Transitioning between underactuated periodic orbits: An optimal control approach for settling time reduction

In underactuated systems, a transition between two periodic orbits is generally characterized by slow convergence. This is due to the fact that the unactuated degree of freedom (DoF) hinders the state of the system to enter the domain of attraction of the target orbit close to the fixed point of the Poincaré Map. In this paper, we introduce an optimal control algorithm to reduce the settling time of transitions between periodic orbits of underactuated walking robots. This is achieved by utilizing the hybrid zero dynamics (HZD) framework to express the feasibility condition of the transition which can be imposed as an inequality constraint in the proposed optimal control problem. In addition, the cost function penalizes deviations from the fixed point of the target periodic orbit in the zero dynamics manifold while at the same time all dynamic and kinematic assumptions are treated as constraints. Furthermore, high magnitude torques are also penalized. The numerical results show that the proposed methodology can indeed improve the settling time compared to the transition methodology usually found in the bibliography and at the same time provide a feasible and smooth motion.