Introduction. Research on theoretical problems in robotics looks ahead to a future generation of robots substantially more autonomous than present robotic systems, whose algorithmic and software capabilities remain rather primitive. The capabilities which this research aims to create can be grouped into three broad categories: sensing, planning, and control. Of these three, planning involves the use of an environment model to carry out significant parts of a robot's activities automatically.

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... aim is to allow the robot's user to specify a desired activity in very high level, general terms, and then have the system fill in the missing low-level details. For example, the user might specify the end product of some assembly process and ask the system to construct a sequence of assembly substeps; or, at a less demanding level, to plan collision-free motions which pick up individual subparts of an object to be assembled, transport them to their assembly position, and insert them into their proper places. Studies in this area have shown it to have significant mathematical content; tools drawn from classical geometry, topology, algebraic geometry, algebra, and combinatorics have all proved relevant. This work relates closely to work in computational geometry, an area which has also progressed very rapidly during the last few years. Statement of the problem. In its simplest form, the motion planning problem can be defined as follows. Let B be a robot system consisting of a collection of rigid subparts (some of which may be attached to each other at certain joints, while others may move independently) having a total of fc degrees of freedom, and suppose that B is free to move in a two-or three-dimensional -spaee4^amidst-a=eolleetion=of=obstacle8=wh^ system. The motion planning problem for B is: Given an initial position Z\ and a desired final position Z<i of B, determine whether there exists a continuous