Discontinuous feedback stabilization of a class of nonholonomic systems based on Lyapunov control

This paper deals with the problem of controlling a class of nonholonomic systems, first order systems. The first order systems are systems where the input vector fields and the first level of Lie brackets between them span the tangent space of the state space. We derive a discontinuous state feedback law for the systems by extending Lyapunov control. The input vector is constructed by multiplying the gradient vector of a Lyapunov function by a matrix which is composed of a negative definite symmetric one and a skew-symmetric one. The designed controller makes the desired point the only stable equilibrium point, and the controlled system converges to the desired point from almost all initial points. The proposed controller is applied to two examples of first order systems, a two wheeled vehicle and a spacecraft composed of two rigid bodies.