In this paper, we generate gaits for mixed systems, that is, dynamic systems that are subject to a set of nonholonomic
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 variableinertia 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.
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