Generation of Leg-Like motion and Limit Cycles with an Underactuated Two DOF Linkage

Biped walking is an application where following a predefined trajectory is entirely not critical as long as the trajectory allows the foot to clear the ground and is periodic. Since a biped need not follow a trajectory exactly, instead of using a fully actuated leg, an underactuated leg might also serve the purpose. By suitable distribution of masses, the dynamics of the system can be made to be linear and by addition of springs the underactuated system can be made controllable. These design conditions make the system amenable to mathematical tools like differential flatness. The ability of such an underactuated system to generate leglike motion is investigated in this paper. A fully actuated system can be forced to execute any trajectory but for an underactuated system it is difficult to prove the existence of limit cycles analytically. Differential flatness allows generation of a family of limit cycles. From this family, one can further optimize a criterion by means of numerical optimization. The system studied is a two dof gravity balanced linkage. Each of the two joints has a torque spring attached to it. The spring at the knee joint ensures controllability of the system and the equilibrium position of the hip spring is taken as the input. The trajectories obtainable with and without the periodicity conditions are investigated. The energy consumption of this underactuated system is compared with the energy consumption of a fully actuated case. Planning is done using the differential flatness property of the system with numerical optimization as a tool to generate flat output trajectories. Such a strategy for generation of limit cycles and approximate trajectories can be generalized to other underactuated systems