Nonholonomic Motion of Rigid Mechanical Systems from a DAE Viewpoint

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NONHOLONOMIC MOTION OF RIGID MECHANICAL SYSTEMS FROM A DAE VIEWPOINT

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NONHOLONOMIC MOTION OF RIGID MECHANICAL SYSTEMS FROM A DAE VIEWPOINT Patrick]. Rabier University of Pittsburgh Pittsburgh, PA Werner C. Rheinboldt University of Pittsburgh Pittsburgh, PA

Society for Industrial and Applied Mathematics Philadelphia

Copyright ©2000 by the Society for Industrial and Applied Mathematics. 1098765432 I All rights reserved. Printed in the United States of America. No part of this book may be reproduced, stored, or transmitted in any manner without the written permission of the publisher. For information, write to the Society for Industrial and Applied Mathematics, 3600 University City Science Center, Philadelphia, P 19104-2688.

Library of Congress Cataloging-in-Publication Data Rabier, Patrick. Nonholonomic motion of rigid mechanical systems from a DAE viewpoint / Patrick J. Rabier,Werner C. Rheinboldt. p. cm. Includes bibliographical references and index. ISBN 0-89871 -446-X (pbk) I. Dynamics, Rigid. 2. Differential-algebraic equations-Numerical solutions. I. Rheinboldt,Werner C. II Title.

QA86I.R332000 5l5'.35-dc2l

registered trademark.

99-053790

Contents Preface

vii

1 Introduction

1

2 The 2.1 2.2 2.3

Gauss Principle for Mass Points Formulation Justification Variable Transformations

9 9 14 17

3 The 3.1 3.2 3.3

Configuration Space of a Rigid Body Rigid Body Representation The Use of Quaternions Planar Rigid Bodies

19 19 21 26

4 Unconstrained Rigid Bodies 4.1 Background 4.2 A Trivialization of TS3 4.3 Second-Order ODE of Motion 4.4 The Euler-Jacobi Operator on Manifolds 4.5 Energy Formulation 4.6 Planar Rigid Body Motion

29 29 32 34 37 41 43

5 Constrained Rigid Bodies 5.1 Kinematic Constraints 5.2 Geometric Constraints 5.3 Multibody Systems 5.4 The Planar Case

45 45 53 55 58

6 DAE Formulation in Linear Spaces 6.1 A DAE Existence Result 6.2 The Case of Nonmaximal Rank

61 61 65

v

vi

CONTENTS

7 DAE Formulation on Manifolds

7.1 7.2 7.3 7.4 7.5 7.6

73

Existence Theory on Manifolds A Projected Derivative Operator Nonmaximal Rank in the Manifold Case Reduction to DAEs on Euclidian Space Characterization of State Spaces ODE Formulations

73 77 80 82 90 92

8 Computational Methods

8.1 Computations on Manifolds 8.2 A DAE Solver for Kinematic Constraints 8.3 Outlook on Computational Approaches

97

.

97 101 105

9 Computational Examples

109

Appendix.

133

9.1 Some Introductory Planar Examples 9.2 Three-Dimensional Problems Submanifolds

109 115

References

137

Index

139

Preface Over the past two decades, both the theory and the numerical analysis of differential algebraic equations (DAEs) have received considerable attention. The interest in such systems has been sustained by the increasing awareness of their pervading relevance in science and engineering. In fact, it is now widely recognized that many physical problems, which, in their simplest forms, are described by ordinary differential equations (ODEs), actually arise as DAEs in their more general variants. Unlike their academic counterparts, these DAEs can rarely be reduced to ODEs by explicit side calculations, and the numerical analysis of DAEs has to aim at the development of methods in which both the differential and algebraic components are treated numerically. Arguably, a prototypical example of a DAE is the one stemming from the constrained motion of a system of mass points. In that setting, Newton's law of motion gives rise to the differential part of the equations while the constraints provide the algebraic part. Of much greater practical importance is the motion of a system of rigid bodies that is subjected to kinematic constraints that depend on the configuration variables as well as the velocities but, of course, not on the accelerations. It is the primary purpose of this monograph to show that such motion, also, can be modeled by a DAE formulation, provided only that the "correct" variables are used. Since nothing can really be new that addresses the motion of rigid bodies, we shall not venture to say that this is a new result, but, at least for general kinematic constraints, it is certainly not widely known. The skeptical reader may reflect upon the fact that even in the case of a single, unconstrained, three-dimensional rigid body, the classical equations of motion involving the angular velocity vector look like ODEs but actually turn out to be of integrodifferential type. The existence of an equivalent formulation as a second-order ODE on the configuration space is well known to geometers but is hardly familiar to general mechanics audiences. We hope that this work will contribute something toward bridging this gap. For (systems of) rigid bodies subjected to kinematic constraints we are not aware that any other exposition of the material presented here is available. Although our discussion centers on the DAE formulation, several other issues must be investigated in some depth to provide a sound and complete justification of the DAE model. This includes, in particular, the development of a generalized Gauss principle of least constraint, a study of the effect of the failure of full-rank conditions, and a precise characterization of the state spaces. The value of the DAE formulation rests chiefly upon the premise that it should be more accessible than others to effective numerical treatment. Accordingly, we found it vii

viii

PREFACE

essential to include in this monograph some evidence that the computational solution of the DAEs obtained here indeed can be handled satisfactorily. This is hardly obvious, since these DAEs not only involve the notorious difficulties due to their being of index at least two, but also their size and complexity invariably require special consideration of the particular features of the problem in order to devise reasonably efficient solution procedures. In line with this, it cannot be our intent to address here the multitude of questions that arise in the development of optimal numerical methods for these DAEs; in fact, at this time, this should still be viewed as a research topic. We are indebted to Edward Haug, Glaus Fiihrer, and Caren Tischendorf for numerous insightful comments on an earlier version of our manuscript. University of Pittsburgh Pittsburgh, PA

Patrick J, Rabier Werner C. Rheinboldt

Chapter 1

Introduction The classical problem of determining the motion of a system of mass points or rigid bodies subjected to both external forces and constraints is central to modern rnultibody mechanics. The history of this subject is associated with a long list of famous names, including Leibniz, d'Alembert, Maupertuis, Euler, Lagrange, Gauss, Hamilton, and many others. By the end of the nineteenth century the general form of the equations of motion was considered to be well understood, at least under simplifying physical assumptions such as ideal constraints. Nevertheless, the study of related issues has continued throughout the twentieth century, notably as regards various aspects of problems incorporating more general constraints. In the second half of this century computational considerations opened a new set of questions. The availability of a correct formulation of the problem does not necessarily mean that the formulation is complete or optimal, and, as always, optimality is a relative notion. Today, the challenge in the analysis of rnultibody dynamics is to compute an approximate solution of systems comprising a large number of components, certainly large enough to discourage any attempt at obtaining solutions in analytic form. In this context, the optimality of a formulation is measured by how well it lends itself to an efficient numerical treatment, and the possibility of producing close-form solutions for special cases no longer carries much weight. More than a century ago H. Hertz introduced the concepts of holonomic and nonholonomic constraints. In essence, in his work, a constraint is a linear form in the virtual displacements of the system and is called holonomic if it represents an integrable form and nonholonomic otherwise. More modern as well as general is a definition used, e.g., by Haug [14] and Neimark and Fufaev [17]. Under this definition a condition on the system is a holonomic constraint if it can be expressed as an equation in the configuration variables, and it is called nonholonomic if it depends also on the time or any velocity component. This terminology will be used occasionally here although it is not always ideal or sufficient. More generally, we will follow the convenient and, more or less, traditional approach of calling a condition that depends only upon time and the configuration variables a geometric constraint. Thus, in particular, a holonomic constraint is a geometric one. Similarly, since the name "nonholonomic" may sometimes appear to exclude 1

2

PATRICK J. RABIER AND WERNER C. RHEINBOLDT

the holonomic case, we found it desirable to define a kinematic constraint to include not only nonholonomic but also holonomic constraints in the above sense. We remark also that some authors (see, e.g., Roberson and Schwertassek [24]) observed that, in practice, there appear to be no examples of nonholonomic systems in which the velocities occur nonlinearly in the constraints. Beside the fact that there are some such—albeit possibly not entirely convincing—examples (see [17]), we found that, from a conceptual as well as a theoretical viewpoint, there is no advantage to restricting the above concepts to reflect this observation. It has been known for a long time that the equations of constrained motion of a system of mass points can be written in the form of a system of second-order DAEs. Although less widely known, a similar DAE formulation exists for systems of rigid bodies subjected to holonomic constraints. A thorough exposition of that formulation was given by Haug [14]. But for more general constraints, such as nonholonomic constraints or a mixture of both types of constraints, and especially also in the presence of nonconservative forces, no precise and numerically suitable derivation of such a DAE formulation appears to be available for systems of three-dimensional rigid bodies. The aim of our work leading to this monograph was to provide such a theoretically sound derivation. Of course, ever since Hertz's work, nonholonomic dynamical systems have been the topic of many studies, and it would be too time-consuming to attempt any historical survey here. For some time, authors concerned themselves with the question of whether the motion of a nonholonomic system can be derived, say, from Hamilton's principle and similar related considerations (see, e.g., Rund [25]). Another body of work concerns the so-called geometry of dynamics. If, say, the kinetic energy of the system is a positive definite quadratic form in the velocity components, then this form can be used to define a Riemannian metric on the state space, and the concepts of Riemannian geometry allow for a detailed characterization of the properties of the system (see, e.g., Synge [29], [30], [31]). These results are difficult to apply in practical physical problems and certainly do not lend themselves readily to computations. Among other results in the area we mention only the theory of constrained Hamiltonian systems developed by Dirac [9], [10], which will have to remain outside of our considerations. When considered in the general mechanics literature, nonholonomic problems are usually treated only for conservative force fields, that is, under the assumption that the problem can be recast as one of finding constrained critical points of some energy functional. The nonconservative case turns out to call for a "correct" choice of variables, and attempts to circumvent this have led to a variety of theoretically and computationally unsatisfactory constructs, such as, for instance, the so-called quasi coordinates (see, e.g., Neimark and Pufaev [17]). These brief comments already point to some of the major aspects we encountered when we began this work. The question of the appropriate choice of the state variables arises already in the absence of constraints. For almost two centuries it has been standard practice to write the equations for the unconstrained motion of a single rigid body in the form

Here u is the position of the center of mass and u the angular velocity vector of the

CHAPTER 1.

INTRODUCTION

3

body. But, in general, the force F and moment N are specified in terms of the time t and the state variables; that is, they are defined on R1 x TC, where C = R3 x 5O(3) is the configuration space of the rigid body involving the group 50(3) of rotations in R3 and TC is the tangent bundle of C. Yet note that all the configuration space variables cannot be recovered from (u,u>) without time integration of the angular velocity vector,1 In other words, (1.1) represents, in general, an integrodifferential equation and not an ODE. This shows that, in spite of being widely recognized in the literature as "the" equations of unconstrained motion of a rigid body, (1.1) is nonetheless inadequate from the viewpoint of the above numerical adequacy criterion, except when F and N depend only on t and u or in the much simpler case of planar motion when the angular velocity is indeed a state variable. Of course, it is well known, especially in the dynamical systems literature (see, e.g., Abraham and Marsden [1]), that there exists an ODE formulation of the equations of unconstrained motion on the space C. But this formulation raises numerical questions. In fact, for numerical purposes, this ODE must then be transformed into one on a linear space which, in turn, requires either the use of charts of C or of an embedding of C into R m with sufficiently large m. The use of charts of C introduces numerical complexities, while the embedding technique is excessively costly. Indeed, the natural vector space containing SO(3) is R9 = R 3x3 ; whence m = 12 should be chosen, resulting in a "waste" of six dimensions since C is only six dimensional. It is well known that 5O(3) embeds into the smaller space R6, but this and similar embeddings (there exists even one into R5) are not canonical, which introduces new complications. These objections may be viewed as petty in the case of a single rigid body, but they become very real in the only case of practical interest, namely, of multibody systems involving a large number of interconnected bodies. In the presence of constraints, the earlier raised question of the proper derivation of the equations of nonholonomic motion is a central concern. For some time it has been a well-accepted fact that any fundamental dynamical principle, from which the motion of a nonholonomically constrained mechanical system is derived, cannot be one of the variational principles in the accepted sense (see, e.g., [25, p. 358]). Instead of any of these variational principles, one of the differential principles of mechanics has to be considered. In the case of constrained systems of mass points, a natural choice here is the Gauss principle of least constraint which, for that case, has been extensively explained and justified by both mathematical and physical considerations. The Gauss principle leads to a Lagrange multiplier (DAE) formulation via the standard Lagrange multiplier theorem. But, when we turn to systems of rigid bodies, it appears that, typically in the literature, one encounters either only a comment to the effect that the Gauss principle is "easily generalized" to larger classes of systems (see, e.g., Sparschuh and Hagedorn [28]) or, alternately, one finds that the Lagrange multiplier formulation is simply used without any further justification or comment. The latter approach conveys the strange idea that the Lagrange multiplier approach requires justification in the case of the constrained motion of a system of mass points but that, having been supported in that case, it applies "of course" to the problems involving rigid bodies. •'For a discussion of this point see Remark 4.2.

4

PATRICK J. RABIER AND WERNER C, RHEINBOLDT

Yet the truth is that no Lagrange multiplier formulation has any value until it is proven to account for some constrained extremal property (or, more generally, to characterize constrained critical points). Despite all claims, an attempt to extend the Gauss principle to more general systems runs into various difficulties. In fact, when—as is natural in the mechanics literature— one starts from the equations of unconstrained motion in the form (1.1), it soon becomes evident that no generalization of Gauss's principle can be justified on the basis of (1.1) alone. Indeed, in the case of mass points, the Gauss principle is a modification of the relationship between the external force and the acceleration, that is, a modification of Newton's law. What makes the Gauss principle easy to understand and to justify mathematically (beyond its physical validity) is that Newton's law for mass-point systems involves only the configuration space variables and their derivatives. The difficulty in trying to support any modification of (1.1) by some generalized Gauss principle is that, in sharp contrast to the planar case, the angular velocity vector is neither a configuration space variable nor a derivative of one. This turns out to make it virtually impossible to base directly upon (1.1) a sound justification of some variant of the Gauss principle. In other words, in the case of three-dimensional rigid body systems subjected to general classes of constraints, one is led again to the earlier indicated requirement for a thorough analysis of the correct state space formulation. The aim of this monograph is to provide such a clarification of the state space concept and to derive a generalized Gauss principle of least constraint for the motion of rigid body systems under general mixed kinematic and geometric constraints and without any limitations on the nature of the external forces. This will show that in all these cases the equations of constrained motion of systems of rigid bodies and mass points can be given in the form of a second-order DAE. These results complement those of Haug [14] for the holonomic case. But, of course, while it will be necessary to point out how our results incorporate the holonomic case, this case will have to remain peripheral to this presentation. Indeed, it would far exceed the framework of this monograph to survey the vast literature on the theoretical and computational aspects of holonomic dynamical systems. While several parts of our work certainly make no claim to novelty, many aspects of the DAE theory in Chapters 6 and 7 appear to be new. Moreover, some of the ideas used in our presentation are not easily found in the literature, such as, for instance, the convenient use of quaternion algebra introduced in section 3.2. Some readers may even be surprised to see how this can bring considerable clarification and simplification to the description of rigid body motion. In addition, we show that our formulation can also be transformed into an efficient computational method for the solution of the DAEs modeling the motion of multibody systems in the case of nonholonomic constraints. Altogether, we hope this monograph will be helpful to readers interested in the DAE aspects of multibody motion under general classes of constraints not discussed elsewhere. Chapter 2 introduces the classical Gauss principle of least constraint for mass-point systems with nonholonomic (or holonomic) constraints. Everything here is standard, including the exposition, but, of course, this material is only intended to provide an essential background for the further developments. In Chapters 3 and 4, we reinvestigate the unconstrained motion of a single rigid body,

CHAPTER 1.

INTRODUCTION

5

which is hardly a new topic. It is shown how the system (1.1) can be reformulated in terms of the configuration space variables to obtain a second-order ODE on the configuration space. As indicated, essential use is made of the properties of quaternion algebra with the double advantage of helping to establish the connection with the Lie group structures of S3 and 5O(3) (the starting point of the "abstract" mathematical treatment, not covered here) and of simplifying the calculations in Haug [14]. No use is actually made of Lie group theory. All the results are obtained with the lowest level of mathematical sophistication possible. Only section 4.4 is more demanding to the reader unfamiliar with the spirit of "abstract" differential geometry. Then in Chapter 5, the equations for the constrained motion of rigid bodies are derived as a modification of the unconstrained case. Here the aforementioned generalized version of the Gauss principle of least constraint is developed and shown to provide again the Lagrange multiplier formulation used throughout the literature but without any limitations placed upon the nature of the external forces (e.g., the system need not be conservative). As with mass points, there is no need to distinguish between geometric and kinematic constraints, the former being simply a special case of the latter. For ease of presentation the material is first developed in sections 5.1 and 5.2 for the case of a single rigid body. Then in section 5.3 the extension to multibody systems is sketched. This reflects our viewpoint that when the problems involving a single body are sufficiently well understood, there is no difficulty in generalizing everything to the case of several rigid bodies. Chapters 6 and 7 introduce some new material necessary to round out the prior theory. In section 6.1 a general existence and uniqueness result for the solutions of DAEs is proved under assumptions satisfied by the equations obtained in the earlier chapters. Aside from being conceptually clearer, the generalized Gauss principle reveals a crucial item invisible to the Lagrange multiplier formulation, namely, that such a formulation is meaningful only when some full-rank condition holds. This leads to a discussion in section 6.2 of the question of how the problem should be handled when the full-rank condition fails to hold as is the case, for instance, when the motion is constrained by pure geometric constraints or, more generally, mixed geometric and kinematic constraints. In such cases, the original constraint G = 0 has to be replaced by a new constraint G = 0 that satisfies the necessary full-rank condition and for which G = 0 implies that G = 0. This is a known approach in the case when G can be split in the form G = (G\,Gy) such that G\ = 0 and G% = 0 represent a geometric and kinematic constraint, respectively. The hypothesis of the existence of such a splitting is unsatisfactory, even though it holds for many concrete problems. Indeed, mathematically, the constraint G = 0 is equivalent to any constraint of the form © o G — 0 involving an arbitrary diffeomorphism Q. Yet the existence of a splitting G = (G\, GI) in no way implies the existence of a corresponding splitting for &oG. Our approach does not presuppose the existence of any splitting and applies equally with G or 8 o G and thereby emphasizes the intrinsic role of the process of transforming G = 0 into a new constraint G = 0. There is a specific procedure to discover, but no canonical choice for, this modified constraint function G, and hence the need arises for showing that the particular choice of G has no effect on the solution. This property of obvious theoretical and practical importance is one of the topics of

6

PATRICK J. RABIER AND WERNER C. RHBINBOLDT

Chapter 7. More specifically, the first four sections of this chapter generalize the theory developed in Chapter 6 to DAEs on manifolds. This begins with a generalization of the existence theory of section 6,1 to such DAEs and then continues, in particular, with an extension of the earlier results for problems where the full-rank condition fails to be satisfied. These results, in turn, provide the needed, definitive clarification of the state space concept of a system of constrained rigid bodies and/or mass points. Section 7.5 discusses some examples that show that the general state space definition coincides with the intuitive definitions introduced in Chapters 3 and 5 in the case of simpler constraints. In section 7.6, we discuss alternate ODE formulations of the equations of motion. The first one is valid in general and is obtained by elimination of the Lagrange multiplier by a procedure generalizing a special case discussed in section 2.1 for mass-point systems. The second formulation is available only in the case of holonomic constraints and consists of a further reduction as a second-order ODE on the constraint manifold. The existence of such a special form of the equations of motion represents the only case in our theory where the assumption that the constraints are holonomic does make a difference. By necessity Chapter 7 is more technical in nature and may be bypassed by readers mainly interested in the standard case when the aforementioned full-rank condition is satisfied. While computational methods for the widely known Euler-Lagrange equations arising as models of holonomic systems have been studied extensively, there appears to be practically no literature on the numerical solution of problems involving kinematic or more general mixed constraints. In fact, not even the correct form of the corresponding equations in the three-dimensional case appears to be widely known. It would exceed our framework to pursue here potential avenues for developing methods for the solution of the latter problems. Instead, in Chapter 8 we restrict ourselves to the discussion of one effective method for the solution of the DAEs arising in problems with kinematic constraints. In line with our differential geometric setting, this method is based on the use of local parametrizations of the constraint manifolds and represents a special case of the methods given by Rheinboldt [23] for a general class of algebraically explicit DAEs. These methods utilize a general toolkit, called MANPACK, given in [22], for computation on manifolds that are implicitly defined as solution sets of submersions. A summary of selected algorithms from MANPACK for the construction of local parametrizations and their derivatives is given in section 8.1. Then section 8.2 presents the design of the indicated method for solving the DAEs of motion involving general kinematic constraints. Since this single method is not meant to be the definitive answer to the computational solution of this broad class of problems, we indicate in section 8.3 some further directions that may lead to the development of other methods for this class of DAEs. In Chapter 9 we present computational results obtained with the method of section 8.2 for several examples of systems with nonholonomic constraints. The aim here was t exhibit the applicability of the method as well as its effectiveness. While planar examples are well understood and are only peripheral to our presentation, their computational solution is by no means trivial. Accordingly, we consider first in section 9.1 three planar problems, two of which have known exact solutions that illustrate the very satisfactory performance of the algorithm. The third example is a model of a commercially available transmission that translates the constant rotational motion of a smooth driveshaft into a linear motion. We kept this model in a planar form since any added complexity appeared

CHAPTER 1.

INTRODUCTION

7

not to justify the small gains in numerical insight. In section 9.2 we turn to threedimensional problems that reflect the theory developed here. The first two examples concern the motion of a ball rolling, without sliding, on a rotating platter and in the inside of an open cone. The third example is a system of three cylinders rolling on each other. Finally, the last two problems involve rolling disks, namely, first a disk rolling by itself on a supporting plane and thereafter a two-wheeled cart rolling on an inclined plane. For ease of reference we collect in the Appendix some basic definitions and results— without proofs—for submanifolds of finite-dimensional spaces in a form readily applicable for computations. The material included here should be essentially adequate for a general understanding of the major part of the monograph, excluding section 4.4 and Chapter 7, It is not our purpose to discuss any matter arising from the lack of regularity. Accordingly, we assume once and for all that the mappings involved throughout this monograph are all of class C°°. Warning. The following notation is used repeatedly throughout this monograph: If M is a manifold and x E M, the vectors of the tangent space TXM are usually denoted by x. Thus, the notation ± does not imply the existence of a curve x(t) such that x = dx/dt. This notation has the advantage of making it clear that the vector x is in some way associated with the point x. It is also consistent with the fact that, eventually, the slot occupied by x will be replaced by x ( t ) = (dx/dt)(t) for some curve x(t). On the other hand, in other places, x will represent the derivative dx/dtof some curve x(t). Whichever notation is used should be clear from the context. For instance, in a relation such as (A(x)x,x) > 0 V x € TXM, it should be obvious that the first notation is used, and when reference is made, say, to the ODE x = f(t,x,x), it should also be clear that x, x, and x represent the derivatives of order 0, 1, and 2, respectively, of some curve x(t). The reader will be occasionally reminded of the use of this notation, which turns out to have many more advantages than inconveniences if attention is paid to the context in which it appears. In addition, we have frequently inserted the warning "omitting i-dependence" or something to that effect whenever x and x should be understood as derivatives of a curve x(t). The notation x for a vector of TXM. is also used when Ai = R n , in which case x represents just an arbitrary vector in R n . This is not an additional convention but is simply consistent with the classical identification T±R™ = Rn for x € R n .

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Chapter 2

The Gauss Principle for Mass Points 2.1

Formulation

Throughout the historical development of theoretical mechanics, optimality principles have played a fundamental role. Among the early contributors to the field these optimality concepts often reflected teleological as well as theological viewpoints. For instance, in 1747 Maupertuis formulated his principle2 that "nature chooses among all possible motions that which reaches its goal with the least amount of action," and he supported it by noting that "it reflects best the wisdom of the creator." The optimality principles assumed a more mathematical form only some time later, beginning with the work of Euler and Lagrange, In the mathematical development of the theory we have to distinguish between variational formulations and differential principles. The foremost example of the first is Hamilton's variational principle, while the second group includes Newton's law of motion as well as the two principles associated with the names of d'Alembert and Gauss. There is a major conceptual difference between these two approaches. The variational principles consider a "path" in the configuration space representing the motion of the mechanical system during a finite time interval and compare it with other "neighboring" paths. Moreover, by nature, variational principles presuppose the existence of an energy function underlying the physical phenomena. On the other hand, the differential principles consider only the state of the system at a given time and provide information about the optimal direction of change. Our presentation here is based on Gauss's principle and some generalizations. Its original version was published by Gauss in 1829 (see [11]) and was formulated there for the constrained motion of a system of mass points. We begin with a brief introduction to this principle. There was, at the time, a major dispute over whether the priority for this concept belonged to Leibniz dating to 1707.

9

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PATRICK J. RABIER AND WERNER C. RHEINBOLDT

Consider a system of I > 1 mass points with constant, positive masses mi,... , m^, which, at time t, are located at the points xi,... ,xg € R3. Hence, the configuration of

the system at time t is specified by the vector x = x(t) := ( x j , . . . , x J ) J £ R3f, and the

time t, together with the configuration vector x(t) and the velocity vector x(t), defines the state ( t , x ( t ) , x ( t ) )of the system at that time. The collections of all admissible configuration vectors and states are the configuration space Co and state space <5>o of unconstrained motion, respectively; that is,

For the dynamics of the system, suppose now that the external force acting on the jth mass point, j — 1,2, ...,£, is defined by a mapping Fj : SQ —> R3. With the vector F :— (Fi ,..., F/)T e R3£, the unconstrained motion of the system is then governed by Newton's law where M = diag(mi/3,... ,771^/3) is the mass matrix and /a denotes the identity in R3. Here Newton's determinacy principle is affirmed by the standard existence theory for the solution of ODEs, which asserts that the solution of (2.2) is uniquely determined by a given initial state (to,xg,x0) £ S$. Now suppose that at any time the state of the system of mass points is constrained by a relation of the form

In other words, (2.3) restricts the states to the constrained state space

and accordingly (2.3) may be called a state constraint. However, as indicated in Chapter 1, when G depends upon the velocity x we shall refer to (2.3) as a kinematic (or nonholonomic) constraint. It is important to note that the suitability of the set S in (2.4) as a constrained state space depends upon certain rank conditions. Such rank conditions will be encountered shortly, but it is only in Chapter 7 that a systematic procedure for identifying the correct constrained state space will be discussed. Until then our definitions may appear somewhat arbitrary. For any motion of the system satisfying (2.3) it follows by differentiation with respect to t that the acceleration x(t) is constrained by the linear equation

This implies that when the system is in the state (t, x, x) 6 S, then its acceleration x has to belong to the affine subspace

CHAPTER 2.

THE GAUSS PRINCIPLE FOR MASS POINTS

11

Obviously, there is no reason to expect that the acceleration M~1F(t,x,x), required by (2.2) for the unconstrained motion, will belong to A(t,x,x), The Gauss principle of least constraint is an amendment to Newton's law addressing constrained systems of mass points. In essence, it requires the equations of motion (2.2) to remain satisfied as far as the constraint (2.3) permits. Gauss [11] formulated his principle (loosely translated) as follows: Consider the motion of a system of material points that are coupled in some way and bound by some type of external restraint. Then, at each time, this motion agrees as closely as possible with the free motion; that is, it occurs under the least constraint where the measure of the constraint, to which the entire system is subjected, is defined as the sum, of the products of the mass of each particle and the square of the deviation of that point from its free motion. Some comments about the mathematical argument used by Gauss to explain this principle are deferred to section 2.2 below. In essence, the principle requires that when the system is in any given state (t,x,x) € S, then the distance between the unconstrained acceleration M~lF(t,x,x) and the admissible acceleration y 6 A(t,x,x) has to be a minimum for y = x. While Gauss does not give an explicit formula, it is unquestionable that he used the following weighted norm:

where {-, •) denotes the Euclidian inner product of R3^ and || • ||2 is the Euclidian norm. Note that, in our setting, the mass matrix M is positive definite arid hence the same holds for M1'2. Obviously, as the Gauss principle itself, this choice of the norm can only be justified by its physical relevance. Thus, in our notation, the Gauss principle stipulates that, in the presence of the constraint (2.3) and for the system in any given state (t,x,x) e S, the acceleration x of the system has to be a solution of the minimization problem

It is clear that (2.8) has a unique minimizer whenever A(t,x,x) is nonempty. By (2.6) this is certainly the case when the full-rank condition

holds, for then A(t, x, x) is an affine subspace of R3f. Otherwise, if the rank of D±G(t, x,x) is less than k, then A(t, x, x) = 0 will be the rule rather than the exception. For that reason, the kinematic constraint (2.3) will be assumed to satisfy (2.9). Then the minimization problem (2.6) has a unique solution x. The way to handle the problem when (2.9) fails will depend largely upon more refined properties and is rarely discussed in the literature. An important special case in the applications (incorrectly referred to as being "the most general case") is investigated by Zeidler [34]. We shall return to these aspects in Chapter 6.

12

PATRICK J. RABIER AND WERNER C. RHEINBOLDT

For the finite-dimensional minimization problems (2.8) and under the full-rank condition (2,9), the Lagrange multiplier theorem, Theorem A.5, applies. Prom it we obtain that, for the minimizer x of (2.8) at the state (t, x, x) € 5, there exists a Lagrange multiplier A G Rfe such that (x, A) solves the system

This has to hold for any state (t,x(t),x(t)) € S assumed by the system of mass points during the constrained motion and defines uniquely the acceleration x(t) at time t. Obviously ( t , x ( t ) , x ( t ) ) € S; that is, equivalently, G(t,x(t),x(t)) = 0 for t in some open interval implies that x(t) € A(t,x(t),x(t)), Hence, it follows from (2.10) that, assuming (2.9), the constrained motion of the system satisfies the DAE

The Lagrange multiplier A(t) can be calculated explicitly in terms of the current state (t, x(t), x(t)), thereby reducing equation (2.11) to a second-order ODE in R3€. In fact, by

(2.9), DxG(t,x,x')M~lD;!;G(t,x,x)~!~€ £(R,fe) is invertible becauseM~lis symmetric

positive definite. Thus, by multiplying the first equation (2.11) by DxG(t,x,x)M~l and using (2.5), we obtain (suppressing the explicit dependence on t)

Because the initial state (to,x(£o)>i(£o)) is assumed to be in 5, (2.5) is equivalent to (2.3) and, since (2.5) was just used to obtain the function A, it follows that the system (2.11) reduces to the following ODE in RM:

Conversely, by reversing the above steps, it is straightforward to check that the solution of (2.12), subjected to the consistent initial condition

also solves the system (2.11). These results hold for systems with constraints of the form (2.3) under the rank condition (2.9). It is plain that such a condition may hold only if the mapping G(t,x,x) does depend upon the variable x, that is, in the indicated terminology, if (2.3) is a kinematic constraint. Condition (2.9) is not satisfied when the motion is constrained by a relation of the form

CHAPTER 2.

