Discontinuous feedback control of a planar rigid body with an unactuated degree of freedom

We study the problem of controlling a specific mechanical system consisting of a rigid base body with an unactuated internal degree of freedom. The key assumptions are that the translational and rotational motion of the base body can be completely controlled by external forces and moments, while the internal degree of freedom is unactuated. It was shown in our previous work (1998) that although the system is not asymptotically stabilizable to a given configuration using a time-invariant continuous feedback, it is strongly accessible and small-time locally controllable at any equilibrium. We construct a time-invariant discontinuous feedback law to control the system to the desired configuration with exponential convergence rates. A simulation example is included to demonstrate the results