Soft nonholonomic constraints : theory and applications to optimal control -

Abstract

This thesis presents a general methodology for tackling the fundamental problem of chattering, which arises in the time-optimal control solutions of the dy- namically extended Dubins car. The proposed approach stems from an insight that chattering arises due to a weakness in the mathematical model of the system dynamics in the optimization problem. Namely, the Dubins car model assumes that the nonholonomic constraints are ideal and satisfied at all times. By relax- ing the no-skidding constraint, we prove that a more realistic dynamical model inherently rules out chattering segments from the optimal control solution. This proposed methodology is the first approach that eliminates chattering by ad- dressing its root cause, namely the order of the singular segments of the optimal control solution. Relaxing the no-skidding constraint is shown to reduce the order of the resulting singular segments from 2, which cannot be concatenated to bang segments, to 1, which are commonly known to cause no chattering when joined with bang segments. To relax the nonholonomic constraints in a manner that does not impede optimal control analysis within the framework of Pontryagin’s Principle, this thesis proposes and develops a new model for skidding, which we refer to as soft nonholonomic constraints. These are nonholonomic constraints that account for skidding but at the same time resemble the ideal no-slipping and no-skidding constraints, in the sense that they are frictionless. In other words, these pro- posed constraints strike a balance between maintaining the properties of the ideal constraints, while allowing for an overall lateral motion. The validity of soft nonholonomic constraints as a skidding model is demonstrated by means of comparing it to an experimentally-validated skidding model from the literature. Moreover, the proposed methodology of relaxing the no-skidding constraint to eliminate chattering is shown to be generally applicable to a class of dynamical systems, which are referred to as