Time-scaling control for an underactuated biped robot

This paper presents a control law for the tracking of a cyclic reference trajectory by an under-actuated biped robot. The robot studied is a five-link planar robot. The degree of under-actuation is one during the single support phase. The control law is defined in such a way that only the geometric evolution of the robot is controlled, not the temporal evolution. To achieve this objective, we consider a time scaling control. But, for a given reference path, the temporal evolution during the geometric tracking is completely defined and can be analyzed through the study of the dynamic model. A simple analytical condition is deduced that guarantees convergence towards the cyclic reference trajectory. This condition implies temporal convergence after the geometric convergence. This condition is defined on the cyclic reference path. The control law is stable if the angular momentum around the contact point is greater at the end of the single support phase than at the beginning of the single support phase.