Optimal gait synthesis of a seven-link planar biped

In this paper, we carry out the dynamics-based optimization of sagittal gait cycles of a planar seven-link biped using the Pontryagin maximum principle. Special attention is devoted to the double-support phase of the gait, during which the movement is subjected to severe limiting conditions. In particular, due to the fact that the biped moves as a closed kinematic chain, overactuation must be compatible with double, non-sliding unilateral contacts with the supporting ground. The closed chain is considered as open at front foot level. A full set of joint coordinates is introduced to formulate a complete Hamiltonian dynamic model of the biped. Contact forces at the front foot are considered as additional control variables of the stated optimal control problem. This is restated as a state-unconstrained optimization problem which is finally recast, using the Pontryagin maximum principle, as a two-point boundary value problem. This final problem is solved using a standard computing code. A gait sequence, comprising starting, cyclic, and stopping steps, is generated in the form of a numerical simulation.