THE GAUSS PRINCIPLE FOR MASS POINTS

13

In contrast to (2.3), condition (2.13) does not involve the velocity and, in accordance with the terminology of Chapter 1, it will be called a geometric constraint (and. by some authors, also a time-dependent holonomic constraint). We shall usually reserve the term holonomic constraint for the time-independent case when the constraint is placed solely upon the configuration space. The original definition of a holonomic constraint by Hertz also includes those that are reducible (i.e., equivalent) to the form (2.14). The essential point now is that any geometric constraint necessarily induces a kinematic constraint for the system. In fact, for any motion of the system satisfying (2.13), it follows by differentiation with respect to t that Conversely, when (2.15) holds for t in an open interval J C R1 and we have G(to, x(to)) — 0 for some to € J~, then, by the integral mean value theorem, it follows that G(t, x ( t ) ) = 0 for t € J. In other words, the kinematic constraint (2.15) is equivalent to the geometric constraint (2.13), provided the latter holds at least at some time. For the system with the kinematic constraint (2.15) the previous results can be applied directly. Evidently the rank condition (2.9) for G is here and, if it holds, the motion has to satisfy the DAE

As just noted, for any motion satisfying G(to,x(to)) = 0 for some time to, the differentiated kinematic constraint (2.15) is equivalent to the geometric constraint (2.13). Hence it follows from (2.17) that any such motion has to satisfy the DAE

This system is usually called the Euler-Lagrange equation for the system of mass points constrained by (2.13). Remark 2.1. Instead of relying on the earlier results for kinematically constrained systems we can also apply the Gauss principle directly to a system constrained by (2.13). In fact, consider the constrained state space

By differentiation it follows that when the system is in the state (t, x, x) € <$, the acceleration x has to be in the affine subspace

14

PATRICK J. RABIER AND WERNER C. RHEINBOLDT

With this the Gauss principle again stipulates that in the presence of the constraint (2.13) and with the system in the state (t,x,x) 6 5, the acceleration x is a solution of the minimization problem (2,8). Obviously, A(t,x,x) is a nonempty linear subspace when the rank condition (2.16) holds. Then the Lagrange multiplier theorem applies again and, as before, we conclude that the constrained motion of the system satisfies the DAE

This DAE requires the consistent initial condition (to, #o, XQ) € S and, as noted, the tw constraints in (2.21) are equivalent when G(to,x(to)) = 0 for some time to- Hence, with such an initial condition, one of the constraints G(t, x) = 0 or DxG(t, x)x + DtG(t, x) = 0 can be dropped. By doing so we recover either the system (2.17) or the system (2.18). From a numerical viewpoint, the initial value problem (2.18) turns out to be preferable over the one obtained by retaining the differentiated constraint (2.15) (and adding, of course, the explicit requirement that G(tg,x(to))).= 0 for some to)- In fact, while the two reduced systems are mathematically equivalent, this equivalence does not carry over to the discretizations used in computations. This may result in a potentially substantial "drift" of the computed points; in other words, the numerical solution rapidly fails to satisfy the constraint (2.13), even approximately. This is a well-known and serious shortcoming when the system (2.17) is used. In contrast, the formulation (2.18) does not display the same deficiency if discretized properly.

2.2 Justification Just as Newton's law, the Gauss principle of least constraint is a physical law and hence cannot be justified by purely mathematical arguments: its adequacy in describing physical phenomena must be its primary justification. At the same time, consistency with other independently known physical laws may be used to support the specific form of the principle. Gauss {11] himself supported his proposed principle by justifying it on the basis of the principle of virtual work. In fact, he begins with the following observation: A s is well known, the principle of virtual velocities3 changes all statical problems into mathematical ones which, in turn, by the d'Alembert principle for the dynamics are formulated as statical problems. Thus it is in the nature of the matter that there cannot be a new fundamental principle for the theory of motion and equilibria which in substance is not already contained in those two and derivable from them. Without entering into any details of the principle of virtual work, we discuss here only the salient points connecting it with the Gauss principle. Physically, the kinematic 3 This terminology was introduced by Bernoulli but later dropped in favor of the current "principle of virtual work."

CHAPTER 2.

THE GAUSS PRINCIPLE FOR MASS POINTS

15

constraint (2.3) is equivalent to a constraining force Fo(t,x,x), and the motion of the system is equivalent to the unconstrained motion characterized by the equation

in agreement with Newton's law. In any virtual displacement with velocity v, the constraining force F(;(t,x(t),x(t))produces the (instantaneous) work(Fc(t,x(t),x(t)),v). The only velocities compatible with. (2.3) are the vectors v G kerD;j;G(i,£(t),i(t)). Indeed, neglecting higher-order terms, as is customary when dealing with infinitesimal variations, we have

which vanishes if and only if v €kerD xG(t,x(t),x(t)). The constraint (2.3) is called ideal if the constraining force FQ produces no work in any admissible virtual displacement, that is, if

In other words, the constraint (2.3) is ideal if the constraining force FG is always orthogonal to the nullspace of D±G, Prom this it follows that the assumption that the constraint (2.3) is ideal is equivalent to the Gauss principle of least constraint. In fact, from (2.23) we obtain (assuming the full-rank condition (2.9)) that

where A = A(i) e R fc . By substitution of (2.24) into (2.25), it follows that

Of course, this relation alone is not sufficient to characterize the motion since A is not known and, as before, the condition (t, x, x) €. S must be added to obtain a well-posed problem. Doing this produces once again the system (2.11) obtained earlier via the Gauss principle. Remark 2.2. It is important to emphasize that the definition of an ideal constraint is that the constraining force Fo(t, x, x) produces no work on any admissible virtual displacement. This does not imply that the constraining force produces no work on the actual motion. Indeed, Fc(t,x,x) produces no work on the actual motion if and only if {Fc(t,x,x),x) = 0. Under the full-rank condition (2.9) it follows from (2.24) that this amounts to x e kerDxG(t,x,x), which does not hold in general. However, in an important case the constraining force does not produce any work on the actual motion, namely, in the case (2.14) of holonomic constraints. As observed earlier, in the holonomic case the geometric constraint G(x) — 0 induces the kinematic constraint G(x,x) = 0 where G(x,x) ',= DG(x)x (the autonomous form of (2.15)). Therefore, it is G (and not G) that is involved here, and indeed DxG(x, x)x = DG(x)x — G(x, x) = 0. The fact that nonholonomic constraining forces may produce work on the actual motion

16

PATRICK J. RABIER AND WERNER C. RHEINBOLDT

is consistent with the observation that, in practice, nonholonomic constraints may be realized as friction forces, as shown by Neimark and Pufaev (17, p. 233],4 The procedure used here to derive the Gauss principle rests on the empirical concept of admissible virtual displacements. Intuition based on that concept may well dictate other definitions leading to incorrect formulations, and it is rather fortunate that Gauss's justification of his principle is far more convincing. The assumption that the constraint (2.3) is ideal has been considered tantamount to a definition of a "mechanical system" (see Sommerfeld [27, p. 49]), On the other hand, this assumption raises the question about the possible existence of physical situations when the hypothesis of ideal constraints— hence the Gauss principle as well—may be inadequate or at least not optimal. A remarkable aspect of Gauss's principle is its independence of the physical origin of the constraints (2.3), Thus by considering specific forms of constraints it becomes possible to provide further justifications for the principle and, in particular, also for the choice of the norm (2.7), which eventually led us to the formulation (2.11). An important special case is that of the holonomic constraints (2.14), which, as we noted in Remark 2.2, possess additional properties. In that case, under the fullrank condition rank DG(x) = k, the submersion theorem, Theorem A.I, ensures that S := G-^O) = {x € R3* : G(x) = 0} is a (31 -fc)-dimensionalsubmanifold of R3e. Hence, with the tangent bundle TS of S, the constrained state space is now <S = R1 x TS, consistent with (2.19), since G is independent of t. Suppose now that the external forces F derive from a potential U — W(x), that is, that F(t,x,x) = F(x) = VU(x), This is the case typically addressed in many introductory texts. Then the system possesses a Lagrangian

and the restriction £|TS of £> to TS is the Lagrangian of the unconstrained motion of the I mass points. Now, Hamilton's variational principle (which is independent of the Gauss principle) requires that at any time t the state ( t , x ( t ) , x ( t ) ) € R1 x TS of the system must be a critical point of its Lagrangian, which means that

By definition of £ this is equivalent to

Thus, at all times, the vector Mx(t) — VW(x(t)) must lie in the orthogonal complement of Tx(t)E — ker DG(x(t)) and, therefore, have the form

4 This contradicts a claim of Caratheodory [6] that nonholonomic constraints cannot be realized as friction forces. The flaw in this argument is discussed in [17].

CHAPTER 2.

THE GAUSS PRINCIPLE FOR MASS POINTS

17

for some Lagrange multiplier A(t) 6 R fe . Thus, for holonomic constraints (2.14) and when the external forces derive from a potential W, arguments independent of the Gauss principle lead to the DAE

which, for this case, is equivalent to (2.18). But the choice of the norm (2.7) was important in deriving the system (2.18), because the specific form of the Lagrange multiplier term depends upon the inner product being used. In fact, any choice of a norm in R3f different from (2.7) will indeed produce a system different from (2.18) and hence different from (2.25), thereby leading to a contradiction to Hamilton's principle. This justifies the choice of the norm (2.7) when the constraints have the form (2.14) and the external forces derive from a potential. Obviously, a general principle should not only be independent of the form of the constraints but also of the form of the external forces. Besides, the minimization problem (2.8) always takes place at fixed admissible state (i, x(t),x(t)), and for such a fixed choice it is impossible to tell whether or not the force vector F(t,x(t),x(t)) comes from a field that derives from a potential. Thus the above arguments support the choice of the norm (2.7) in all cases.

2.3

Variable Transformations

Since the configuration space of a mass point is the ordinary space R3, the use of the natural Cartesian coordinates in R3f to specify the position of the I mass points of our system is convenient in most cases. Yet one may certainly conceive of situations when other variables are being used, e.g., because their choice facilitates calculations. This raises the question about the behavior of the system (2.11) under changes of variables. The same question will present itself again later when mass points are replaced by rigid bodies and, in that setting, it will turn out to be more important because a wide assortment of variables characterizing the configurations of these systems can and will be used in practice. Let T(x,x) := ^(Mx,x) be the kinetic energy associated with the system of mass points. Then the system (2.11) can be written in the form

Here and in what follows, we purposely ignore the fact that the kinetic energy T is independent of x, whence DXT = 0. Consider now a local diffeomorphism $ : H31 —> R3£ under which the external force F, the constraint function G, and the kinetic energy T are transformed into the functions

18

PATRICK J. RABIER AND WERNER C. RHEINBOLDT

Then a straightforward calculation5 shows that

or, equivalently, that

Thus, by setting x — $(£)> % = £*$(£)£ m (2-26), and by multiplying both sides by D$(£)T, we obtain

where we have used the earlier noted equivalence between G = 0 and G — 0 along with Z?jG(U,0 = D ±G(t,$(S),D3>(^)D$(S). This shows that the system (2.26) is almost invariant under the change of coordinates x = $(£)> except for the need to multiply F(t,£,£) by D(£) T . Thus, even in cases when both the kinetic energy of the system and the constraint can be expressed directly in terms of the variable £, that is, without any explicit knowledge of the local diffeomorphism x — 3>(£)i this local diffeomorphism is nevertheless needed to obtain the termD$(0 T F(t,e,0An important exception arises when the external forces F derive from a potential function U : R3^ —> R1, that is, when F := F(x) = VU(x). In that case, we have %) = F($(0) = VW(*(0). and hence ^*(0T^(0 = V%) where %) := W(*(0). In fact, with the Lagrangian £(x,x) := T(x,x) —U(x), the system (2.26) becomes

Thus, with £(£,£) := £($(£), £>$(£)£), the system (2.28) has the form

This shows that when both the Lagrangian and the constraint can be expressed directly in terms of the variable £, then the system (2.29) requires no explicit knowledge of the local diffeomorphism x = $(£). This is a very convenient and widely used fact in the case of conservative systems, but it is essential to note that, in general, it is not valid in the nonconservative case. In other words, the treatment of a nonconservative problem requires the use of the "Cartesian" variable x or, equivalently, of the explicit form of the transformation x = $(£)• 5

See section 4.5 for further details.

Chapter 3

The Configuration Space of a Rigid Body 3.1

Rigid Body Representation

A rigid body is thought of as being composed of a continuum of mass points that are constrained not to move relative to each other. This assumption reflects the observation that, while no actual body is ever perfectly rigid, its deformations are often negligibly small. In order to specify the geometry of a rigid body B in R3, a body-fixed, orthonormal coordinate system (frame) (eg,e|,eg) is chosen in B with its origin at the center of mass (centroid) 75. In addition, a time-invariant orthonormal reference frame of R3 is introduced (e.g., the frame defined by the natural basis vectors e1, e2, e3 of the space). At any time t, and in terms of the reference frame, let u(t) e R3 denote the position of fBand

the linear operator transforming the body-fixed frame into the reference frame, that is, RB -- e1, i — 1,2,3. Here, as usual, GL(3) denotes the general linear group of R3,(t)el that is, the group of all linear isomorphisms of R3, and O(3) is the orthogonal group of R3. Then, in the reference frame, any point P of B at time t has the coordinates

involving a unique vector y = (yi,2/2j2/3) T 6 R3i which has as components the coordinates of P in the body-fixed frame. Therefore, if XQ e R3 represents the coordinates in the reference frame of the body-point P at some time t0, then the position of P at time t in the reference frame is given by

19

20

PATRICK J. RABIER AND WERNER C. RHEINBOLDT

Since the motion of B in R3 precludes reversals of the orientation of the body-fixed frame, the determinant of the operator R(t) e 0(3) must have constant sign at all times t. It is no restriction to assume that det R(t) = I and, therefore, that R(t) 6 50(3), where

is the special orthogonal group of R3 or, more simply, the rotation group of R3. Thus, at any time t the geometric location of B is fully known when the pair ( u ( t ) , R ( t ) ) € R3 x 50(3) is given. In other words, R3 x 50(3) constitutes the configuration space of the unconstrained motion of a rigid body in R3. Because the product of any two operators in 0(3) (or 50(3), respectively) is again in 0(3) (or 50(3)), and the same is true for their inverses (transposes), both 0(3) and 50(3) are subgroups of (7Z/(3) under multiplication. As an open subset, GL(3) is a submanifold of £(R3) and multiplication and inversion are smooth maps on GL(3). Naturally, these operations will remain smooth when restricted to 0(3) and 50(3). Both 0(3) and 50(3) are three-dimensional submanifolds of GL(3). For 0(3) this follows from the submersion theorem, Theorem A.I. In fact, the mapping H : £(R3) >-+ £5(R3),6 H(X) = X^X - I3, is smooth, and satisfies DH(X)Y = YTX + XrY. If Q € H~l(0), that is, Q € 0(3), then, for any Z & £S(R3) we have DH(Q)Y = Z with Y = ^QZ. Thus the derivative of H is surjective on the zero set 0(3) of H, and hence 0(3) is a three-dimensional submanifold of £(R 3 ) since dim £(R3) = 9 and dim £S(R3) — 6. As an open subset of 0(3), the same is true for 50(3). Actually, 50(3) is the connected component of 0(3) containing the identity 1% (see, e.g., Bredon W). It turns out that, topologically, 50(3) has a nontrivial, "twisted" structure that causes various problems, including numerical ones. This structure precludes 50(3) from being embedded in R 4 . It is possible to embed 50(3) into R6 (see Whitney [33]), but, as noted earlier, this as well as other similar embeddings do not appear to be computationally efficient. Only the canonical embedding 50(3) <—> R9 is easy to use in practice, but, of course, at the expense of tripling the dimension. An alternative is to use charts on the three-dimensional manifold 50(3), but this option comes with the necessity of monitoring which chart is being used at any given time and thereby creates bookkeeping problems when a system of several rigid bodies is considered. Another possibility may be the application of the well-known fact that the matrix representation of any Q 6 50(3) (in the canonical basis of R3) can be written as a product

involving the Eider angles QI, a^, 0:3. Note that this does not amount to the use of a chart. In fact, given Q, the three angles are not always uniquely determined since, for 6

As usual £ s (R n ) C £(R n ) denotes the n(n + l)/2-dimensional linear subspace of all self-adjoint operators where Rn is equipped with its canonical Euclidian structure.

CHAPTER 3.

THE CONFIGURATION SPACE OF A RIGID BODY

21

example, for a^ = 0 there are infinitely many different choices of a\ and 0:3 producing the same rotation. Nevertheless, provided these limiting cases are taken into account, the decomposition can be implemented by means of Givens rotations (see, e.g., Golub and Van Loan [12, p. 209]). The utilization of this approach does not appear to have been explored in the mechanics literature.

3.2

The Use of Quaternions

In order to overcome the indicated numerical difficulties we show now that, modulo a simple convention, the configuration space R3 x SO(3) may be replaced by R3 x S3 where S3 := {x € R4 : ||x||2 = 1} is the unit sphere in R 4 for the Euclidian norm || • ||2This requires establishing a relationship between S3 and SO(3), a task greatly simplified by the use of elementary properties of quaternion algebra. In the complex matrix ring C 2 x 2 consider the four linearly independent matrices7

which satisfy the relations J2 = K2 = L2 = -I, JK = -KJ = L, JL = -LJ = K, KL = -LK = J. With these matrices, we form the four-dimensional real linear space H consisting of all 2 x 2 matrices of the form

If Y = y»I + yi J + ysK + y\L is another matrix in H, then

From this it follows readily that H is not only a four-dimensional real linear space but also a noncommutative algebra, that is, an associative, noncommutative ring under matrix addition and multiplication. It is called the (Hamilton) quaternion algebra. The definition (3.4) suggests that we identify H with R 4 . More specifically, we shall use R1 x R3 ~ R 4 and, in line with (3.5), define on that space the (quaternion) product

where x x y 6 R3 is the standard vector (cross) product on R3. Clearly then, R1 x R3 becomes a noncommutative algebra that, with some abuse of language, we again call H. The "star" notation for quaternion multiplication is not standard but is necessary here to avoid future ambiguities. 7

In the physics literature the matrices (3.3), multiplied by — i, are called the Pauli spin matrices.

22

PATRICK J. RABIER AND WERNER C. RHEINBOLDT

Clearly, (1,0) € H is the unit of the multiplicative group. Moreover, since the product (xo,0) * (yo,y) = (xoyo,xoy) is equivalent to scalar multiplication, it is natural to call the set of elements of the form (xo,0) the real axis of H and to identify (xo,0) with the real number XQ. The "complementary" elements (0,x) £ H are often called pure quaternions. For that reason vectors in R3 are identified with pure quaternions via the canonical embedding R3 ~ {0} x R3 c H, just as real numbers are identified with a subset of C via the canonical embedding R ~ R x {0} C C. Note that by (3.6) we have p* q = ( — ( x , y ) , x x y) whenever p — (0,x) and q — (0,y). Hence

For any p — (XQ,X) 6 H, the conjugate element is p = (x0,-x). Obviously, this conjugation operation satisfies the rules p + q — p + q, p~*~q = q * p for all p, q £ H. Moreover, by (3.6) we have p * p = (xjj + ( x , x ) , 0 ) and the nonnegative number \p\ = (XQ + (x, x)) 1 / 2 is called the modulus of p 6 H. This also shows that any nonzero element p has the multiplicative inverse

and hence that H is a (skew) field. Clearly, when the vector space R1 x R3 underlying H is identified with R4 then the modulus corresponds to the Euclidian norm of R4. Accordingly, S3 = {p e H : \p\ = 1} is called the unit sphere of H. For any p € S3 we introduce the real linear mapping

It is easily checked that

and, in particular, that p*q*p = (2/0,0) for q = (yo,0), and p*q*p 6 R3 for q £ R3. In other words, under the mapping (3.9) the real axis and the space of pure quaternions are both invariant. Accordingly, for any p € S3 we may introduce the real linear mapping From (3.9) it follows for p € S3 that

Thus, by the definition (3.11), we have \Rpq\ = \q\ for p € 53 and q 6 R3, which means that Rp 6 £(R3) is orthogonal and hence belongs to 0(3). Since p 6 S3 can be connected to 1 € S3, Rp can be connected to RI — I3 £ SO(3), which, by the earlier noted connectedness of SO(3), implies that Rp € 5O(3). From (3.10) we conclude that, for any p = (XQ,X) 6 S3 and q € R3, the mapping .Rp € 50(3) satisfies

CHAPTER 3.

THE CONFIGURATION SPACE OF A RIGID BODY

23

If the natural basis vectors e1, e 2 , e3 of R3 are substituted for q in (3.12), then we obtain the matrix representation of Rp

which is frequently encountered in mechanics books. It is typically derived by means of the following geometric characterization of the rotation defined by the vector p. For any p G 53 we may introduce the angle a € [0, TT] and unit vector v € R3 by

Then we have XQ = cos ^a and x = usin |a, and Rp represents a rotation about an axis defined by v, with a rotation angle a, that is measured positive in a counterclockwise direction in the plane perpendicular to v. In this connection the four components of p e S3 are often called Euler parameters (see, e.g., Haug [14]). Of course, they should not be confused with the earlier mentioned Euler angles. Some authors apply instead the term Cayley-Klein parameters (see, e.g., Condon [8]). For any given operator A € 50(3) a vector p = (XQ,X] 6 S3 can be explicitly calculated such that Rp = A. In other words, the mapping

is surjective. A nonconstructive proof will be given in Theorem 3.3. Here we use this result to show how p can be calculated from A. To begin with, identify A with its matrix representation (a.^) in the canonical basis of R3. By substituting 1 — x$ + x\ + x\ + x| in each of the diagonal elements of (3.13) we obtain for the squares XQ, x\, x2,, x\ the linear system

The 4 x 4 matrix on the left is its own inverse modulo the factor 1/4; whence

All four terms on the right are nonnegative. In fact, with A\ = A G 50(3), the three matrices AZ = A diag(l, -1, -1), A$ — A diag(-l, 1, -1), A\ — A diag(-l, —1,1) also belong to 50(3). Evidently then, (3.16) can be written as x\ = (1/4)(1 + Tr(Ajt)), fc = 0 , . . . , 3 .

24

PATRICK J. RABIER AND WERNER C. RHEINBOLDT

Now recall that the eigenvalues of any matrix B € SO(3) consist of 1 and a pair of conjugate complex numbers of modulus one, whence Tr(S) = 1 + 2 cos 9 for some 9, and, therefore, 1 + Tr(B) = 2 + 2cos0 > 0. This proves that (1/4)(1 + Tr(A k)) > 0, k — 0 , . . . , 3, and thus that the system (3.16) determines XQ, x\, x2, £3 up to their signs. At least one of these components must be nonzero and, once its sign has been chosen, the sign of the other nonzero components can be determined by checking the consistency with the as yet unused relations deriving from (3.13) for the off-diagonal elements of A. By combining symmetric terms of A in (3.13) we obtain, for any k, j = 1,2,3, k ^ j, and the corresponding complementary third index I ^ k,j, the relations

Therefore, if, say, x^ ^ 0 for some k = 1,2,3 and the sign of x^ has been chosen, then the signs of Xj, j ^ k, follow directly from the relation (3.17(i)), while that of XQ is provided by (3.17(ii)). Analogously, if XQ ^ 0 and the sign of XQ has been chosen, then the signs of the other three components are determined by the relations (3.17(ii)). Clearly, this procedure is incomplete since it does not guarantee the simultaneous solvability of equations (3.16) and (3.17). In fact, since (3.16) already determines the components Xi up to their signs, it might be the case that equations (3.17) arising from equating the off-diagonal entries lead to an unsolvable system, and it is by no means clear that it is not so. The reason that this does not occur is due to the surjectivity of the mapping p H-» Rp. More specifically, in Theorem 3.3 below we show that for any A e SO(3) there are exactly two vectors p 6 S3 such that
On H multiplication on the left by a fixed p € H defines the real linear operator

This mapping is bijective for p ^ 0 since, for any r e H, we have p * q = r if and only if q = (l/\p\)p * r. Furthermore, it follows at once from (3.6) that

CHAPTER 3.

THE CONFIGURATION SPACE OF A RIGID BODY

Lemma 3.2. The mapping (p of (3.15) is a local diffeomorphism

25 of 53 to 50(3).

Proof. In order to prove that the derivative (tangent map)

is invertible, it suffices to show that it is one to one. It follows from (3.11) that for p £ 53 and any r £ TPS3, D(p(p)r is the real linear mapping

If D(p(p)r = 0 for some r € TP53, then r * q * p — —p * q * f for every q £ R 3 . Since q — —q, this means that p* q * f = p* q*f and therefore that p * q * f must be on the real axis of H for every q & R 3 . By (3.20) we have p * q £ Tp53, and since the left multiplication operator (3.19) is bijective, this implies that s * f must be real for every s £ Tp53. Hence the real linear mapping

has rank at most one. But, as in the case of (3.19), we see that for r / 0 the mapping (3.21) is injective and hence has real rank three. This is a contradiction; whence r = 0. D Theorem 3.3. The mapping
A(t) £ 50(3) with A(0) = /, A(l) = A. By a standard continuation argument, using ip and the connectedness of S3, we can "lift" this path to a (smooth) path t i—> p(t) £ 6'3 such that 50(3) of (3.15) induces a two-sheeted covering

between the tangent bundles. More precisely, we have $(p',r') = $(p, r) for (p,r) £ T53 if and only if (p1, r') = ±(p, r).

26

PATRICK J. RABIER AND WERNER C. RHEINBOLDT

Proof. The relation (p',r') = $(p,r), then ip(p) = ip(p') and D(po)- Since, by Lemma 3.2, if in (3.15) is a local diffeomorphism, the curve R(t) in SO(3) induces a unique curve p(t) in S3 such that R(t) = (p(p(t))(= R p(t)) and p(to) = PQ. Thus, the curve (u(t),p(t)) in R3 x S3 characterizes the motion in precisely the same manner as does the curve (u(t), R(t))- In other words, as announced at the beginning of this section, we can indeed use instead of R3 x 50(3), as the configuration space of the rigid body. Evidently, Co is a submanifold of R3 x R4, but, since by Theorem 3.3 the curve (u(t), — p(t)) also characterizes the same motion, it always has to be understood that both (u,p) € Co and (u, — p) € Co correspond to the same configuration of the body. With this, (t,x(t),p(t),x(t),p(t)) is the state of the body at time t and hence, by Definition A.4 of tangent bundles and Corollary 3.4, the state space is

where, again, it is understood that (t, (u,p), (u,p)) and (t, (u, —p), (u, —p)) inR 1 xT(R 3 x S3) correspond to the same state. Remark 3.2. Let $ be the two-sheeted covering (3.22) between TS3 and TSO(3). Then for any mapping / defined on TSO(3), it follows from Corollary 3.4 that the composite map F = /o$ on TS3 satisfies F(-p, -r) = F(p, r) for p € S3 and r € TPS3. Conversely, if F is a smooth mapping defined on T53, then it is straightforward to check that this condition is sufficient for F to factor as a smooth mapping on TSO(3). Since T(R3 x S3) = R3 x TS3 and T(R3 x 50(3)) = R3 x TSO(3), this shows that a smooth mapping F defined on TS3 factors as a smooth mapping on T(R3 x 5O(3)) if and only if

holds.

3.3

Planar Rigid Bodies

Whenever possible, the modeling of a body as a two-dimensional object should always be expected to bring simplifications, although the absence of a third dimension may

CHAPTER 3.

THE CONFIGURATION SPACE OF A RIGID BODY

27

sometimes create ambiguities. As regards rigid body motion, the topic of interest in this monograph, no such ambiguity arises when not only is the body two-dimensional but also its motion is confined to the Euclidian plane R2. The terminology "planar rigid body" will henceforth be used to summarize both hypotheses. We shall reexamine the procedure developed earlier in this chapter and show that it does indeed lead to considerable simplifications. Similar simplifications will be highlighted in the subsequent chapters. We emphasize that it is important to revisit the procedure entirely, as opposed to specializing the earlier results to the case of planar motion. Otherwise, no substantial simplification occurs (see Remark 3.3 below). In other words, there is no particular value in viewing the set 50(2) of plane rotations as a subset of 50(3) and applying the "induced" 50(3) theory. Indeed, the noticeable simplifications arise precisely from the fact that the universal covering of SO(2) (i.e., R 1 ) is fundamentally different from the universal covering of 50(3) (i.e., 53). Therefore, while the procedure remains the same, namely, the replacement of the configuration space by its universal covering, an independent treatment is needed to obtain a truly simpler theory. As in the three-dimentional case, every point P of a moving planar rigid body B has coordinates x(t) = (^i(0> xi(t)) in some (orthonormal) reference frame given by x(t) = u(t) + R(t)y, where u(t) — (ui(t), u^(t)) is the position of the center of mass 75, y = (2/1,2/2) are the coordinates of P in a chosen (orthonormal) body-fixed frame, and R(t) is the rotation transforming the body-fixed frame into the reference frame. As a result, the configuration space of B is the space R2 x SO(2). In section 3.1 we saw that some difficulties arising from the structure of 5O(3) could be easily bypassed by replacing the configuration space R3 x 50(3) by R3 x 5s, at only the expense of having to identify the two representations (u,p) and (u, —p). While it may be argued that the geometry of 50(2) (i.e., the unit circle 51) is far less complicated than that of 50(3), the geometry of the universal covering of 50(2), namely, R1, is unquestionably even simpler. The occurrence of the real line as the universal covering space of 5O(2) is due to the well-known fact that in any fixed orthonormal frame a rotation is represented by a 2 x 2 matrix of the form

The mapping

provides the desired covering (after identifying the elements of 50(2) with their matrices in a given frame). Note that the covering (3.26) has infinitely many sheets since ip(a + 2kir) = ip(a) for every a e R1 and k € Z. This contrasts sharply with the covering of 50(3) by 53 being two sheeted and stresses the fact that the two coverings are not related.

28

PATRICK J. RABIER AND WERNER C. RHEINBOLDT

The foregoing remarks show that the configuration space of a planar rigid body can be chosen as R2 x R1 provided that all the configurations (u,a + 2fc7r), k € Z, are identified. Naturally, as argued in Corollary 3.4, we find that the covering (3.26) induces a covering given by il>(a,r) — (i/>(a),'!/>'(a:)r) for every pair (a,r) 6 R1 x R1. The converse is also true: If tp(a,r) — i/>(a',r'), then certainly a' = a + 2/c?r for some k € Z. In that case, tp'(a) — ip'(a'); whence ip'(a)r = i/)'(a')r' if and only if tjj'(a)r = t()'(a)r'. Since

this implies r = r'. This says that the state space So = R1 x T(R2 x SO (2)) of the planar rigid body B can be identified with S0 = R1 x T(R2 x R1) = R1 x (R2 x R1) x (R2 x R),

provided that the states (t,(u,a),(ii,a}) and (t, (u, a + 2fc?r), (u, d)) are identified for every k £ Z. The analogue of Remark 3.2 is that a mapping F defined on R1 x (R2 x R1) x (R2 xR 1 ) represents a mapping on So if and only if F(t, (u,a), (r, £)) = F(t, (u,a+ 2fc7r), (r, C)) for every k e Z and every (t, (u, a), (r, £)) € R1 x (R2 x R1) x (R2 x R1). In other words, F must be 2?r-periodic in the a variable. Remark 3.3. It is informative to check what is obtained when the planar problem is viewed as a special case of the three-dimensional theory. Recall that (3.13) represents the matrix of the rotp e SO(3) induced by the vector p = (x0,xi,X2,x3)T € S3 ination R the orthonormal frame e1, e2, e3. Suppose that Rp € SO(2); i.e., Rp represents a rotation in a given two-dimensional subspace of R3, say, spanje^e 2 } to fix ideas. This implies that, in (3.13), x\xz — XQX^ — x^x^ + XQXI — 0, which happens if and only if x0 = £3 = 0 or x\ = #2 — 0- However, the matrix (3.13) obtained in the first case does not induce a rotation in span{ei,e2J since its restriction to this plane has determinant —1. Hence, elements of SO(2) viewed as the group of rotations of span{e1,e2} are obtained if and only if p = (x0,Q,0,X3)r and p E S3, i.e., x% + x\ — 1. In other words, the rotations of spanje1^2} correspond to the points p in the unit circle C of the (xQ,X3}-pla,ne. This correspondence is not one to one since p and — p always induce the same rotation. Actually, if p = (cos/3,0,0,sin/3) T , it follows from (3.13) that the matrix of Rp restricted to spanje^e 2 } is

i.e., the rotation of angle 2/3. This shows that the mere restriction of the three-dimensional theory to the two-dimensional case replaces SO(2) = Sl by the circle C = S1, and its only effect is to (needlessly) "double" the domain of the variable without bringing any concomitant improvement. D

Chapter 4

Unconstrained Rigid Bodies The topic of this chapter has a venerable history, but our presentation may be found less standard than might at first be expected. There are several approaches to the study of the unconstrained motion of a rigid body. An older, classical one, used in many textbooks of mechanics, leads to the well-known equations (1.1) of motion involving the angular velocity vector. As noted earlier, it is rarely pointed out explicitly that this standard formulation looks like an ODE but, as will be discussed below, may not be one. More modern, but also more abstract, is the "mechanics on Lie groups." developed, e.g., in Abraham and Marsden [1]. This approach uses the "natural" configuration space R3 x 50(3) and does characterize the motion by means of a second-order ODE on R3 x 50(3). However, for numerical purposes this ODE appears to be suboptimal, especially when the problem of interest concerns not just one but n > 2 rigid bodies, and hence the configuration space becomes (R3 x 50(3))". Here, we show that the unconstrained motion of a rigid body can be characterized by a second-order ODE on R3 x 53, whose formulation involves only readily available quantities. The next chapter extends this theory to the motion of a constrained rigid body, in which case we obtain, analogously, an equally explicit DAE on R3 x 53 or, alternatively, on R3 x R 4 . The latter DAE corresponds to the formulation used by Haug [14], but our treatment incorporates various aspects of conceptual and practical value and, in particular, the use of more general nonholonomic constraints. Moreover, it provides a tight connection with the aforementioned abstract Lie group approach.

