Steering nonholonomic mobile robots using a trajectory interception approach

Abstract

Employing the example of a class of wheeled mobile robots, a simple and systematic approach to the construction of stabilizing feedback control for nonholonomic systems is presented. Two types of robotic models are considered: models with the degree of mobility δ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</inf> = 1, and the degree of steeribility δ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</inf> = 1, and models characterized by (δ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</inf> ,δ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</inf> ) = (1,2). The controllability Lie algebras for both models are infinite dimensional. For feedback design purposes, the original models are first approximated by ones whose respective controllability Lie algebras are finite dimensional. A time varying stabilizing feedback law is constructed for each of the simplified models by employing their respective Lie bracket extensions. The resulting feedback laws can be regarded as compositions of standard stabilizing feedback control laws for the extended systems and periodic continuations of parametrized solutions to certain open-loop, finite horizon trajectory interception problems, stated in logarithmic coordinates of flows. An adequately large stability robustness margin for the extended controlled systems can always be insured, and is shown to guarantee that the constructed feedback laws are stabilizing not only for the approximate but also for the original models. This approach does not rely on a specific choice of a Lyapunov function, and does not require transformations of the models to chained forms.