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Über dieses Buch

The primary emphasis of this book is the modeling, analysis, and control of mechanical systems. The methods and results presented can be applied to a large class of mechanical control systems, including applications in robotics, autonomous vehicle control, and multi-body systems. The book is unique in that it presents a unified, rather than an inclusive, treatment of control theory for mechanical systems. A distinctive feature of the presentation is its reliance on techniques from differential and Riemannian geometry.

The book contains extensive examples and exercises, and will be suitable for a growing number of courses in this area. It begins with the detailed mathematical background, proceeding through innovative approaches to physical modeling, analysis, and design techniques. Numerous examples illustrate the proposed methods and results, while the many exercises test basic knowledge and introduce topics not covered in the main body of the text.

The audience of this book consists of two groups. The first group is comprised of graduate students in engineering or mathematical sciences who wish to learn the basics of geometric mechanics, nonlinear control theory, and control theory for mechanical systems. Readers will be able to immediately begin exploring the research literature on these subjects. The second group consists of researchers in mechanics and control theory. Nonlinear control theoreticians will find explicit links between concepts in geometric mechanics and nonlinear control theory. Researchers in mechanics will find an overview of topics in control theory that have relevance to mechanics.

Inhaltsverzeichnis

Frontmatter

Modeling of mechanical systems

Frontmatter

1. Introductory examples and problems

Abstract
To motivate the kinds of problems we consider in this book, we use this introductory chapter to present some simple examples for which the problems are fairly easily understood. The short presentation in this chapter is more informal than will be encountered in the remainder of the book. We draw the examples in this chapter from three loosely defined collections of physical systems: aerospace and underwater vehicles, robotic manipulators and multi-body systems, and constrained systems. These examples will give us an opportunity to discuss a variety of topics, and motivate the introduction of appropriate mathematical tools.
Francesco Bullo, Andrew D. Lewis

2. Linear and multilinear algebra

Abstract
The study of the geometry of Lagrangian mechanics requires that one be familiar with basic concepts in abstract linear and multilinear algebra. The reader is expected to have encountered at least some of these concepts before, so this chapter serves primarily as a refresher. We also use our discussion of linear algebra as a means of introducing the summation convention in a systematic manner. Since this gets used in computations, the reader may wish to take the opportunity to become familiar with it.
Francesco Bullo, Andrew D. Lewis

3. Differential geometry

Abstract
As can be seen from the title of this book, our treatment of mechanical control systems is geometric, more precisely, differential geometric. In this chapter we review the essential differential geometric tools used in the book. Since a thorough treatment of all parts of the subject necessary for our objectives would be even more lengthy than what we currently have, we essentially present a list of definitions and facts that follow from these definitions. The treatment we give, while comparatively brief, is not as fast-paced as was the case in Chapter 2. In this chapter we spend more time on examples that illustrate the main ideas. Nevertheless, a reader who knows no differential geometry can expect to invest some time in learning the subject, and might find the review in this chapter too hasty. In this case, additional references (see below) will be helpful in providing context and further examples. Readers having a passing acquaintance with differential geometry may wish to quickly scan this chapter to see if there are any major topics that are unfamiliar to them.
Francesco Bullo, Andrew D. Lewis

4. Simple mechanical control systems

Abstract
The preceding two chapters were preparatory material, and were presented fairly hastily, with the reader being directed to references for further details. In this chapter, we begin the presentation of the material that can be considered the core of the book. The aim in this chapter is to provide the methodology for going from a physical problem to a mathematical model. What is more, the mathematical model we consider uses the tools described in Chapter 3, particularly the affine connection formalism of Section 3.8. We are extremely systematic in our presentation, far more so than is the norm. This has the disadvantage of making the presentation lengthy. To this end, we should mention that it is not normal, when modeling a mechanical system, to go systematically through the steps we outline in this chapter. Indeed, someone experienced with the process can perform many of the steps we describe “just by looking at the system.” The systematic presentation we provide has two advantages to counterbalance its length.
Francesco Bullo, Andrew D. Lewis

