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Chap7-Modelling_ControlOfNonholonomicMechanicalSystems -...

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Chapter 7 MODELING AND CONTROL OF NONHOLONOMIC MECHANICAL SYSTEMS Alessandro De Luca and Giuseppe Oriolo Dipartimento di Informatica e Sistemistica Universit` a degli Studi di Roma “La Sapienza” Via Eudossiana 18, 00184 Roma, ITALY { deluca,oriolo } @dis.uniroma1.it Abstract 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 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 actuators. On the other hand, both planning and control are much more difficult than in conventional holonomic sys- tems, and require special techniques. We show first that the nonholonomy of kinematic constraints in mechanical systems is equivalent to the controllability of an associated control system, so that 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 conditions for partial or complete non- holonomy, so as to devise a classification of nonholonomic systems. Several kinematic models of nonholonomic systems are presented, including examples of wheeled mobile robots, free-floating space structures and redundant manipulators. We introduce then the dynamics of nonholonomic systems and a procedure for partial linearization of the corresponding control system via feedback. These points are illustrated by deriving the dynamical models of two previously considered systems. Finally, we discuss some general issues of the control problem for nonholonomic systems and present open-loop and feedback control techniques, illustrated also by numerical simulations.
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7.1 Introduction Consider a mechanical system whose configuration can be described by a vector of generalized coordinates q ∈ Q . The configuration space Q is an n -dimensional smooth manifold, locally diffeomorphic to an open subset of IR n . Given a trajectory q ( t ) ∈ Q , the generalized velocity at a configuration q is the vector ˙ q belonging to the tangent space T q ( 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).
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