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T. Marcucci, M. Gabiccini, and A. Artoni. "A Two-Stage Trajectory Optimization Strategy for Articulated Bodies With Unscheduled Contact Sequences." IEEE Robotics and Automation Letters 2.1 (2017): 104-111.

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In this paper, we propose a two-stage strategy for optimal control problems of robotic mechanical systems that proves to be more robust, and yet more ecient, than straightforward solution strategies. Specifically, we focus on a simplified humanoid model, represented as a two-dimensional articulated serial chain of rigid bodies, in the tasks of getting up (sitting down) from (to) the supine and prone postures. Interactions with the environment are integral parts of these motions, and a priori unscheduled contact sequences are discovered by the solver itself, opportunistically making or breaking contacts with the ground through feet, knees, hips, elbows, and hands. The present investigation analyzes the e ects on the computational performance of: (i) the explicit introduction of contact forces among the optimization variables, (ii) the substitution of undesired contact forces with geometric constraints that prevent interpenetrations, and (iii) the splitting of the planning problem
into two consecutive phases of increasing complexity. To the best of our knowledge, these tests represent the only quantitative analysis of the performances achievable with di fferent solution strategies for optimization-based, whole-body dynamic motion planning in the presence of contacts.