This paper studies properties of homogeneous systems in a geometric, coordinate-free setting. A key contribution of this paper is a result relating regularity properties of a homogeneous function to its degree of homogeneity and the local behavior of the dilation near the origin. This result makes it possible to extend previous results on homogeneous systems to the geometric framework. As an application of our results, we consider finite-time stability of homogeneous systems. The main result that links homogeneity and finite-time stability is that a homogeneous system is finite-time stable if and only if it is asymptotically stable and has a negative degree of homogeneity. We also show that the assumption of homogeneity leads to stronger properties for finite-time stable systems.

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Geometric homogeneity with applications to finite-time stability

Provides a summary of recent developments in control of nonholonomic systems. The published literature has grown enormously during the last six years, and it is now possible to give a tutorial presentation of many of these developments. The objective of this article is to provide a unified and accessible presentation, placing the various models, problem formulations, approaches, and results into a proper context. It is hoped that this overview will provide a good introduction to the subject for nonspecialists in the field, while perhaps providing specialists with a better perspective of the field as a whole. The paper is organized as follows: introduction to nonholonomic control systems and where they arise in applications, classification of models of nonholonomic control systems, control problem formulations, motion planning results, stabilization results, and current and future research topics.

Developments in nonholonomic control problems

Chain form systems have recently been introduced to model the kinematics of a class of nonholonomic mechanical systems. The first part of the study is centered on control design and analysis for nonlinear systems which can be converted to the chain form. Solutions to various control problems (open-loop steering, partial or complete state feedback stabilization) are either recalled, generalized, or developed. In particular, globally stabilizing time-varying feedbacks are derived, and a discussion of their convergence properties is provided. Application to the control of nonholonomic wheeled mobile robots is described in the second part of the study by considering the case of a car pulling trailers. >

Control of chained systems application to path following and time-varying point-stabilization of mobile robots

The feasibility problem is studied of achieving a specified formation among a group of autonomous unicycles by local distributed control. The directed graph defined by the information flow plays a key role. It is proved that formation stabilization to a point is feasible if and only if the sensor digraph has a globally reachable node. A similar result is given for formation stabilization to a line and to more general geometric arrangements.

Necessary and sufficient graphical conditions for formation control of unicycles

This book presents methods to study the controllability and the stabilization of nonlinear control systems in finite and infinite dimensions. The emphasis is put on specific phenomena due to nonlinearities. In particular, many examples are given where nonlinearities turn out to be essential to get controllability or stabilization. Various methods are presented to study the controllability or to construct stabilizing feedback laws. The power of these methods is illustrated by numerous examples coming from such areas as celestial mechanics, fluid mechanics, and quantum mechanics. The book is addressed to graduate students in mathematics or control theory, and to mathematicians or engineers with an interest in nonlinear control systems governed by ordinary or partial differential equations.

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Control and Nonlinearity

The systems considered in this work are composed of weakly coupled, linear and essentially nonlinear (nonlinearizable) components. In Part I of this work we present numerical evidence of energy pumping in coupled nonlinear mechanical oscillators, i.e., of one-way (irreversible) channeling of externally imparted energy from the linear to the nonlinear part of the system, provided that the energy is above a critical level. Clearly, no such phenomenon is possible in the linear system. To obtain a better understanding of the energy pumping phenomenon we first analyze the dynamics of the underlying Hamiltonian system (corresponding to zero damping). First we reduce the equations of motion on an isoenergetic manifold of the dynamical flow, and then compute subharmonic orbits by employing nonsmooth transformation of coordinates which lead to nonlinear boundary value problems. It is conjectured that a 1:1 stable subharmonic orbit of the underlying Hamiltonian system is mainly responsible for the energy pumping phenomenon. This orbit cannot be excited at sufficiently low energies. In Part II of this work the energy pumping phenomenon is further analyzed, and it is shown that it is caused by transient resonance capture on a 1:1 resonance manifold of the system.

Energy pumping in nonlinear mechanical oscillators : Part I : Dynamics of the underlying Hamiltonian systems

This paper focuses on the problem of exponential stabilization of controllable, driftless systems using time-varying, homogeneous feedback. The analysis is performed with respect to a homogeneous norm in a nonstandard dilation that is compatible with the algebraic structure of the control Lie algebra. It can be shown that any continuous, time-varying controller that achieves exponential stability relative to the Euclidean norm is necessarily non-Lipschitz. Despite these restrictions, we provide a set of constructive, sufficient conditions for extending smooth, asymptotic stabilizers to homogeneous, exponential stabilizers. The modified feedbacks are everywhere continuous, smooth away from the origin, and can be extended to a large class of systems with torque inputs. The feedback laws are applied to an experimental mobile robot and show significant improvement in convergence rate over smooth stabilizers.

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Exponential stabilization of driftless nonlinear control systems using homogeneous feedback

This study quantifies the dynamics of actuation for the temporally forced, round gas jet injected transversely into a crossflow, and incorporates these dynamics in developing a methodology for open loop jet control. A linear model for the dynamics of the forced jet actuation is used to develop a dynamic compensator for the actuator. When the compensator is applied, it allows the jet to be forced in a manner which results in a more precisely prescribed, temporally varying exit velocity, the RMS amplitude of perturbation of which can be made independent of the forcing frequency. Use of the compensator allows straightforward comparisons among different conditions for jet excitation. Clear identification can be made of specific excitation frequencies and characteristic temporal pulse widths which optimize transverse jet penetration and spread through the formation of distinct, deeply penetrating vortex structures.

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The actively controlled jet in crossflow

This article analyzes the global dynamical behavior of sim plified hopping robot models that are analogous to Raibert's experimental machines. We first review a one-dimensional ver tical hopping model that captures both the vertical hopping dynamics and nonlinear control algorithm. Second, we present a more complicated two-dimensional model that includes both forward and vertical hopping dynamics and a foot placement algorithm. These systems are analyzed using a Poincare return map. In this approach, issues of stability and global dynamical behavior are reduced to the study of the fixed points of this map. For the one-dimensional model, a closed-form return map is obtained. For the two-dimensional model, we derive an exact return map based on the first integrals of motion. Because this map can only be constructed numerically, we also derive an analytical approximation to the return map based on perturba tion methods. The approximate return map is shown to closely predict the behavior of the exact map for ...

Periodic Motions of a Hopping Robot With Vertical and Forward Motion

In this paper the authors make a contribution to the analysis of nonholonomic systems with exponential rates of convergence. A key idea is the use of control laws which render the closed loop system homogeneous with respect to a dilation. The analysis is applied to nonholonomic systems in power form and consists of two steps. The first step is a reduction to an invariant set and then the application of an averaging result. The averaging theorem is a stability result for C/sup 0/ homogeneous order zero vector fields. >

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Robert T. M'Closkey

Papers

Energy pumping in nonlinear mechanical oscillators : Part I : Dynamics of the underlying Hamiltonian systems

Exponential stabilization of driftless nonlinear control systems using homogeneous feedback

The actively controlled jet in crossflow

Periodic Motions of a Hopping Robot With Vertical and Forward Motion

Nonholonomic systems and exponential convergence: some analysis tools