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In this paper, we explore the potential of trajectory optimization for unspecified contact sequences as a tool to identify optimal gaits and motions for legged robots. This work is based on a recently proposed method that states the mechanical dynamics in a floating base description, makes the ground contact forces part of the free variable vector, and implements the requirement that a foot is either on the ground or that the corresponding contact force is zero via a set of complimentary conditions. We introduce an algorithmic improvement that uses higher order integration for states that are continuous through collisions and thus increases the accuracy of the obtained solutions. The benefits of the proposed changes are evaluated with the models of a 1D hopper and a 2D bipedal robot, and we additionally compare our results with analytic solutions and an established multiple shooting implementation. The proposed method was able to automatically discover walking and running as the most energetically economic ways of locomotion for a conceptual biped that is moving at different speed. It additionally discovers an elastic walking gait that is used at intermediate velocities.
Xi et al. (Mon,) studied this question.