Motion planning for variable inertia mechanical systems

In this paper, we generate gaits for mixed systems, that is, dynamic systems that are subject to a set of non-holonomic constraints. What is unique about mixed systems is that when we express their dynamics in body coordinates, the motion of these systems can be attributed to two decoupled terms: the geometric and dynamic phase shifts. In our prior work, we analyzed systems whose dynamic phase shift was null by definition. Purely mechanical and principally kinematic systems are two classes of mechanical systems that have this property. We generated gaits for these two classes of systems by intuitively evaluating their geometric phase shift and relating it to a volume integral under well-defined height functions. One of the contributions of this paper is to present a similar intuitive approach for computing the dynamic phase shift. We achieve this, by introducing a new scaled momentum variable that not only simplifies the momentum evolution equation but also allows us to introduce a new set of well-defined gamma functions which enable us to intuitively evaluate the dynamic phase shift. More specifically, by analyzing these novel gamma functions in a similar way to how we analyzed height functions, and by analyzing the sign-definiteness of the scaled momentum variable, we are able to ensure that the dynamic phase shift is non-zero solely along the desired fiber direction. Finally, we also introduce a novel mechanical system, the variable inertia snakeboard, which is a generalization of the original snakeboard that was previously studied in the literature. Not only does this general system help us identify regions of the base space where we can not define a certain type of gaits, but also it helps us verify the generality and applicability of our gait generation approach.