Motion planning in cluttered environments is a weakness of current robotic technology. While research addressing this issue has been conducted, few efforts have attempted to use minimum distance rates of change in motion planning. Geometric influence coefficients provide extraordinary insight into the interactions between a robot and its environment. They isolate the geometry of distance functions from system inputs and make the higher-order properties of minimum distance magnitudes directly available. Knowledge of the higher order properties of minimum distance magnitudes can be used to predict the future obstacle avoidance, path planning, and/or target acquisition state of a manipulator system and aid in making intelligent motion planning decisions. Here, first and second order geometric influence coefficients for minimum distance magnitudes are rigorously developed for several simple modeling primitives. A general method for similar derivations using new primitives and an evaluation of finite difference approximations versus analytical second order coefficient calculations are presented. Two application examples demonstrate the utility of minimum distance magnitude influence coefficients in motion planning.

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