A practical exact motion planning algorithm for polygonal objects amidst polygonal obstacles
Let I be a 2-dimensional polygonal rigid object (with m edges) moving amidst polygonal obstacles E (with n edges) and let P init and P end be two free placements of I, where the interior of I does not intersect E. We investigate here the problem of finding a continuous motion of I from P init to P end , such that during this motion the interior of I does not intersect E, or to establish that no such motion exists. This problem is an instance of the well known ”Piano Movers' Problem”. We have shown in  that it is possible to compute an exact description of free space in time O(m3n3log(mn)). We show in this paper that, using this description, a motion can be found in time O(m3n3). The actual complexity of our algorithm inmany practical situations is much smaller. In particular, for the so called situation of local bounded complexity often encontered in robotics, the complexity of computing free space is O(nlogn) and the complexity of planning a motion is O(n). The method has been implemented and experimental results are discussed.
Unable to display preview. Download preview PDF.
- AVNAIM F., BOISSONNAT J.D., Simultaneous containment of several polygons, 3rd ACM Symp. on Computational Geometry, Waterloo (June 1987).Google Scholar
- AVNAIM F., BOISSONNAT J.D., Polygon placement under translation and rotation, LNCS N. 294 Springer-Verlag pp.322–333 (1988). To appear also in RAIRO Informatique théorique et applications 1988.Google Scholar
- BROOKS R.A., Solving the find-path problem by good representation of free space, IEEE Trans. on Systems, Man and Cybernetics, Vol. SMC-13 pp.190–197, (March–April 1983).Google Scholar
- CHANDERJIT BAJAJ, MOH T.T., Generalized unfoldings for shortest paths. The international journal of Robotics Research vol. 7 number 1 ISSN 0278-3649 MIT press (Feb. 1988).Google Scholar
- FAVERJON B., Obstacle avoidance using an octree. In Proceedings of IEEE Int. Conference on Robotics and Automation (March 1988).Google Scholar
- FREDMAN M., TARJAN R.E., Fibonacci heaps and their uses uin improved network optimization problems. In Proc. 25th IEEE FOCS, pp.338–346, (1984).Google Scholar
- KEDEM K., SHARIR M., An efficient motion planning algorithm for a convex polygonal object in 2-dimensional polygonal space, Tech. Rept. No 253, Comp. Sci. Dept., Courant Institute, (Oct. 1986).Google Scholar
- LOZANO-PEREZ T., BROOKS R.A., A subdivision algorithm in config uration space for findpath with rotation, IEEE Trans. on Systems, Man and Cybernetics, Vol. SMC-15 No 2, pp.224–233 (1985).Google Scholar
- POLLACK R., SHARIR M., SIFRONY S., Separating two simple polygons by a sequence of translations, Discrete and Computational Geometry 3:pp.123–136 (1988).Google Scholar
- PREPARATA F.P., SHAMOS M.I., Computational geometry: an introduction, Springer-Verlag, (1985).Google Scholar
- SCHWARTZ J.T., SHARIR M., On the piano movers' problem: I. The special case of rigid polygonal body moving amidst polygonal barriers, Comm. Pure Appl. Math., Vol. XXXVI, pp.345–398 (1983).Google Scholar
- SIFRONY S., SHARIR M., A New Efficient Motion Planning Algorithm for a Rod in Two-Dimensional Polygonal Space, Algorithmica 2, pp. 367–402 (1987).Google Scholar