Optimal trajectories for a quadruped robot with Trot, Amble and Curvet gaits for two energetic criteria

In this paper, optimal cyclic reference trajectories are designed for three gaits of a quadruped robot, the curvet, the amble, and the trot, taking into account the actuators characteristics. The gaits are composed of stance phases and instantaneous double supports. The principle of virtual leg is used to obtain simpler dynamic model describing the motion of the quadruped. The impact phases are modeled by passive impact equations. For the curvet the step is composed of two different half steps. For the amble and trot gaits two following half steps are symmetrical. The optimization problem is solved with an algebraic optimization technique. The actuated joint evolution is chosen as a polynomial function of time. The coefficients of the polynomial functions are optimization parameters. The quadruped studied has non-actuated ankles. The kinetic momentum theorem permits to define the evolution of this non-actuated variable in function of the actuated variables. Two energetic criteria are defined: a torque cost and an energetic cost. The first is represented by the integral of the torque norm and the second by the absolute value integral of the external forces work. The two criteria are calculated for a displacement of one meter. During the optimization process, the constraints on the ground reactions, on the validity of impact, on the torques, on the joints velocities and on the motion velocity of the robot prototype are taken into account. Simulation results are presented for the three gaits. All motions are realistic. Curvet is the less efficient gait with respect to the criteria studied. For slow motion, trot is the more efficient gait. But amble permits the fastest motion with the same actuators.