Bipeds’ trajectories and control during walking are closely coupled with the contact force distribution in time and space as an indeterminate problem. Therefore, generating the motion of redundant bipeds in presence of unilateral contact is usually formulated as a nonlinear constrained optimization problem. The optimal walking motion must be solved in terms of trajectories, control, contact status (i.e., when, where, and whether a foot is in contact), and contact response (i.e., ground reaction forces). The solution for this problem requires predictive methods within the general optimal motion planning framework. However, there is a lack of fully predictive methods that can concurrently solve for all the above mentioned unknowns. This represents an important challenge in the simulation, design, analysis, and control of general robotic systems. A novel approach for the optimal motion planning of multibody systems with contacts is developed, based on a Sequential Quadratic Programming (SQP) algorithm for Nonlinear Programming (NLP). The complete formulation is presented and demonstrated with numerical experiments on a simple planar biped with the assigned task of one complete step motion in forward progression.

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