Control of the Compass-Gait Walker Using an Enhanced Poincaré Map and via LMI-Based Optimization

In this paper, we introduce a control approach using a quadratic polynomial expression of the controlled Poincaré Map to actively stabilize the passive walking motion of the two-degree-of-freedom compass-gait biped walker. The passive gait cycle of the bipedal walker is depicted for a given fixed point. The control of the passive gaits involves firstly the reconstruction of the nonlinear complex dynamics describing the passive bipedal walking into an amendable linear system around the period-1 limit cycle. It involves secondly the determination of the quadratic polynomial expression of the nonlinear controlled Poincaré Map, and finally the identification of its period-1 fixed point. Successively, to stabilize such fixed point, we develop the linearized Poincaré Map, which will be explored to design the feedback gain of the control law. The control problem is cast then into a convex optimization involving a linear matrix inequality (LMI) by maximizing the bound on the nonlinear term in the Poincaré map. Simulation outputs illustrate the efficiency of the adopted LMI-based optimization method in the control of the passive motion of the compass-gait walker.