An improved Udwadia–Kalaba approach for controller design in underactuated mechanical systems

This paper further develops the Udwadia–Kalaba-approach-based view for the study of the controller design of underactuated systems. A challenge issue of the controller design for such complex systems is to implement an effective control input due to the non-full-rank feature of the control force configuration space. It becomes more difficult especially for the situation in which the control constraints are, in general, incompatible with the modeling constraints. In this paper, the modeling constraints are further divided into the natural and underactuated constraints, which can well capture the proper physical descriptions of underactuated systems. The control input that minimizes the control error and cost function can be derived by matrix operations, and then an additional constraint will be designed fully to address the incompatibility between the modeling and control constraints. This allowed us to develop an approach with precise effectiveness, high stability, and good robustness, which is applicable for various typical cases of complex underactuated systems. Finally, several representative numerical examples, including the fixed-point stabilization and trajectory tracking of a mobile robot, and the trajectory tracking of a hovercraft, are presented to demonstrate the proposed method.