Biped Without Feet in Single Support: Stabilization of the Vertical Posture with Internal Torques

We consider a two-link biped, a three-link biped, and a five-link planar biped without feet (with “point feet”). Their ankles are not actuated. The control torques are applied to the bipeds in the inter-link joints only. Then this family of bipeds is under actuated when only one leg tip touches the ground. It is difficult to control the walking of this kind of bipeds because they are statically unstable in single support. For example the vertical posture of these bipeds in single support is an unstable equilibrium state, as an equilibrium state of inverted pendulum. The operations of stabilization for the biped vertical posture, of balancing around this equilibrium posture, using only the inter-link torques, are also difficult. These problems are interesting from the point of view of dynamical stabilization of walking for bipeds with motions in saggital plane or (and) in frontal plane. They are also interesting from the biomechanical point of view. In the chapter, the problem of stabilization of the vertical posture for each mentioned above biped is studied. For each biped, a control law to stabilize the vertical posture is designed. Among the mechanical systems, the under actuated systems, which have fewer controls than configuration variables, represent a great challenge for the control. An active field of research exists, due to the applications of under actuated systems such as aircrafts, satellites, spacecrafts, flexible robots, inverted pendulums, legged robots. The under actuated systems are characterized by the under-actuation degree, which is the difference between the numbers of configuration variables and controls. The under-actuation degree for all our studied bipeds equals one in single support. The control laws to stabilize the vertical posture are designed, using the biped linear models and their associated Jordan forms. The feedback is synthesized to suppress the unstable modes. The restrictions imposed to the torques are taken into account explicitly. Thus, feedback control laws with saturation are designed. It is important for an unstable system to maximize the basin of attraction. Using the Jordan form to design the control law, we can obtain a large basin of attraction of equilibrium state.