4.1

Background

As seen in section 3.1, the motion of a rigid body is characterized by a curve (u(t), R(t)) in R3 x 50(3). Differentiation of R(t)TR(t) = I gives RT R + RT R = 0, which implies that RrR is skew symmetric. For every skew-symmetric K e £(R3) there exists a unique vector w € R3 such that Ky = w x y for all y € R3. Hence, in the canonical basis, K 29

30

PATRICK J. RABIER AND WERNER C. RHEINBOLDT

has the representation

Thus, in our case we have

and the vector w(t) is the angular velocity vector of the rigid body motion at time t. For later purposes, it is important to recall that, in section 3.1, R(t) represents the rotation transforming the body-fixed frame into the reference frame. Hence, in the relation R(t)T R(t)y = y for y € R3, the components of y should be viewed as the coordinates of a point in the body-fixed frame. The same interpretation, namely, that components of vectors in R3 represent coordinates in the body-fixed frame, must then be used in the relation (4.1). As a result, the components of w(t) in (4.1) correspond to the coordinates of the angular velocity vector in the body-fixed frame. In contrast, the differentiation of the identity R(t)R(t)^ = / leads, by the same arguments, to

and the components of the vector u?(£) G R3 represent the coordinates of the angular velocity vector in the reference frame. This follows from the relation

which, in turn, is easily obtained from the fact that the cross product commutes with rotations; that is,

These observations lead to a relation that will be useful in Chapter 9. Let y denote the coordinates of a fixed point P of the body in the body-fixed frame. Then the position x(t) of P at time t in the reference frame is given by (3.1). Since y is time independent, the velocity of P in the reference frame will be ±(t) = ii(t) + R(t)y or, because of (3.1),

By using (4.2) and (4.3) this can be written as

As mentioned in the Introduction, traditionally the equations of rigid-body motion are given in the form

Here u = u(t) £ R3 defines the location of the center of mass 75 of the body; the positive definite matrix M € R 3x3 is the mass matrix; the vector Fr € R3 specifies

CHAPTER 4,

UNCONSTRAINED RIGID BODIES

31

the resultant of the external forces; the symmetric, positive definite matrix J € R 3x3 is the inertia tensor; N e R3 defines the moment of the external forces with respect to 7e; and, of course, u> e R3 is the angular velocity vector (4.1). Most texts on classical mechanics explain how the system (4.5) can be derived from Newton's law of motion. A mathematically complete exposition can be found in Arnold [3]. Since (4.5) describes the motion of a single rigid body B, the mass matrix M is simply M — m/3, where m > 0 is the mass of the body and J3 denotes the identity on R3. The inertia tensor J, actually a linear operator, is obtained as follows: Identify R3 with the afrme space equipped with the body-fixed frame and the body B with an open subset £1 of R3. In other words, z = (z1; z 2 , z 3 ) T e R3 corresponds to a point having coordinates z \i z'2-, zz in the body-fixed frame. For z € R3 we set

where p(y) is the mass density of B at y. Since

it follows that J is symmetric. Moreover,

shows that J is also positive definite. From these observations it follows that, in the expressions Jui and Jui occurring in the second equation of (4.5), the vector u> (and hence also w) should be viewed as a vector with components o>i,u>2,w 3 in the body-fixed frame. This is indeed consistent with the remarks made to that effect earlier in this section. Accordingly, the moment N on the right side of (4.5) should be identified with its components in the same body-fixed frame. Suppose that the system (4.5) has been solved and hence the functions u(t) and u(t] are available. Then relation (4.1) can be used to recover R(t). In fact, an equivalent formulation of (4.1) is that R(t)y = R(i)(uj(t) x y) for every y 6 R3, which, of course, amounts to the same relation holding for three linearly independent vectors y. By using (say) the canonical basis of R3, we obtain

which is a linear ODE for R since o>(£) is known. Thus R(t) is obtained by solving this ODE and the motion x ( t ) of the point P of the body located at u(to) + y for t = to is given by x ( t ) = u(t) + R(t)R(to)~ly. In this respect note that by (4.6) and the relation (w x /3)T = —(u x /s) we have R1 — -(u> x 73).RT. By multiplying both sides by R we see that RR1' + RRr = 0 and hence that RRr is constant. Therefore, RRT = I provided that R(to) € 0(3). In fact, since R(to) G 50(3), it follows from the connectedness of 50(3) that R(t) 6 5O(3). Since ( u ( t ) , R ( t } ) e R3 x 50(3) represents the configuration of the rigid body, its state is described by

32

PATRICK J. RABIER AND WERNER C. RHEINBOLDT

and therefore the force Fr and moment N in (4.5) are functions Fr(t,u,R,u,R) and N(t,u,R,u,R) denned on R1 x T(R3 x 5O(3)). In light of (4.1) the angular velocity vector uj(t) at time t depends only upon R(t) and R(t) at the same time. Accordingly, a special case arises when Fr and N depend upon t, u(t), ii(t), and w(£). In that case the system (4.5) is an ODE. But, conversely, the knowledge of u(t) at time t alone does not permit the calculation of R(t). As explained in Remark 4.2, the recovery of the function R(t) from the function u(t) is possible, but the process requires time integration, which is hardly surprising since w(t) depends on the product -R(t) T R(t). In other words, when considered as a system for the determination of the motion of the body in the configuration space R3 x SO(3), (4.5) is an integrodifferential equation, not an ODE. This appears to be the principal reason why often, in the engineering literature, fictitious variables (the so-called quasi coordinates) come into play, which are obtained by integration of the angular velocity vector. These quasi coordinates are difficult to justify mathematically and, at least from a theoretical standpoint, they turn out to be an unnecessary substitute for reformulating (4.5) directly in terms of the configuration variables. As mentioned earlier, Lie group theory and differential calculus on manifolds lead to a formulation of (4.5) as a second-order ODE on R3 x SO(3) whose only shortcoming, besides a higher level of abstraction, is an inadequate consideration of numerical issues. Our approach here bypasses the use of Lie group theory and leads to a simple secondorder ODE in R3 x S3. Naturally, all the ingredients of the Lie group treatment are present, but they are made explicit (at little or no expense to brevity) and hence become rather straightforward. Of course, the existence of a formulation of the equations of motion as a second-order ODE on R3 x S3 is a priori guaranteed by their formulation as a second-order ODE on R3 x 50(3) together with the fact, established in Theorem 3.3, that 53 is a covering space of SO(3). But, instead of pulling back the equations on R3 x 5O(3) to R3 x S3, we derive the desired ODE on R3 x S3 directly from the standard formulation (4.5). This is consistent with our intent of addressing this monograph to an audience familiar with this standard formulation but perhaps not with the abstract Lie group approach.

4.2

A Trivialization of TS3

The possibility of writing the equations of motion as an ODE on R3 x S3, announced at the end of the previous section, depends crucially upon the property that the tangent bundle of S3 is trivial. More generally, it is well known that the tangent bundle of any Lie group is trivial, a feature that follows from the existence of an action of the group on itself by multiplication. Instead of relying upon such an abstract result, we shall construct here an explicit trivialization of TS3. Since a trivialization is needed to write down the ODE and to solve it computationally, there is a clear justification for working out this trivialization carefully. To say that TS3 is trivial means that there is a smooth diffeomorphism TS3 —> 3 S x R3 of the form (p, q) i-» (p,H(p)q), where H(p) 6 £(TPS3,R3) is an isomorphism for every p <5 S3. The reader with limited familiarity with vector bundles may take

CHAPTER 4.

UNCONSTRAINED RIGID BODIES

33

this as a definition. Since TPS3 = {p}x C H, there is reasonable hope of obtaining H(p) as the restriction to {p} of a real linear mapping, still denoted by H(p), from H into R3. In other words, the trivialization question is resolved if a smooth mapping H : S3 -> £(H, R 3 ) can be found such that H(p)l{p}± € GL^p^, R3) for every p e S3. This is exactly the topic of this section and, in fact, we shall construct H : H —> £(H, R3) satisfying the desired property. In light of the earlier remark that tangent bundles of Lie groups are trivial because of the action of the group on itself by multiplication, it should come as no surprise that we first examine properties of multiplication-related operators. Consider, for p 6 H, the real linear operator

so that 7i(p) corresponds to the left-multiplication operator (3.19) with p replaced by p. It thus follows from our remarks in section 3.2 that when p / 0, then H(p) is bijective and, since l~t(p)(p~l * q) = q for any q e H, the inverse is H(p)~l — H(p~l). For any p € S3, it follows from |W(p)
Note that, by (3.6) and with p :— (XQ,X) & H,

whence, in the canonical basis of R 4 . H(p) has the matrix representation

(4.11)

With the natural projection TT : R1 x R3 —> R3 onto the second factor, we introduce the reduced real linear operator

By (4.10) H(p) is given by

and thus its matrix representation (in the canonical basis) consists of the last three rows of (4.11). From (4.13) it follows immediately that

34

PATRICK J. RABIER AND WERNER C. RHEINBOLDT

Moreover, for p ^ 0, we have H(p)q = 0 if and only if p*q is real; that is, x0y-y0x—xxy = 0, which holds exactly if x^y — yox = 0 (recall that x x y is orthogonal to both x and y and x x y = 0 when x and y are collinear). Thus, for p ^ 0, we find that

By (3.6) it is obvious that

and (4.16(i)) implies that If now p, r € H and q € R3, then we have where the second equality follows from q € R3 and the third from (4.9). This holds for every r 6 H so that H(p)Tq — W(p)T
In particular, for p € S3 we obtain H(p)H(p)T = I, which, in turn by (4.15), implies that H(p)TH(p) is the orthogonal projection onto {p}x; that is, where Pp denotes the orthogonal projection onto {p}-1 = TPS3.

4.3

Second-Order ODE of Motion

In this section we establish the desired equations of motion of an unconstrained rigid body in terms of the configuration space Co = R3 x 53 and hence the state space SQ = R1 x T(R3 x S3). For this recall from Remark 3.2 that any mapping on the tangent bundle T(R3 x S3) factors as a mapping on the tangent bundle T(R3 x 50(3)) if and only if the condition (3.25) holds. In particular, we assume this for the resultant force Fr and the moment N acting on the body; that is, we require that

The desired result can then be formulated as follows.

CHAPTER 4.

UNCONSTRAINED RIGID BODIES

35

Theorem 4.1. Let (u(t},p(t)) be a curve in R3 x 53 representing the motion of a rigid body B with mass matrix M and inertia tensor J subjected to the resultant force Fr : SQ —> R3 and moment N : SQ —+ R3 (with respect to its center of mass) satisfying the conditions (4.20). Moreover, let H(p) be given by (4.12) for p € S3. Then the following hold: (i) The angular velocity vector satisfies

(ii) The equation of motion (4.5) is equivalent to the second-order ODE 8 on R3 x S3:

Proof, (i) Since the curve (u(t),p(t)) represents the motion of the rigid body, the rotation R(t) in (3.2) is given by R(t) = if>(p(t)), where
Prom (4.23) and R(t)^ = R(t)~l it follows that (suppressing the explicit dependence on t]

R Rq = p * p * g * p * p + p * p * g * p * p = p * p * < j ' + g * p * p

sincep*p = \p\2 = 1. Because of q € R3 and hence q — —q, we have q*p*p = p*p* q = -p * p * q and thus R1 Rq — 2ir(p * p * q), where, as before, it : R1 x R3 —> R3 denotes the projection onto the second factor. From (4.18) it follows that p*q — H(p)'Tq, so that by the definition (4.12) of H(p),

The curve p(t) lies in S3 and hence the vectors p(t) and p(t) are always orthogonal. Thus p * p € R3 by (4.16(i)) and hence n(p * p * q) = (p * p) x q by (3.7). Next, we have p*p = H(p)p by (4.17), and (4.24) yields

Since R1Rq = LJ x q by definition of the angular velocity vector, the first equality in (4.25) along with (4.14(i)) shows that (4.21) holds. (ii) By (4.21) and the second equality in (4.25) we have ui x q = 2H(p)H(p)Tqfor every q € R3. By letting q = Ju — —2JH(p)p, this gives

Since p g S3 and hence p ^ Q, the operator H ( p ) v is one to one by (4.15). Hence (4.22) implies that 2JH(p)p - 4H(p)H(p)~T JH(p)p = N(t,u,p, ii,p) or, equivalently, by (4.26) 8

It will facilitate comparison with the constrained case in Chapter 5 not to simplify by dividing by 2.

36

PATRICK J. RABIER AND WERNER C. RHEINBOLDT

By differentiating (4.21) we find that u = 2H(p)p + H(p)p = 2H(p)p since H(p)p = 0 by (4.14(ii)). Therefore (4.27) is just the moment equation JCj + w x Jui = N in (4.5). This shows that the system (4.22) implies the equations of motion (4.2). The converse is obtained by reversing the above steps. Equations (4.22) are the analogue of Newton's law for the motion of a rigid body. Actually, they are Newton's law, because they reduce to the system (4.5) which in turn is usually derived from Newton's law itself. Remark 4.1. It must be emphasized that the second equation of (4.22) does not characterize the acceleration vector p but only its tangential component. In fact, by (4.15(iii)) the term on the right is in TPS3 and, on the left side, the operator H(p)'r JH(p) is an isomorphism of TPS3 = {p}1- because, by (4.15(ii)), H(p) has rank three. Since H(p)H(py = /, by (4.19), the left side is also

and therefore determines uniquely the tangential component H(p)TH(p)p of the acceleration vector. Recall indeed that H(p)r H(p) is the orthogonal projection onto {p}T (see once again (4.19)). The normal component of p(t) is already known from the fact that the curve lies in S3. In fact, it equals — \p(t)\2p(t), as follows directly by differentiation of the identity |p(t)|2 = 1. Remark 4.2. The curve p(t) in S3 can be recovered from the angular velocity vector (j(t) as follows: Since \p\ = 1 and hence (p,p] = 0 we obtain from (4.17) that H(p)p = p*p 6 R3. In view of (4.21) and the definition (4.12) of H(p) this implies that Since p € S3 we know that *H(p) is orthogonal, so that

This is a first-order ODE for p. Given p(to) = po € S3, it has a unique solution in R4. This solution lies in S3. In fact, for any p 6 53 and q 6 7r(H) we have and hence the right side of (4.28) induces a vector field on S3. This corroborates the earlier statement that the configuration variable p can be recovered only by time integration of the angular velocity vector. Of course, once p is known, the rotation R in (3.2) is given by R = f ( p ) , that is, by (3.11). Note that (4.28) also shows how initial conditions on p can be obtained from initial conditions on u. Recall that we showed in Chapter 3 how po € S3 can be recovered from RQ € 5O(3) in (3.2). Remark 4.3. The operators related to left multiplication introduced earlier do not suffice to recover the rotation Rp in (3.11). To do so we must consider the right multiplication operator by p; that is,

CHAPTER 4.

UNCONSTRAINED RIGID BODIES

37

which has the matrix representation (compare with (4.11))

For p e H the real linear mapping q e H H-> p * g * p is thus the mapping K,(p)H(p)q. For p 6 -S3 it follows from (3.11) that the rotation Rp is the restriction of this mapping to R3 (identified with the pure quaternions) and hence

with H(p) given by (4.12) and the operator K(p) € £(H,R 3 ) defined by K(p)q = ir(IC(p)q) = -rr(q * p ) , where, as before, ir denotes the projection TT : R x R3 —> R3. The matrix of K(p) in the canonical bases of R3 and R4 corresponds to the last three rows of (4.29). Since that of H(p) consists of the last three rows of (4.11), it is readily checked that (4.30) gives again the formula (3.13) for the matrix representation of Rp. The linearity of H(p) and K(p) with respect to p also yields a convenient formula for the derivatives of Rp with respect to p:

These results are important in concrete applications, and their usefulness will be demontrated in Chapter 9. Note also that, because of Rp = R~l = -Rj, it follows from (4.30) that Rp — H ( p ) K ( p ] T . This decomposition is not the same as (4.30) since H(p) ^ K(p) and K(p) ^ H(p), as the matrix representations reveal at once. D

4.4

The Euler-Jacobi Operator on Manifolds

The material in this section assumes a somewhat deeper background in analysis than the previous chapters. The section may be bypassed by readers mainly interested in the computational aspects of the theory. Our discussion about mass-point systems in sections 2.2 and 2.3 exhibited the longknown fact that the Euler-Jacobi operator Dx — -jrtD± plays a central role in the formulation of equivalent forms of the equations of motion that involve the various energies associated with the system. Since the configuration space of a system of t unconstrained mass points is the space R™, n = 3f, with tangent bundle TR" = Rn x R™, there is, in that case, no need to explain the meaning of (Dx — ^Dx)C(x(t),x(t)) when x(t) is a curve in Rn and L : TRn —> R1 is a smooth functional. Indeed, if £ = £(x,x) where x and x denote arbitrary vectors in Rm (and hence x does not refer to any specific time derivative), then the expressions DXL and D±C are well-defined partial derivatives, and likewise •jjDj;£(x,x) makes obvious sense after the substitution x — x(t), x = x(t), However, the configuration space of a rigid body is a manifold (namely, R3 x SO(Z) or 3 R x S3, depending upon which point of view is taken) and hence the energies associated

38

PATRICK J. RABIER AND WERNER C. RHEINBOLDT

with the rigid body will be functionals defined on the tangent bundle of that manifold. It can safely be anticipated that equivalent formulations of the equations of motion involving these energies will require the use of an operator Dx — ^D±, but, as explained below, such an operator no longer has any obvious meaning. Let M be a smooth m-dimensional manifold and £ : TM —> R1 a smooth functional. We shall once again write C — C(x,x) where x € M and x 6 TXM.. If x is held fixed, the functional x e TXM H-> C(x}±] € R1 is well defined and hence Dx£(x,x) makes sense as the derivative of that functional. Unfortunately, the same reasoning does not apply for defining Dx£(x,x), for it would require fixing x € TXM and considering the partial functional x 6 M >-> £(x,x). This does not make sense since the specification of x € TXM presupposes that x € M itself has been fixed and hence that x can no longer be viewed as a free variable. Thus, the difficulty here is that a given vector x £ TXM does not lie in TXM for x ^ x, nor can it be associated with any vector of TXM in some canonical way,9 even if x is confined to an arbitrarily small neighborhood of x. For M = Rm this difficulty disappears since TxRm = {x} x Rm can be identified with TjRm = {x} x Rm by simply exchanging x and x. This operation is defined for all x,x e Rm without requiring any arbitrary choices (whence it may be referred to as "canonical"). In spite of the problem just mentioned, it is possible to make sense of the entire Euler-Jacobi operator Dx — j^Dx without having to define Dx and jiDx separately. As is often the case when dealing with features attached to manifolds, we introduce a definition involving the choice of a chart and next verify that the result is independent of that choice. Let x = x(t), t e J7", be a curve on M. defined on an open interval J C R1 and t0 € J & fixed point. After shrinking J if necessary, it is not restrictive to assume that x(J] C Um where Um C M. is some open neighborhood of x(to). In particular, let Um be the domain of a chart ~1 : Um -» Rm (so that 0 : 4>~l(Um) -» M is a parametrization of M. near x(to)). With any given v € TX^0)M. we set

where

for every (£,£) £ 4>-l(Um) x R m . In order to verify that the right side of (4.32) is independent of the parametrization <j> we must compare the results obtained from two different choices fii and fa. Equivalently, we must show that (4.32) is a formula (as opposed to a definition) when M (and hence Um) is an open subset of Rm (and hence DXC and DXL have their usual meanings as partial derivatives), and <j> = <j>^1 o fa is the change of coordinates. This verification is routine, but we include the details for completeness. 9

It is always possible to exhibit a bijective correspondence between TXM and T^M., but, in general, this cannot be done without involving arbitrary choices such as charts, bases, etc.

CHAPTER 4.

UNCONSTRAINED RIGID BODIES

39

In particular, with

this becomes

Since Di£+(S,£)w = D±£(($,D(S)S)D(t)w, and differentiating with respect to t, that

we find, by setting f - ^(t), C - £(*),

Now the right side of (4.32) is obtained by subtraction of (4.34) for t = t0 from (4.33). Clearly the terms containing D2(f> cancel and the result is exactly (4.32), as claimed. Note that by its very definition (Dx£ — -^D±£)(x(to),x(to')) is an element of the dual T*,t -.M of Tx(to)M. Note also that the same procedure does not permit us to define Dx£(x(to),x(to)) since the cancellation of the terms containing Z?20 is crucial to the above justification of (4.32). This may be a little confusing since, in light of our remarks above, Dx£(x,x) is well defined and hence -^Dx£ is also well defined after the substitution x = x(t), x = x(t) is made. It thus seems possible to define Dx£(x,x) as the sum [DX£--^DX£] + ^D±£. The reason why this is meaningless is as follows: For x € M and x <= TXM, we have Dx£(x,x) e £(TXM,T&) = TXM. Hence, the curve DxC(x(t),x(t)} is a curve on T*.M and we have (^Dx£)(x(to),x(to)) € T( x ( (o ) j , i .( (o ))(T*A / l). This cannot be added to Dx£ - £tDx£}(x(t0),x(t0)) e T*(tg)M. In other words, the term ^Dx£ in the expression [Dx£ — ^Dx£] does not have its natural meaning. In fact, within this expression, j-tD±L has no more individual meaning than Dx£ does, and only Dx£ — -j^Dx£ makes sense. While the definition (4.32) does not require any embedding of the manifold M, in some Euclidian space R n , the existence of such an embedding provides an alternate way of calculating (Dx£ — -^Dx£}(x(t0),±(to)), at least when an extension of £ to TR" = Rn x Rn is readily available. In fact, by using the affine space x(to) + Tx(<0).A/f to specify a parametrization of M. near x(to), we may arrange it so that £(to) — %(to)> D(£(to)) = I, and hence £(io) = x(to) and v = w in (4.32). The two curves x(t) and £ ( t ) , t €. J~, viewed as curves in R", have contact of order two at to; that is, lk(*) - CWII = O((t ~ *o) 2 ) - This is easily seen to imply that the right side of (4.32) coincides with Dx£(x(to),x(to))v — ^[Dx£(x(t),x(t))v]^t=to, whereEEEenotes now the

40

PATRICK J. RABIER AND WERNER C. RHEINBOLDT

extension of C to R" x R" and DXC and DXC, refer to the partial derivatives of L. In other words,

where on the right side v (viewed as a vector of R" rather than of T x ( to )M C R") is held fixed during the ^-differentiation. Formula (4.35) is often easier to use than the original definition (4.32), although it is, of course, limited to the case when M. is a submanifold of R n . Remark 4.4. The Euler-Jacobi operator Dx - ^D± is historically well known to arise naturally in the variation of the integral

when the curve x(t] with fixed endpoints x(a), x(b) is varied on M. This fact will not be used later on, but, in light of the results in the subsequent section, it is the link between the equations of motion and Hamilton's variational principle. For simplicity we clarify the occurrence of (Ac - ^D±)C only in the case when M = Rm. The set of (say) C2 curves from [a, 6] into Rm is a Banach space E in the obvious way, and the subset consisting of all curves with fixed endpoints x(a), x(b) is an affine subspace of E. The derivative of the integral (4.36) with respect to the curve x(t) is the mapping

If attention is confined to curves with fixed endpoints, then y lies in EQ = {y € E : y(a] = y(b) = 0}, and hence integration by parts yields

Thus the variation (or directional derivative) of the integral (4.36) is given by

Remark 4.5. If £ does not depend upon x (and hence is a functional on M), then, in (4.32), L* does not depend upon £ and we have (DXC - ±Dx£)(x,x) = D£(x), the derivative of L : M —> R.

CHAPTER 4.

4.5

UNCONSTRAINED RIGID BODIES

41

Energy Formulation

With the help of the material developed in the previous section, we provide now a formulation of the equations of motion (4.22) involving the kinetic energy of the rigid body. As in section 4.3 the notation <S + R1 x T(R3 x S3) will be used. Theorem 4.2. Let ( u ( t ) , p ( t ) ) be a curve in R3 x S3 representing the motion of a rigid body B with mass matrix M and inertial tensor J subjected to the resultant force Fr : So —> R3 and moment N : SQ —* R3 (with respect to the center of mass) satisfying the conditions (4.20). Then (i) the kinetic energy of B is given by

(ii) the equations of motion (4.22) are equivalent to

for every (z,() € T (u , p) (R 3 x S3) = R3 x {p}±. Proof, (i) As is well known, the kinetic energy is given by T :— ^(Mu, it) -I-^{JLJ.LJ}; whence (4.37) follows by using for w the expression (4.21) obtained in Theorem 4.1(i). (ii) Let t0 be fixed and ( 2 ,0 e T (u(to)Xto)) (R 3 x S3) = R3 x {p^o)}1- The kinetic energy T of (4.37) makes sense not only for (u,p, u,p) in T(R3 x S3) but also in T(R3 x R 4 ) = (R3 x R 4 ) x (R3 x R 4 ). Hence, with £~T,x = (u,p), x = (u,p), and v = (z,C), the left side of (4.38) for t = t0 can be calculated by (4.35), holding (z, <) 6 R3 x {p^o)}^ fixed during the t-differentiation as explained in section 4.4. Since io is arbitrary, we may return to the simpler notation to — t, keeping in mind that ^-differentiation should not be performed on the pair (2, C) even though £ G {p(t)}'L. Because H(p) is real linear in p and J is symmetric, we obtain (suppressing the explicit dependence on t)

From (4.14) it follows that H(p)p = 0, H(p)C, = -H((,)p, and H(p)p = -H(p)p. Thus the above relation becomes

42

PATRICK J. RABIER AND WERNER C. RHEINBOLDT

and equation (4.38) may be rewritten as

for every (z, C) € R3 x {p}-1. Clearly this shows that

for every £ 6 {p}~L- Since the first equations are the same in both the systems (4.22) and (4.39), it suffices to prove that the second equations are identical as well. In (4.39) we have

By (4.19), H(p)TH(pis the orthogonal projection onto {p}x. Hence, since £ 6 {p}X, we have £ = H(p)TH(p)£ and therefore

Together with (4.40) this shows that the second equation in (4.39) is just

for every £ 6 {p}±• Since H(p)T maps into {p}x by (4.18), this relation is equivalent to

which is exactly the second equation in (4.22). Theorem 4.2 is especially useful when the forces applied to the rigid body are conservative. This means that the resultant force Fr = Fr(u,p) and moment N = N(u,p) derive from a potential U : R3 x S3 —> R1 via the relations

Here the notation VpU(u,p) refers to the gradient of the functional p & Ss >—> U(u,p) € R1, so that Vp U(u,p) e TS3. If U extends as a functional UE : R3 x R4 -* R1, then Vp U(u,p) in (4.41) is the orthogonal projection of the gradient of this extension with respect to p; that is, Vp U(u,p) = H(p)TH(p)Vp UE(u,p}. Remark 4.6. Compliance with the "factorization" condition (4.20) requires that U(u, —p) = U(u,p) for every pair (u,p) € R3 x S3, and then U factors as a functional on R3 x 5O(3).

CHAPTER 4.

UNCONSTRAINED RIGID BODIES

43

In the case of (4.41) the equations of motion (4.38) simply read

A further simplification arises from noticing that, since U does not depend on u and p, we have by Remark 4.5 that

and hence equation (4.38) takes the form

where £ is the Lagrangian

and, of course, the kinetic energy T is given by (4.37). The usefulness of the formulation (4.42) of the equations of motion in the conservative case is due to the fact that the operator .D(u,p) — 3£-D(«,p) is independent of the specific chart used for the calculations. This independence was seen to be involved in the very definition of the operator in section 4.4. The practical value of this property is, of course, that it makes it possible to find the correct equations of motion without having to use the configuration variables u and p. In fact, any set of variables specifying u and p can be chosen instead, and there is no need to know the explicit transformation relating (u,p) and the chosen variables. This opportunity is widely used in the examples given in mechanics textbooks. As we shall see, this simplification also remains available for constrained problems, but, as for systems of mass points discussed in Chapter 2, it will be valid only in the case of conservative forces. Otherwise, the formulation (4.22) must be applied.

4.6

Planar Rigid Body Motion

The aspects discussed in this chapter reduce to very little in the case of planar rigid body motion. In particular, the linear structure of the configuration and state space, that is, R2 x R1 and R1 x (R2 x R1) x (R2 x R 1 ), respectively (see section 3.3), makes it unnecessary to develop any material along the lines of section 4.4. The rest is very much standard and elementary, but we give a quick exposition to highlight a few aspects that will be called upon later. When R(t) is a curve in 5O(2), relation (4.1) remains valid for y e R 2 provided that 2 R is embedded in R 2 x R1 and the vector ui(t) has the form (0,0,u;o(£)) T . In that case, the third component of u>(t) x y is always 0 (since y = (2/1,2/2,0)). This property merely reflects the fact that SO(2) can be viewed as a subset of 5O(3). After choosing an orthonormal frame in R 2 , the rotation R(t) can be identified with (p(a(t)) where a ( t ) is a curve in R 2 and (p is the covering (3.26) of SO(2) by R1. It

44

PATRICK J. RABIER AND WERNER C. RHEINBOLDT

is straightforward to check that (4.1) holds if and only if u(t) = (0, 0,d(t)) T , i.e., if wo(t) = a(t). This fully justifies the terminology "angular velocity" for w 0 (i). Since w(t) = (0,0,wo(i)) T and hence <j(t) = (0, O,WO(*)) T I RHW vector Ju> in (4.5) has the form Jw = (0,0, JOU>O)T , where

p(y) is the mass density of B, and 0 is the open subset of R2 occupied by the planar body B. (This is consistent with the definition of J given at the beginning of section 4.1.) Also, Jui = (0,0, JO^O) T is collinear with w; whence w x Jw = 0 in (4.5). This is meant to say that the second equation in (4.5) is simply JOCJQ = NO, where N = (0,0, A^o)T is the moment of the external forces with respect to the center of mass 7^3. By using w0 = a, it follows that (4.5) reduces to the simple system

Thus, for the motion of the planar body B, we obtain a second-order ODE in R2 x R1 in a completely straightforward way. As explained in section 3.3, for consistency both the resultant force F and moment NO must be 2?r-periodic in the a variable.

Chapter 5

Constrained Rigid Bodies In this chapter, we derive the equations of motion of a constrained rigid body, or system of rigid bodies, in direct continuation of what was done in Chapter 4 in the unconstrained case. Our goal is to obtain a primary DAE formulation, namely, the system (5.49) below or (5.18) if only one body is involved. In addition, several other versions of these systems are given for cases involving certain special features. Specifically, while the main system (5.49) is a DAE on the configuration space, and hence a DAE on a manifold, the system (5.51) is a simpler DAE in Euclidian space in the case when extensions of the constraint function and of the force and moment fields are readily available. Alternately, geometric constraints lead to a special form of (5.49) given by the system (5.56). Although we set up the equations of motion, we do not address here the existence of solutions for the corresponding systems. This will be done later in Chapters 6 and 7, where, among other things, existence theorems for initial value problems are proved for DAEs in Euclidian space (Chapter 6) and manifolds (Chapter 7).

5.1

Kinematic Constraints

We will retain here the same notation developed in Chapter 4 for the unconstrained motion of a rigid body B. But, for the sake of brevity, it is useful to introduce the simplification where the subscript "m" is a reminder that Fm represents a moment force. With this the equations of unconstrained motion (4.22) assume the more concise form

The unconstrained state space is SQ = R1 x T(R3 x S3), with the understanding that (t, (u,p), (it,p)) and (t, (u, -p), (ii, —p)) in R1 x T(R3 x S3) correspond to the same state. 45

46

PATRICK J. RABIER AND WERNER C. RHEINBOLDT

For the subsequent discussion, we will always assume that, as in Theorem 4.1 and in line with Remark 3.2, the resultant force Fr in (5.2) and the moment N in (5.1) satisfy the conditions

Suppose now that the motion of B is subjected to a constraint of the form

where the constraint function

is assumed to satisfy, in line with (5.3),

Hence the state space is restricted to

and, of course, now the system (5.2) no longer describes the motion and must be modified to account for the constraining force exerted by (5.4). In the mechanics literature, it often appears to be taken as a physical principle that, as in the case of a system of mass points, the resultant force and moment produced by the constraint (5.4) are accounted for by a Lagrange multiplier term involving the derivative D^&G of the constraint function G. However, unlike for mass points, in the case of a rigid body (or system of such bodies; see section 5.3) and of a kinematic constraint (5.4), this approach does not appear to be backed up by some optimality principle, such as the Gauss principle of least constraint. In other words, it is disregarded that a Lagrange multiplier formulation is meaningless unless it is shown to translate a constrained minimization property. Our first goal will be to show that a generalized Gauss principle does indeed stand behind this procedure. Before going further, there is a need to address a notational issue that may cause conceptual errors in connection with the derivative D^^G of G. The definition (5.5) of G implies that the partial derivative with respect to p is a mapping DpG(t,u,p,ii,p) € £(T p S 3 ,R fc ) and hence is defined only on TPS3 = {p}'L- In turn, this means that its transpose maps into the dual T*S3 of TP53. In practice, the constraint is often defined on the larger space R1 x T(R3 x R4) or, at least, on some open set containing R1 x T(R3 x 53). Then DpG(t,u,p,u,p) maps R4 into Rfe and therefore the range of its transpose is a subspace of the dual (R4)* of R4. It is obvious that for A* e (Rfe)* the element DpG(t,u,p,u,p)JA* 6 T*S3 is the restriction to TPS3 C R4 of DpG(t,u,p,u,p)\* 6 (R 4 )*- This becomes less transparent when, as is customary, the spaces (R4)* and (Rfc)* are identified with R 4 and R fc , respectively. Recall that this identification is made via the Riesz representation theorem and the canonical inner product. In this process, T*S3 is identified with TP53 and restriction of linear forms becomes orthogonal projection of their representatives. In other words, DpG(t,u,p, u,p) T

CHAPTER 5.

