On the Synthesis of Stable Walkover Gaits for the Acrobot

We introduce a novel method of producing stable walking gaits for the acrobot using virtual holonomic constraints (VHCs). Using this method, we produce a new stable walking motion for the acrobot, the walkover gait. In this gait, the swing leg rotates counterclockwise up and over the stance leg, as opposed to the standard compass gait where the swing leg rotates clockwise and overlaps with the stance leg partway along the motion. The walkover gait is found by searching for a VHC enjoying certain properties, among them the requirement of producing a stable hybrid limit cycle corresponding to walking. Key to the proposed approach is the recently developed notion of the virtual constraint generator (VCG), a control system on the configuration manifold of the robot whose solutions are all possible VHCs up to reparametrization. A systematic procedure is presented for the synthesis of VHCs achieving the desired walking motion for the acrobot, culminating in an optimal control problem for the VCG, solved numerically to produce the gait. Theoretical characterizations are given for the feasibility of the gait generation problem.