Stable Walking Gaits for a Three-Link Planar Biped Robot With One Actuator

We consider a benchmark example of a three-link planar biped walker with torso, which is actuated in between the legs. The torso is thought to be kept upright by two identical torsional springs. The mathematical model reflects a three-degree-of-freedom mechanical system with impulse effects, which describe the impacts of the swing leg with the ground, and the aim is to induce stable limit-cycle walking on level ground. The main contribution is a novel systematic trajectory planning procedure for solving the problem of gait synthesis. The key idea is to find a system of ordinary differential equations for the functions describing a synchronization pattern for the time evolution of the generalized coordinates along a periodic motion. These functions, which are known as virtual holonomic constraints, are also used to compute an impulsive linear system that approximates the time evolution of the subset of coordinates that are transverse to the orbit of the continuous part of the periodic solution. This auxiliary system, which is known as transverse linearization, is used to design a nonlinear exponentially orbitally stabilizing feedback controller. The performance of the closed-loop system and its robustness with respect to various perturbations and uncertainties are illustrated via numerical simulations.