CONSTRAINED RIGID BODIES

47

is viewed as an element of £(R fc ,T p 5 3 ) or, alternatively, as an element of £(R f c ,R 4 ), and pG(t,u,p,u,p)r € £(Rfc,Tp53) is the product of the orthogonal projection onto TPS3D with DpG(t,u,p,u,p)T £ £(R f c ,R 4 ) (in this order). Since it is the former concept that is relevant here, it must be kept in mind that composition with the orthogonal projection oPS3 = {p}-1- must be used whenever DpG(t,u,p,u,p)T may be understood as anntoT element of £(R f c ,R 4 ) (which, in practice, is almost always the case). No confusion arises if G is carefully distinguished from its extension, say. GE, outside R1 x T(R3 x S3). We will return to this point later. As discussed in Chapter 2, the idea of the Gauss principle is that, at any fixed state, the acceleration of the constrained body must be as close to the unconstrained acceleration as the constraint (5.4) permits. In this connection note again, as discussed in Remark 4.1, that the second equation (5.2) does not characterize the acceleration vector p but only its tangential component. Suppose now that the curve (u(t),p(t)) € R3 x S3 represents a motion of B satisfying the constraint (5.4). In line with the above remark, we split the acceleration component p(t) into its normal and tangential parts

with respect to S3. As noted, by differentiation of jp(i)| 2 = 1, we have pN(t) = -\p(t)\ p(t}. Hence, by differentiating the constraint condition (5.4) with respect to t, we find that for any state s = ( t , u ( t ) , p ( t ) , u ( t ) , p ( t ) ) ,

where we refer to Remark 5.1 below for an explanation of the two, apparently undefined, terms DpG(s)p and DpG(s)p. It follows that, when the motion is assumed to satisfy an initial condition ( t o , u ( t o ) , p ( t o ) , u ( t o ) , p ( t o ) ) 6 <S, then (5.9) is equivalent to the constraint (5.4) since the latter is recovered by time integration of (5.9). The condition (5.9) implies that, for any state s — (t, u,p, u,p) € S, the part (U,PT) £ R3 x TP53 of the acceleration has to be contained in the affine space

Remark 5.1. In formula (5.9) as well as in (5.10) the term DpG(s)p is not defined since, as in the setting of section 4.4, DpG(s) does not exist (because p is the base variable of the bundle TS3). Furthermore, DpG(s)p makes no sense either since DPG acts only on the elements of TPS3 = {p}1. This means that in (5.9) and (5.10) it is necessary to use a (local)term extension GE of G to R1 x (R3 x R 4 ) 2 instead of G to make sense of the (combined) term Since the affine space A*(s) in (5.10) will be used below to justify a generalized version of the Gauss principle, it is important to make sure that the term (5.11) is independent

48

PATRICK J. RABIER AND WERNER C. RHEINBOLDT

of the extension of G near the state s = (t,u,p, ii,p). For this consider a smooth curve C(r) on S3 such that C(0) = p and C(0) = p. We have

where, on the right, GB denotes any (local) extension of G. Since £ lies on S3 the normal component Gv(0) is just — |C(0)|2p = ~|p|2P; whence

where Cr(0) € TPS3 is the tangential component of £(0). Since Cr(0) £ TPS3, we have D p G B (sXr(0) = DpG(s)Cr(0). Thus DpG(s)£T(Q) is independent of the extension of G. Moreover, because (, lies on S3, G(£,U,C(T),U,£(T)) is also independent of such an extension, and hence the same is true of its derivative at r = 0. This proves our claim. With this, it is now natural to generalize the Gauss principle by requiring that at any fixed state s = (t,u,p,ii,p) £ S the part (u,pr) of the acceleration is obtained as the solution of the minimization problem

3 where [H(p)r JH(p)}-1 is the inverse of H(p)r JH(p] € GL(T PS ) (see Remark 4.1). Clearly, the norm has to have appropriate physical relevance. The norm

may be called the "energy norm" of the body, and it is indeed the only choice that is consistent with the Lagrange multiplier rule of the mechanics literature. As in Chapter 2, the minimization problem (5.12) is well posed if the partial derivative D(u,p)G = DuG®DpG <= £(R3 x TpS3,Rk) of the constraint mapping (5.5) satisfies the full-rank condition

since then the affine space A*(s) in (5.10) with s = (t,u,p,u,p) is, of course, nonempty. Evidently the condition (5.14) does imply that G must be a kinematic constraint. Hence, in that case, it follows again by the Lagrange multiplier theorem, Theorem A.5, that for the minimizer (U,PT) of (5.12) at the state s = (t,u,p,u,p) € S there exists a Lagrange multiplier A € Rfc such that (u,pT,X) solves the system

CHAPTER 5.

CONSTRAINED RIGID BODIES

49

Here once again the invertibility of H(p)TJH(p) as a linear operator onPS3 is T used (see Remark 4.1), and the pG(s)T is the transpose of the operator DpG(s) €operator D £(T P S 3 ,R 4 ) after the identifications T*S3 = TPS3, (R fe )* = R fc . This has to hold for any state 5 assumed during the constrained motion of B. We multiply the first and second equations of (5.15) by M and 4H(p)T JH(p), respectively. Since keiH(p)Tt JH(p] = span {p}, this gives, for the decomposed acceleration (5.8), H(p)rJH(p)pT = H(p)TJH(p)p. By (4.19) H(p)'[H(p) is the orthogonal projection onto TpS3. Thus it is redundant, but, in view of our earlier discussion, safer, to use the relation for it then does not matter whether, on the right of (5.16), DpG(s)T is understood as a linear operator with values in R4 or with values in TPSZ. Correspondingly, a similar precaution is to formulate the full-rank condition (5.14) as

Thus, when, in (5.15), the moment force Fm is replaced by its definition (5.1), we obtain the following result. Theorem 5.1. Suppose that the rigid body B, with mass matrix M and inertia tensor J, is constrained by (5.4) where the constraint function G of (5.5) satisfies the rank condition (5.17) and has the factorization property (5.6). Let the curve t i—> (u(t),p(t)) € R3 x S3 represent the motion of B under the action of the resultant force Fr : S —» R3 and moment N : S -* R3 (with respect to its center of mass), defined on the state space (5.7), for which (5.3) holds. Then the generalized Gauss principle requires the existence of a curve t *-> A(t) € R fc such that t >-> (u(t), p ( t ) , \(t)) satisfies the DAE

Proof. The full-rank condition (5.17) validates the use of the Lagrange multiplier theorem to translate the extremal property of the generalized Gauss principle. Together with the definition (5.1) of the moment force Fm this leads to the desired conclusion. For the computational solution of (5.18) some local parametrizations of the sphere S3 have to be used. Of course, such parametrizations are explicitly known and easy to obtain. But they can also be avoided altogether if we assume that we are given a constraint function

which is defined on some open set containing R1 xT(R 3 x S3) and represents an extension of the original G. Similarly, suppose that extensions

50

PATRICK J. RABIER AND WERNER C. RHEINBOLDT

of the resultant force and moment are given and that, as before, the factorization conditions

hold. We can now reformulate the system (5.18) on R3 x S3 as an equivalent system on 3 R x R4 by introducing an additional constraint equivalent to p 6 S3. The "obvious" condition \p\ = 1 represents a geometric constraint. Hence we follow the approach used in section 2.1 and replace it with the kinematic constraint pTp = 0 obtained by differentiation. Of course, for any curve satisfying the latter constraint we necessarily have \p\ = 1 provided only this condition holds at some point. Thus the state space (5.7) is characterized by

Equivalently. we replace the constraint mapping GE by the augmented constraint function

We do not incorporate the component \p\2 — 1 since, as observed earlier, \p\ = 1 holds as a result of prp = 0 as soon as it is satisfied at one point and, hence, is essentially redundant. The reason why we retain pTp but discard \p\2 - 1 will become clear in Chapter 7, where this choice will appear as a special case of a general procedure. The hat notation for F is also chosen for consistency with Chapter 7. For any state s = (t,u,p,ii,p) € <S, the constrained acceleration (u,p) attained by the motion has to be contained in the affine space

Clearly, the linear operator defining this space will be surjective if the safe-guarded fullrank condition (5.17) holds. With this we obtain now the following equivalent equations of motion. Theorem 5.2. Let Fr, N, and G have extensions FE, NE, and GE, respectively, to some open subset o/R1 x T(R3 x R4) containing R1 x T(R3 x S3). Assume further that the factorization conditions (5.21) hold. Then we have the following: (i) The full-rank condition (5.17) holds if and only if the mapping (5.23) satisfies the full-rank condition

CHAPTER 5.

CONSTRAINED RIGID BODIES

51

(ii) Assume that either of the two equivalent full-rank conditions in (i) above holds. Then the system (5.18) is equivalent to the system in Euclidian space

involving a new scalar Lagrange multiplier n = /J,(t), provided that attention is confined to solutions of (5.26) with initial condition10 p(io) = Po € S3. Proof, (i) For (v, q) € R3 x R4 we have

Suppose first that the full-rank condition (5.17) holds, and let h G R fc , r 6 R1. For every q S {p}"1, we have p1 (rp + q) = T, and by (5.17) the pair (v, q) 6 R3 x {p}± may be chosen so that D^G(s)v + DpG(s)q = h - TDpG(s)p. Since GE is an extension of G, this is just DuGE(s)v + DpGE(s)q = h-DpGE(s)p. Thus for q := rp + q it follows from (5.27) th(iltp)f(s)(v,q) = (h,r) and therefore that (5.25) holds.at D Conversely, if (5.25) is valid, that is, if the mapping (5.27) is surjective. then, given h e R fe , there exists (v,q) e R3 x R4 such that D(u,p)T(s}(v,q) = (/i,0). From (5.27) we obtain prq = 0, that is, q <E {p}-1. as well as DuGE(s)v + DpGE(s}q = h. Since (v,q) £ R3 x {p}"1, this shows that the mapping D^,p)GE(s)^3x^± is surjective and, hence, that (5.17) holds. S3. Then, clearly, (ii) Let (u(t),p(t),\(t),fjL(t))be a solution of (5.26) with p (0 ) €t the curve p is iand, hence, ( t , u ( t ) , p ( t ) , u ( t ) , p ( t ) ) is in S for all times (see (5.22)). But on S the extended mappings (5.19), (5.20) are the same as the mappings Fr, N, and G used in (5.18). Therefore, the first equations in (5.18) and (5.26) are the same. Since H(p)TH(p)H(p)T= H(p)rH(p)1H(p)p = 0 by (4.19), the second equation of (5.18) is obtained by multiplying the second equation of (5.26) by H(pYH(p). Thus, altogether, ( u ( t ) , p ( t ) , A(£),ju(t)) solves (5.18). Conversely, suppose that (u(t),p(t),X(t)) solves the system (5.18), where G,Fr, and N may as well be replaced by their extensions. The relation pTp = 0 holds since the curve p is in S3, and it suffices to show that fj,(t) € R1 can be chosen so that the second equation in (5.26) holds. Since H(p(t))TH(p(t)') t projects onto {p(t)}x (see (4.19)) and ( u ( t ) , p ( t ) , A(t)) solves (5.18), we need only to show that fj,(t) € R1 can be found so that the projection along p(t) of the second equation in (5.26) is satisfied, that is. that

10 This initial condition implies that p(t) 6 S3. However, note that DpGE(t,u,p,u,p)TX£ TPS3 = {p}-1- because Dp refers to the derivative with respect to p in R 4 , not merely to p in TPS3. Consequently, ^ need not be zero. All the terms other than fjtp and DpGE(t, u, p,u,p)T\ in the second equation of (5.26) are in {p}-1.

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PATRICK J. RABIER AND WERNER C. RHEINBOLDT

This actually gives the desired formula for ^(t), and the proof is complete. We point out the difference between the terms SH(p)r t JH(p)p in equation (5.26) and 8H(p)rH(p)H(p')'TJH(p')pin (5.18). The projection H(p)1'H(p) may be omitted (and has been) in (5.26) because the Lagrange multiplier fi takes care of adjusting the "normal" component along p. The kinetic energy T of the body continues to be given by formula (4.37) since, as should be expected, the arguments used in the proof of Theorem 4.2(i) are independent of the motion being free or constrained. This justifies the terminology "energy norm" for the norm (5.13). In the conservative case, when there is a potential U : R3 x S3 —> R1 and corresponding Fr(u,p) = VuU(u,p) and H(p)TN(u,p) = V p W(u,p), it follows by the arguments used in section 4.5 that the equations of constrained motion (5.18) now take the form

where £.(u,p,u,p) = T(u,p,ii,p) —U(u,p) is the total energy. This system is formally identical with the system (5.28) in the case of mass points and enjoys exactly the same invariance property under changes of coordinates. If, in place of G, we work with an extension GE to some open subset of R1 x T(R3 x 4 R ) as in (5.19), then, consistent with earlier warnings as regards the interpretation of DpG(u,p,u,p)T, we observe that in (5.28), D(u,p)G(u,p,u,p)T\ must be understood as the pair

On the other hand, from (5.26) we obtain in that case

where D^,p)GE(u,p,u,p) € £(R3 x R^R*) is the derivative of the extension GE of G. Hence, in (5.29) the component DpGE(u,p, u,p) T need not, and should not, be multiplied by H(p)T H(p) since the correcting term p,p takes care of the necessary adjustment. Since (5.29) can also be written as

with F defined by (5.23), this system has the same invariance properties as (5.28) with respect to changes of coordinates.

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5.2

CONSTRAINED RIGID BODIES

53

Geometric Constraints

By analogy with the results of section 2.1 for systems of mass points, it should be clear how the results of the previous section apply to the motion of the rigid body B subjected to a geometric constraint G(t, u,p) = 0. A more general case will be discussed in section 7.1. For simplicity we use the notation x = (u,p) € R3 x S3 and hence assume that the motion of the body is constrained by

Since (u,p) and (u, -p) represent identical configurations of B, we assume, in line with (5.6), that Hence, as in section 2.1, we replace this geometric constraint by the kinematic constraint G(t,x,x) = 0 where x = (u,p) and G is the mapping

This mapping satisfies the full-rank condition (5.14) if and only if

Since G(t,x(t)) — 0 implies that G(t,x(i),x(i)) = 0 whenever t i—> x(t) is a curve in R3 x S3, any motion of the rigid body B constrained by G ~ 0 is constrained by G = 0. Thus, assuming (5.33), we obtain from the generalized Gauss principle with G replaced by G that a motion of B must comply with the system (5.18) with G replaced by G. Because of

this proves the following result. Theorem 5.3. Suppose that the rigid body B, with mass matrix M and inertia tensor J, is constrained by (5.30), where the constraint function G satisfies the rank condition (5.33) and has the factorization property (5.31). Let the curve t H-> (u(t),p(t)) € R3 x S3 represent the motion of B under the action of the resultant force Fr : S —+ R3 and moment N : S —> R3 (with respect to the center of mass) for which (5.3) holds. Then the generalized Gauss principle requires the existence of a curve t <—> \(t) such that t >-». (u(t),p(t),X(t)) satisfies the DAE

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PATRICK J. RABIER AND WERNER C. RHEINBOLDT

Conversely it is obvious from (5.32) with x = x(t), x = x ( t ) that any solution of (5.34) such that G(to,x(to)) = 0 for some time to continues to satisfy G(t,x(t)) = 0 as long as the solution is denned. In practice, it is, of course, usual to work with a constraint

extending G and defined on some open set containing R1 x R3 x S3. As in the case of Theorem 5.2 we then replace G of (5.32) by the augmented constraint function

This leads to the following result. Theorem 5.4. In Theorem 5.3, let GE in (5.35) and F f , NE in (5.20) be extensions of the functions G, Fr, N, respectively, to some open subset o/R1 x T(R? x R4) containing R1 x T(R3 x S3). Assume further that the factorization conditions (5.21) hold. If the mapping (5.36) satisfies the full-rank condition

then the system (5.34) is equivalent to the system in Euclidian space

provided that attention is confined to solutions of (5.38) with initial conditions p(io) ++ Po e S3. These arguments apply to a much broader class of constraints G(t,x,x) = 0 that fail to satisfy the full-rank condition (5.14). In fact, suppose that (5.14) fails to hold for G but that there exists some other mapping G : R1 x T(R3 x £3) —* R fc for which G(t,x(t),x(t)) = 0 whenever t H-> x ( t ) is a curve in R3 x S3 satisfying G(t,x(t),x(t)) = 0. If, now, the full-rank condition (5.14) holds for G, then the motion of B under the constraint G = 0 must also comply with G — 0, and it follows from the Gauss principle that this motion must be governed by the system (5.18) with G replaced by G. The question, when such a mapping G can be found, happens to have a general answer as will be discussed in section 7.1 in a more abstract setting appropriate for these considerations. The case of geometric constraints turns out to be one for which it is simple to obtain a formula for G, namely, (5.32). The general case is hardly that simple, but a formula does remain available. Moreover, the mapping G satisfies the condition that G(t,x(t),x(t)) = 0 as soon as G(t,x(t),x(t)) - 0 and G(t0,x(t0),x(to)) = 0 for some to.

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Remark 5.2. Our rank conditions in this chapter are not very economical. In Theorem 5.1 the generalized Gauss principle requires only that rank D±G(t,x.x) = k on G~ i (0) since only the points of G~1(Q) are relevant in a problem subjected to the constraint G = 0. In Theorem 5.3 the same statement can be made with G, given by (5.32), replacing G. But here note that the solutions of interest must satisfy G(t,x,x) = 0 as well as G(t, x, x) — 0, and, hence, only the points of the set G~l (0) n G~' (0) are relevant. It is at those points that the generalized Gauss principle, and, therefore. Jfche full-rank condition for DiG(i,x,x), is needed. The same remark is valid whenever G exists with the above-mentioned general properties. As it turns out, close attention to the points where the full-rank condition for D±G must hold will be crucial for the construction of G in section 7.1 because we will see that G^ 1 (0) n G~ 1 (0) can be determined before G is known. Naturally, Theorem 5.3 is valid when, in particular, the constraint G is holonomic, that is, when G = G(x) is also independent of t. In [21] a somewhat different approach for that case is given that involves the use of the second fundamental tensor of the manifold G -1 (0) and shows that the resulting DAE reduces globally to a second-order ODE on the manifold G~ 1 (0). This approach will be summarized in the second part of section 7.6.

5.3

Multibody Systems

So far, in Chapter 4 and sections 5.1 and 5.2, we confined our attention to the case of a single rigid body. All these results, including, of course, the generalized Gauss principle, can be easily extended to the case of a system of more than one rigid body. There should be no need to repeat the four theorems proved in sections 5.1 and 5.2; instead we will provide only the corresponding four DAE formulations. Consider a multibody system consisting of i > 1 interrelated rigid bodies where iii £ R3 is the position of the center of mass of the ith body and ( u , , p i ) belongs to the configuration space R3 x S3. In other words, the configuration vector of the multibody system is

and the unconstrained configuration space becomes Co = R3^ x (5 3 ) f . Let Mj be the mass matrix and J, the inertia tensor of the ith body. Then for the multibody system we define the composite mass matrix and inertia tensor as the block diagonal matrices

Correspondingly, let FiiT and TV, be the resultant force and moment (with respect to its center of mass) of the ith body, i = 1,... ,£, all of which are assumed to satisfy the

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PATRICK J. RABIER AND WERNER C. RHEINBOLDT

conditions

where &j := ( 0 , . . . , 0, -1,0,..., 0) and -1 is the jth entry. Then the resultant force and moment of the system are defined as

and

respectively. In addition, we introduce the block-diagonal matrix

where each diagonal block is a 3 x 4 matrix given by the last three rows of (4.11). The system is assumed to be subjected to a kinematic constraint

and we require the full-rank condition

and, of course, also the factorization condition

where, as in (5.41), <jj := (0,..., 0, — 1 , . . . , 0). The state space of the system is then

Now, with an entirely analogous proof, we arrive at the following equations of motion generalizing those of Theorem 5.1:

where H(p) is given by (5.44) and A e Rfe is a Lagrange multiplier.

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57

When, in place of (5.45), we have a constraint

where GE is an extension of G, and FE, NE are extensions of Fr and N, respectively, then Theorem 5.2 also extends easily to systems of rigid bodies and we obtain the following DAE:

where A e R fc and /j. € Rf are Lagrange multipliers and p is the block diagonal matrix with 4 x 1 blocks11

and where it is again assumed that our attention is confined to solutions of (5.51) with initial condition p(to) = p0 € (S3)e. The results of section 5.2 are also easily extended to the case of t > I rigid bodies. Again we give here only the resulting DAEs and do not repeat the theorems. In the case of a constraint

satisfying the condition

we replace G by

and obtain from (5.49) the DAE

which is formally identical to (5.34). The comments following Theorem 5.3 regarding more general constraints that fail to satisfy the rank condition, and also Remark 5.1, can be repeated verbatim. When, in place of (5.53), a constraint

11

This p is not to be confused with the conjugate of p in section 3.2.

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PATRICK J. RABIER AND WERNER C. RHEINBOLDT

is used that extends G and is denned on some open set containing R1 x R3^ x (S3)e, then we obtain the DAE corresponding to (5.38):

where again FE and NE are extensions of Fr and N, respectively, the operator p is given by (5.52), and it is assumed that the solution satisfies the initial condition p(to) = po €

(s*y.

The system (5.58) is essentially the one obtained by Haug in [14, p. 444], except that he differentiates the equations DuGE(t,u.p)u + DpGE(t,u,p)p + DtGF(t, u, p) = 0 and pTp = 0 to obtain an invertible system for (it,p, A, /it). A variant of this procedure will be discussed in the first half of section 7.6. Remark 5.3. The redundancy introduced as a safeguard in (5.16) is repeated in systems l (5.49) and (5.56). In fact, H(p)rH(p)DpG(t,u,p,u,p)TXandH(p)TH(p)D pG(t,u,p)' ~X T T can be replaced by DpG(t,u,p.it,p) A and DpG(t, u,p) A, respectively, if it is kept in mind that the domain of definition of G is indeed assumed to be R1 x T(R3 x S3). The term H(p)^H(p) was incorporated to preempt the error that would occur if an extension GE of G were used instead of G. In all cases, the term H(p)TH(p) is needed in H (p)^ H (p} H (p) JH(p)p appearing in (5.49) and (5.56). It is not not needed in (5.51) or (5.58) because the Lagrange multiplier jj. makes the necessary adjustment. Not only systems of rigid bodies, but also mixed systems of mass points and rigid bodies, can be handled in an analogous manner. If the tth body is only a mass point, there is no corresponding variable Pi and no corresponding momentum balance equation. In the case of (. mass points, the results of Chapter 2 are recovered again.

5.4

The Planar Case

The very simple form of the equations of unconstrained motion of a single planar rigid body, obtained in (4.40), i.e.,

where Fr and A^o are 27r-periodic in the a variable, is similar to the form of the equations for mass points obtained in Chapter 2. They even involve a variable x = (it, a) £ R3, and hence the only formal differences in the case of mass points are the use of the mass matrix diag(M, M, Jo) instead of M/s and the 27r-periodicity requirement. Accordingly,

CHAPTER 5.

CONSTRAINED RIGID BODIES

59

all the arguments in Chapter 2 can be repeated verbatim to handle the case when the motion is subjected to a constraint of the form

where, consistently with section 3.3, G is 27f-periodic in the a variable. Evidently, the same considerations apply to a system of t planar rigid bodies with (unconstrained) configuration space

so that the configuration vectors have the form

with Ui 6 R2, a.i € R, 1 < i < I. Admissible forces, moments, and constraints acting on such a system must be 2?r-periodic in each of the variables QI, . . . , ag. The complete analogy with the case of mass points should make it unnecessary to discuss planar rigid motion any further, until examples are given in Chapter 9.

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Chapter 6

DAE Formulation in Linear Spaces 6.1

A DAE Existence Result

The results of section 2.1 and Chapter 5 led to equations of motion in the general form of a DAE

where x 6 Rn and, consistent with previous notation, (t, x, x) denotes an arbitrary point of R1 x R" x Rn while (t, x ( t ) , x ( t ) ) will refer to a curve t H-> x ( t ) with x(t) = (dx/dt)(t). The coefficient functions

in (6.1) are assumed to be defined and smooth in some appropriate open sets. Four of the principal cases, included in the DAE formulation (6.1), are as follows: (i) Systems of t mass points subjected to the kinematic constraint (2.3) and governed by the system (2.11). Here we have n = 3£, K = k, A(x) = M, and F = G as given in (2.3). (ii) Systems of I mass points subjected to the geometric constraint (2.13) and governed by the system (2.17). Here we have again n = 3f, K = k, A(x) — M, and F = G as given in (2.15). (iii) Systems of i rigid bodies subjected to the kinematic constraint (5.50) and governed by the system (5.51). Here we have n = 7£, K = k +1, x = (u,p) with u and p as in (5.39), x — (ii,p), and

61

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PATRICK J. RABIER AND WERNER C. RHEINBOLDT

Moreover, F in (6.1) is given by

with p as in (5.52) and GE extending G in (5.45). (iv) Systems of I rigid bodies, subjected to the geometric constraint (5.53), governed by the system (5.56). Here we have again n = 71, K = k +1, x = (u,p) with u and p as in (5.39), x = (u,p), and A(x) as in (6.3). Moreover, F in (6.1) is now

with p as in (5.52), GE extending G in (5.53), and

Similarly, the corresponding formulations of the equations of a combination of mass point and rigid bodies are also easily seen to be governed by a DAE of the form (6.1). However, some of the formulations obtained in Chapter 5, such as those in Theorems 5.1 or 5.3 and their generalizations to systems of rigid bodies, do not satisfy the condition (6.2) since they are defined only on the tangent bundle of a manifold. This case will be examined in Chapter 7. In (i) and (ii) the diagonal mass matrix M and hence also A(x) is symmetric positive definite. In (iii) and (iv), the block diagonal matrices M and J of (5.40) are again symmetric positive definite, but now the symmetric matrix A(x) in (6.3) has only rank 61 and is only positive semidefinite. In all four cases the Lagrange multiplier formulation was justified only under the assumption that the constraint function satisfies a full-rank condition which, for (6.1), translates into the requirement that Since, by (4.15(i)) and (5.52), we have ker H(p) = rge p, it then follows immediately that in (iii) and (iv), A(x) is positive definite on keTD±T(t,x,x). These observations form the basis of the following general solvability result for DAEs of the form (6.1) irrespective of their physical origin. Theorem 6.1 (local existence and uniqueness). Suppose that, in the DAE (6.1), the coefficient functions (6.2) are defined and smooth in some neighborhood of a point (t0,x0,±0) € R1 x Rn x Rn and that

Then there exists a 6 > 0 such that (6.1) has a unique solution (x(t), A(£)) for \t — to\ < 6 that satisfies the initial condition

Along this solution the full-rank condition (6.6) holds for \t — t(,\ < S.

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63

Proof. From (6.7(iii)) it follows that rank D±T(t,x,x) — K for all (t,x,x) in some open neighborhood of (
Moreover, r(t,x(£),x(t)) = 0 must, hold on J and, by differentiation, we obtain

Note that 7r(to, XQ, Xo) is a linear isomorphism of ker D±T(t, x, x) onto ker D±r(to, XQ, XQ) for (t,x,x) near (to,Xo,xo) in R1 x R™ x Rn. Thus, for a sufficiently small interval J, (6.9) is equivalent to

We claim that, for (t,x,x) near (t 0 ,xo,x 0 ), the linear operator

is a linear isomorphism. By continuity, it suffices to show this for (t,x,x) = (to,xo,xo). Thus suppose that z € R" satisfies 7r(to,Xo,xo)A(xo)z = 0 and D j F (to, XQ 1X0)2 — 0. The latter equation means that z € ker Z?jr(to,xo,xo), and hence the former yields

where we used the symmetry of the orthogonal projection 7r(to,xo,xo). But then we must have z = 0 by (6.7(i)). This shows that, as desired, for (t,x,x) = (t0,xo,io) the operator (6.12) is injective and hence bijective. With this we see that for a sufficiently small interval J the combined system of equations (6.10) and (6.11) can be solved for x(t) as a smooth function of (t,x(t),x(t)), say, Hence, on a sufficiently small interval J our assumed solution x = x(t) of the DAE (6.1) has to solve the second-order ODE (6.13) on Rn. Since the initial conditions (6.8) specify the solution of (6.13) uniquely, it follows that the solution x(t) of (6.1) must be unique. Thus, because DxF(t,x(t),x(t))^ is injective, A(t) is also unique. Conversely, we claim that the (unique) solution of the ODE (6.13) satisfying the initial conditions (6.8) must solve (6.1). Indeed, since the right side of (6.13) is the solution of the system (6.10), (6.11), this solution must also satisfy the relations (6.9) and (6.10).

