Rank deficiency and superstability of hybrid systems with application to bipedal robots

The objectives of this paper are to study the rank properties of flows of hybrid systems, show that they are fundamentally different from those of smooth dynamical systems, and consider applications that emphasize the importance of these differences. In contrast with smooth dynamical systems, the rank of a solution to a hybrid system, a hybrid execution, is always less than the dimension of the space on which it evolves and falls within easily-computed and possibly distinct upper and lower bounds. Our main contribution is the derivation of conditions for when an execution fails to have maximal rank, i.e., when it is rank deficient. Given the importance of periodic behavior in many hybrid systems applications, for example in bipedal robots, these rank deficiency conditions are applied to the special case of periodic hybrid executions. Our secondary contribution is the derivation of superstability conditions for when a periodic execution has rank equal to 0 and is therefore completely insensitive to perturbations in initial conditions. The results are illustrated in application to a planar kneed biped.