| Cruse's model of leg coordination (CCM) was derived to account for gaits and gait transitions in arthropods (analogous to, e.g. walk → trot → gallop in some quadrupeds). It has also been adapted to control locomotion in a series of hexapod robots. CCM is a systems-level, kinematic model that abstracts key physiological and dynamical properties in favor of tractability. A key feature is that gaits emerge from interaction among pairs of legs as effected by a set of coordination mechanisms acting at discrete points in time. We represent CCM networks as systems of coupled hybrid oscillators. Gaits are quantified by a temporal (discrete) phase vector. System trajectories are polyhedral, hence solvable over finite-time, but the presence of the switching automaton renders infinite horizon properties harder to analyze. Via numerical and symbolic simulations, we have mapped out the synchronization behavior of CCM networks of various topologies parametrically. We have developed a section-map analysis approach that exploits the polyhedral geometry of the hybrid state space. Our approach is constructive. Once switching boundaries are appropriately parameterized, we can extract periodic orbits, their domains of admissibility and stability, as well as expressions for the period of oscillation and relative phase of each cycle, parametrically. Applied to 2-oscillator networks, our approach yields excellent agreement with simulation results. A key emergent concept is that of a virtual periodic orbit (VPO). Distinguished from admissible periodic orbits, VPOs do not correspond to any in the underlying hybrid dynamics. However, when stable and close to being admissible, they are canonical precursors for a class of nonsmooth bifurcations and predictive of long transient behavior. Last, we take into consideration the possibility and difficulties of extending our approach to larger networks and to related oscillator-like hybrid dynamical systems with polyhedral trajectories. |
