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This paper presents a control law for the tracking of a cyclic reference path by an under-actuated biped robot. The robot studied is a five-link planar biped. 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 biped configuration is controlled, but not the temporal evolution. To achieve this objective, we consider a parametrized control. When a joint path is given, a five degree of freedom biped in single support becomes similar to a one degree of freedom inverted pendulum. The temporal evolution during the geometric tracking is completely defined and can be analyzed through the study of a model with one degree of freedom. Simple analytical conditions, which guarantee the existence of a cyclic motion and the convergence towards this motion, are deduced. These conditions are defined on the reference trajectory path. The analytical considerations are illustrated with some simulation results.

This article describes a walking vehicle with two telescopic legs that was designed at the Institute of Mechanics at Moscow University. The vehicle is driven by two DC drives. Its con trol system comprises a computer, hardware servosystems, and power amplifiers. An adaptive algorithm has been developed for control of vehicle locomotion. This algorithm is based on the tracking of the commanded path. A part of the path is planned by the computer in advance using a mathematical model of the vehicle. The rest is built up during walking of the vehicle based on the information on its state. In the ex periments, the designed control algorithm was successfully implemented.

On the one hand, impulsive torque at the beginning and end of the motion and ballistic motion between these two instants give energetically optimal motion for a pendulum. On the other hand many authors suppose that in human walk, muscle activities alternate with some periods of relaxation. From these two arguments, we assume that a motion defined by ballistic motion and impulsive control at double support instant will lead to an energetically economical trajectory.Optimal configurations are defined. Then smooth changes on the reference trajectory are proposed to obtain a trajectory which can be followed with finite torques. The physical constraints on the reaction forces to avoid take off or sliding of the robot, and saturation on the torques produced by the actuators are explicitly taken into account.

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