Abstract not found

this paper is as follows: in Section 2, we collect some mathematical preliminaries from the literature on controllability of nonlinear systems and on classification of free Lie algebras. These are drawn from classical references in control theory [4, 17, 18, 36, 40] and Lie algebras [15, 43]. In Section 3, using some outstanding results of Brockett on optimal steering of certain classes of systems as motivation [5], we discuss the use of sinusoidal inputs for steering systems of first order, i.e., systems where controllability is achieved after just one level of Lie brackets of the input vector fields. Section 4 attempts to expand the domain of applicability of these results to more complex systems, where several orders of Lie brackets are needed to obtain the full Lie algebra associated with the input distribution. The 4 MURRAY AND SASTRY

We propose a general strategy for solving the motion planning problem for real analytic, controllable systems without drift. The procedure starts by computing a control that steers the given initial point to the desired target point for an extended system, in which a number of Lie brackets of the system vector fields are added to the right-hand side. The main point then is to use formal calculations based on the product expansion relative to a P. Hall basis, to produce another control that achieves the desired result on the formal level. It then turns out that this control provides an exact solution of the original problem if the given system is nilpotent. When the system is not nilpotent, one can still produce an iterative algorithm that converges very fast to a solution. Using the theory of feedback nilpotentization, one can find classes of non-nilpotent systems for which the algorithm, in cascade with a precompensator, produces an exact solution in a finite number of steps. We also ...

We propose a general strategy for solving the motion planning problem for real analytic, controllable systems without drift. The procedure starts by computing a control that steers the given initial point to the desired target point for an extended system, in which a number of Lie brackets of the system vector fields are added to the right-hand side. The main point then is to use formal calculations based on the product expansion relative to a P. Hall basis, to produce another control that achieves the desired result on the formal level. It then turns out that this control provides an exact solution of the original problem if the given system is nilpotent. When the system is not nilpotent, one can still produce an iterative algorithm that converges very fast to a solution. Using the theory of feedback nilpotentization, one can find classes of non-nilpotent systems for which the algorithm, in cas...

We describe sufficient conditions, extending earlier work by Kurzweil and Jarnik, for a sequence of inputs u j = (u j 1 ; : : : ; u j m ) 2 L 1 ([0; T ]; IR m ) to be such that, for every m-tuple (f 1 ; : : : ; f m ) of smooth vector fields, the trajectories of x(t) = P m k=1 u j k (t)f k (x(t)) converge to those of an "extended system" x(t) = P r k=1 v k (t)f k (x(t)), where the new vector fields fm+1 ; : : : ; f r are Lie brackets of the original f k 's. Using these conditions, we can solve the inverse problem: given a trajectory fl of the extended system, find trajectories of the original system that converge to fl. This is done by means of a universal construction that only involves a knowledge of the v's. These results can be applied to solve the problem of approximate tracking for a controllable system without drift. 1. Introduction In this paper we report a number of results on the relation between the solutions of an equation x(t) = m X k=1 u k (t)f k (x(...

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We illustrate the use of the techniques of modern geometric optimal control theory by studying the shortest paths for a model of a car that can move forwards and backwards. This problem was discussed in recent work by Reeds and Shepp who showed, by special methods, (a) that shortest path motion could always be achieved by means of trajectories of a special kind, namely, concatenations of at most five pieces, each of which is either a straight line or a circle, and (b) that these concatenations can be classified into 48 three-parameter families. We show how these results fit in a much more general framework, and can be discovered and proved by applying in a systematic way the techniques of Optimal Control Theory. It turns out that the "classical" optimal control tools developed in the 1960's, such as the Pontryagin Maximum Principle and theorems on the existence of optimal trajectories, are helpful to go part of the way and get some information on the shortest paths, but do not suffice ...

In this paper we address the constructive controllability problem for drift-free, leftinvariant systems on finite-dimensional Lie groups with fewer controls than state dimension. We consider small (ffl) amplitude, low-frequency, periodically time-varying controls and derive average solutions for system behavior. We show how the pth-order average formula can be used to construct open-loop controls for point-to-point maneuvering of systems that require up to (p \Gamma 1) iterations of Lie brackets to satisfy the Lie algebra controllability rank condition. In the cases p = 2; 3, we give algorithms for constructing these controls as a function of structure constants that define the control authority, i.e., the actuator capability, of the system. The algorithms are based on a geometric interpretation of the average formulas and produce sinusoidal controls that solve the constructive controllability problem with O(ffl ) accuracy in general (exactly if the Lie algebra is nilpotent). The methodology is applicable to a variety of control problems and is illustrated for the motion control problem of an autonomous underwater vehicle with as few as three control inputs.

We study length-minimizing arcs in sub-Riemannian manifolds (M;E;G) whose metric G is defined on a rank-two bracket-generating distribution E. It is well known that all length-minimizing arcs are extremals, and that these extremals are either "normal" or "abnormal." Normal extremals are locally optimal, in the sense that every sufficiently short piece of such an extremal is a minimizer. The question whether every length-minimizer is a normal extremal remained open for several years, and was recently settled by R. Montgomery, who exhibited a counterexample. But Montgomery's geometric optimality proof depends heavily on special properties of his example and still leaves open the question whether abnormal minimizers are an exceptional phenomenon or a common occurrence. We present an analytic technique for proving local optimality of a large class of abnormal extremals that we call "regular." Our technique is based on (a) a "normal form theorem," stating that, locally, a regular abnormal e...

This paper addresses the optimal control and selection of gaits in a class of nonholonomic locomotion systems that exhibit group symmetries. We study optimal gaits for the snakeboard, a representative example of this class of systems. We employ Lagrangian reduction techniques to simplify the optimal control problem and describe a general framework and an algorithm to obtain numerical solutions to this problem. This work employs optimal control techniques to study the optimality of gaits and issues involving gait transitions. The general framework provided in this paper can easily be applied to other examples of biological and robotic locomotion. KEY WORDS---optimal control, robotic locomotion, geometric mechanics, locomotive gaits 1.