Stability Analysis and Control of Nonholonomic Systems with Potential Fields

Abstract

This paper presents a theoretical analysis on the stability of nonholonomic systems when a potential field method is applied to them, and shows a numerical example for a nonholonomic underwater vehicle among obstacles. Nonholonomic systems with a potential function have an infinite number of equilibrium points, because the motion of the systems cannot always be generated exactly along the gradient vector of the potential function. By utilizing the component of the input that does not increase or decrease the potential function, the equilibrium points other than the critical points of the function can be destabilized, if the controllability of the systems is satisfied with the first-order Lie brackets of input vector fields. A time-invariant controller is proposed based on the theoretical analysis on the stability of equilibrium points, and applied to an underwater vehicle among obstacles. When the potential function has saddles as its critical points, the potential function is modified to be time-varying near the saddles in order to prevent the system from being trapped in the saddles. Numerical simulation results demonstrate that the underwater vehicle with the proposed controller converges to the desired point without collision with obstacles.