We present kinetic data structures for detecting collisions between a set of polygons that are moving continuously. Unlike classical collision detection methods that rely on bounding volume hierarchies, our method is based on deformable tilings of the free space surrounding the polygons. The basic shape of our tiles is that of a pseudo-triangle, a shape sufficiently flexible to allow extensive deformation, yet structured enough to make detection of self-collisions easy. We show different schemes for maintaining pseudo-triangulations as a kinetic data structure, and we analyze their performance. Specifically, we first describe an algorithm for maintaining a pseudo-triangulation of a point set, and show that the pseudo-triangulation changes only quadratically many times if points move along algebraic arcs of constant degree. In addition, by refining the pseudo-triangulation, we show triangulations of points that only change about O(n 7 / 3 ) times for linear motion. We then describe an algorithm for maintaining a pseudo-triangulation of a set of convex polygons. Finally, we extend our algorithm to the general case of maintaining a pseudo-triangulation of a set of moving or deforming simple polygons.
A kinetic data structure for the maintenance of a multidimensional range search tree is introduced.This structure is used as a building block to obtain kinetic data structures for two classical geometric proximity problems in arbitrary dlmensions: the first structure maintains the closest pair of a set of continuously moving points, and is provably efficient. The second structure maintains a spanning tree of the moving points whose cost remains within some prescribed factor of the minimum spanning tree.
We design a kinetic data structure for detecting collisions between two simple polygons in motion. In order to do so, we create a planar subdivision of the free space between the two polygons, called the external relative geodesic triangulation, which certifies their disjointness. We show how this subdivision can be maintained as a kinetic data structure when the polygons are moving, and analyze its performance in the kinetic setting.
Probcibilistic road-map (PRM) planners have showngreat promise in attacking previously infeasible motion planning problems with many degrees of freedom. Yet when such ci plnnner fails to jind a path, it is not clear that no path exists, or that the planner simply did not scirnple adequately or intelligently the free part of the con$guration space. We propose to attack the motion planning problem from the other end, focusing on disconnection proofs, or proofs showing that there exists no solution to the posed motion planning problem. Just as PRM planners civoid generating a complete description of the configuration space, our disconnection provers search for certcrin special classes ofproofs that are compact and easy to j n d when the motion planning problem is 'obviously impossible,' avoiding complex geometric and combinatorid ccrlculations. We demonstrate such a prover in artion for a simple, yet still realistic, motion planning problem. When it fails, the prover suggests key milestones, or conjgurations of the robot that can then be passed on and used by a PRMplanner: Thus by hitting the motion pleinning problem from both ends, we hope to resolve the existence of a path, except in truly delicate border-line situations.
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