Invariant Manifold of Symmetric Orbits for Globally Optimal Biped Locomotion

This paper proposes novel walking gait generation for a double-pendulum-like biped walking model based on the family of the symmetric orbits. By introducing an involution R associated with the leg switching map, the flow of the saddle-center dynamics of the uncontrolled pendulum possesses time-reversal symmetry with respect to R, and the conjunction of the flow and R forms a family of symmetric orbits parameterized by the orbital energy and the stride. The invariant manifold of the family of symmetric orbits is obtained numerically or approximately using a perturbation method. The proposed methods are evaluated on numerical simulations. By constraining the solution onto the invariant manifold using the control inputs, stable walking gaits are generated in a semi-global manner in simulations. Based on the passivity of the closed-loop system, a robust speed-controlled walking is achieved in a very simple way.