Uniform ultimate boundedness for underactuated mechanical systems as mismatched uncertainty disappeared

We propose to design control for uncertain underactuated mechanical systems. The underactuated mechanical system is to follow prescribed holonomic or nonholonomic constraints. The uncertainty in the system does not in general fall within the range space of the input matrix, which is a major obstacle for control design. To resolve this difficulty, we decompose the uncertainty into matched uncertainty and mismatched uncertainty in a unique manner using the geometric structural characteristics of the system. A control scheme is designed to guarantee uniform boundedness and uniform ultimate boundedness of a constraint-following performance measure. The control is based solely on the matched uncertainty. The mismatched uncertainty turns out to be disappeared, as far as the performance analysis is concerned, since it is orthogonal to the geometric space of interest. For demonstrations, a vehicle/inverted pendulum platform is selected. We charge the system to follow either holonomic or nonholonomic constraint. The simulation shows the system performance, in following the prescribed constraint, is superior.