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PATRICK J. RABIER AND WERNER C. RHEINBOLDT

In particular, (6.9) states that

But every such element can be written in the form F(£,x(i),x(i)) T A for a unique A = A(t) e R K . This shows that A(x(t))x(t} = F(t,x(t),xi;r(t,x(t),x(t))T \(t).(t)) + D Moreover, equation (6.10) ensures that F(i,x(t),x(£)) is constant, and, hence, because of F(£o,x(£o),i(io)) = F(io,xo,x 0 ) = 0, we obtain T(t,x(t),x(t)) = 0. Thus the solution of (6.13) indeed solves (6.1). In Theorem 6.1 the solution x ( t ) (but not X(t)) depends only on the set r~ J (0) near (to,xo, XQ) but not on the specific choice of F as long as rank D±T(to, XQ, XQ) = K. To see this, we need the following (standard) result. Lemma 6.2. Let g, h : Rm —> R* be two smooth functions defined on some open neighborhood of the origin o/R m and suppose that g(Q) — h(Q) = 0, g~ 1 (0) = /i-1(0), and that both Dg(0) and Dh(0) map onto R K . Then there exists a smooth function if? = t^(u, v) : Rm x RK —> RK such that ^(u, 0) = 0, Dvi[>(u, 0) is an isomorphism of R K , and h(u) = tlj(u,g(u)) foru in some open neighborhood of the origin o/R m . Proof. With no loss of generality assume that ker£>g(0) = R m ~ K x {0} and write u € Rm as u = (wi,u 2 ), HI € R m ~ K , u2 € R K . The mapping Q(u) := (ui,g(u)) is an origin-preserving local diffeomorphism of R m ; hence h = ( / i o 0 ~ 1 ) o 0 in some neighborhood of the origin of R m . Let ip(u, v) := (h o Q~I)(UI, v), so that

For v = 0 we have ^(^,0) = (h o 0 -1 )(ui,0) and 0~ 1 (ui,0) = (ui,U2), where u2 is characterized by the condition 3(111,^2) = 0. Thus, (h o 0~ 1 )(u 1 ,0) = /i(ui,U2) = 0 since g~1(0) = /i~ : (0). This shows that ^(u,0) = 0. For w 6 RK we have Dv^(Q)w = Dh(Q)DQ~l(Q)(0,w) and D©" 1 ^)^,^) is the (unique) vector z € {0} x RK such that Dg(Q)z = w. Hence Dv^(0)w = 0 if and only if Dh(Q)z = 0. Since ft"1^) = g'1^) and both g and h are submersions at 0, we find that kerDg(O) = ker£>/i(0) (equal to the tangent space of h~1(0) — g~l(0) at 0); whence Dh(Q) is one to one on {0} x R K . Therefore, Dh(Q)z = 0 implies that z = 0 and hence w = 0. This shows that Dvip(0,0) is an isomorphism of R K , so that Dvi^(u,Gi) remains an isomorphism for u in some neighborhood of the origin. Theorem 6.3. In Theorem 6.1 the curve x(t) (but not A(t)) is unchanged ifT is replaced by any other mapping F : R1 x Rn x R" —» RK such that F~1(0) and F~ 1 (0) coincide in a neighborhood o/(to,^o>io) and that rank DiF(to,xo,io) = «• Proof. By Lemma 6.2 there exists a smooth mapping

such that (i) ip(t,x, x, 0) = 0, (ii) Dvijj(t, x, x, 0) is an isomorphism of R14, and (hi) f (t, x, x) = V>(t, x, x, F(t, x, x)) for (t, x, x) in a neighborhood of (t0, XQ, x 0 ) in R1 x R" x

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65

R™. From (i), D±^(t,x,x,0) = 0. In particular, for ( t , x , x ) £ r~l(0) = f-^O) it follows that For (t,x,x) = (tf,,xo,Xd) we have ker D±T(td,XQ,xo) — ker D±r(tc,,xo,xo); whence the condition (6.7(i)) is valid with F replaced by F. The same result is obvious for (6.7(ii)) and (6.7(iii)). This shows that Theorem 6.1 holds with F replaced by f. Let (x(t), A(t)) denote the corresponding solution. Since F-^O) = F^O), we have T ( t , x ( t ) , x ( t ) ) = 0. Thus, dropping the t-dependence for simplicity, it follows from (6.14) that

Hence the pair ( x ( t ) , D v i p ( t , x ( t ) , x ( t ) , 0) T A(t)) solves the system (6.1). Since we required that x(
6.2

The Case of Nonmaximal Rank

The full-rank condition (6.6) clearly plays a central role in the solvability result of section 6.1, and it was also critical for our application of the Gauss principle in sections 2.1, 5.1, and 5.2. This raises the question about what can be said when it fails to hold. It is essential to recall that the full-rank condition applies only to kinematic constraints. In fact, we saw in connection with geometric constraints that the required rank condition does not hold for the original constraint but only for the induced kinematic constraint. This is a special case of a more general procedure of replacing the original constraint F = 0 by a modified constraint F = 0 that satisfies the full-rank condition (6.6) and for which F = 0 implies that F = 0. This procedure carries over to other related types of problems as, for instance, those involving motions under mixed geometric and kinematic constraints: If FI is nontrivial, that is, if it maps, say, into R K ] , n\ > 1. then the geometric constraint FI(£, x) — 0 induces again a kinematic constraint. Thus we have to consider the motion under the kinematic constraint:

Here Fi(t,x(i),i(i)) = 0 is equivalent to T i ( t , x ( t ) ) = 0 provided that T i ( t o , x ( t o ) ) = 0 holds for some t0. For the kinematic constraint F = 0 we require now that

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has full rank, in which case the theory will carry over by replacing F by F in (6.1). This approach for the case (6.15) is used, e.g., by Zeidler [34]. There are ways of generalizing the above procedure when there is no obvious splitting of the form (6.15). The most natural way is to obtain such a splitting after performing a change of variables. This will work when the rank is constant but not full. However, changes of variables are inconvenient because their local and arbitrary nature may conceal the intrinsic or global validity of the properties they help to establish. Accordingly we present now an apparently new approach of modifying the constraint without having to involve any change of variable or assuming any splitting as (6.15). For given F : R1 x R™ x R" -+ R*, K < n, denote by £(F) the set

so that, in particular, S(F) C F 1(0). The precise identification of S(F) will be central for later considerations. Suppose that the rank of Z?iF(t, x, x) is constant on some open neighborhood V of £(F) in R1 x R" x R"; that is, more specifically, suppose that we have

This constant rank condition implies the existence of projection operators P(t,x,x) onto rge DxT(t,x,x) depending smoothly upon (t,x,x) e V. For instance, orthogonal projections will have this property, but there is no reason to confine attention to that special choice. The operator Q := I — P is then a smooth projection onto some complement of rge -DiF and hence also

represents a smooth mapping on V. Lemma 6.4. A smooth curve x(t) in Rn defined for t in some open interval J C R1 satisfies T ( t , x ( t ) , x ( t ) ) = 0 on J if and only if it satisfies

at some point to € J. Proof. In this proof x, x, and x stand for x(t), x(t), and x(t). Suppose first that r(t, x, x) — 0; whence

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and, by differentiation.

Since Q(t,x, x) projects onto a complement of rge DxT(t,x,x), we always have

and, therefore, multiplication of (6.21) by Q(t,x,x) gives

By our definition (6.18), the relations (6.20) and (6.23) together show that (6.19(i)) holds and, of course. (6.19(11)) is valid for any choice of to. Conversely, suppose that (6.19) is satisfied with some to- Then by multiplying (6.19(1)) by Q we have

which, together with the still valid relation (6.22), implies that

and therefore that

Now, multiplication of (6.19(1)) by P gives P(t,x,x)r(t.x,x) = 0 on J and. by differentiation, we obtain

so that (6.24) becomes

This means that the curve 8(t) := T ( t , x ( t ) , x ( t ) ) solves the linear initial value problem

where B(t) :— —^P(t,x(t),x(t)). Obviously, the unique solution of this system is Q — 0, as desired. By Lemma 6.4, a curve (t,x(t),x(£)) lies in r -1 (0) only if it lies in r-1(0) and the converse is true as soon as (fo,z(£ 0 ),i(*o)) € F~ 1 (0) for some to. This shows the special relevance of the set r-1(0) n f ^(O). Note that

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PATRICK J. RABIER AND WERNER C. RHEINBOLDT

where S(F) is given by (6.16); whence r-1(0) n r-1(0) is independent of the choice of F, that is, of the choice of the projection operator P in (6.18) (and therefore also of Q). A less obvious fact is that, even though DxT(t,x,±] is a fairly complicated expression (involving the second derivative of F), the nullspace ker D^(t, x, x) and the condition rank D±r(t,x,x) = K are very easily expressed in terms of F provided that (t,x,i)eS(r). Lemma 6.5.

Then

which, in turn, is equivalent to

Proof, For h € R n , differentiation of (6.18) yields

where, for convenience, the (t, x, or)-dependency was dropped and the abbreviation D*xT(hi,h2) was used for Dx(DiThi)h'2= D±(DxFh2)hi(reflecting the equality of mixed partial derivatives) and hence where

If (t, x, x) e £(F) C F-^O), we have T(t, x,x) = Q and PD±F = D±T since P projects onto rge D±T. Thus (6.28) simplifies to

Because of (t,x,x) € S(F), it also follows from (6.16) that there is a w S Rn for which D±Yw = -DXF± - AF; whence (D±Qh)(DxTx+ DtT)= -(DiQ/i)DiFw. Since Q projects onto a complement of rge D±T, the identity QD±r = 0 provides, by differentiation, therelation(D±T + QD^Th = 0. Hence we have (DiQtyDjFw^Qh)D -QDljTfaw),which gives

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By substitution into (6.30) we obtain

which shows at once that D±Th = 0 if and only if h £ ker D±r and

We claim that in (6.32) the three terms QD|ir(/i, w), QD^r(x, w), and QD'f±rhvanish. Since h and w play symmetric roles in D|ir(/i,w), it suffices, for the first term, to swap h and w in the procedure used above of differentiating the identity QDXT = 0 with respect to x and thereby obtainingD^T(h.w)xTh and DxTh = 0 because of -(D±Qw)D h 6 kerDjT. For the second term, we use differentiation of the identity Q-D^F = 0 with respect to x, which yields (DxQh}D±£h — —QDx(D±Th)h. For h — x and h £ ker£>iF this shows that QDx(DxTh)x = 0; whence QD2xxr(x, h) = 0 by (6.29). Finally, by differentiation of QDXT — 0 with respect to t and, once again, by the hypothesis h € ker D^T, it follows that QD^Th = 0. At this stage we have obtained from (6.32) that D±Fh — 0 if and only if h e ker D^F and QDxTh = 0, that is, if and only if h e kerD ± r and Z?xF/i e rge D^F. This proves (6.26). For the proof that rank DXT = K if and only if (6.27) holds, note first that rank D±T — K amounts to dim ker DXT — n - K. From (6.26) we have

the latter because of rge D±T D rge QDXT = {0}. Now dim ker [DXT + QDXF] = n - K shows that Dxr + QDXT maps onto RK and vice versa. The following linear algebra lemma shows that this is equivalent to condition (6.27). Lemma 6.6. For A,B 6 £(R",R K ) let P e jC(R K ) be, a projection operator onto rg a and Q:= I - P. Then

Proof. Suppose that rge (A + QB) = RK For given v e R K let h € R n be such that (A + QB)h = Qv. Since QA = 0 and Q2 = Q, we have QBh = Qv, and it follows that Ah = 0, that is, h £ ker A. Because of v = Pv + Qv = Pv + QBh - P(v - Bh) + Bh and P(v — Bh) e rge A, there exists w € R™ such that F(v — Bh) = Aw. Hence we obtain v = Bh + Aw with h € ker A and thus rge B^erA 4- rge A = R K . Conversely, suppose that rge B|kpr A + rge A = R" and let u 6 ~RK. Since Pu e rge A there exists tu 6 Rn such that Pu — Aw. Let /i g ker A and z € Rn be such that Q(v — Bw) = Bh + Az. By multiplying both sides by Q we find that Q(v — Bw) — QBh or, equivalently, that Qv = QB(w + h). Thus it follows that v = Pv + Qv — Aw + QB(w + h) = (A + QB)(w + h) because h e ker^. From Lemma 6.5 it follows that for (t, x, x) € S(F) neither the nullspace of D^T(t, x, x) nor the condition rank D,j.r(t,x,x) — K depends upon the specific choice of F. The desired main result is now an immediate consequence of Lemmas 6.4 and 6.5 and Theorem 6.1 together with (6.25).

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Theorem 6.7. Assume that the constant rank condition (6.17) holds so that the mapping F of (6.18) is well defined and smooth. With (t0jxo,±0) € £(F) suppose that

or, equivalently, rank Dir(t0,x0,x0) = n, and that ( A ( x 0 ) z , z ) > 0 for every nonzero z € ker DxT(t0,xQ,x0) (i.e., nonzero z € kerD-jF^o^o^o) such that DxT(to,Xo,Xo)z € rge Dxr(t0,xo,x0)). Then there exists an open interval J C R1 with to € J such that the problem (6.1) with F replaced by F has a unique solution (x(t), A(i)) defined on J satisfying x(to) = x0, x(to) = XQ. Furthermore, (t,x(t),x(t)) £ S(F); whence T(t, x(t), i(t)) = 0 for every t e J. The value of Theorem 6.7, at least as regards any systems modeled by (6.1), should be clear. Its full significance will become more evident once we prove its generalization to the manifold setting in Chapter 7. The hypotheses of Theorem 6.7 are independent of the choice of the projection P (and hence of Q) used to define F in (6.18). However, since the proof of Theorem 6.7 utilizes Theorem 6.1 with F, different choices of P and Q will lead to different F, which means that F~ 1 (0) does depend upon these choices. Thus, we cannot use Theorem 6.3 to claim that the solution x(t) is independent of P and Q. Nevertheless, this is true as we shall see in Theorem 6.9 below. Lemma 6.8. Let (t,x,x) € S(F). Then the solutions h g R™ of the equation

are independent of the choice o f T . Proof. We use (6.18) to express equation (6.33) in terms of F and the projections P and Q. From the calculation of Dxr(t,x,x)h for (t,x, x) e S(F) in the proof of Lemma 6.4, we obtained in (6.31) that

where all mappings are evaluated at (t, x, x) and w € Rn is any vector such that D^Tw = -DxTx - DtT. A similar procedure yields DXT± and DtT. We find that

and

By (6.34), (6.35), and (6.36) this shows that equation (6.33) has the form

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where C - C(t,x,x,w) € £(R n ,R K ) and D = D(t,x,x,w)6 R" are independent of the choice o f f . In (6.37) wexTx + DtT] = -PDATw = -DATw. Thus (6.37) takes have P[D the form and, since Q projects onto a complement of rge D±T, this amounts to h e w + kerD^F, Q[Ch + D] = 0 or, equivalently, h e w + kerl^r, Ch + D 6 rge D±T. This characterization of the set of solutions of (6.33) is independent of the choice of F. Theorem 6.9. The solution x(t) (but not A(t)) obtained in Theorem 6.7 is independent of the choice o/F. Proof. We must revisit the proof of theorem 6.1 with F replaced by F. keeping in mind that, by Theorem 6.7, we already know that (t,x(t),x(t)) € £(F) for t 6 J. In the proof of Theorem 6.1 with F replaced by F, the solution x ( t ) is characterized as the solution of a second-order ODE x — !/>(£, x, x), where ^ corresponds to ip in (6.13) after replacing F by F everywhere. Therefore, it will suffice to show that ip(t,x,x)is independent of F when (i,x,x) is a point of S(F) in a neighborhood of (to,x0,xo) € S(F). In the proof of Theorem 6.1, •!/>(£, x,x) was obtained by solving the equation

for h. Because such a solution is not unique, h was identified uniquely by requiring, in addition, that

where 7?(t, x, x) denotes the orthogonal projection onto ker D±^(t, x, x). Equations (6.38) and (6.39) are the analogues of equations (6.10) and (6.11) in the proof of Theorem 6.1. (Note that, in that proof, h appears as x, which, of course, it becomes via the equation x = t/j(t,x,x) since the above procedure describes the calculation of h = ^ > ( t , x , x ) . ) Since we already know that ( t , x ( t } , x ( t ) ) G S(F) when x ( t ) is the solution given by Theorem 6.7, our interest is in the case when (t,x,x) belongs to S(F). Thus we can use Lemma 6.8 to assert that the set of solutions of (6.38) is independent of the specific choice of F. In other words, the only possible dependency of h upon F could arise from solving equation (6.39). However, this equation is also independent of F because 7r(t,x, x) and TT(£O,XO,XO) are the orthogonal projections onto kerD^r(t,x,x) and ker DiF(to,Xo,xo), respectively, and, by Lemma 6.5, these nullspaces are independent of the choice of F because (t,x,x) and (to,xo,±o) are in S(F). This completes the proof. Theorems 6.7 and 6.9 clarify fully that, when the equations of constrained motion are given by the system (6.1) where the constraint function F does not satisfy the ful rank condition (6.6), then a constrained motion under the more general constant rank condition (6.17) is given by (6.1) with F replaced by the mapping F of (6.18) and initial condition (to,xo,xo) e S(F). This choice of initial conditions is by no means restrictive since, by (6.25) and Lemma 6.4, every curve x(t) for which T(t,x(t),x(t)) = 0 also satisfies ( t , x ( t ) , x ( t ) ) e E(F) for

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PATRICK J. RABIER AND WERNER C. RHEINBOLDT

all t. In other words, the mapping F is essential for setting up and solving the correct system of equations, while the mapping F is used "only" for the selection of the initial values. Remark 6.1. If the full-rank condition (6.6) fails only at "exceptional" points, such as the points of a hypersurface in R1 x Rra x R n , then the constant rank condition (6.17) also fails, but the situation is entirely different. In fact, the formulation (6.1) is then valid away from these exceptional points, which therefore must be viewed as singularities. Such singularities have features very different from the well-understood "impasse points" (see Rabier and Rheinboldt [19] and [20]), and to the best of our knowledge the behavior of the solutions (x(t},X(t)))))_in their vicinity has not yet been investigated. As a rule, th study of singularities in implicit second-order ODEs is much more difficult than in the first-order case, and there is very little available in the literature. The thesis of Lei [16] contains partial but substantial results in that direction.

Chapter 7

DAE Formulation on Manifolds The results of section 6.2 were based on the setting of the DAE (6.1) for which the coefficient functions are defined on Euclidian spaces. This setting does not include the cases of Theorems 5.1 or 5.3 (or their generalizations to systems of rigid bodies), since there the coefficient functions, and, in particular, the constraint functions, are defined on the tangent bundle of a manifold. This chapter extends the results of section 6.2 to this case. For this we first generalize the existence and uniqueness theorem, Theorem 6.1, to DAEs on manifolds. Then we extend the theory of section 6.2 for problems where the full-rank condition is not valid. This requires a prior analysis of certain derivative operators, which, in turn, allows us to show that, once again, the specific choice of the modified constraint has no effect on the solution. As in section 4.4, the material in this chapter assumes a somewhat deeper background in analysis than the other parts of this monograph. In practice, the manifold M in this chapter will be a product of copies of R3 and 3 R x S3. If at least one copy of R3 x S3 is involved, then M is not "flat." All the issues investigated here arise because of the lack of flatness of Ai. Since S3 is no flatter than any other manifold, there would have been no advantage, especially in clarity, in developing a theory specialized to the case arising in the applications. Furthermore, the generality of the treatment given here has at least one concrete application involving arbitrary manifolds, discussed in the second part of section 7.6.

7.1

Existence Theory on Manifolds

We generalize the results of section 6.1 to DAEs for which solutions are sought on a given m-dimensional manifold M. In other words, we wish to consider now a DAE of the form

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PATRICK J. RABIER AND WERNER C. RHEINBOLDT

with the coefficients

The change of notation for the constraint (G in (7.1) versus F in (6.1)) is motivated by the material in section 7.4, where we will show how to pass from a "G" constraint (on R1 x TM) to a T" constraint (on R1 x Rn x R"). Without further limiting assumptions, the formulation (7.1) requires a substantial preliminary discussion since, for instance, the second derivative x(t) of a curve x(t) on a manifold M. has, in general, no intrinsic meaning. To avoid difficulties of that type, we shall confine our attention to the case when M. is a submanifold of R™. This suffices for all our applications because, say, in Theorems 5.1 and 5.3 we have M = R3 x S3 and hence the embedding M. C R" holds with n = 7. Correspondingly, in the case of the DAEs (5.49) or (5.56) modeling the motion of i rigid bodies, the manifold M. is a product of I copies of R3 x S3 and hence M can be embedded in R7^. The first simplification introduced by the embedding M C R" is that, for every curve x(t) in M, we obtain a well-defined vector x(t) by simply viewing x(t) as a curve in R". Furthermore, a routine check reveals that all the second derivatives arising in Chapter 5 are understood exactly in that sense. Next, our application examples show that for every x € M, the mapping F(t,x,-) should map TXM. C Rn into itself. This property is usually summarized by saying that F is a tangent bundle morphism. It is satisfied by all the choices of F arising in Chapter 5.12 For (t,x,x) € R1 x TM, we have D±G(t,x,x) 6 £(TxM,Hk) and DxG(t,x,x)~r £ C.((TlkY,T*M), where TxM)f is the dual space of TXM. As usual, by means of*M = (T the Riesz representation theorem and with the canonical inner products, the spaces (R*)* and (R")* are identified with Rfe and R", respectively. This produces the identification T*M = TXM. Indeed, if I* e T*M is a linear form on TXM C R", then £* can always be extended to R", albeit not uniquely, and thereby produces an element f* € (Rn)*. This element is identified with t 6 R" via t*(v) = (£,v) for every v € R", where (-,-) denotes the canonical inner product of R n . Therefore, for v € TXM, we have t*(v) = l*(v) = (l,v) = (l,v), where I € TXM is the orthogonal projection of I. It is straightforward to check that t is independent of the extension I* of I* and hence that I* 6 T*M i—> I 6 TXM is the desired canonical identification of T*M with TXM. From all this it follows that DxG(t, x, z)T € £(R fc , TXM) for (t, x, i) e R1 x T X M . In turn, this implies that, for A £ R fc , the sum F(t, x,x) + DxG(t,x,x)r X e TXM is defined without ambiguity. Now note that the acceleration vector x(i) is not confined to any particular subspace of R" by the mere condition that the curve x(t) lies in M. Therefore, in order for the equation A(x)x = F(t,x,x) + DxG(t,x,x)\ in (7.1) to make sense, it is necessary in (7.2(i)) to require that the linear operator A(x) € £S(R"), x € M, maps into TXM. This condition is satisfied by the matrix A(x) arising from the system (5.18) 12 It may be noted that we are here taking some liberty with the terminology: Strictly speaking, it is not F but, for each t e R1, the mapping (x,x) 6 TM <—> (x,F(t,x,x)) 6 TM that is a tangent bundle morphism.

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in Theorem 5.1 where M = R3 x S3; in fact, there, it follows from rge H(p)T = TpS3 = {p}1- for p € S3 by (4.18). More generally, by the same argument, rge A(x) C TXM holds for all the problems of interest here, in particular, also those considered in section 5.3 on multibody systems. Note that the condition rge XM. implies that A(x)TxM CA(x) C T TXM. Since A(x) is symmetric, it then follows that A(x)(TxM)LC (TXM)L;whence (TxM)1- C keiA(x) because xM)-L C rge A(x) C TXM. In the system (5.18),A(x)(T the condition (TXM)^ C ker A(x) is a consequence of H(p)p = 0 for p e S3 and, in fact, f o r p e R4 by (4.14). At this stage, we have all the conditions necessary for making sense of the system (7.1) in the more general setting of submanifolds of R n . Our next task then is to extend to this setting the existence and uniqueness results proved in section 6.1 for the case M. = R n . This will not only establish the existence and uniqueness of the constrained motion of systems of rigid bodies but will also provide us with an approach for characterizing such motions when the original constraint fails to satisfy the necessary full-rank condition. Theorem 7.1 (local existence and uniqueness). Suppose that in the DAE (7.1) the coefficient functions (7.2) are defined and smooth in a neighborhood of (to,Xo,Xo) £ R1 x TM., where F is a tangent bundle morphism. Moreover, assume that

and that

Then there exists a 6 > 0 such that (7.1) has a unique solution (x(t), A(t)) for \t — to\ < 6, which satisfies the initial conditions

Along this solution, the full-rank condition DxG(t,x(t),x(t)) = k holds for \t — to I < o. Proof. Let
M. be a local parametrization of .A/I near XQ, which is defined on an open subset Vm C R m . Moreover, let £o € V m , £o € Rm be the unique points such that 0(£o) = ^o and Z>0(^o)^o = %o- 1fx(t) is any curve in M. with x(to) = XQ, x(tg) = x0, then f(t) := $~l(x(t)) is a curve in Vm with ^(t 0 ) = Co, Co(*o) = Co- Furthermore, we have x(t) = £>0(^(t))^"(t) + D2(j)(£(t))(£,(t))2,so that the system (7.1) may be rewritten as

where For x = (£) the operator -D0(£) is a linear isomorphism from Rm onto TXM; whence D(j)(£)is a liXMtoRm.Therefore, the system (7.6)isnchangednear isomorphism ofT

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PATRICK J. RABIER AND WERNER C. RHEINBOLDT

if the first equation is multiplied by D0(£) T - In fact, recall that, by (7.3) and (7.7), andbecauseFisabundlemorphism,therightsideofthiquationXM,x =sinT 0(0- Since the term D^)1 D±G(t, 0(£), £>0(00T coincides with D^G'i>(t^,£.)T, where ^(t, £, 0 := G(t, 0(£), £0(0£), we obtain the system

which is now in Rm and, therefore, to which Theorem 6.1 potentially applies. In order to check that Theorem 6.1 does apply to (7.8), we must verify the three conditions

By (7.4(i)) we know that

if and only if D0(£o) is a bijection from Rm to TXOM, we have by (7.4(iii))

Now all stated results follow directly from Theorem 6.1. Remark 7.1. The solution x ( t ) (but not A(£)) obtained in Theorem 7.1 is independent of the choice of G, as long as G -1 (0) is unchanged near (to,Xo,Xo) and condition (7,4(iii)) continues to hold. This follows from Theorem 6.3 after transformation of the problem using a parametrization. Theorem 7.1 is tailored to systems such as (5.18) (or its multibody generalization (5.49)). In that setting, the use of Theorem 7.1 requires the explicit knowledge of initial conditions (u(t0),p(tQ)) = (u0,p0) and (u(to), p(t0)) = (u 0 ,po). In this respect, we recall that Remark 4.2 explains how p$ and po can be recovered from two more frequently available data, namely, the rotation RQ specifying the initial position of the body in the reference frame and the initial angular velocity vector WQ. As in section 6.2, we want to consider now also the case when in (7.1) the full-rank condition (7.4(iii)) no longer holds. Then, as before, the constraint G = 0 has to be changed into a new constraint G = 0 and we have to consider the effects of that change.

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A Projected Derivative Operator

As in the previous section, let M be a smooth m-dimensional manifold and consider a mapping G : R1 x TM —*• R fc , as in (7.2(iii)). For reasons that should be clear by now, we want to incorporate into our theory the case when for (t,x,x) € R1 x TM. the derivative DxG(t,x,x) with respect to the variable x has constant, yet not necessarily full, rank. For the case M = R" this was considered in section 6.2, where G = F, and there we also saw the relevance of the operator DxT(t,x,x) for these results. In fact, there is no hope of expecting to develop the desired extensions of the theory to manifolds without using the "derivative" DxG(t,x,x}. Unfortunately, similar to section 4.4, this derivative does not exist except when M = Rn or when G(t,x,x) is independent of ±, since x is the base variable of the bundle TM.. The most important occurrence of Dxr(t,x,x) in section 6.2 is in the term Q7x,x)DxT(t1x,x)x in the definition (6.18) of the mapping F. In this section we(t will show that the absence of a derivative DxG(t,x,x) in the manifold setting does not affect the existence of a well-defined operator replacing Q(t,x,x)DxG(t,x,x). In other words, it is still possible to make sense of the combined term QDXG, even though DXG alone is not defined. In addition, some consequences of this result, needed later, will also be examined. Let (to,xo,±o) € R1 x TM be fixed, so that XQ € M and XQ G TXOM. Moreover, let PO 6 £(R /C ) be a projection operator onto rge DxG(to, ZQ, ±o) and denote by Qo G £(R fe ) the operator Q0 := / - P0- Note that DxG(t0,XQ,xo) is unambiguously defined as the derivative of the mapping x e TXoM i—> G(to,xo,x) € R fc at x — XQ. Let Um c M be the domain of a chart 0"1 containing the point XQ, and (j>: V™ —> Um the corresponding parametrization of M defined on an open subset Vm C Rm. We denote by £o € Vm and £o € Rm the (unique) points for which <j>(£o) = x0, ^50(Co)Co = %$• Every mapping H on R1 x TM (or, more generally, on R1 x TUm) induces a mapping H^ on R1 x Vm x Rm via

L e m m a7.2 of the parametrization 4>. Proof. Let ip be another parametrization with VK^o) = X 0) D^>(r]o)r)o — XQi so that, with the change of coordinates B := (f>~o •!/>, we have G^(t, 77,77) = G^^.O^^B^fi). Differentiation with respect to r\ then implies that

Since, by equation (7.9), DfG^^o^o,^) = DxG(t0,xo,±o)D(f>(^0), QoD^^o^o^o) = 0, and, hence, by multiplying (7.10) by Q0, that

we obtain that

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PATRICK J. RABIER AND WERNER C. RHEINBOLDT

With

it then follows that

as had to be proved. Lemma 7.2 shows that for any point (t, x, x) £ R1 x TM. and any projection P(t, x, x) onto rge DxG(t,x,x), we may define, with Q := I — P, the operator [QDxG}(t,x,x) € C(TxM,Rk) by

arametrization of M. about x such that x = (£) and x = -D0(£) whence Q*(£,£,£) = Q(t,x,x). The (bracketed) notation [QDXG] is meant to emphasize that, generally, Q and DXG cannot be separated, that is, that [QDxG](t,x,x) should be thought as, but is not equal to, Q(t,x,x}DxG(t,x,x), which does not exist since DxG(t, x, x) does not exist. Since in (7.11) we have x = £>(£)£> it follows that

irrespective of the parametrization (/>. We are now in a position to generalize the definition of the set E(F) in (6.16) and begin with a preliminary remark. Lemma 7.3. The set

is independent of the choice of the projection Q(t, x, x) := / — P(t, x, x), where P(t, x, x) projects onto rge D±G(t,x,x). Proof. Since, for a given parametrization 0 with x = (£) and x = £>(£)£, we have

it follows from (7.11) that [QDxG](t,x,x)x + Q(t,x,x)DtG(t,x,x) = 0 if and only if <9*(*,&e)[-De G *(*.60€ + AG*(t,£,C)] = °- But <9*(*.£>0 = Q(*,x,x) projects onto a complement of the range of DxG(t, x, x), and, since DfG^(t, £, |) = DxG(t, x, ±)£>^>(£) and D(£) is an isomorphism of Rm onto TXM, it follows that rge DjG^(t,^,^) = rge D±G(t,x,x). Thus, Q^,£,£)[£> 4 <S"*(i,£,£)£ + AG"^(t,e,0] = 0 means that

which is independent of Q(t, x,x). In view of Lemma 7.3 we may now define the set

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which is independent of the choice of Q(t, x, x). In analogy to (6.17), suppose that

where V is an open neighborhood of E(G) in R1 x TM. This ensures that projection operators P(t,x,x) onto rge D±G(t,x,x) exist, which depend smoothly upon (t,x,x) e V. For instance, orthogonal projections have this property. As before, we set Q(t, x, x) := I - P(t,x,x) and define for (t,x,x) e V:

Obviously, since Q is smooth, the right side of (7.12) is a smooth function of (£,£,£). TherefoxG](t,x,x)x is a smooth function of (t,x,x) e V and also G in (7.15) isre, [QD smooth. A trivial but important consequence of (7.11) and (7.15) is that

which shows that the set G~ 1 (0) n G~l(Q) is independent of the choice of G, that is, ofQ. Remark 7.2. It is important to note that there is no ambiguity in this approach because the definition of [QDxG](t,x,x), and thus of [QDxG](t, x,x)x, requires no smoothness or even continuity of Q with respect to (t, x, x). In particular, Lemma 7.3 holds without any such continuity requirement. This is crucial to defining £(G) in (7.13) independently of Q(t,x,x), which in turn shows that the neighborhood V of £(G) on which the constant rank condition (7.14) holds is independent of Q(t,x,i), and finally permits the definition of [QDxG](t, x, x)x as a smooth function of (t, x, x) when Q itself is smooth. By using (7.12), we may rewrite (7.15) as G(t,x,x) = G^ (£,£,£), where x = <£(£), ± = D(t>(£)£, and & is given by (6.18) with T, P, and Q replaced by G*, P*, and Q*, respectively. By (7.9), this is Since (t,x,x) € G^O) (respectively, G"1^)) if and only if (£,£,£) £ (G*)- 1 ^) (respectively, (G*)- 1 (0)), it follows from (7.16) and (7.17) that

The relations (7.17) and (7.18), together with the arguments of Lemma 7.3, now allow us to recover important properties of G from their counterparts proved in section 6.2 for the case M = Rn. Lemma 7.4. For (t,x.x) 6 S(G) we have

Furthermore, the nullspace kerZ?iG(t,x,x) is independent of the choice of the projections P and Q used in the construction of G, Thus, in particular, rank DxG(t,x,x) is independent of P and Q.

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Proof. Let <j> again be a parametrization of M. and set x = 4>(£), x = £>^(£)£. By (7.9), and since D<j>(£) is an isomorphism of Rm onto TXM, we have ker D±G(t,x,x) = £>
Two different choices for the projections P and Q result in two different choices for the projections P^ and Q* used in constructing G*. By Lemma 6.5, the nullspace ker DfG^(t, £, £) is unaffected by these changes and hence ker D^G(t, x, x) is independent of P and Q. Clearly, this implies the independence of rank DxG(t,x,x). Unlike in the case M = R", the condition rank D±G(t,x,x) = k is not expressed by the (here meaningless) relation (6.27) for F = G. However, this relation continues to be available with F = G^, and this provides a way of checking the condition rank D±G(t,x,x) = k by using local parametrizations. Moreover, Lemma 6.4 is easily generalized to the manifold setting by means of (7.16), (7.17), and Lemma 6.4 itself with F = G*. Lemma 7.5. A smooth curve x(t) in M. defined on some open interval J C R1 satisfies G(t,x(t),x(t)) =0 on J if and only if it satisfies

at some point to G J.

