Planning collision-free, dynamically-balanced movements for humanoid robots is a challenging problem. An effective approach consists of first planning a motion satisfying geometric and kinematic constraints (such as collision avoidance, joint angle limits, velocity limits, etc.) and, in a second stage, modifying this motion so that it respects dynamic balance criteria, such as those relative to the Zero Moment Point (ZMP). However, this approach currently suffers from the issue that the modified motion may give rise to new collisions with respect to the original motion, which can be very costly to deal with, especially for systems with many degrees of freedom and cluttered environments. Here we present an algorithm to modify the motions of humanoid robots under ZMP constraints without changing the original motion path, making thereby new collision checks unnecessary. We do so by adapting the minimum-time path parameterization under torque constraints algorithm of Bobrow et al. to the case of ZMP constraints. In contrast with a previous approach based on finite differences and iterative optimization to find the optimal path parameterization under ZMP constraints, our Bobrow-based algorithm finds this optimal parameterization in a single pass. We demonstrate the efficiency of this algorithm by simulations.
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