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Minkowski sum used for Robot Motion Planning The minkowski sum of the robot (yellow, upper third) and the obstacles in an environment (green) described as polygons is the forbidden area for the robot with respect to some point within it.In the above example, the robot can just barely find its way out to the left of the rightmost column of obstacles.

Minkowski Sums

Abstract

Several algorithms for computing the Minkowski sum of two polygons in the plane begin by decomposing each polygon into convex subpolygons. We examine different methods for decomposing polygons by their suitability for efficient construction of Minkowski sums. We study and experiment with various well-known decompositions as well as with several new decomposition schemes. We report on our experiments with the various decompositions and different input polygons. Among our findings are that in general: (i) triangulations are too costly (although they can be produced quickly, they considerably slow down the Minkowski-sum computation), (ii) what constitutes a good decomposition for one of the input polygons depends on the other input polygon---consequently, we develop a procedure for simultaneously decomposing the two polygons such that a ``mixed'' objective function is minimized, (iii) there are optimal decomposition algorithms that significantly expedite the Minkowski-sum computation, but the decomposition itself is expensive to compute --- in such cases simple heuristics that approximate the optimal decomposition perform very well.

Links

• Eyal Flato and Dan Halperin
  Robust and Efficient Construction of Planar Minkowski Sums
  Abstracts 16th European Workshop Comput. Geom. (CG 2000),
  Eilat, Ben-Gurion University of the Negev, 2000, pp 85-88.
  [BibTex,gzipped PostScript]
  
• Eyal Flato's Master thesis
• Pankaj Agarwal, Dan Halperin, and Eyal Flato.
  Polygon Decomposition for Efficient Construction of Planar Minkowski Sums
  Computational Geometry: Theory and Applications, vol. 21, 2002, pp 39-61.
  A preliminary version appeared in Proc. 8th European Symposium on Algorithms (ESA), Saarbrucken, 2000. Springer LNCS Vol. 1879, 20-31.
  [BibTex,gzipped PostScript]
• Eyal Flato, E. Fogel, Dan Halperin, and E. Leiserowitz
  Exact Minkowski Sums and Applications
  Proc. 18th ACM Symposium on Computational Geometry, Barcelona, 2002, pp 273-274. The Video [BibTex,mpg format, 114MB]

Contact

Eyal Flato