The dynamics of systems with servoconstraints. I

Abstract

The paper discusses the dynamics of systems with Béghin's servoconstraints where the constraints are realized by means of controlled forces. Classical nonholonomic systems are an important particular case. Special attention is given to the study of motion on Lie groups with left-invariant kinetic energy and left-invariant constraints. The presence of symmetries allows one to reduce the dynamic equations to a closed system of differential equations with quadratic right-hand sides on a Lie algebra. Examples are given which include the rotation of a rigid body with a left-invariant servoconstraint -the projection of the angular velocity onto some direction fixed in the body is equal to zero (a generalization of the nonholonomic Suslov problem) -and the motion of the Chaplygin sleigh with servoconstraints of a certain type. The dynamics of systems with Béghin's servoconstraints is richer and more varied than the more usual dynamics of nonholonomic systems.MSC2010 numbers: 34D20, 70F25, 70Q05

A Study of the Controlled Motion of a Four-wheeled Mecanum Platform

“…Conclusions concerning the change in the motion pattern of the system due to the change in its mechanical energy were drawn already by H. Béghin [3, p. 29]. In the work of V. V. Kozlov [12,13] it is shown that the dynamics of systems with servo-constraints is richer and more various than the dynamics of nonholonomic systems.…”

“…Program motions (servo-constraints) can be realized by adjustment of forces within the framework of this theory [3,11,12]. In this case the structure of control actions repeats the structure of the reaction of ideal constraints, the equations of which coincide with the equations of program motion.…”

Historical and critical review of the development of nonholonomic mechanics: the classical period

“…Требование ортогональ-ности реакции является естественным обобщением ситуации, когда частица закреплена на нерастяжимой нити (сферический маятник) так, что в этом случае реакция направле-на вдоль нити. Вообще говоря, выбор сил реакции может существенным образом зависеть от реализации связей [52,53,108].…”

The dynamics of systems with servoconstraints. II

“…Servoconstraints of the first kind (discovered by Béghin) are realized by a suitable choice of controlled forces. Important examples are the classical nonholonomic constraints, which, as pointed out in [2], not only admit a passive realization, but can be implemented by a special choice of active forces. Recently there has been considerable progress in the qualitative and topological description of nonholonomic systems (see, e. g., [51][52][53][54][55][56][57][58]).…”

“…In this paper we will be primarily interested in the dynamics of systems with servoconstraints of the second kind. A qualitative analysis of the motion of systems with servoconstraints of the first kind is presented in [2].…”