Modelling and Control of Nonholonomic Mechanical Systems
The goal of this chapter is to provide tools for analyzing and controlling nonholonomic mechanical systems. This classical subject has received renewed attention because nonholonomic constraints arise in many advanced robotic structures, such as mobile ro
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A. De Luca and G. Oriolo University "La Sapienza", Roma, Italy
Abstract The goal of this chapter is to proviele tools for analyzing and controlling nonholonomic mechanical systems. This classical subject has received renewed attention because nonholonomic constraints arise in many advanced robotic structures, such as mobile robots, space manipulators, and multifingered robot hands. Nonholonomic behavior in robotic systems is particularly interesting, because it implies that the mechanism can be completely controlled with a reduced number of actnators. On the other hand, both planning and control are much more difficult than in conventional holonomic systems, and require special tcchniques. We show first that the nonholonomy of kinematic constraints in mechanical systems is equivalent to the controllability of an associated control system, so tlwt integrability conditions may be sought by exploiting concepts from nonlinear control theory. Basic tools for the analysis and stabilization of nonlinear control systems are reviewed and used to obtain condit.ions for partial or complet.e nonholonomy, so as to devise a classification of nonholonomic syst.ems. Several kinematic modcls of nonholonomic systems are presented, including examples of wheeled mobile robots, free-fioating spacc structures and redundant manipulators. We introduce then thc dynamics of nonholonomic systems and a procedure for partiallinearization of the corrcsponding control system via feedback. These points are illustrated by deriving thc dynamical modcls of two previously considered systems. Finally, we discuss some general issues of the control problem for nonholonomic systems and present open-loop and fcedback control techniques, illustrated also by numerical simulations. J. Angeles et al. (eds.), Kinematics and Dynamics of Multi-Body Systems © Springer-Verlag Wien 1995
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7.1
Introduction
Consider a mechanical system whose configuration can be described by a vector of generalized Coordinates q E Q. The configuration space Q is an n-dimensional smooth manifold, locally diffeomorphic to an open subset of IRn. Given a trajectory q(t) E Q, the generalized velocity at a configuration q is the vector rj belanging to the tangent
space Tq(Q). In many interesting cases, the system motion is subject to constraints that may arise from the structure itself of the mechanism, or from the way in which it is actuated and controlled. Various classifications of such constraints can be devised. For example, constraints may be expressed as equalities or inequalities (respectively, bilateral or unilateral constraints) and they may explicitly depend on time or not ( rheonomic or scleronomic constraints). In the discussion below, one possible-by no means exhaustive-classification is considered. In particular, we will deal only with bilateral scleronomic constraints. A treatment of nonholonomic unilateral constraints can be found, for example, in [1]. Motion restrictions that may be put in the form
h;(q) = 0,
i = 1, ... , k < n,
(7.1)
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