5. Lie groups, systems on groups, and symmetries

Abstract
The previous chapter provides a general framework for modeling mechanical control systems on manifolds. Frequently however, there is additional structure in a mechanical system that can be exploited for analysis and control. This chapter discusses an important class of manifolds, called Lie groups, that arise naturally in rigid body kinematics, as well as the properties of mechanical systems defined on Lie groups or possessing Lie group symmetries.
Francesco Bullo, Andrew D. Lewis

Analysis of mechanical control systems

Frontmatter

6. Stability

Abstract
The analysis of stability of mechanical systems is a classic topic in mechanics and dynamics that has affected a number of mathematical and engineering disciplines. In his classic work, Lagrange [1788] investigated the stability of mechanical systems at local minima of the potential function using energy arguments. In a work widely recognized to be one of the first on control theory, Maxwell [1868] analyzed the stability of certain mechanical governors using linearization. At about the same time, Thomson and Tait [1867] studied the asymptotic stability of mechanical systems subject to dissipative forces also via linear methods. Finally, Lyapunov [1892] developed the key elements of a stability notion and of stability criteria applicable to a broad class of nonlinear systems; this work laid the foundations for modern stability theory. We present the Lyapunov Stability Critera in Theorem 6.14. So-called invariance principles were later developed to establish stability properties of dynamical systems on the basis of weaker requirements than those required by Lyapunov’s original criteria. Early work on invariance principles in stability is due to Barbashin and Krasovskiĭ [1952]; LaSalle presented his Invariance Principle in [LaSalle 1968]. Recent influential works on stability include [LaSalle and Lefschetz 1962], [Hahn 1963, 1967] and [Chetaev 1955]. Nowadays, stability theory is a cornerstone of dynamical systems and control theory; examples of modern treatments in nonlinear control monographs include [Khalil 2001, Sastry 1999, Sontag 1998], and dynamical systems references include [Arnol’d 1992, Guckenheimer and Holmes 1990, Hirsch and Smale 1974, Merkin 1997].
Francesco Bullo, Andrew D. Lewis

7. Controllability

Abstract
One of the fundamental problems in control theory is that of controllability. Indeed, many design methodologies rely on some hypotheses that concern controllability. The problem of controllability is essentially one of describing the nature of the set of states reachable from an initial state. In the development of this theory, there are two properties that arise as being important. The first is the property of “accessibility,” which means that the reachable set has a nonempty interior. The treatment of accessibility we present follows the approach of the fundamental paper of Sussmann and Jurdjevic [1972]. Results of a related nature are those of Krener [1974] and Hermann and Krener [1977]. The property of “controllability” extends accessibility by further asking whether the initial state lies in the interior of the reachable set. The matter of providing general conditions for determining controllability is currently unresolved, although there have been many deep and insightful contributions. While we cannot hope to provide anything close to a complete overview of the literature, we will mention some work that is commensurate with the approach that we take here. Sussmann has made various important contributions to controllability, starting with the paper [Sussmann 1978]. In the paper [Sussmann 1983], a Lie series approach was developed for the controllability of control-affine systems, and this approach culminated in the quite general results of [Sussmann 1987], which incorporated the ideas of Crouch and Byrnes [1986] concerning input symmetries.
Francesco Bullo, Andrew D. Lewis

