Finite time stabilization of a perturbed double integrator - Part II: applications to bipedal locomotion

Orbital stabilization of a bipedal robot. The robot walk is composed of single support phases separated by impacts. The underactuation degree of the robot is one during the single support phase. The generalized positions are assumed to be the only available measurements of the robot state. The proposed synthesis procedure is constituted by several successive designs which are developed step by step. At the first step, the generalized position θ of the virtual leg that would correspond to a three-link biped with no knees is viewed as a time substitution θ(t) and the desired path is obtained as a function of θ rather than that of time. Second, the underactuated orbital stabilization problem is reduced to stabilization of the geometric configuration evolution of the robot with a supplementary control input, being the second order time derivative θ̈ of the virtual leg position. The original orbital stabilization problem is thus decoupled to a path stabilization of five double integrators, controlled independently. At the third and fourth steps, the so-called twisting and supertwisting algorithms, which are well-recognized for their finite time stability and robustness properties, are modified to present, respectively, the state feedback controller and velocity observer, developed for the finite time stabilization of a double integrator. Finally, the resulting position feedback synthesis is composed to orbitally stabilize the five-link bipedal robot. Performance and robustness issues of the enforced biped cyclic gate are illustrated in a numerical study.