Decoupled limbs yield differentiable trajectory outcomes through intermittent contact in locomotion and manipulation

Abstract

When limbs are decoupled, we find that trajectory outcomes in mechanical systems subject to unilateral constraints vary differentiably with respect to initial conditions, even as the contact mode sequence varies. OrganizationWe begin in Sec. 2 by specifying the class of dynamical systems under consideration, namely, mechanical systems subject to unilateral constraints. Sec. 3 imposes conditions on the system dynamics and trajectories that enable us in Sec. 4 to report that trajectories vary differentiably with respect to initial conditions, even as the contact mode sequence varies. d k=1 (D k M m + D m M k − D M km ) [JK13, Eqn. 30]. 3 We let 2 n = {J ⊂ {1, . . . , n}} denote the power set (i.e. the set containing all subsets) of {1, . . . , n}.

Connecting Gaits in Energetically Conservative Legged Systems

“…Without A3, a Saltation matrix [18] may become discontinuous [17], which directly propagates to discontinuities in Φ T and the associated tangent space. For legged systems, continuity in the Saltation matrix can be ensured under certain conditions [24]. Turning these conditions into systematic modeling guidelines, or finding ways to connect gaits despite these discontinuities are avenues for future work.…”

“…during flight and stance, respectively. The constraint (23) fixes the leg length to l o during flight, whereas equation (24) implements the assumption of no sliding during stance (with a horizontal contact point position x c ). For the continuous dynamics in equations (1), we have…”

Nonsmooth Optimal Value and Policy Functions in Mechanical Systems Subject to Unilateral Constraints

“…We emphasize that the nonsmoothness in Fig. 4 arises from the nonsmoothness in the underlying system dynamics (1), as the functions in (14) and (15) are smooth.…”

“…The seemingly benign equations in (1) can yield dynamics with a range of regularity properties. This issue has been investigated elsewhere [1], [14], [15]; here we focus specifically on how the design of a robot's mechanical and control systems affect (non)smoothness of trajectory outcomes.…”