7.3

Nonmaximal Rank in the Manifold Case

We return again to the setting and notation of section 7.1 and assume now that the mapping G of (7.2(iii)) satisfies the constant, but not necessarily full, rank condition (7.14); that is,

where V is an open neighborhood of the set S(G) in R1 x TM. Recall that the set £(G) of (7.13) is independent of the projection Q appearing in the definition. The condition (7.14) ensures that the (smooth) mapping G in (7.15) is defined on V and, hence, if desired, can be extended to all of R1 x TM. Recall, moreover, that the smoothness of G depends upon the smoothness of P which cannot be continuous if (7.14) fails to hold. By Lemma 7.5 and Theorem 7.1 with G replaced by G, and since £(G) = G -1 (0) n G-1(0) by (7.16), we obtain at once the following generalization of Theorem 6.7. Theorem 7.6. Suppose that the constant rank condition (7.14) holds so that the mapping G in (7.15) is well defined and smooth. Let (to,xo,x0) € £(G) and assume that

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rank DxG(t0,x0,xo) = k and ( A ( x 0 ) z , z ) > 0 for every z € kerDxG(t0,x0,x0), z ^ 0. ITien i/?,ere is an open interval J C R1 with to & J such that the problem (7.1) with G replaced by G and A, F satisfying the conditions of Theorem 7.1 has a unique solution (x(t),X(t)) defined on J and satisfying x(t0) = x0, ±o(to) = io- Furthermore, ( t , x ( t ) , ± ( t ) ) € £(G); whence G ( t , x ( t ) , x ( t ) ) = 0 for every t <E J. Lemma 7.4 shows that the hypotheses of Theorem 7.6 are independent of the choice of G defined by (7.15). As in the case M = R™, we then obtain the following extension of Theorem 6.9. Theorem 7.7. The solution x(t) (but not A(i)) obtained in Theorem 7.6 is independent of the choice of G. Proof. Once again let 0 : Vm —> M be a parametrization of A4 near XQ defined on some open subset Vm C R m and £0 6 Vm, £0 e Rm the unique points for which x o-fj 7-1), G is replaced by G, this system reduces to (7.8) with G* replaced by G*. Since G* = G* by (7.17), two different choices for G thus lead to two different choices for G*. But, by Theorem 6.9, the corresponding solution £(£) of the system (7.8) with either choice of G$ and £(to) = £o ; £(^o) = £o is independent of that particular choice. Since the solution obtained in Theorem 7.6 is x(t) = <j>(£(t)), it is independent of the choice of G. The significance of Theorem 7.6 regarding the constrained motion of a rigid body is easy to infer from the comments at the end of section 5.2, but there are some interesting subtleties that could not be mentioned before. Let B be a rigid body subjected to a constraint G = 0, where G : R1 x T(R3 x 3 5 ) —> R fc and rank DxG(to,XQ,Xo) = k at some point (£0,2:0, .TO) € G~ 1 (0). Then the generalized Gauss principle ensures that the motion of B beginning at the "initial" state (£o,£o,£o) is characterized by the system (7.1) with appropriate A(x), F(t.x,x) as in Theorem 5.1 (in particular, x — (u,p),x = (it,p)) and, of course, with the initial conditions x(to) = XQ, x(to) = XQ. Suppose now that rank D±G(t,x,x) = r < k for (t,x,x) in a neighborhood of (to,z 0 ) xo) and that (t0,x0,x0) € £(G). Recall that the set E(G) of (7.13) depends intrinsically upon G. Then G is well defined by (7.15) in the neighborhood of (to, XQ, XQ, and every motion of B that is compatible with the constraint G = 0 must also be compatible with the constraint G = 0. This is (i) of the "only if" part of Lemma 7.5. It also shows that there exists no motion of B originating from the state (to,x0,x0) that is compatible with G = 0 unless (t0,x0,Xo) € G- 1 (0)nG~ 1 (0). By (7.16), G-l(Q)riG-1(0) coincides with S(G), the very set in the vicinity of which G was constructed. This discussion shows that, when rank .D±G(£o,£o,io) — k, the motion of B starting at (to,xo,±0) 6 E(G) must be characterized by the system (7.1) with G replaced by G. Thus, the correct equations of motion, when rank D±G is constant but not full, are implied by what they are when rank D±G is full: No extra physical argument is needed and everything follows from the generalized Gauss principle. Now Theorem 7.6 asserts

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that the solutions of the system (7.1) with G replaced by G and with the initial state (to,xo,xo) 6 £(G) do comply with the original constraint G —0. Theorem 7.7 settles the fundamental question of the uniqueness of the motion constrained by G = 0 and originating from the state (to,xo,xo). This is by no means an obvious issue, nor a usual one, since the equations of motion corresponding to the constraint G = 0 are not directly expressible in terms of G but, rather, in terms of G, and there is no canonical choice for G. By proving that the specific choice of G has no effect on the solution, Theorem 7.7 answers the uniqueness question in the only way possible for such a problem. Last, the fact that the hypotheses made in Theorem 7.6 are also independent of the choice of G (see Lemma 7.4) completes the justification of the procedure of replacing G by G in (7.1) to obtain the correct equations of motion whe rank D±G is constant but not full in the neighborhood of some point of £(G). While our discussion here focuses on the case of a single rigid body, everything extends verbatim to systems of rigid bodies or mass points or both: only the manifold M. changes. The case of a geometric constraint G(t,x*) = 0 considered in section 5.2 is certainly a simple example where the full-rank condition for D±G fails to hold since D±G = 0. But, as noted, more general examples are provided by systems of rigid bodies where some bodies are subjected to geometric constraints and others to kinematic ones. It was this wide range of examples that prompted our development here of the general theory for the rank-deficient case. Remark 7.3. Theorem 7.7 gives only a partial answer to the dependence of the solution in Theorem 7.6 upon G and G,j3ince it shows only that, for given G, the solution does not depend upon the choice of G. This does not address the dependence of the solution upon G and, in fact, this issue is not solved by Remark 7.1 unless r = k. However, it is true that the solutions of Theorem 7.6 do not depend upon G as long as S(G) is unchanged and the constant rank condition (7.14) continues to hold (with the same r). The problem can be reduced to the case M. = R" via parametrizations and then handled by the subimmersion theorem rather than the implicit function theorem used in the proof of Theorem 6.3 for r = k. The proof is technical and is omitted here.

7.4

Reduction to DAEs on Euclidian Space

The problems of section 7.1 were posed on a submanifold M of R" and, by means of local parametrizations, reduced to problems on R m , m = dim.M, of the form (7.8). This is appropriate for theoretical considerations but may. introduce complications when it comes to the numerical implementation. Accordingly, we describe now a procedure by which our systems on M. can be transformed into systems on Rn without the use of any local parametrization. This procedure represents a simple generalization of our approach in Chapter 5 when passing from the formulation (5.18) in Theorem 5.1 to (5.26) in Theorem 5.2. We use the same notation as in section 7.1. An m-dimensional submanifold M. C Rn is always described, at least locally, as the zero set of a submersion g : Rn —> R n ~ m . In

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the case when M — R3 x S3 C R3 x R4, this is even true globally when one works with x — (u,p), g(u,p) : — \p\2 - 1. The same holds when M is a product of copies of R3 x 53. Hence, in the applications of interest here, it is not restrictive to assume that

where g is a submersion on M. = g~ 1 (0) and thus we have rank Dg(x) = n — m for every x e M. As before we continue to require that A(x) & £S(R") satisfies

but, in addition, we also assume that A(x) has a smooth extension AE(x) defined for every x € R n . Likewise, we assume that the tangent bundle morphism F : R1 x TM —> R extends as a smooth mapping FE : R1 x R" x Rn -> Rn and that G : R1 x TM -> R fc , k <m = dim.M, extends smoothly to GE : R1 x Rn x Rn -» R fe ; that is,

A solution (x(t), A(t)) of the system (7.1), with the curve x(t) lying in M, can also be viewed as a solution of the same system with x(t) as a curve in R n constrained by the condition g ( x ( t ) ) = 0. Thus, to solve the original problem, we may consider solving (6.1) in R" with A and F replaced by AE and F£, respectively, and with the constraint function F given by

However, things are not as simple since F in (7.22) never satisfies the full-rank condition DiF(i. x. x] = n—m+k at any point. In the previous section we developed a procedure for handling constraints that fail to satisfy the full-rank condition, and, in fact, for the case of R", the same but simpler procedure was discussed in section 6.2. For this discussion the simpler variant will suffice. Suppose that Since D±G(t, x, x) = D±GE(t, x, x)\T M , it follows that rank Dj..GF'(t, x, x) = k for every (t,x,x) € G-^O). It is then straightforward to check that the set E(F) of (6.16) (with F given by (7.22)) is because TM — {(x, x) € Rn x R" : g(x) = 0, Dg(x}± — 0}. Moreover, the relation D±T(t,x,x) = (DxGE(t,x,x),Q) shows that rank D±r(t,x,x) = k is constant on some open neighborhood V of S(F) in R1 x Rn x R n . The conditions now hold for constructing a mapping F via formula (6.18), and we may even choose P(t,x,x) independent of (t,x,x), that is, as the natural projection R n - m x R f c ^ R f c along the first factor. Then Q(t,x,x) = I-P(t,x,x) is the projection R n-m x R fc _j R n-m aiong ^ ne second factor and we obtain

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An explicit example of such a mapping F is the formula (5.23) in the case when M = R3 x S\ x = (u,p), and g(u,p) := |(b|2 - 1). From (7.23) and (7.24), and because TXM = k e r D g ( x ) for every x e M, it follows that Let (t0,x0,±0) € £(F) be fixed. By (7.24), we have G(tQ,x0,±0) = 0. Moreover, (7.23) gives i&nk D±G(to,Xo,xo) = k. Therefore, conditions (7.4(ii)) and (7.4(iii)) hold with G. The remaining condition (7.4(i)) will be met by assuming that

With this, Theorem 7.1 applies and ensures that the system (7.1) with initial conditions x(t0) = x 0 (e M), x(t0) = i0(e TXOM) has a unique solution (x(t),X(t)) with x ( t ) lying in M.. By (7.25), the characterization TXOM = ker Dg(x0) and the relation

provide that kerDxG(to,xo,xo) = ker DxT(to,x0,±o). By (7.27), we obtain that

By (7.24), (7.26), and (7.28), and since rank D±T(t,x,x) = k on the open subset V, Theorem 6.7 ensures that the system (6.1) with A, F, and F replaced by AE, FE, and F, respectively, and with initial conditions x(t0) = x0 and X0(t0) = XQ has a unique solution (x(t), A(i)) with x(t) in R". The following theorem shows that the notation x(t) is consistent with that used above when calling (x(t), A(t)) the solution of (7.1) with x(t) lying in M.. Theorem 7.8. With the splitting A(t) = (A(*),//(*)) € Rfc x R n ~ m , the pair (x(t),\(t)) is the (unique) solution given by Theorem 7.1. Proof. Since (x(t),X(t)))),solves (6.1) with A, F, and F replaced by AB, FE, and f, respectively, we have, omitting t-dependence for notational convenience, that

andx(to) = x0, x(t0) = XQ, (t0,xo,xo) € S(F). Theorem 6.7 ensures that T(t,x(t),x(t)) + 0; whence g(x(t)) = 0 and therefore x(t) € M by (7.19) and G(t,x(t),x(t)} = 0 since GB 1 = G. Thus, x(t) lies in M. and satisfies the constraint (7 = 0. In particular, \R xTM

AE(x(t)) = A(x(t)) and FE(t,x(t),x(t)) = F(t,x(t),x(t)) by (7.21). For the completion of the proof, it remains to show that the first equation in (7.29) entails

since it is indeed the x-derivative of G that is involved in Theorem 7.1.

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Thus, the first equation in (7.29) becomes

By the hypothesis (7.20), A(x) maps into TXM for x 6 M. Moreover, F is a bundle map; whenceF(t,x(t),x(tx(t}M.On the other hand,Dg(x)maps into(TXM)Lfor))£T x e MXM = kerDg(x) and rge Dg(x)T = [ker Dg(x)]-L. Thus, bymultiplyingsincimT (7.31) by the orthogonal projection onto TXM, we obtain

Here the last term is the tangential component (equal to the orthogonal projection) of D±GE(t,x,x)'rXontoTXM, discussion of the special case at the beginning of section 5.1). Thus, (7.29) holds and the proof is complete. A similar procedure can be used in the more general setting of Theorem 7.6, that is, when condition (7.23) fails. Here (7.15) and (7.11) indicate that the calculation of the mapping G in the manifold setting requires the use of parametrizatioiis and is rather complicated. Thus in this case a formulation of the problem in Euclidian space does offer a particular advantage. However, such a formulation in Euclidian space cannot be obtained with an arbitrary smooth extension GE of G. This contrasts with the case when condition (7.23). that is, condition (7.4(iii)) of Theorem 7.1, holds. In order to highlight this fact further, we proceed in reverse order and first describe a problem in Euclidian space that yields solutions to a problem on M essentially similar to the one resolved in Theorem 7.6. We assume that the mapping GE satisfies

on an open neighborhood V of E(G £ ) n (R1 x TM] in R 1 x R n x Rn (and not merely in R1 x TM). Here, T>(GE) is given by (6.16) with F replaced by GE, while F continues to be defined by (7.22). We thus have

so that V is also an open neighborhood of E(F) in R1 x Rn x Rn and

since rge Dxr(t,x,x) = rge DxGE(t,x,x) x {0} C Rk x R n ~ m . Then, it is obvious how a projection P(t,x,x) e £(Rk) onto rge D±GE(t,x,±) yields the linear projection P(t,x,x) x 0 from ~Rn~~m x R fc onto rge DxT(t, x, x), and, similarly, how Q(t,x,x) = I — P(t, x, x) provides the projection Q(t, x.x)xl onto some complement of rge DxT(t, x, x). Thus, we obtain a mapping F derived from F via formula (6.18) in the form

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where GE is given by (6.18) with F replaced by GE. From (7.33) and the characterization TM = {(x,x) € R" x R™ : g(x) = 0, Dg(x}± = 0} it is clear that

which therefore leads to the assumption

Now let (t0,x0,x0) e R1 x TM. Since

it follows from Theorem 6.7 that, when

then the system (6.1), with A, F, and F replaced by AE, FE, and F, respectively, has a unique solution (x(t), A(t)) for which x(to) = XQ, x(t0) — ±o and which also satisfies (t,x(t),x(t)) € S(F). In particular, we have T(t,x(t),x(t)) = 0 and thus g(x(t)) = 0; that is, x(f) e M, as well as GE(t,x(t),x(t)) = G(t,x(t),x(t)) = 0. Below, we will show that, under our assumptions, the hypotheses of Theorem 7.6 are satisfied and that, when A(t) is split in the form A(t) = (A(<),//(*)) e R* x Rn~m, then (x(t), A(t)) is the (unique) solution provided by Theorem 7.6. In both steps, we need the following technical lemma whose proof will be postponed until the end of this section. Lemma 7.9. Under the stated assumptions the following two relations hold: (i) E(G) C E(GE) n (R1 x TM). (ii) I/rank DxGE(t,x,x) = rank DxG(t,x,x) for all (t,x,x) 6 V n (R1 x TM), and P(t,x,x) e £(R fe ) is a projection onto rge DxGE(t,x,±) for (t,x,x) £ V, then, with Q:=I-P,

Lemma 7.10. Under the stated assumptions the hypotheses of Theorem 7.6 are satisfied. Proof. First, we must show that rank D±G is constant on some open neighborhood of E(G) in R1 x TM and hence that (7.14) holds. By Lemma 7.9(i), it suffices to show that this is true on some open neighborhood of S(G£) n (R1 x TM} in R1 x TM. In the following all the mappings are assumed to be evaluated at a point (t,x,x) € S(GE) n (R1 x TM) = E(F) (see (7.33)). By (7.36) and (7.37) we haveAf = ra n—m+k and hence by Lemma 6.5 with Q replaced by Qx.1, rank (DiF + (Qx/)£> a; r) = n - m + k. Since D±T + (Q x I)DXT = (D±GE + QDXGE, Dg(x}} by (7.22), this means that rank (DxGE+QDxGE,Dg(x)) = n-m+k, i.e., dimkeT(D±GE+QDxGE,Dg(x)) = m-k. Since kerDg(x) = TXM, this also reads dimker^iGjf^+Q-DsGjf,^) = m-k.

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Thus rank xGfT M + QDxGfT M) = k at every point (t, x, x) 6 E(G£) n (R1 x TM)(D and hence on an open neighborhood VM of E(G£) (^(R1 x TM) in R1 x TM since k is thee maximum possible rank. Therefore we have rank (DXGE + QDXGE)\TTM — ^> where the mappings are evaluated at (t,x,x) € VM. This implies that rge DXGET^M = rge DXGE, for, given 2 e R n , there exists h 6 TX.M such that DxGEh+QDxGBh = DxGEz; whence DxGBh = DxGEz by multiplying both sides by P. Thus, rank DxGfT^M = rank DXGE on VM. But, since D^G^ M — D±GE, this amounts to rank DXG = rank DXGE on VM; whence rank DXG = r on V.M by (7.32), after shrinking VM if necessary, to ensure that VM C V. We now prove that

By the demonstrated constancy of rank DXG on V», G is well defined by the formula (7.15). The above proof also showed that rank DXG — r := rank DXGE on VM- Since rge DXG C rge DXGE, this implies that rge DXG = rge DXGE. so that Q|v^ is a (smooth) projection onto a complement of rge DXG and therefore can be used for the calculation of G. On VM, the conditions required in Lemma 7.9(ii) are satisfied. Therefore, by (7.15) and (7.39), and, since GE = G on R1 x TM, we have

which, of course, is simply the relation G = GE\vM since GE is given by (6.18) with F replaced by GE. By the openness of VM in R1 x TM, this implies that D±G — D±GE\TtM on VM and thus, because of E(G) C VM, that

Now (7.40) follows from (7.41) and the hypothesis (7.37) since, by Lemma 7.9(i), E(G) C S(G E ) n (R1 x TM). Actually, the relation G = GE\\>M, proved above, together with (7.33) and S(G) = G-1^) n G-HO) as well as S(G£) = (G^)- 1 ^) n G^'^O) (see (7.16) and (6.25) with F = GE, respectively) shows that

(However, here (7.32) and (7.37) are used via arguments involved earlier in the proof to justify (7.42). In general, only Lemma 7.9(i) is true.) By (7.42), the point (t 0 , x 0 ,x 0 ) 6 E(G £ )n(R 1 xTM) is actually in S(G). From (7.40) it also follows that rank D±G(t0, x0, XQ) = k. Finally, the last property to check for ensuring the applicability of Theorem 7.6 is that (A(xo)z.z) > 0 for z 6 ker DxG(t0,x0,±0), z -£ 0. But, due to (7.41), it is immediate that this is the earlier assumed condition (7.38).

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Remark 7.4. Conversely, the hypotheses of Theorem 7.6 do not imply the validity of the condition (7.32) except when r — k. Thus, condition (7.32) is the additional requirement necessary to transform the problem from one on M. to one on R n . It merely says that not every smooth extension GE of G can be used. Otherwise, such a transformation is not possible. Before proving Lemma 7.9, we complement Lemma 7.10 by generalizing Theorem 7.8 as follows. Theorem 7.11. With the splitting X(t) = ( \ ( t ) , f i ( t } ) £ R f e xR n ~ m , the curve (x(t),X(t)) is the unique solution given by Theorem 7.6. Proof. We proceed as in the proof of Theorem 7.8. As noted after (7.38), we have

whence x(t] lies in M and GE(t,x(t),x(t}) = 0, so that G(t,x(t),x(t)) = 0. Thus, it suffices to show that (x(t), A(t)) solves the equation

for then the uniqueness part of Theorem 7.6 (available by Lemma 7.10) proves our claim. The relation with F given by (7.35) yields

Next, by using AE(x(t)} = A(x(t}} and FE(t,x(t),±(t)) = F(t,x(t),x(t)} since x(t) lies in M, and by multiplication by the orthogonal projection onto TXM — kerDg(x), we obtain where [D±GE(t, x, x) T A]r is the tangential component of DxGE(t, z,i) T A and hence equals (D^GE(t,x,x)\TiM)r X. Now, by the relations (7.43), (7.42), and (7.41), we have D±GE(t,x,x)lTiM = D±G(t,x,x), and (7.44) follows. For completeness, we now prove Lemma 7.9. Proof of Lemma 7.9. (i) The characterization of S(G) in section 7.2 uses local parametrizations and a reduction to the case when M. is an open subset of Rm. Naturally, any suitable parametrization can be used. In particular, we can apply the parametrization induced by a local diffeomorphism of Rn = Rm x R n ~ m that maps (locally) R m x {0} onto M. This reduces the problem to the case when M. is an open subset of R m , GE has the form GE(t,xi,x2,xi,X2) relative to the splitting R" = Rm x R n ~ m , and G(t,x,x) = GE(t,xi,0,xi,0) with x = (n,0), x = (ii,0).

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In that setting, to say that ( t , x , x ) € £(G) means that

and that i.e., that there exists hi & Rm for which

On the other hand, to say that (t,xi,X2,xi,X2) € T,(GE) means that G B (i,xi,x 2 l xi,x 2 ) = 0 arid that

i.e., that there exist (/ii,/i 2 ) e Rm x R n ~ m such that

Because

it is clear that when (7.45) holds, then (7.46) holds with x 2 = 0, £'2 = 0, and /i2 = 0. Thus {(t,x,x) € £(G),x = (x!,0),x = (x - i,0)} implies that (t,xi,0,xi,0) € £(G B ) and thus ( t , x , x ) € E(G E ). It is obvious that E(G) C R1 x TA1 and hence, as claimed, E(G) c S ( G £ ) n R 1 xTAi. (ii) For (t, x, x) e Vn(R J xTX) the condition rank D±GE(t, x, x) = rank D.iG(i, x, x) together with G = G^_ lxTjVf show that

Hence the projection P(t, x, x) can be viewed either as a projection onto rge D^GE(t, x, x) in Rfe or as a projection onto rge DxG(t,x,x) in R fc . Thus, the operator [QD xG](t,x,x..) makes sense for (t, x, x) e V n (R1 x TM). Since [QDXG] is defined via local parametrizations, we may proceed as in the proof of part (i) above, thereby reducing the problem to the case when M is an open subset of Rm C R™ and GE = G £ (i,xi,x 2 ,x 1 ,i 2 ), G(t,x,x) = GE(t,xi,0,±i,Q) where x = (xi,0), x = (xi,0). In that case, the definition of [QDxG](t,x,x) is

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while, since TXM = Rm x {0} and x = (ii,0), x = (zi,0),

Thus [QDxG](t,x,x) = Q(t,x,x)DxGE(t,x,x)\TIM>

an

d the proof is complete.

Remark 7.5. The results of this section can be summarized as follows: Let M C R™ be characterized by M. = g~l(0), where g : Rn —» R ra ~ m satisfies the condition rank Dg(x) = n — m at every point x € M. Assume that for x € M, A(x) € £5(R") satisfies the condition rge A(x) C TXM of (7.20). Furthermore, assume that F — F(t,x,x) is a bundle morphism, that is, that F(t,x,x) £ TXM for (t,x,x) € R1 x TM. Finally let G : R1 x TM —> R fc . Now suppose that A, F, and G have (available) extensions to R1 x Rn x R", denoted by AE, FE, and GE', with AE(x) e £ 5 (R n ) for x e R". Assume that the condition (7.32), i.e.,

holds on some open neighborhood V of £(G £ ) in R1 x R™ x R". Then a mapping G~£ : R1 x R™ x Rn -> R fc can be defined by (6.18) and can be extended arbitrarily outside V. If the condition (7.37) holds, that is, if rank£»iGS(i,x,x) = k

V (t,x,x) € S(G£),

then the mapping T(t,x,x) := (GE(t,x,x),Dg(x)x) (7.36): rank Dxf(t,x,x)=n-m + k

satisfies the full-rank condition

V(t, x, x) e £(F) = T,(GE) n (R1 x TM) = S(G).

Now let (to,xo,io) £ S(F) = S(G) be any point such that (A^rro)-^ z) > 0 for every 2 e keri'iF^o^o^o) = kerZ)iG B (to>xo)^o)|T, M> z ^ 0- Then the problem (6.1) with ^4, F, and F replaced by AE, FE, and F, respectively, has a unique solution (x(t), A(t)) such that x(to) = XQ, i(io) — ^Oi and that, with the splitting A(£) = (A(t),/u(i)), the curve (x(t),X(t)) is the unique solution provided by Theorem 7.6, which involves the original mappings A, F, and G and not their extensions AE, FE, and GE. If r = k in (7.47), then Theorem 7.11 coincides with Theorem 7.8 and G^ =GE above. For r < k in (7.47) the hypotheses made in this summary are stronger than those needed for the validity of Theorem 7.6 since they put limitations upon the choice of the extension GE. But without them the solution (x(i),A(t)) of Theorem 7.6 can no longer be obtained by the above procedure of extending the problem to Euclidian space. Fortunately, this theoretical limitation does not appear to be a serious one in the applications to rigid body motion.

7.5

Characterization of State Spaces

The material developed in this chapter helps to clarify the concept of a state space for a system of rigid bodies and/or mass points. Let M. C R" be a submanifold representing

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the configuration space of such a system, so that M is a product of copies of R3 x S1'3 and/or R 3 . Let the system be subjected to a constraint of the form G(t.x,x) = 0, where G : R1 x TM —> R fc , k < m — dim M. The essence of the results of this chapter is that, under general conditions, the only states (t,x,x) 6 R1 x TM compatible with the constraint are the points of (t,x.±) of the set £(G) introduced in (7.13), which depends intrinsically upon G. In fact, the motion of the system satisfying the constraint G = 0 under any external force F = F(t,x,x) can only occur from a state (to,xo,±o) € S(G), in which case the motion x ( t ) has the property that ( t , x ( f ) , x ( t ) ) € S(G) at all times. The conditions for E(G) to represent all the possible states of the system during the motion are simply that

where V is an open neighborhood of E(G) in R1 x TM., and that any mapping G denned from G by (7.15) satisfies the full-rank condition

Indeed, in Lemma 7.4 this condition was seen to be independent of the choice of G. Thus, when (7.48) and (7.49) hold, it makes sense to call S(G) the state space S of the system. Recall that (see (7.16))

and that, by (7.15) with Q = 0,

This definition of S coincides with the definitions introduced in Chapters 3 and 5 in the case of rather simple constraints G = 0. To corroborate this, we briefly review a few examples. (a) Mass-point systems. Here M = R3£. For r = k in (7.48) we have S(G) = G^O) by (7.50) and (7.51). The set G -1 (0) was indeed our choice for the state space S in (2.4) (where
which is S in (2.19). (b) Rigid body. In Chapter 3 we saw that the state space of a rigid body B is M — R3 x S3 provided that (u,p) and (u,—p) correspond to the same physical configuration. As a result, if G : R1 x TMm—> R* is used to constrain B, and if r — k in (7.48), then, as in (a) above, we find that

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With x — (u,p), x = (u,p), this is the state space S of (5.7). A geometric constraint (r = 0) leads to

Note that in this case DXG exists because G is independent of x and, because Q = I, [QDXG] = [DXG] in (7.12) coincides with DXG. Clearly, the general procedure for the calculation of S(G), as well as the correct conditions, that is, (7.48) and (7.49), under which £(G) can legitimately be called the state space S of the system, make it possible to characterize S for much more complicated systems where intuition and common sense would be of limited help. It should also be observed that the "natural" choice of R1 x T(R3 x S3) for the state space of an unconstrained rigid body is also provided by this method in spite of the fact that the constraint involved in that case has no physical meaning. This is shown in the next example. (c) Unconstrained rigid body. The configuration space R3 x 53 of an unconstrained rigid body is characterized by {(u,p) G R3 x R4 : \p\ — 1 = 0}, and hence

may be viewed as a constraint on M := R3 x R 4 . Clearly, for this G we have r = 0 in (7.48) and G(t,u,p,u,p) := 2pTp by (6.18) with T = G. Since G~l(Q) = R1 x R3 x S3 and, obviously, rank D^,p}G(t,u,p,u,p) — k = 1 on G~ 1 (0), the condition (7.49) holds on G-^O) and hence on E(G) C G-^O). Thus

Since there is no "physical" body with configuration space R3 x R4, the constraint |p|2 — 1 = 0 has no physical meaning. Yet the result that R1 x T(R3 x 53) should be the state space of an unconstrained rigid body is the (physically) correct one.

7.6

ODE Formulations

The equations of constrained motion given in Chapter 7.1, or in more general, abstract form by the DAE (7.1), can sometimes be manipulated to yield new formulations. For instance, in the case of mass-point systems, we showed in section 2.1 that the Lagrange multiplier A can be explicitly calculated as a known function A = A(i, x, x) of the state of the system. This procedure leads to equations of motion represented by the ODE (2.12) instead of the DAE (2.11). This approach can be extended to systems of rigid bodies and, in fact, to general DAEs of the form (7.1) on a submanifold M of R n . As in section 7.1 we assume that A : M —> £S(R"), F : R1 x TM —* Rn is a tangent bundle morphism and G : R1 x TM —> R fc . We further assume that the full-rank condition

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holds and that These assumptions, or at least minor variants, were already involved earlier, e.g., in Theorem 7.1. In addition, we require that A(x) is positive definite on TXA4,

which, together with (7.53), implies that

Since A(x] is symmetric, it follows from (7.55) that

Note that (7.54) is stronger than condition (7.4(i)) required in Theorem 7.1. From these conditions it follows that A(x)\TxM G GL(TXM] for every x G M. We spt

Suppose now that x — x(t), A = X(t) solves the system (7.1), where, of course, x(t) is a curve on M. By (7.56) we have A(x(t}}x(t] = A(x(t}}xT(t], where XT(£) e TX^M. is the tangential component of x(t). Thus, dropping the t-dependence for simplicity, we obtain and hence, with (7.57),

As a result, we have

By (7.52) and (7.57) the operator

is invertible; whence

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As in Remark 5.1 we use a local extension GE of G to make sense of the partial derivative DxG(t,x(t),x(t))x(t). Then differentiation of the identity G(t,x(t),x(t)) = 0 yields

By writing x = XT + XN-, where, of course, XN represents the normal component of x, we obtain

But XN depends only upon x and x via the formula

where Vx denotes the second fundamental tensor of the manifold M at x 6 M. (see, e.g., Choquet-Bruhat, DeWitt-Morette, and Dillard-Bleick [7] and also Rabier and Rheinbold [18] for an elementary description and computational aspects). Hence, by substituting (7.61) into (7.60) and the result into (7.59), we obtain a representation

of A in terms of a known function A : R1 x TM —> R fc . With this equation (7.58) becomes As we shall see below, equation (7.63) is a second-order ODE on .A/I, which has a unique solution for given initial condition

Moreover, if x0 and ±0 in (7.64) are chosen so that G(t,xo,±0) — 0, then the solution of (7.63) satisfies the constraint G(t, x(t),x(t)) = 0 at all times. The simplest way to see this is to recall that the DAE (7.1) with the same initial condition has a unique solution by Theorem 7.6, and, because (7.62) gives a formula for A in (7.1), this solution must also solve the ODE (7.63). The uniqueness of the solution of (7.63) thus implies that it coincides with the solution of (7.1) and hence satisfies the constraint G = 0. In order to see that (7.63) is indeed an ODE on M with a unique solution satisfying the initial condition (7.64) we define

so that H : R1 x TM -+ R" is a bundle morphism. Let 4>: Vm -> M (m = dimM) be a local parametrization of M near x 0, and £o 6 V m , £o 6 Rm be such that (£o) — ^o>

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Z)0(Co)Co = x0. If x(t) is a curve on M with x(to) = XQ, x(to) = XQ, then £(£) := 0"1 (#(£)) is a curve in Vm with £(£o) = Co, C(^o) = Co- Furthermore, we have x ( t ) = D0(C(£))C(t) and x(t) = Zty(£(t))£(t) + D 2 0(C(£))(C(£)) 2 . Since £><£(£(*)) maps onto Ta;(t)Ai, the tangential component £T(£) is given by

where, for every x e A4, H(x) is the orthogonal projection onto TXM and depends smoothly upon x £ M. Thus, with the notation of (7.65), the equation (7.63) becomes

which, together with the initial condition £(to) = £o> £(£o) = Co, is a second-order ODE in RTO with a unique solution. Except perhaps for the full-rank condition (7.52), all the hypotheses listed at the beginning of this section that ensure the reducibility of the DAE (7.1) to the ODE (7.63) will hold when (7.1) accounts for the constrained motion of a rigid body or a system of rigid bodies and/or mass points. This can be readily verified from the explicit form of the corresponding operator A(x) (see sections 5.2 and 5.3, respectively). Naturally, when the constraint function G satisfies the constant rank condition

where V is an open neighborhood of E(G) in R1 x TM (with E(G) denned by (7.13)), and if where G is given by (7.15), then, as we already explained in section 7.3, the equations of constrained motion are given by a DAE of the form (7.1) with G replaced by G. Due to (7.66) this DAE reduces to a second-order ODE on A4 of the form (7.63). Consistent with Theorem 7.6, the relevant initial conditions for this ODE must satisfy the requirement (to,XQ,xo) 6 S(G); that is, by (7.16), G(t 0 ,x 0 ,x 0 ) = 0 and G(t0,xo,x0) = 0. Another alternate formulation is possible when, on some submanifold M. of R n , the DAE (7.1) has the form

and G : M. —> Rfc satisfies the full-rank condition

Naturally, (7.67) is obtained when in (7.1) G is replaced by G(t,x,x) = G(x,x) :— DG(x)x. The condition (7.68) ensures that the set C := G-1(0) is a submanifold of M.

