, 165 gait construction, 36 construction algorithm, 37 kinematic, 28, 63 HILLY, 71 center approximation, 73 equations of motion, 72 HILLY 1, 74 gear shift, 74 HILLY 2, 78 hybrid system, 23 inch-worm, 46, 54, 92, 112 equations of motion, 48, 54, 55 mode, 55 spike force, 56 input, 159 interaction with environment, 8 kinematical theory, 26, 28, 63, 64, 139 danymical feasibility, 29 validity, 48, 56 locomotion peristaltic, 6, 137, 138 serpentine, 7 locomotion system, 6 bidirectional, 144

Index mass, 151 center of, 151 total, 151 mass point, 5, 156 chain of, 157 mode, 20 dynamical characterization, 30 kinematical characterization, 30 of worm dynamics, 22 sequence, 39 switching, 23, 49 model, 2 mathematical, 2 physical, 2 momentum angular, 152, 154 linear, 20, 152, 154 motion, 4 principles of, 152 motion system, 5 multibody system, 5 norm, 161 output, 159 reference, 161 point material, 3 point mechanics, 4 position, 4, 15 power balance of, 24, 62, 103, 106 of external propulsive forces, 24, 62 of the actuators, 24, 62 relative degree, 164, 165 strict, 165 shape, 15 simplest guy, 46 SLIDY, 91, 96–98 snake, 7 spike, 8, 13, 59, 138, 139, 141, 144 active, 15, 22, 28, 61 feasible force, 31 force, 19 non-active, 17

Index of finite strength, 31 resutltant force, 31 strength of active, 41 SPIKY, 13, 91, 97 dynamical, 93, 94 kinematical, 93 STICKY, 91, 94–97 stiction, 8, 84, 85 mathematical model, 85 STOKY, 99 equations of motion, 100 power balance, 103, 106 tracking, 110 λ-, 110, 161 approximate, 110, 161 asymptotic, 161 control, 161 error, 111, 161 uncertainty, 109 undulatory locomotion, 9, 46 velocity, 4 generalized, 15 worm, 6, 7 continuous model, 58 active spike, 61 center of mass, 60 central equation, 62 change of shape, 60 configuration, 62 dynamics, 61 equations of motion, 61, 64 kinematic drive, 63 kinematics, 59 local strain, 60, 62 motion, 59 strain wave, 66 discrete model, 14 center of mass, 19 dynamics, 16, 22, 29 equations of motion, 18 kinematical theory, 26 kinematics, 14 mode, 22 worm-like locomotion system, 6

195