Simultaneous Leg Impacts Lead to a Differentiable Flow

Multi-contact dynamical systems are generally thought to have non-smooth properties. Surprisingly, recent work [arXiv:2102.10702] found that for a class of common mechanical systems, the flow will be continuously differentiable even at a multiple collision event with arbitrarily many contacts. We present experimental validation of this finding on a 3-legged hopping robot with springy legs by showing that all 6=3! hopping impact maps are indistinguishable from one another. Using motion capture data of foot position and body pose, we estimate derivative maps for each ordering of foot impacts with the ground. We then compare their pairwise distances on the Grassman Manifold. Finally, we compare prediction errors between the estimated maps. The results suggest that multi-contact locomotion may be easier to model than previously assumed.