8. Low-order controllability and kinematic reduction

Abstract
The discussions of controllability in Chapter 7 are of a quite general nature. However, as we saw in Exercise E7.14 (see also Example 7.22 for a control-affine example), even these very general results can fail to provide complete characterizations of the controllability of a system, even in rather simple examples. In this chapter we provide low-order controllability results that are quite sharp. These are extensions of results initially due of Hirschorn and Lewis [2001], and considered in the context of motion planning by Bullo and Lewis [2003b]. One of the interesting features of these results is that their hypotheses are “feedback-invariant,” a notion that we do not discuss here; see [Lewis 2000a] for an introduction in the setting of affine connection control systems. Interestingly, these controllability results are related to a notion called “kinematic controllability” by Bullo and Lynch [2001], and considered here in Section 8.3. While the controllability results of Section 8.2 have more restrictive hypotheses than those of Chapter 7, it turns out that the restricted class of systems are those for which it is possible to develop some simplified design methodologies for motion planning. These methodologies are presented by way of some examples in Chapter 13. The emphasis in this chapter is on understanding the sometimes subtle relationships between various concepts; the design issues and serious consideration of examples are postponed to Chapter 13.
Francesco Bullo, Andrew D. Lewis

9. Perturbation analysis

Abstract
In this chapter we apply perturbation methods to analyze the trajectories of forced affine connection control systems. We consider two types of control signals: periodic, large-amplitude, high-frequency signals, which we refer to as oscillatory; and small-amplitude signals. For both classes of signals, we perform a perturbation analysis that predicts, with some specified level of accuracy, the behavior of the resulting forced affine connection system.
Francesco Bullo, Andrew D. Lewis

A sampling of design methodologies

Frontmatter

10. Linear and nonlinear potential shaping for stabilization

Abstract
Our first chapter on control design is concerned with somewhat classic methods for stabilization. The techniques described here involve linearization methods and potential energy shaping methods. Although these techniques are somewhat limited in scope, they are valuable because they provide some insight into the problem of stabilization, both generally and for mechanical systems. Readers can then use the experience gained from this chapter to more easily read the literature on stabilization, some of which is discussed in Section 10.5.
Francesco Bullo, Andrew D. Lewis

11. Stabilization and tracking for fully actuated systems

Abstract
In this chapter we design feedback control laws for simple mechanical control systems with dissipation. We assume that the system is fully actuated and that the control set is unbounded throughout the chapter. Our approach builds on the proportional-derivative control designs presented in the previous chapter, but we are able to obtain stronger results by exploiting the full actuation. We provide a comprehensive solution to the problems of stabilization of controlled equilibrium configurations and tracking of reference trajectories.
Francesco Bullo, Andrew D. Lewis

12. Stabilization and tracking using oscillatory controls

Abstract
In this chapter we design oscillatory control laws to solve stabilization and tracking problems for forced affine connection control systems. The objective is to exploit the averaging analysis obtained in Chapter 9 for the purpose of control design. In particular, we shall present results on stabilization and tracking that are applicable to systems that are not linearly controllable. As in the perturbation analysis in Chapter 9, we shall consider smooth systems.
Francesco Bullo, Andrew D. Lewis

13. Motion planning for underactuated systems

Abstract
In Chapter 8 we saw that it was possible to come to a fairly complete understanding of low-order controllability of affine connection control systems, and that this sort of controllability was closely related to the notion of kinematic controllability. Implicit in these observations is that kinematic controllability is useful for motion planning for affine connection control systems. In this chapter we illustrate this by looking in detail at a couple of nontrivial examples. The material in this chapter has been developed only recently in the research literature. As mentioned in Chapter 8, the idea of a kinematic reduction is inspired by the work of Arai, Tanie, and Shiroma [1998], and Lynch, Shiroma, Arai, and Tanie [2000]. A first formal definition was given in [Bullo and Lynch 2001], while more complete work followed in the papers [Bullo and Lewis 2003a, Martínez, Cortés, and Bullo 2003b]. The kinematic reduction results of Section 8.3 provide a means of reducing the order of the dynamical systems being considered from two to one. The idea of lowering the complexity of representations of mechanical control systems can be related to numerous previous efforts, including work on hybrid models for motion control systems [Brockett 1993], oscillatory motion primitives [Bullo, Leonard, and Lewis 2000], consistent control abstractions [Pappas and Simić 2002], and maneuver automata [Frazzoli, Dahleh, and Feron 2003]. For a general introduction to motion planning, we refer to [Latombe 1991].
Francesco Bullo, Andrew D. Lewis

Backmatter

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