Control and Stability Analysis of Limit Cycles in a Hopping Robot

A new stabilizing nonlinear controller for the vertical motion of an electrically actuated hopping robot is introduced and analyzed. The approach starts by finding reference limit cycles from the "passive dynamics" of a mass-spring system. The controller then modulates the system dynamics via the leg actuator during the stance phase to force the system trajectory to converge to this reference limit cycle. The controlled system dynamics is a continuous yet nonsmooth vector field. A piecewise-continuous Lyapunov function and the general forms of Lasalle\´s invariance and domain of attraction theorems are used to prove global asymptotic stability of the desired limit cycles. It is also shown that the derived passive dynamic cycles are indeed the positive limit sets of the controlled system. The rate of convergence can be adjusted but is limited by actuator constraints.