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of dimension m — k, m — dim M, usually called the constraint manifold of the problem.. The corresponding set S(G) in (7.13) is simply

As we saw, in order for (7.67) to account for the holonomic constraint G(x) = 0, the initial condition (£o,xo,xo) must be chosen in S(G) and hence such that G(XQ) — 0 and DG(x0)i0 = 0. In that case, the DAE (7.67) is simply

If x(f) is a curve in C, then the acceleration splits as x ( t ) = xr(t) +XAT(£), where XT and Xjv are the tangential and normal components of x. Although this notation is the same as the one used earlier, we emphasize that now "tangential" and "normal" are understood relative to the manifold C and not just to M. Since xw(t) — V£,t-,(x(t),x(t)), where V£ denotes the second fundamental tensor of C at x € C, we may rewrite (7.70) as

Since DG(x) maps into Rfe and TXC = ker DG(x) for every x £ C, observe that the operator DG(x)T e £(Rk,TxM) actually maps into (T^C)1- n TXM. Therefore, if Hc(x) € C(TXM) denotes, for any x e C, the orthogonal projection onto TXC, then multiplication of the first equation (7.71) by Uc(x) yields

Since XT € Tx(t)C, the left side of (7.72) coincides with Uc(x)A(x)Iic(x)xT. Hence, whenever the operator Hc(x)A(x)Hc(x) € L(T XC) is invertible, then (7.72) can be transformed into a second-order ODE on C. This invertibility property will hold for every x e C and every submanifold C of M if, as before, the operator A(x) 6 £s(Hn))ssatisfies rgej4(x) C TXM. and { A ( x ) z , z ) > 0 for every z € TXM, z ^ 0, and every x € M. As pointed out earlier, these conditions hold for problems arising from constrained rigid body motion (and also systems that possibly incorporate mass points). In summary, under the stated assumptions, the DAE (7.67) reduces to the secondorder ODE on C:

Note that the condition x e C need not be incorporated into (7.73) because it is redundant as soon as (7.73) is used with initial conditions x(
Chapter 8

Computational Methods Currently there appear to be no reliable general-purpose algorithms available on which production-level software may be based for solving the DAEs of the form (6.1) or, more generally. (7.1), arising as models of problems involving kinematic or mixed kinematic and geometric constraints. In this chapter we show that a class of algebraically explicit DAEs considered by Rheinboldt [23], based on local parametrizations, includes a method that can be adapted to the construction of an algorithm for solving the DAEs covered by Theorem 6.1. This algorithm is meant to show that it is indeed possible to construct an effective method for solving this class of DAEs. In section 8.3 we indicate some possible further directions of work that may lead to the development of other methods for the DAEs discussed in the previous chapters.

8.1

Computations on Manifolds

The DAE methods developed in [23] were based on a package of supporting algorithms, called MANPACK, for computing local parametrizations on submanifolds of R™ that are implicitly defined by local submersions (see Theorem A.I). This section provides a brief overview of some of the relevant MANPACK algorithms, as given in [22]. The presentation is independent of the earlier results. It should be noted that the computational tasks associated with implicitly defined manifolds differ considerably from those arising in connection with manifolds defined in explicit, parametric form as in, e.g., computational graphics. In fact, unlike in the latter case, for implicitly defined manifolds the algorithms for determining local parametrizations and their derivatives still need to be made available. Throughout this section it is assumed that F : R" -» R K , 0 < K < n, is a smooth mapping and a submersion on its nonempty zero set

Thus £ is a d — (n - K)-dimensional submanifold of Rn. Definition A.2 recalls the concept of a local parametrization of such submanifolds. The following result provides 97

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PATRICK J. RABIER AND WERNER C. RHEINBOLDT

a basis for the computational evaluation of such a local parametrization. We henceforth denote by the canonical injection that maps Rd isomorphically to {0} x Rd. Theorem 8.1. Under the stated assumption about F, let U e £(R d ,R") be an isomorphism from R6* onto a d-dimensional linear subspace T C R™. Then the mapping

is a local diffeomorphism

on an open neighborhood of xc € £ in R" if and only if

If (8.3) holds at xc, then there exists an open set Ud ofRd such that the pair (Ud,$), defined with the mapping (j) = H~1 o j : Ud —> R", is a local parametrization ofS near xc. Proof. If DH(xc)h = 0 for some h e R n , then DT(xc)h = 0 and U^h = 0; whence h £ TxcS and, because

also h € T x . By (8.3) this implies that h = 0. Conversely, for any nonzero h € TxcSnT-1 we have DT(xc)h = 0, and (8.4) requires that UTh = 0, which together imply that DH(xc)h = 0. Hence, if (8.3) holds, then there is some open neighborhood Un of xc in R" such that H is a diffeomorphism from Un onto the open set H(Un] in R™. Evidently, the set # ( £ n W n ) = H(Un) n ({0} x R d ) is open in {0} x Rd and Ud = j-lH(Y.r\Un) is an open subset of Rd. This shows that <j> = H~l o j maps Ud onto the open subset £ nUn of £. Since H maps £ r\Un into {0} x R d , the inverse j~l o H^nu» of 4> is well defined and both and its inverse are continuous maps. Hence (j) is a homeomorphism of Ud onto S nW n . Since both H~* and j are immersions, the same is true for (j). Thus, altogether, 0 is a local d-dimensional parametrization of S near xc. Note that (8.3) is equivalent either to ker DT(xc)nTA- = {0} or to rge £>r(x c ) T nT = {0}. We call a d-dimensional linear subspace T C Rn a coordinate subspace of S at xc € £ if (8.3) holds. At any point xc & S an obvious choice for a coordinate subspace is T = TxcS, which we will sometimes identify as the tangential coordinate space of £ at that point. Theorem 8.1 readily becomes a computational procedure for local parametrizations by the introduction of bases. On R™ and Rd the canonical bases will be used and we assume that the vectors u 1 ,... ,ud € Rn form an orthonormal basis of the given coordinate subspace T of S at xc. Then the matrix representation of the mapping U is the n x d matrix with the vectors w 1 , . . . , ud as columns. We denote this matrix by Uc. It is advantageous to shift the open set Ud such that 0(0) = xc. Now, componentwise, the nonlinear mapping H of (8.2) assumes the form

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By definition of 0 we have

Thus the evaluation of x = 0(y) for given y 6 Ud requires finding zeros of the nonlinear mapping

For this a chord Newton method

works well in practice. Here we used the fact that, because (8.3) is assumed to hold at x° e S, the derivative DH(x°} is nonsingular. By standard results, it follows that there exists a 6 > 0 such that for ||y||2 < 6/2 the process (8.8) converges to a unique zero x* of Hy in some neighborhood of x° and thus gives x* = (f>(y). For the special choice x° = xc + Ucy, ||y||2 < <5/2, the iterates satisfy 0 = C/J (xk — x °] ~ V — Uc (%k — ^o)) which implies that the process can be applied in the form

with a y-dependence only at the starting point. This shows that, for any y near the origin of R d , the algorithm of Table 8.1 produces the point x = (y) in the local parametrization (Ud,
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PATRICK J. RABIER AND WERNER C. RHEINBOLDT

computation of the nullspace of Z. A simple one is based on the LQ-factorization (with row pivoting)

Here P is a K x K permutation matrix; L is a K x re, nonsingular, lower triangular matrix; and Q is an n x n orthogonal matrix partitioned such that Qi and Qi are n x K and n x d matrices, respectively. Then, clearly, the d columns of Q^ form the desired orthonormal basis of T. This justifies the algorithm in Table 8.2. Table 8.2: Algorithm COBAS. Input: {Z} compute row-pivoted LQ-factorization (8.10) of Z; for j = l , 2 , . . . , d do V? :=Q2ei; return: Obviously, when the tangential coordinate system is used at zc, then COBAS can be applied with the Jacobian matrix DT(xc) as the matrix Z. In that case, GPHI simplifies considerably if the LQ-factorization (8.10) of DT(xc) is also applied for solving the corrector equation DH(x°)w = q. Then, for each step, only a K x K, instead of an n x n, lower-triangular system has to be solved. For details we refer to [22]. In general, the matrix Z used in Table 8.2 is constructed from the Jacobian matrix DT(xc). For example, it is frequently important to work with coordinate subspaces T that contain a specific canonical basis vector, say, e", of R™. Often the reason for this is that the independent variable xv represented by this vector is of a special nature, as, for instance, in our setting here, when xu corresponds to the time variable in a nonautoiiomous DAE. Evidently, in order to ensure that ev 6 T we have to replace the I'th column of Dr(xc) by a zero column. This leads to the algorithm in Table 8.3. Table 8.3: Algorithm GNBAS. Input: form Z by zeroing column v of DT(x°); by COBAS compute orthonormal basis Uc of ker W; Output: Of course, we require that the constructed matrix Z still has maximal rank K — dim T^-. In order to verify this during the computation of the basis of the nullspace, we may replace the LQ-factorization of Z in COBAS by the singular value decomposition (SVD) and then apply scaling and standard rank tests. The details should be evident. Let (W d ,0) be a local parametrization of S near x° e S. Then by Theorem A.7 the pair (U x R d , ($>, D0)) is a local parametrization of the tangent bundle T£ near (x0,v) for any v € TXoT,. For the evaluation of this parametrization of T£ we need now an algorithm for the computation of the derivative D<j> of 0.

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Let T be a coordinate subspace of E at x° and, as before, suppose that on Rn and R the canonical bases are chosen and that the columns of the n x d matrix Uc form an orthonormal basis of the given coordinate subspace T at x°. Then by differentiation of (8.6), it follows that d

Since the Jacobian of H is nonsingular at x c , this shows that at any the derivative D(j)(y) can be computed as follows in Table 8.4. Table 8.4: Algorithm DGPHI. Input: compute LU-factorization of A : — for j — 1 , . . . , d do: Output:

solve Azj = ed+j;

By further differentiation of (8.11) we obtain

which, because of the nonsingularity of D#(0(y)), defines D24>(y)(v1,v2) uniquely. The algorithm of Table 8.5 for computing D2(j> assumes that the vector w — D2r(x)(u1,u2) has already been computed for given x = 0(y) and ul = D4>(y)vl, i = 1, 2. Table 8.5: Algorithm D2GPHI. Input:

compute the LU-factorization of A := solve Az Output:

8.2

A DAE Solver for Kinematic Constraints

In this section we utilize the algorithms of section 8.1 in the development of the mentioned algorithm for solving DAEs of the form (6.1); that is,

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PATRICK J. RABIER AND WERNER C. RHEINBOLDT

under the conditions of Theorem 6.1 and with N = In + 1. Thus, in particular, by (6.7(iii)) the submersion theorem, Theorem A.I, ensures that E := r -1 (0) is a submanifold of Rn of dimension d = N — K. The proof of Theorem 6.1 showed that the DAE can be locally reduced to an equivalent ODE. This suggests that by applying a standard ODE solver to a suitable sequence of these local ODEs it should be feasible to compute a solution of the original DA However, while the proof of Theorem 6.1 is constructive, it Involves the use of certain projections that are not readily computable. In fact, it is easily seen that these constructions require, in essence, calculations with certain local parametrizations of the manifold E defined by the constraint function. Instead of attempting to translate the earlier proof into a numerical procedure, we sketch here a different proof of Theorem 6.1 that exhibits directly the steps of the desired computational algorithm and that, in particular, reflects the use of the parametrization algorithms discussed in section 8.1. For simplification we introduce the notation

where as before x and later also x are not derivatives but denote vectors in R n . Then (8.13) can be written as the first-order DAE

For any solution (v(t),X(t)) of (8.15) we noted repeatedly that the curve v = v(t) also satisfies the differentiated constraint DT(v(i))v(t) = 0 and therefore that the solution must remain in the set

Evidently, for any (vc,w°, A c ) 6 II we have

as well as A(xc)xc = F(vc) + £>iI>c)TAc. For given (vc,wc, A c ) e Ef let (Ud,(f)) be a local parametrization of E near vc. Then, because of (v c , wc) E TS, Theorem A.7 asserts that (Ud x Rd, (<^>, D
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For any fixed y e Ud consider the ./V-dimensional linear system

The operator K(0) is invertible. In fact, by (6.7(i)), K(0)h = 0 for some (hl,h2) e Rd x RK implies that D(j)(Q)hl = 0 and h2 — 0; whence h1 = 0 by the injectivity o D$(Q), and therefore h = 0. Thus, after shrinking Ud if needed, the operator K(y) of (8.17) will be nonsingular for any y € Ud and hence the mapping

is well defined. Consider now the local initial value problem

where the prime denotes the derivative operator d/ds and the initial condition corresponds to our assumption that 0(0) = vc. By standard ODE theory, (8.19) has a unique solution y : J >—> Ud on an open interval J containing the origin. By definition of <j> and the construction (8.17), (8.18) of * we obtain

Now set v(s) — (y(s)) and A(s) — ip2(y(s)). Then it follows from equation (8.19) that D(y(s))ipi(y(s)) = D(y(s))y'(s) — v'(s) and hence that (8.20) can be rewritten as

Comparing this with the notation (8.14) we see that the first component equation (corresponding to the original variable t) has the form t'(s) = 1. The initial condition j/(0) — 0 together with 0(0) = vc gives -u(O) — v°; whence, in particular, 4(0) = tc. Thus the first equation of (8.21) requires that t(s) = tc + s and hence that the operators d/dt and d/ds are the same. But then (8.21) is exactly the DAE (8.15). Moreover, it follows readily that, by our choice of (vc, wc, A c ) G II, this solution must satisfy the initial conditions

In other words, we proved that for any (vc, wc, A c ) 6 II there exists a local solution of the DAE (8.15) satisfying (8.22). Conversely, let t 6 J i-> (v(t},X(t)), 0 6 J be any solution of (8.15). Then as noted, we necessarily have (v(t),i>(t),X(t)) € II for t e J, and, hence, the solution can satisfy the initial condition (8.22) only if (vc,wc, A c ) e II. For any t0 € J write v° = v(t0), w° = w(ta), A° = A(t 0 ), and let (Ud,(j>) be a local parametrization of S near v°. Then the mapping ^ of (8.18) is well defined on Ud

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PATRICK J. RABIER AND WERNER C. RHEINBOLDT

(possibly after shrinking this neighborhood). For the local curve y : J i—» Ud, y = <j> we have v(t) = (y(t))y(t), and y(0) = 0, which shows that

1

ov,

Since the matrix on the left equals K ( y ( t ) ) and hence is nonsingular for y(t) 6 Ud, this implies that ^>(y) = (y(t),X(t)) and, hence, that the local curve is a solution of the initial value problem (8.19). Thus the DAE initial value problem (8.15), (8.22) with (vc,wc, Xc) € n is locally equivalent to the initial value problem (8.19). This shows that we can solve the local ODE to obtain the solution of the (global) DAE. Of course, for this we have to work, in general, with several, appropriately constructed, local parametrizations of E to obtain the solution of the DAE on a prescribed interval. Any standard ODE solver requires an algorithm for evaluating, at any y e Ud, the right side of (8.19), that is, the derivative vector y. This is accomplished here by the algorithm of Table 8.6. It assumes that the current local parametrization of S near xc is available and uses as input, besides the local vector y, the corresponding global point v = 4>(y). Note that as byproducts we also obtain the derivative v and the algebraic variable A. Table 8.6: Algorithm DYNH. Input:

use GPHI with y, xc, Uc, DH(xc) to get v = (j>(y); evaluate use DGPHI to compute D4>(y)', evaluate solve the linear system Output:

For the overall solution algorithm recall that only the initial conditions x(t0) = x0, x(to) = XQ satisfying the consistency condition T(to,xo,±o) = 0 are required and that consistent conditions for x(to] and A(io) will be determined by the process. With this, our solution algorithm DAENH assumes the form given in Table 8.7 below. It retains, where possible, a computed local parametrization over several steps of the ODE solver. The local basis at v G S is constructed by means of the algorithm GNBAS of Table 8.3 with index v — 1. Hence, because v contains, by (8.14), the time t as first component, GNBAS preserves the t variable. This explicit availability of t allows for a test of whether a prescribed terminal time has been reached. In Table 8.7 we simply work with a generic, user-supplied termination test, called t-test, which is assumed to return t-test := 'true' when the process is to be terminated. For the solution of the local ODEs (8.19) we use a user-prescribed ODE solver denoted by @[DYNH], which calls on the algorithm DYNH of Table 8.6 for the evaluation of the right side of the local ODE. More specifically, 6[DYNH](y, h) e Rd is the "next" point generated by this solver for the current point

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Table 8.7: Algorithm DAENH. Input: {t,x,p,h,km&x}; mode = init, k = 0; while mode = init then vc := (t,x,p);

evaluate use GNBAS with v — 1 to construct basis Uc at v°; set up DH(xc) y = (0,0); mode = step; while mode = step then use DYNH with y, vc, Uc, DH(xc) to get y, v, A; if DYNH failed then mode = init; break; Output: if t-test = 'true' then return success; if k > /cmax then return failure; take local ODE step: y = " [DYNH](y, h); if step unacceptable then mode = init; break; fc:=fc+l; endwhile; endwhile. y e Ud and stepsize h. The method is expected to generate a local error estimate for testing whether the step from y to 0[DYNH](y, h) is accepted or has to be repeated with a smaller h. A certain number of such step reductions causes the mode indicator to be set to init. In addition, we also set mode = init when the iteration methods in DYNH or GPHI converge poorly or not at all or when in GPHI, in the case of acceptable convergence, the distance between the computed point and the starting point of the process becomes too large. The algorithm DAENH represents an adaptation of the algorithms DAEN2 and DAEQ2 of [23]. As the latter two algorithms, DAENH has been implemented effectively with various local ODE solvers 0[DYNH] of different order. In particular, for the numerical experiments discussed in Chapter 9 we used the explicit Dormand-Prince Runge-Kutta DOPRI 5(4). In other work we also used several stiff methods such as RADAU and SDIRK.

8.3

Outlook on Computational Approaches

As noted earlier, much work is still needed in the computational analysis of the DAEs arising as models of rigid multibody problems with general kinematic or mixed kinematic and geometric constraints. Standard DAE software, such as the widely used code DASSL (see Brenan, Campbell,

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PATRICK J. RABIER AND WERNER C. RHEINBOLDT

and Petzold [5]) are designed for index-one DAEs and are certainly not useable for the production solution of the DAEs (6.1), which, typically, have index two,13 not to mention (7.1). There exists a large amount of literature on the study of computational algorithms for holonomic problems and we cite here only the books by Brenan, Campbell, and Petzold [5] and Hairer and Wanner [13], where other references can be found. Much of this literature concentrates strictly on the computational solution of the Euler-Lagrange equations (of index three)

In other words, except possibly for illustrative examples, there is little connection with the underlying mechanical problems. While this may be reasonably acceptable for most planar problems, in the case of systems of three-dimensional rigid bodies our discussion in the previous chapters shows that the details of the modeling aspects have to be considered carefully. In other words one cannot presume simply that such questions as, for instance, the handling of the condition p 6 S3, have been resolved already. Broadly speaking, the methods for solving (8.23) may be subdivided into two classes. The methods of the first class are characterized by their use of some local parametrization of the constraint manifold r -1 (0). Wehage and Haug [32] appear to have been the first to propose the use of certain specific parametrizations for the computational solution of (8.23), and, since then, this has been taken up by several authors. The extension of the approach to certain general classes of algebraically explicit DAEs by Rheinboldt [23] provided the basis of our algorithm in section 8.2 for the DAEs (6.1). The methods of the second class work with (8.23) or the DAEs obtained from that system by replacing the geometric constraint F(x) = 0 by its induced kinematic constraint DT(x)x = 0 or, alternately, by the acceleration constraint

As we noted before, the resulting DAEs are mathematically equivalent provided consistent initial conditions are used. But, in general, this equivalence is not preserved by the computational solution procedures. For the systems involving the induced kinematic (respectively, the acceleration) constraint, this exhibits itself in a drift of the computed approximations away from the geometric constraint manifold F"1(0) (respectively, from the tangent bundle Tr-1(0)). The computational methods of the first class work, in essence, with these modified DAEs but introduce various techniques to ameliorate or suppress the drift phenomenon. For details we refer again to the cited books and the original literature. It appears to be entirely feasible to apply analogous approaches also for the solution of DAEs of the general form (6.1). In contrast, it appears to be highly unlikely that the techniques underlying the methods of the second class for (8.23) can be applied to the DAE on a manifold M. of the 13 We shall not enter into any discussion of the index definition since it plays no further role in this presentation.

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form (7.1). Under the conditions of Theorem 7.1, it is clearly feasible to reduce this DAE locally to the form (6.1) via local parametrizations of the manifold A4. Moreover, since in practice, this manifold is usually a product of several copies of R3 x 513, such loca parametrizations can be computed relatively easily and without recourse to the MANPACK algorithms. The situation becomes more complex in the case of mixed kinematic and geometric constraints when, a priori, the full-rank condition of Theorem 7.1 fails to hold. Yet, again, as shown in section 7.4 a local parametrization approach still remains feasible, at least, in theory. However, in all these cases it is to be expected that any efficient software implementation has to be based directly on the details of the mechanical formulation rather than on a general DAE in the form (7.1). In other words, the software has to incorporate much of the mechanical modeling process. An example for this is the DADS system of Haug [14] for multibody systems subjected to geometric constraints. On the basis of a detailed description of the parts of the mechanical problem and their interconnections, DADS constructs the DAE corresponding to (5.58) in terms of the variables u and p. For details of the solution algorithms used in the system we refer to [14]. In particular, in [14] procedures are also indicated that apply a solver for the computation of approximations in (u,p) but allow work during each step in terms of the variables u and u; and their derivatives. The approach is similar to that discussed in the first half of section 7.6, and hence it appears to be feasible to adapt these algorithms also to systems of the forms (5.51) and (5.58).

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Chapter 9

Computational Examples This chapter presents several computational examples of systems with kinematic or mixed kinematic and geometric constraints. The main intent is to show that the algorithm DAENH of section 8.2 provides an accurate and effective tool for solving such problems. Of course, examples with pure geometric constraints are not considered since they are widely available in the literature and, as we noted before, this case is not of central concern in this monograph. For ease of presentation only fairly small model problems were chosen. By nature this resulted in a selection of problems which are either planar or for which simple models in terms of special variables, such as certain angles, etc., are readily available. Since, in the latter case, the forces involved in these problems are generally conservative, it appeared to us artificial, if not pedantic, to rewrite each of them in terms of the ( u , p ) variables. But for several of these three-dimensional problems it is shown that there are some natural techniques that can often be applied to arrive at a model in terms of the (u,p) variables. Generally, of course, for larger and more complex problems than those chosen here it is rarely possible to use special variables, and then a consistent use of the (u,p) formulation becomes a necessity. However, as noted before and discussed in [14], for the computation it then becomes essential to develop the models directly from a description of the mechanical problem, that is, from a characterization of the various bodies and their interconnections. The development of a software system for that purpose is certainly outside the framework of this monograph. In all examples, the canonical basis {el, e2,e3} of R3 is chosen as the reference frame.

9.1

Some Introductory Planar Examples

In the context of the theory presented here, planar problems are of little interest since they are essentially of the same type as the classical mass-point problems discussed in section 2.1. However, from a computational viewpoint these planar problems are certainly not at all trivial and, in fact, they can and do exhibit most of the difficulties encountered in the 109

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PATRICK J. RABIER AND WERNER C. RHEINBOLDT

three-dimensional cases. Accordingly, we present in this section three planar examples that provide an initial illustration of the behavior of our computational method. In particular, the first two choices represent special cases of a simple single body problem for which exact solutions are known that allow for a direct comparison with our computed results. The third example involves a practically interesting multibody problem, which we present here in a simplified, planar version. While it would not be difficult to expand this example into a more realistic three-dimensional model, it appeared to us that the added complexity hardly justified the small gains in numerical insight. The first two examples concern the classical model problem of the motion of a rigid body that "skates" on a tilted plane. This supporting plane is chosen to be span {e1, e2} and the direction of the gravitational force is — sin /Je1 — cos /3e3. In other words, we consider a plane that is tilted around the e2-axis by a fixed angle 0 from the horizontal. At any time, the (single) sharp-edged skate is assumed to point in a specific direction and to be in contact with the plane at exactly one point. The problem is planar in the sense of section 3.3 if we assume that the point of contact is also the body's center of gravity. Let (ui, 112)T be the point of contact and

As shown in Neimark and Fufaev [17], it is readily verified that the Lagrangian has the form where m is the mass and J is the moment of inertia of the body with respect to e3. Thus as discussed in sections 2.3 and 4.1 it is here possible to work directly with the variables HI, u^, and (p. The problem becomes dimensionless with the substitutions

where r is a reference length. Then we obtain, in line with (4.42), the DAE

For our first example we assume that the supporting plane is horizontal, that is, that /? = 0. Then it is easily verified that under any initial conditions

that are consistent with the constraint (9.1), the solution of (9.3) with /3 = 0 and under

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the assumption
For the computations we used u° = 0, (p° — 0, ii° = (—4,0) T , <^° = 2; whence a — 2 and the solution is

In this case the contact point of the skate describes a circle centered at (0, — 2) T with radius two. The computations with the algorithm DAENH of Table 8.7 were performed in double precision using the floating-point processor of a Pentium-Pro chip. A relative tolerance of 10~8 was chosen for the ODE solver. The maximum absolute error of the seven components of the computed solution after 10 revolutions was 0.4 x 10~7. For the second example we consider (9.3) with /? ^ 0 and under the assumption that at t = 0 there is only an angular velocity. In other words we have the initial conditions

Then it is easily verified that (9.3) has the exact solution

For the computations we used a — 1 and f3 = arcsin 1. Then as shown in Figure 9.1 the point of contact traces a cycloid in the {e1,e2}-plane with its cusps at the points (0, fc-7r/2)T, k = 0,1, 2 , . . . . At these cusps the velocity becomes zero, the skate comes to a stop, and the motion reverses itself, that is, changes the sign of its direction. The accuracy of the computed solution is of the same order of magnitude as in the first case. More specifically, again with the relative tolerance of 10~8 for the ODE solver, the maximum absolute error of the seven components of u ( t ) , u(t), X(t) equaled about 0.2 x 10~7 over about 10 loops of the cycloid. Thus in both examples the performance of DAENH is certainly highly satisfactory. As the third example we consider a simplified model of a commercially available transmission (see [15]), which translates the constant rotational motion of a smooth driveshaft into a motion along the axis of the shaft. The effect is similar to that of a worm drive, i.e., of a nut moving along a screw-threaded shaft, but here, in effect, the thread can be continuously varied. The transmission consists of a box containing an assembly of three coupled rings placed around the shaft. The inner surface of these rings is prepared to ensure good frictional contact with the shaft. The rings are arranged such that their symmetry axes form an angle ±

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PATRICK J. RABIER AND WERNER C. RHEINBOLDT

Figure 9.1: Path of skate on inclined (iti,U2)-plane. of a lever,
with the e2-axis. The shaft is assumed to rotate with a constant velocity c. Then the translation of the rotation into a motion along the axis of the shaft can be modeled by the condition ur - ctany = 0. Since if is required to be small, this will be simplified to

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Now we attach the aforementioned three bodies. In practice, the transmission is in a box to which the other equipment is then connected. Irrespective of the specific physical arrangement, we simply assume that the shock absorber of mass ms has its center of gravity at (us, 0)T and is connected to the ring body by means of an elastic spring with spring constant 7 and a damper with damping constant 6.

Figure 9.2: Schematic of the roll-ring model. In Figure 9.2 the box containing the transmission ring is not shown and the spring and damper simply connect the shock absorber with a vertical line through the center of gravity of the ring body. From the center of gravity of the shock absorber hangs a planar double pendulum consisting of two rigid bars. The pivot pin of the first bar is at the center of gravity of the shock absorber, and the second bar swings around a pivot at the other end of the first bar. The bars are characterized by their lengths i\, 1% and masses m\, m^- Then their moments of inertia are

The arrangement of the four bodies is schematically shown in Figure 9.2. Since the model is planar, we can work with the configuration vector

At a given time, let v,w € R2 be the location of the centers of gravity of the two bars, and a\ and a? the angles formed by the axes of the bars with the direction —e2.

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PATRICK J. RABIER AND WERNER C. RHEINBOLDT

Then the two pivots are modeled by the holonomic constraints

For the complete model this has to be combined with the nonholonomic constraint (9.8); in other words the complete model involves mixed constraints. In accordance with the analysis of section 6.2 we introduce the kinematic constraint

It is now straightforward to see that the equations of motion are of the form

where A — diag (mr,Jr,ms,mi,mi,Ji,m2,m2,J2) and and N is the external moment applied to the ring body when its axis is tilted. For the computation we use the data

and construct a moment function N(t) for driving the transmission as follows: In the piecewise constant function

each one of the eight discontinuities at to = 0.5,1.0,1.5,2.0,5.5,6.0,6.5,7.0 is replaced by a cubic spline that is nonzero for \t — t0\ < 0.1, and then the moment is computed as N(t) = av(t) with a scale factor a = 1.0 x 10~3. Now for the consistent initial conditions

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Figure 9.3: Dynamic, behavior of roll-ring model. the computed behavior of the variables
9.2

Three-Dimensional Problems

In this section we turn now to problems in three dimensions, which require the full theory of the previous chapters. As a first example we consider a homogenous ball rolling without sliding on a horizontal platter that rotates with a constant angular velocity around a normal direction. The supporting platter is supposed to be contained in the plane span {e1, e 2 } with e3

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PATRICK J. RABIER AND WERNER C. RHEINBOLDT

as its axis of rotation. Let Q be the (constant) angular velocity of rotation of the platter and assume that the ball has radius r. We derive the equations of motion in terms of the variables (u,p), that is, in the form of the system (5.26). In the reference frame, let u = (ui, U2j u 3) T be the ball's center of gravity. Then the condition of contact is uz = r and, if it holds, x — (ui, u^, 0)T is the point of contact. Let u be the angular velocity of the ball in the body-fixed frame; then by (4.4) the velocity of x, in the reference frame, is given by The no-slip condition requires that, in a frame fixed in the platter, the velocity of the contact point in the plane of the platter must be zero. Since any given point y = (yi,y2;0) T of the platter has the velocityfi(—2/2,2/1, 0)T in the reference frame, it follows from (9-14) that

By (4.21) and (4.30) we obtain (dropping the t-dependence)

where we used the fact that, because of pTp = 0, the projection H(p)TH(p) onto {p}~L maps p into itself. Now recall that the matrix representation of K(p) (in the canonical bases) is given by the last three rows of (4.29). Thus the no-slip constraints (9.15) can be written as

For the equations of motion (5.26) we have in this case Fr = 0, N = 0, M — mis, and J = J(,/3, where m and J;, are the mass and moment of inertia of the ball. With the obvious extensions for Fr, N, and G we then obtain

I

The term H(p)T JH(p)p on the right of the second equation of (5.26) is here proportional to p and was simply subsumed in the term p,p. The problem becomes dimensionless with the following substitutions:

where, of course, r is the radius of the ball and T a reference time. For the computations we used fl = 2, J = 0.6 and the consistent initial conditions

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The initial spin ensures that the trajectory of the ball is an ellipse with focal point at the origin. The computation proceeded through slightly more than four orbits and the resulting (overlapping) trajectory in the (e1, e2)-plane is shown in Figure 9.4. Of course, under different initial conditions the ball can fly out along a hyperbolic arc.

Figure 9.4: Trajectory of ball on rotating (m, U2)-plane. There are numerous other rolling-ball examples in the literature. Among these we select here a ball of radius r rolling, without sliding, on the inside of a cone, which in the reference frame is defined by

The opening angle 7, 0 < 7 < ^TT, of the cone is fixed. Here it is natural to introduce the polar coordinates (p, a) so that any point on the cone is specified by

When the Eulerian angles (p, ip, 0 are used to characterize the rotation of the ball, the configuration vector becomes y = (p, a, ,#) T and, as shown by Neimark and Fufaev [17], the problem has the Lagrangian

At the point of contact consider a tangential coordinate system that is transformed into

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PATRICK J. RABIER AND WERNER C. RHEINBOLDT

the reference frame by the rotation

Let (jj denote the angular velocity vector of the ball in this tangential frame. Then the conditions of rolling without sliding have the form

By using the well-known relations between w and the derivatives of the Eulerian angles (see, e.g., again [17]) and transforming the results into the reference frame by means of (9.22), the constraints (9.23) become

The formulation becomes dimensionless by the substitutions

Then, in accordance with the discussion of section 4.5, it follows from (9.21) and (9.24) that the equations of motion have the form

with

and

Let 7 = 1; then the initial conditions

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are consistent and provide the ball with an initial spin. The computational results in Figure 9.5 show that the resulting trajectory is similar to that of the skate in Figure 9.1. The spin causes the ball to roll up briefly until the upward component of the velocity becomes zero, and the direction reverses itself. The ball then rolls downward in an arc until the position of the spin axis causes the trajectory to move upward again and the cycle repeats itself.

