Rank properties of poincare maps for hybrid systems with applications to bipedal walking

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

“…This is easily confirmed by computing Dφ t (x) = D x φ t (x) = Φ(t), which is nonsingular. The total derivative of the flow to a section φ τ : U 0 → V ⊂ S, on the other hand, is [4], [7], [8] …”

“…where x 0 ∈ U 0 , x 1 = φ τ (x 0 ), Id n is the n × n identity matrix and h defines the local section S. It was shown in [4] that the rank of (2) is equal to the dimension of the local section S, and that flows to sections satisfy: (S1) For any local section S of c(t 0 ) there exists a sufficiently small neighborhood U 0 of c(t 0 ) such that φ τ (U 0 ) ⊂ S. (S2) By Theorem 1 and Corollary 1 of [4], there exists a local…”

“…The following result is the extension of Theorem 4 in [4] to arbitrary, nonperiodic hybrid systems and executions, and so we omit the proof for reasons of space.…”

“…For example, the existence and uniqueness properties of solutions of hybrid systemscalled hybrid executions -are not the same as for smooth systems [1], [2]; therefore, one may not regard the stability of hybrid system equilibria in the same way as the stability of smooth system equilibria [3]. Recent work [4] has also shown that Poincaré maps for hybrid systems are fundamentally different from Poincaré maps for smooth systems.…”

“…The first contribution of the present work is the extension of the results in [4] to arbitrary, non-periodic hybrid executions. In particular, we show that the rank of an execution will always fall between possibly distinct upper and lower bounds, and that the upper bound is always less than the dimension of the space on which the execution evolves.…”