Equations of Motion for Constrained Multibody Systems and their Control

F E Udwadia

doi:10.1007/s10957-005-7507-8

Abstract

This paper presents some recent advances in the dynamics and control of constrained multibody systems. The constraints considered need not satisfy the D’Alembert principle and therefore the results are of general applicability. They show that, in the presence of constraints, the constraint force acting on the multibody system can always be viewed as made up of the sum of two components whose explicit form is provided. The first of these components consists of the constraint force that would have existed were all the constraints ideal; the second is caused by the nonideal nature of the constraints, and though it needs specification by the mechanician who is modeling the specific system at hand, it has a specific form. The general equations of motion obtained herein provide new insights into the simplicity with which Nature seems to operate. They point toward the development of new and novel approaches for the exact control of complex multibody nonlinear systems.

Highlights

The general problem of obtaining the equations of motion of a constrained discrete mechanical system is one of the central issues in multibody dynamics
The explicit equations of motion obtained by Udwadia and Kalaba (Ref. 7) provide a new and different perspective on the constrained motion of multibody systems
Equation (10) leads to the following new fundamental principle of motion for constrained multibody mechanical systems: “The motion of a discrete dynamical system subjected to constraints evolves, at each instant in time, in such a way that the deviation in its acceleration from what it would have at that instant if there were no constraints on it, is the sum of two M-orthogonal components; the first component is directly proportional to the extent e to which the accelerations corresponding to its unconstrained motion, at that instant, do not satisfy the constraints, the matrix of proportionality being M−1/2B+; the second component is proportional to the given n-vector c, the matrix of proportionality being M−1/2(I − B+B)M1/2.”

Summary

Introduction

The general problem of obtaining the equations of motion of a constrained discrete mechanical system is one of the central issues in multibody dynamics. They introduce the notion of generalized inverses in the description of such motion and, through their use, obtain a simple and general explicit equation of motion for constrained multibody mechanical systems without the use of, or any need for, the notion of Lagrange multipliers Their approach has allowed us, for the first time, to obtain the explicit equations of motion for multibody systems with constraints that may be: (i) nonlinear functions of the velocities, (ii) explicitly dependent on time, and (iii) functionally dependent. Their equations deal only with systems where the constraints are ideal and satisfy the D’Alembert principle, as do all the other formulations/equations developed so far With the help of these equations, we provide a new fundamental, general principle governing constrained multibody dynamics

Fundamental Equation

Discussion and Conclusions