Motion Planning in the Presence of Moving Obstacles

This paper investigates the computational complexity of planning the motion of a body B in 2{D or 3{D space, so as to avoid collision with moving obstacles of known, easily computed, trajectories. Dynamic movement problems are of fundamental importance to robotics, but their computational complexity has not previously been investigated. We provide evidence that the 3-D dynamic movement problem is intractable even if B has only a constant number of degrees of freedom of movement. In particular, we prove the problem is PSPACE-hard if B is given a velocity modulus bound on its movements and is NP hard even if B has no velocity modulus bound, where in both cases B has 6 degrees of freedom. To prove these results we use a unique method of simulation of a Turing machine which uses time to encode con gurations (whereas previous lower bound proofs in robotic motion planning used the system position to encode con gurations and so required unbounded number of degrees of freedom). We also investigate a natural class of dynamic problems which we call asteroid avoidance problems: B, the object we wish to move, is a convex polyhedron 88{K{0458, DARPA/ARO Contract DAAL03{88{K{0195, and NASA/CESDIS Contract 550{63 NAS 5{30428 URSA. The work was completed while the rst author was on sabbatical at the Laboratory for Computer Science, MIT, Cambridge, MA. Work by the second author was supported in part by a grant from the U.S.-Israeli Binational Science Foundation, by the O ce of Naval Research Contracts N00014-82-K-0381, N00014-87-K-0129, and N00014-90-J-1284, and by National Science Foundation Grants DCR-83-20085 and CCR-89-01484. A preliminary version of the paper has appeared in the Proceedings of the 26th IEEE Symposium on Foundations of Computer Science, 1985.