Differentially Flat Design of Bipeds Ensuring Limit Cycles

For bipedal walking, a set of joint trajectories is acceptable as long as it satisfies certain overall motion requirements, such as: 1) it is repetitive (limit cycles); 2) it allows the foot to clear ground; and 3) it allows the biped to move forward. Since the actual trajectory followed by a biped is not as important, a biped having some unactuated joints can also meet these motion requirements. Furthermore, due to physical constraints, a biped cannot have an actuator between the foot and the ground. Hence, it is underactuated during the phase when the foot is rolling on the ground. Besides underactuation, a bipedal robot has nonlinear dynamics and impacts. In general, it is difficult to prove existence of limit cycles for such systems. In this paper, a design methodology that renders planar bipedal robots differentially flat is presented. Differential flatness allows generation of parameterized limit cycles for this class of planar nonlinear underactuated bipeds. Sequential quadratic-programming-based numerical optimization routines are used to optimize these limit cycles while satisfying the motion constraints. The planning and control methodology is illustrated by a two-link biped.