On computability of fine motion plans

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Abstract
It is shown that fine motion plans in the LMT framework developed by T. Lozano-Perez, M. Mason and R. Taylor (1984) are computable, and an algorithm for computing them by reducing fine motion planing to an algebraic decision problem is presented. Fine-motion planning involves planning a successful motion of a robot at the fine scale of assembly operations, where control and sensor uncertainty are significant. It is shown that, as long as the envelope of trajectories generated by the control system can be described algebraically, there is an effective procedure for deciding if a successful n-step plan exists. The proposed method makes use of recognizable sets as subgoals for multistep planning. These sets are finitely parameterizable, and it is shown that they are the only sets that need be considered as subgoals. Unfortunately, if the full generality of the LMT framework is used, finding a fine-motion plan can take time double exponential in the number of plant steps.