Figure 9.5: Path of ball inside cone in (a, p)-coordinates. We used in this example the special variables p, a, (p, •(/>, 9. In analogy to the previous example it is fairly straightforward to model the problem in the (u, p) variables, as shown below. Let u be the center of gravity of the ball in the reference frame and u = (u 1 ,u 2 ,0) T be its projection onto the {e1,e2}-plane. In analogy to (9.22) we introduce the vectors

forming an orthonormal basis of R3. Since u T 6 2 = 0, we have u = (u T 6 1 )6 1 + (w T 6 3 )6 3 , where x — (u T fe 3 )6 3 lies on the cone and u — x = (UTbl)bl is a normal vector. Thus the ball is in contact with the cone exactly if

When this condition holds then

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PATRICK J. RABIER AND WERNER C. RHEINBOLDT

Figure 9.6: Three rolling cylinders. is the point of contact and, as in the previous example, we can use (4.4) and (9.16) to obtain the velocity of the point of contact in the reference frame as since x — u = —rbl by (9.26). The no-slip condition now requires that the component of x(t] parallel to the tangent plane span {b2, b3} at x must be zero, that is, omitting the t-dependence,

With F® = (0,0, — 77M7COS7)T and N = 0, the equations of motion (5.26) are now fully determined. Note that in this formulation we have, besides the two no-slip conditions (9.29), also the contact condition (9.26) as an explicit constraint. In the previous form (9.25) the contact was implicit in the choice of the variables. As a third three-dimensional example we consider a system of three cylinders as shown in Figure 9.6. The two lower cylinders have identical radius and roll on a horizontal plane, while the third one rolls on top of them. All cylinders are assumed to be sufficiently long to ensure that in the time interval under consideration the top cylinder does not slip off the lower ones. Let span {el,e2} again be the supporting plane and denote by r\ and r2 the radii of the lower two and the upper cylinder, respectively. Because the cylinders are rolling without slipping, the angle a between the axes of the lower cylinders has to remain fixed. Thus, as shown in Figure 9.6, it is no loss of generality to assume that the axis of one of the lower cylinders is parallel to the e1-axis. Of course, the angle 9 between el and the axis of the upper cylinder will vary with time. Let x = (xi,X2,X3) T denote the center of gravity of the upper cylinder. Then the condition of contact with the supporting plane requires that £3 — 1r\ + r^. We follow here Neimark and Fufaev [17] and use as special variables the coordinates xi, #2,

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the angle 0, and the angles of rotation fa, fa<, and tp of the two lower and the upper cylinder, respectively. In other words the configuration is characterized by the vector

j/ = (xi,x 2,0,ip,fa,ip2)T-

Then the Lagrangian of the system is

where m and J", J% are the mass and the principal central moments of inertia, respectively, of the upper cylinder, and J1, J2 are the moments of inertia of the lower cylinders about some generator line. For the points of contact between the upper and lower cylinders we obtain

and vl = (0,2ri^i,0) T and v2 = (-2risinaV>2,2ricosm/>i,0) T , respectively, are the velocities of these two points of contact. Evidently the angular velocity of the upper cylinder in the reference frame is uj = (<£> cos $,<,:> sin #,$) T . Thus, in agreement with (4.4), and (4.3), the no-slip conditions must be

which gives the constraint

Note that there should have been six scalar no-slip conditions but two of them are trivial zero-relations. With (2.29) it is now straightforward to generate from (9.30) and (9.31) the equations of motion in the form of the DAE

For the computations we chose the constants

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PATRICK J. RABIER AND WERNER C. RHEINBOLDT

Figure 9.7: Paths of the contact points and of the center of gravity of the upper cylinder. and the consistent initial conditions

Figure 9.7 shows the computed projections onto the (x^a^-plane of the paths zl(t), z2(t) of the contact points, and of x(t). In addition, the axis-line of the upper cylinder at times t = 0.0,2.7,3.72,4.8 are shown. The upper cylinder first rolls in the direction of the positive £i-axis with decreasing axis angle 9, then at t = 2.7 it reverses itself and rolls backward until t = 3.7 where it begins again to roll in the positive Zi-direction. From t = 2.7 on, B increases monotonically. It is not difficult to model the three-cylinder problem in terms of the (u,p)-variables. Before sketching this, we briefly discuss the constraint arising from one cylinder with radius r rolling, without sliding, on a horizontal plane span {e1, e2}. For this let (u,p) € R3 x S3 denote the configuration variable of this cylinder. For simplicity we assume that the cylinder is homogenous, so that u(t) remains on the cylinder-axis at all times. Contact of the cylinder with the plane implies that ^3 = r and the condition of rolling without sliding requires the axis of the cylinder to have a constant direction. With no loss

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of generality, suppose that the axis-direction coincides with e1 at time t = 0 and that the reference frame {e^e^e 3 } and the body-fixed frame coincide at that time. At times t the direction of the axis is given by Rplel = -Rje1; whence the constraint Rpel = e1. By (3.13) (with Xi — pi, 1 < i < 4) this is equivalent to y>2 = PJ, — 0. But these conditions do not prevent the cylinder from sliding in the axis direction. To keep this from happening we must require that the velocity of one contact point, say, x = (w 1 ,u 2 ,0) T , vanishes at all times. Then, arguing as in the example of a ball rolling on a platter (but now with ft — 0), we find that ui = 0, and u^ = 2r(pip(> - popi) — 0. (Compare with (9.17).and note that p2 = p3 = Q because of p2 = Pz — 0.) Thus altogether we obtain the five constraints

It is informative to observe that, in view of \p\2 = 1 and p% — p% = 0. we may set po = cosijj, pi — sinip, in which case the last constraint of (9.33) becomes u\ when considering the first lower cylinder (so that r = r\). The reason why here the "natural" variable is lip and not tj) is consistent with Remark 3.3. With Po = cos-)/;, pi = sin?/;, the rotation Rp is the rotation of angle 2i/> in the {e1,e2}-plane. Obviously, the first three constraints of (9.33) are geometric and hence must be differentiated. This gives the kinematic constraints

Remark 9.1. The case of a nonhomogenous cylinder is slightly more complicated. If the center of mass is not on the axis, let WQ denote the coordinate of some material point on the axis at time t — 0 in the cylinder-fixed frame. Then v(t) = u(t] + Rp(t)w0 are the coordinates of the same material point at time t in the reference frame. Thus the constraint (9.34) holds with v in place of u. Since v — u + Rpwo and Rp = K(p)H(p)T (see (4.30)) we obtain

in place of the first equation in (9.34). The other two equations are unchanged. We return now to the modelling of the three-cylinder problem in terms of the variables (u ,pl) € R3 x 53, i = 1,2,3. Here the indices 1,2,3 and the terminology "first" and "second" cylinder refers to the two lower ones, while the "third" is the upper cylinder. Let T :-- {e^e^e3} denote a reference frame such that span {el,e2} is the horizontal plane. As in Figure 9.6 suppose that el is parallel to the axis of the first cylinder. Then, as discussed above, the condition of rolling without sliding for that cylinder is accounted for by the constraint (9.34) with u = u1, p — pl. In order to apply the same result to i

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PATRICK J. RABIER AND WERNER C. RHEINBOLDT

the second cylinder, which, in general, is not parallel to the first one, let Q be a rotation of the reference frame around e3 such that Qre1 is in the direction of the axis of the second cylinder. The corresponding change of coordinates affects only the first equation in (9.34) but it has to be kept in mind that under this coordinate transformation Rpz and K(p2)H(p2)~r are changed to QrRp2Q and QT K(p2)H(p2)TQ,m.respectively. With this we obtain the constraints expressing the rolling conditions for the two lower cylinders in the reference frame f in the form of ten scalar equations. Suppose now that a € R3 represents the direction of the axis of the upper cylinder at time t = 0; whence Rl3a is its direction at time t. The condition that the upper cylinder rests on the lower two is accounted for by the requirement that two distinct points on its axis, say u3 and u3 + R\a, remain at the distance 1r\ + TI of the horizontal plane (assuming homogeneity). This yields two scalar geometric constraints which, of course, need to be differentiated. It remains to obtain the constraint that the upper cylinder rolls without sliding on the lower two. As before, this requires the calculation of the two contact points in the reference frame J-, which shall be denoted again by z1 and z2. Their third coordinates equal 1r\ while their first two coordinates are the points of intersection of the projected axis of the upper cylinder with the projected axes of the lower two. These points are easily calculated since the three axes have the parametric equations

The velocity of the material point in the upper cylinder that coincides with z1 at time t is (9.35) while the velocity of the corresponding material point in the first cylinder is (9.36) The no-slip condition of z1 requires equating the velocities in (9.35) and (9.36) where it should be noted that the third components of both of these equations vanish; whence we obtain two, and not three, scalar constraints. In fact, when the material point in the upper cylinder reaches z1, its third coordinate achieves a minimum and hence its velocity component in the direction e3 vanishes. In the case of (9.36), the third coordinate of the material point achieves a maximum. Naturally, similar considerations apply to z 2 , of course, with Q T R p iQ in place of Rpz. As a result the (u, p)-formulation of the problem comprises 16 scalar constraints and 18 variables. If we view p1, p2, p3 as vectors in R4, the number of variables becomes 21 and now, with the three added constraints (p4)Tp* = 0, i = 1,2,3, (see (5.51)), we have 19 constraints. Since there are no external forces or moments, the constrained system is given by the DAE (5.51) with FE = 0 and NE — 0. Given the size of the system, the full-rank condition for the constraints can only be checked numerically. For the next problem, we consider a single sharp-edged disk (or wheel) rolling on a rough plane where by "sharp-edged" we mean that the disk cannot slide sideways. This

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is a frequently addressed problem in the literature; in fact, it is often used as the prime example for showing that a kinematic constraint need not imply any restrictions on the configurations (see again Neimark and Fufaev [17]). As before we use span {e1, e 2 } as the supporting plane and —e3 as the direction of the gravitational force. The body-fixed frame {a1, a 2 , a3}, with origin at the center of gravity of the disk, is chosen such that a1 is the rotation axis, and a 2 and a3 span the plane of the disk. A critical assumption is that the disk is never parallel to the supporting plane, that is, that the planes spanja^a 2 } and spanfe^e 2 } intersect in a unique line spanned by 6 = (e3 x al}/\\e* x a 1 ^. If this holds and the disk is in contact with the plane, then there is a unique point of contact x = (xi,X2,0) T . The position of the disk can now be characterized by x\, x^, and three angles, namely, the angle (p between e1 and b, the angle 9 between e3 and the plane of the disk, and the angle ip between a? and the radius from the center of gravity to the point of contact. In other words, the configuration is characterized by the vector y — (xi,X2,tp,d,ij))~r. Here we assume that\0\ < ir/2/to ensure that the disk is not flat. on the supporting plane. Since the disk is assumed to have a sharp edge, the velocity x of the point of contact must always be in the direction of the vector £>; that is, we have the nonholonomic constraints

where r is the radius of the disk. The center of gravity is at the point

With this we see that the system has the Lagrangian (see [17, p. 102])

Here m denotes the mass of the disk and Ji, J% its principal moment of inertia, respectively. Once again, for the computation it is useful to substitute the dimensionless quantities

Since the system is conservative, we obtain then the equations of motion in the form

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PATRICK J. RABIER AND WERNER C. RHEINBOLDT

where a straightforward calculation shows that

with

and

As in the first two examples of this section, it is easy to formulate this problem in the (u,p)-variables. Let P e £(R3) denote the orthogonal projection onto spanfe1^2}. By considering the symmetry-plane spanje3^1} of the system as shown in Figure 9.8 we observe that the condition of contact is

whence the condition of contact becomes

If the contact condition (9.39) holds, then it also follows easily from the figure that the point of contact is

Now let (j be the angular velocity of the disk in the body-fixed frame. Then, as before, in the reference frame, the velocity of the point of contact x is, by (4.4),

where, as in (9.16), RpU = 2K(p)p. The no-slip condition requires that Px(t) = 0. Hence using (9.40) we obtain the no-slip constraints

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Figure 9.8: Symmetry plane of the disk system. Finally, again using Figure 9.8 we note that gravity induces the moment

With this, most of the information is available for obtaining the DAE (5.26) for the disk system in the (u,p)-variables. For this observe that by (4.30), in terms of (u,p), the vector a1 becomes Similarly the moment N can be expressed in the body-fixed frame; we will omit the details. Finally, note that the geometric constraint (9.39) (in terms of ( u , p ) variables) must still be differentiated. This problem shows that the case of mixed constraints in the (u,p)-formulation may arise even when a single rigid body is considered. In the study of equations (9.38), an important aspect is, of course, the stability of the solutions. It is well known, for instance, that the motion of a rolling disk can be stable only when the velocity is sufficiently large. Stability studies are outside the topic of this monograph. We only note that, typically (see again [17]), these studies address the stability of two classes of special solutions of (9.38) which are a good source of computational examples. The first of these two classes is that of the trivial rectilinear motions of the contact point with constant velocity

for which it is interesting to study the effect of small perturbations. As an example, we used Ji = 0.35, J2 = 0.5, 00 = 0.3 and introduced a small tilt 0° = 1.0 x 10~5; in other

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PATRICK J. RABIER AND WERNER C. RHEINBOLDT

Figure 9.9: Terminating disk path in (iri,X2)-plane. words we solved (9.38) with the initial conditions

Figure 9.9 shows the resulting path traced by the contact point in the supporting plane. As expected, it is initially almost a straight line, but the tilt of the disk is increasing. Due to the moment induced by gravity this tilting motion accelerates, which in turn causes the trajectory to curve away from the straight path. Shortly thereafter the tilt angle passes through 7r/2 and accordingly we terminated the calculation at that point. Note that (9.38) does not reflect the condition \Q\ < ir/2, which was essential for the derivation of the system. Yet it turns out that, as in our example, it is not rare to encounter points along a solution where \9\ = n/2 and hence where the solution becomes physically meaningless. Of course, the computational algorithm will not "see" any of these points and will continue beyond them although often with very erratic results. Thus, during the computation we have to monitor the value of the tilt angle 6 and terminate the calculation when \8\ has passed through Tr/2. The second class of special solutions arising often in stability studies is that of the circular motions

which exist under the assumption that the equation

has a solution q, 00 £ (-7r/2,7r/2). Once again we consider the effect of certain perturbations on these solutions. For this let Ji := 0.35, J2 '•— 0.5, #o : = 0.25. Then (9.43) is a quadratic equation in q that has a

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positive solution q :— q° = 2.17080452. We solve (9.38) with various perturbed values of g, that is, with the initial data

For q — q° the resulting trajectory in the supporting plane is a circle of radius l/q° centered at the origin. It is interesting that the solution behavior is rather stable under changes of q. For q = q° + A with A = 0 , 1 0 ~ 7 , . . . , 10~2,0.08, Table 9.1 shows, besides A and q, the deviation from circularity

where the maximum is taken over all computed points during a ^-interval of about 10 orbits. The computed trajectories spiral away from the circle fairly slowly. Only when beginning with q = q° + 0.08 did we encounter a more erratic trajectory, namely, that shown in Figure 9.10. Table 9.1: Perturbation of circular disk motion.

A 0.0 1.0(-7) 1.0(-6) 1.0(-5) 1.0(-4) 1.0(-3) 1.0(-2) 0.08

q

£

2.17080452

6.660(-8)

2.1708046 2.1708055 2.1708145 2.1709045 2.1718045 2.1808045 2.2508045

5.948(-7) 7.223(-6) 7.198(-5) 7.160(-4) 6.834(-3) 5.239(-2) 3.743(-l)

As the final problem we consider the motion of a two-wheeled cart on an inclined plane in a slight generalization of a related problem given by Neimark and Fufaev [17]. As before, we use span {e^e2} as the supporting plane and (sin/3,0, -cos/3)T as the direction of the gravitational force. Both wheels have the same radius r > 0 and are assumed to have sharp edges. As sketched in Figure 9.11, the following variables will be used for the specification of the current configuration of the cart: (i) The point of intersection (xi, £2,r ) T between the symmetry axis of the cart with the axle on which the wheels are mounted, (ii) the angle 6 between e1 and the symmetry axis of the cart, and (iii) the angles of rotation (j)f and <pr of the left and right wheel, respectively (each measured from a fixed radius vector). Thus the configuration is characterized by the vector y = (xi,X2,d,£,r)r• The center of gravity is assumed to be on the cart's symmetry axis at a fixed distance

130

PATRICK J. RABIER AND WERNER C. RHEINBOLDT

Figure 9.10: Perturbed circular disk motion in (x1, x2)-plane.

from (xi, X2, r) T and is to remain always at a distance r from the supporting plane. This can be ensured, for example, by introducing a support under the pole of the cart that slides freely on the supporting plane. No vertical displacements of the cart body are allowed. Let a > 0 be the half-length of the axis. Then the conditions of rolling without sliding require for the left and right wheels the following preservation conditions for the velocities:

In addition, the absence of lateral sliding demands that

Clearly, the constrained system is conservative and has the Lagrangian

Here m is the mass of the entire cart, Jc is the moment of inertia of the cart when it rotates as a whole about the axis with direction e3 passing through (xi,X2,r) T , and Jw is the axial moment of inertia of each wheel. For the computation it is useful to work with the dimensionless configuration vector

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Figure 9.11: Two-wheeled cart. Correspondingly we introduce the dimensionless quantities

Hence, in accordance with (4.42), we obtain the equations of motion

where

For the computation we used a = 0.5, Jc = 1, Jw — 0.5, g — 0.3, and the following consistent initial conditions

The computed path of the point (xi,£2) T in the supporting plane is shown in Figure 9.12. By (9.52) the initial velocity vector (ii,£2) T has a positive first component and hence points upward. The cart begins to move in that direction, but then gravity takes over and forces it to roll backward down the slope. Since here the turning velocity Q is positive, the path turns gradually upward again until, just slightly above

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PATRICK J. RABIER AND WERNER C. RHEINBOLDT

Figure 9.12: Motion of two-wheeled cart in (o;i,a:2)-plane. the e2-axis, it reaches a cusp point where the motion stops and the cart begins to roll downward once more. Thus the motion of the cart is similar to that of the skating body in Figure 9.1. But since <^;(0) and (^r(0) are distinct, the two wheels will retain different angular velocities throughout the motion. This causes the downward swings to have different depths depending on whether the cart is rolling forward or backward.

Appendix

Submanifolds This appendix collects some basic definitions and results on submanifolds of finitedimensional spaces in a form that is readily applicable for computations. For proofs we refer, e.g., to Abraham, Marsden, and Ratiu [2]. Throughout this appendix, the dimension n of the ambient space is given, d is an integer with 0 < d < n, and p denotes a positive integer or oo. Recall that, when G : E —> Rm is of class Cp on an open subset E C R n , then G is called an immersion or submersion at a point z0 € E if its first derivative DG(xo) € £(R n ,R m ) is a one to one mapping or a mapping onto R m , respectively. More generally, G is an immersion or submersion on a subset S C E if it has that property at each point of 5. Note that these definitions require n < m for G to be an immersion and n > m for it to be a submersion. Clearly, if n > m and DG(xo) has maximal rank m, then G is a submersion at XQ £ E. Definition A.I. A subset M C R" is a d-dimensional Cp-submanifold o/R™ if M is nonempty and for every point XQ 6 M there, exists an open neighborhood U of XQ in Rn and a submersion G : U i-> Rn~d of class Cp such that M n U = G~1(Q). The following result is frequently used. Theorem A.I (Submersion Theorem). Suppose that for the Cp mapping G : E C R" H+ R m , n > m > 0, on the open set E, the set M = G^O) = {x & E : G(x) = 0} is not empty and G is a submersion on M. Then M. is an (n — m)-dimensional Cp-submanifold o/R". An essential property of manifolds is the concept of a local parametrization. Definition A.2. Let M be a nonempty subset o/R n . A local d-dimensional Cp parametrization of M is a pair (U, (U) is an open subset of M. (under the topology induced by the standard topology ofRn) and 0 is a homeomorphism oflA onto (£•/) • (ii) 0 is an immersion on U. If XQ & M and (U, $) is a local d-dimensional Cp parametrization of M such that XQ € 4>(U), then (U,<j>) is called a local d-dimensional Cp parametrization of M near XQ. 133

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PATRICK J. RABIER AND WERNER C. RHEINBOLDT

The close relationship between submanifolds and local parametrizations is provided by the following result. Theorem A.2. A nonempty subset M C R" is a d-dimensional Cp-submanifold o/R™ if and only if for every XQ € M. there exists a local d-dimensional Cp parametrization of M. near XQ. The following result shows that, when M is a d-dimensional C^-submanifold of R™, then any local Cp parametrization of M is necessarily d-dimensional and hence we may suppress for them the specification of the dimension. Theorem A.3. Let M be a d-dimensional Cp-submanifold o/R". // (14,(f>) is a local k-dimensional Cp parametrization of M, then k = d. Moreover, if(Ki,(f>i) and (^2,^2) are two local Cp parametrizations of A4, then ip^1 ° (f>i is a Cp-diffeomorphism ofV\ := ^OM^i) n <^(W2)) onto V2 := ^(^(Wi) n 0 2 (W 2 )). If (U, <j>) is a local Cp parametrization of a d-dimensional Ctp-submanifold M of R", the pair (^(W),^" 1 ) is called a local chart of M. Moreover, if (Ui,\) and (^2,^2) are two local Cp parametrizations of M,2then the mapping 4>^ l °i1denned on (j>il(^i(Ui)r\ ^2(^2)) is a change of local coordinates. The second part of Theorem A.3 asserts that changes of local coordinates are Cp-diffeomorphisms. Moreover, it can be shown that when (U,
(A.I) where (L(, 0) is any local Cp parametrization of M. near XQ. Theorem A.4. If V is an open neighborhood of XQ in R™ and G : V i-> Ttn~d is a submersion at XQ such that G~l(0) = M. n V, then TXOM — ker DG(XQ). Let / : M —> R1 be a function on a given submanifold M. of R n . Then a point XQ & M is a critical point of / if for some local parametrization (U, (/>) of M. near XQ, the function / o <j> is differentiable at £0 = ^~ I (XQ) and D(f o <^>)(£o) = 0. It is easily seen that this condition is independent of the choice of the local parametrization (U, ) of M. near XQ. Critical points include local extrema of / on M.. If E C R" is an open subset and h : E —» R1 is differentiable, then the gradient V/i(x) of h at x e E is the unique vector in Rn such that (V/i(x), v} = Dh(x)v for every v £ R n , where {-, •} denotes the Euclidian inner product on R n . With these definitions, Theorem A.4 together with the submersion theorem, Theorem A.I, leads to the frequently used Lagrange multiplier theorem.

APPENDIX.

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135

Theorem A.5 (Lagrange Multiplier Theorem). Assume that for the C1 mapping G : E C Rn >—> R771, n > m > 0. on the open set E, the set M = G~l(0) is not empty and G is a submersion on M.. Let h : E C Rn >—»• R1 be a Cl functional on E and XQ 6 M. a critical point of h. Then there exists AQ G Rm such that (XQ, AQ) is a critical point of the functional h : E x Rm >—> R 1 , h(x, A) = h(x) + (G(x), A); that is, (XQ, AQ) is a solution of the system

Definition A.4. Let M be a d-dimensional Cp-submanifold o/R™. The subset TM — Uxe.MKx} x TXM]}].of R™ x R™ is the tangent bundle of M and, for every x € M, {x} x TXM is the fiber ofTM above x. Since TxRn = R" for every x € R n , we have TR™ = R" x R™. Thus, the tangent bundle of a submanifold M. of R™ appears as a subset of TRn. The following result shows that when M. is sufficiently smooth, then this subset is a submanifold of TRn. Theorem A.6. Let M be a d-dimensional Cp-submanifold o/R n and p>2. Then TM is a Id-dimensional Cp'1-submanifold o/TR" = R" x Rn = R2n. The next result addresses the construction of local parametrizations of TM. from local parametrizations of M • Theorem A.7. Let M be a d-dimensional Cp-submanifold of R" with p > 2, and x0 e M. Then for any local Cp parametrization (U,<j>) of M near x0, the pair (U x R rf , (0,D)) is a local Cp~l parametrization ofTM near (XQ,V) and any v & TXOM. The inverses of the local parametrizations (U x Rr, ($, D<j>)) of TM obtained in Theorem A.7 are called (tangent) bundle charts.

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Foundations of Mechanics, 2nd ed.,

Ben-

[2] ABRAHAM, R., MARSDEN, J. E., AND RATIU, T., Manifolds, Tensor Analysis, and Applications. 2nd ed., Springer-Verlag, New York, 1988. [3] ARNOLD, V. I., Mathematical Methods of Classical Mechanics. Springer-Verlag, New York, 1978. [4] BREDON, G. E., Topology and Geometry, Springer-Verlag, New York, 1993. [5] BRENAN. K. E., CAMPBELL, S. L., AND PETZOLD, L. R., Numerical Solution of InitialValue Problems in Differential-Algebraic Equations, North-Holland, New York, 1989; reprinted as Classics in Applied Mathematics 14, SIAM, Philadelphia, 1996. [6] CARATHEODORY, C., Der Schlitten, Z. Angew. Math. Mech., 13 (1933), 71-76. [7] CHOQUET-BRUHAT, Y., DE\VITT-MORETTE, C., AND DILLARD-BLEICK, M., Manifolds, and Physics, Vol. I, North-Holland, Amsterdam, 1982.

Analysis,

[8] CONDON, E. U., Kinematics. In Handbook of Physics, E. U. Condon and H. Odishaw, eds.. McGraw-Hill, New York, 1958. [9] DIRAC, P. M. A., Generalized Hamiltonian dynamics, Canad. J. Math., 2 (1950), 129-148. [10] DIRAC, P. M. A., Generalized Hamiltonian dynamics, Proc. Roy. Soc. London Ser. A, 2462 (1958), 326-332. [11] GAUSS, K. F., Uber ein neues allgemeines Grundgesetz der Mechanik, J. Reine Angew. Math. (Crelle), 4 (1829), 25-28. [12] GOLUB, G. H. AND VAN LOAN, C. F., Matrix Computations, 2nd ed., Johns Hopkins Univ. Press, Baltimore, MD, 1989. [13] HAIRER, E. AND WANNER, G., Solving Ordinary Differential Equations II: Stiff Differential-Algebraic Problems, 2nd ed., Springer-Verlag, New York, 1996.

and

[14] HAUG, E. J., Computer Aided Kinematics and Dynamics of Mechanical Systems, Vol. I, Allyn and Bacon, Boston, MA, 1989. [15] JOACHIM UHING, KG, GMBH AND Co. Uhing-Linearantrieb, Firmenprospekt 07d 03.91, 1991. [16] LEI, X., Singularities of a Sheet Metal Stretching Problem and Quasilinear Second Order Ordinary Differential Equations, Ph.D. thesis, University of Pittsburgh, Pittsburgh, PA, 1996. 137

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[28] SPARSCHUH, S. AND HAGEDORN, P., On the Gauss principle in the numerical integration of mechanical systems. In Real-Time Integration Methods for Mechanical Systems Simulation, E. J. Haug and R. C. Deyo, eds., NATO ASI Series F, Vol. 69, Springer-Verlag, New York, 1990, 293-300. [29] SYNGB, J. L., On the geometry of dynamics, Philos. Trans. Roy. Soc. London Ser. A, 226 (1926), 31-106. [30] SYNGE, J. L., Geodesies in non-holonomic geometry, Math. Ann., 99 (1928), 738-751. [31] SYNGE, J. L., Geometrical mechanics and de Broglie waves, Cambridge Univ. Press., Cambridge, UK, 1954. [32] WEHAGE, R. A. AND HAUG, E. J., Generalized coordinate partitioning for dimension reduction in analysis of constrained dynamic sytems, J. Mech. Design, 104 (1982), 247255. [33] WHITNEY, H., The self-intersections of a smooth n-manifold in 2n-space, Ann. of Math., 45 (1944), 220-248. [34] ZEIDLER, E., Nonlinear Functional Analysis and Its Applications, Springer-Verlag, New York, 1985.

Index DeWitt-Morette, C., 94 differential principles, 9 Dillard-Bleick, M., 94 Dirac, P. M. A., 2

Abraham, R., 3, 29, 133 angular velocity, 2, 30 Arnold, V. I, 31 Bredon, G. E., 20 Brenan, K. E., 105, 106 bundle morphism, 94

embedding, 3, 20 energy norm, 48, 52 Euler angles, 20 parameters, 23 Euler-Jacobi operator, 37, 38 Euler-Lagrange equation, 13 external force, 17, 18, 31

Campbell, S. L., 105, 106 Caratheodory, C., 16 Cayley-Klein parameters, 23 center of mass, 19, 30 Choquet-Brubat, Y., 94 computational drift, 14 Condon, E. U., 23 configuration space, 3, 10, 20, 26, 29, 34, 55, 91 conservative force, 42, 43, 130 constant rank condition, 66, 70, 79-81, 91, 95 constrained motion, 49 constraining force, 15 constraint geometric, 13, 50, 53, 54, 65 holonomic, 1, 13, 15-17, 96 ideal, 15 kinematic, 2, 4, 10-12, 48, 50, 53, 56, 65 manifold, 96 mixed, 65, 114 nonholonomic, 1, 4, 10, 16 state, 10 coordinate subspace, 98-101 covering, 25, 26 critical point, 134

frame, 19 friction, 16 Fufaev, N. A., 1, 2, 16, 110, 117, 120, 125, 129 full-rank condition, 11, 13, 16, 48, 56, 65, 71, 72, 75, 91, 92, 95 Gauss principle, 3, 4, 9, 11, 14, 47 Gauss, K. P., 9 general linear group, 19 generalized Gauss principle, 4, 46, 48, 49, 53, 81 Golub, G. H., 21 gradient, 134 Hagedorn, P., 3 Hairer, E., 106 Hamilton's principle, 9, 16, 17, 40 Haug, E. J., 1, 4, 5, 23, 24, 29, 106, 107 Hou, M., 112 immersion, 133 inertia tensor, 31, 55 initial condition, 12, 14, 51, 62, 71, 75, 96, 103

DAE

existence, 62, 75, 102 formulation, 12-14, 17, 49, 52, 53, 56, 61, 95, 101 139

INDEX

140 kinetic energy, 17, 41, 43, 52 Lagrange multiplier, 12, 17, 46, 48, 51, 56, 92 multiplier theorem, 3, 12, 14, 48, 49, 134 Lagrangian, 16, 18, 43, 110, 117, 121, 125, 130 Lei, X., 72 local parametrization, 6, 49, 75, 97, 106, 133, 135 Marsden, J. E., 3, 29, 133 mass matrix, 10, 11, 30, 55 mass-point system, 10 moment, 31 multibody systems, 55 Neimark, J. I., 1, 2, 16, 110, 117, 120, 125, 129 Newton's law, 4, 9-11, 14, 31, 36 method, 99 orthogonal group, 19 Pauli spin matrices, 21 Petzold, L. R., 105, 106 planar motion, 3, 27, 43, 58 potential, 16, 18, 42, 52 principle of virtual work, 14 quasi coordinates, 32 quaternion algebra, 4, 21 conjugation, 22 modulus, 22 multiplication, 21 pure. 22 Rabier, P. J., 72, 94, 96 Ratiu, T., 133 Rheinboldt, W. C., 72, 94, 96, 97, 106 Riesz representation theorem, 46, 74 rigid body, 19, 29 Roberson, R. E., 2 rotation group, 20 Rund, H, 2 Schmidt, T., 112

Schwertassek, R., 2 second fundamental tensor, 55, 94, 96 Sommerfeld, A., 16 Sparschuh, S., 3 special orthogonal group , see rotation group 20 state space, 10, 26, 34, 56, 90 subimmersion theorem, 82 submanifold, 133 submersion, 133 submersion theorem, 16, 20, 97, 102, 133 Synge, J. L., 2 tangent bundle, 16, 26, 32, 135 bundle morphism, 74 space, 134 tangential coordinate subspace, 98 total energy, 52 trivialization, 32 Uhing transmission, 111 unconstrained motion, 2, 4, 10, 15, 16, 29 Van Loan, C. P., 21 variational principles, 9 virtual displacement, 15, 16 velocities, 14 Wanner, G., 106 Wehage, R. A., 106 Whitney, H., 20 Zeidler, E., 11

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