Geometric Motion Planning Analysis for Two Classes of Underactuated Mechanical Systems

In this paper we generate gaits for two types of underactuated mechanical systems: principally kinematic and purely mechanical systems. Our goal is to specify inputs in the form of gaits, that is, a sequence of controlled shape changes of a multi-bodied mechanical system that when executed would produce a desired change in the unactuated position or orientation variables of the entire mechanical system. In other words, we want to indirectly control the unactuated degrees of freedom of the mechanical system utilizing a controlled “internal” shape change. More precisely, in this paper we develop a gait evaluation tool which easily measures the change of position, computed in a body-attached coordinate frame, due to any closed curve in the shape space. This evaluation tool is simple enough that we can use it to generate gaits or to design curves that move the mechanical system along a desired direction. Finally, we verify that this gait analysis technique applies to two seemingly different classes of mechanical systems, purely mechanical and principally kinematic systems, and unify the gait generation problem for both classes. KEY WORDS—gait generation, motion control, underactuated robots, non-holonomic motion planning, generalized momentum, purely mechanical systems, principally kinematic systems, Stokes’ theorem