An Approach to the Dynamics and Control of Uncertain Multi-body Systems

This paper deals with the initial development of a methodology for controlling real-life, multi-body dynamical systems in the presence of uncertainties in our knowledge of their exact physical nature as well as uncertainties in the ‘given’ forces acting on them. We assume that both these uncertainties can be time-varying, yet bounded. The methodology is developed in two steps. In the first step an exact control of the so-called nominal system—our best assessment of the real-life physical system and the forces acing on it—is developed. This control is inspired by results from analytical dynamics and involves no linearizations and/or approximations of the nonlinear, nonautonomous system. It also minimizes a suitable norm of the trajectory errors at each instant of time. In the second step, a continuous controller using a generalized sliding mode approach is developed that guarantees that the real-life system tracks, without any chatter and to within pre-specified error bounds, the control requirements that are imposed on the nominal system. Thus an explicit closed-form control for an uncertain, nonlinear, nonautonomous multi-body system that satisfies trajectory control requirements placed on the nominal system (to within pre-specified error bounds), is obtained. An example of a triple pendulum in which there are uncertainties both in the description of its physical parameters and in the description of the gravity field in which it moves is considered. The example demonstrates the simplicity and efficacy of the approach when there are uncertainties both in the description of a dynamical system and in the given forces to which it is subjected.