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## CS B659: Principles of Intelligent Robot MotionCollision DetectionProbabilistic

Parisi, Jim, Morning Drive Host has reference to this Academic Journal, PHwiki organized this Journal CS B659: Principles of Intelligent Robot MotionCollision DetectionProbabilistic RoadmapsHow to test as long as collisionCollision Detection MethodsMany different methodsIn particular: Grid method: good as long as many simple moving objects of about the same size (e.g., many moving discs with similar radii)Closest-feature tracking: good as long as moving polyhedral objectsBounding Volume Hierarchy (BVH) method: good as long as few moving objects with complex in addition to diverse geometry

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Grid MethodSubdivide space into a regular grid cubic of square binsIndex each object in a binGrid MethodRunning time is proportional to number of moving objectsUseful also to compute pairs of objects within some distance (vision, sound, )Closest-Feature Tracking (M. Lin in addition to J. Canny. A Fast Algorithm as long as Incremental Distance Calculation. Proc. IEEE Int. Conf. on Robotics in addition to Automation, 1991) The closest pair of features (vertex, edge, face) between two polyhedral objects are computed at the start configurations of the objectsDuring motion, at each small increment of the motion, they are updated Efficiency derives from two observations:The pair of closest features changes relatively infrequentlyWhen it changes the new closest features will usually be on a boundary of the previous closest features

Closest-Feature Test as long as Vertex-VertexVertexVertexApplication: Detecting Self-Collision in Humanoid Robots (J. Kuffner et al. Self-Collision in addition to Prevention as long as Humanoid Robots. Proc. IEEE Int. Conf. on Robotics in addition to Automation, 2002)BVH with spheres: S. Quinlan. Efficient Distance Computation Between Non-Convex Objects. Proc. IEEE Int. Conf. on Robotics in addition to Automation, 1994. BVH with Oriented Bounding Boxes: S. Gottschalk, M. Lin, in addition to D. Manocha. OBB-Tree: A Hierarchical Structure as long as Rapid Interference Detection. Proc. ACM SIGGRAPH ’96, 1996. Combination of BVH in addition to feature-tracking: S.A. Ehmann in addition to M.C. Lin. Accurate in addition to Fast Proximity Queries Between Polyhedra Using Convex Surface Decomposition. Proc. 2001 Eurographics, Vol. 20, No. 3, pp. 500-510, 2001.Adaptive bisection in dynamic collision checking: F. Schwarzer, M. Saha, J.C. Latombe. Adaptive Dynamic Collision Checking as long as Single in addition to Multiple Articulated Robots in Complex Environments, manuscript, 2003.Bounding Volume Hierarchy Method

Enclose objects into bounding volumes (spheres or boxes) Check the bounding volumes first Decompose an object into twoBounding Volume Hierarchy Method Enclose objects into bounding volumes (spheres or boxes) Check the bounding volumes first Decompose an object into two Proceed hierarchicallyBounding Volume Hierarchy Method Enclose objects into bounding volumes (spheres or boxes) Check the bounding volumes first Decompose an object into two Proceed hierarchicallyBounding Volume Hierarchy Method

BVH is pre-computed as long as each objectBounding Volume Hierarchy MethodBVH in 3DCollision DetectionTwo objects described by their precomputed BVHs

Collision DetectionAASearch treeAACollision DetectionAASearch treeAACollision DetectionCCCBBCBBAASearch tree

Collision DetectionCCCBBCBBAASearch treeGDVariantSearch treeAACollision DetectionPruning discards subsets of the two objects that are separated by the BVsEach path is followed until pruning or until two leaves overlapWhen two leaves overlap, their contents are tested as long as overlap

Search Strategy in addition to HeuristicsIf there is no collision, all paths must eventually be followed down to pruning or a leaf nodeBut if there is collision, it is desirable to detect it as quickly as possible Greedy best-first search strategy with f(N) = d/(rX+rY) [Exp in addition to the node XY with largest relative overlap (most likely to contain a collision)]Recursive (Depth-First) Collision Detection AlgorithmTest(A,B)If A in addition to B do not overlap, then return 1If A in addition to B are both leaves, then return 0 if their contents overlap in addition to 1 otherwiseSwitch A in addition to B if A is a leaf, or if B is bigger in addition to not a leafSet A1 in addition to A2 to be As childrenIf Test(A1,B) = 1 then return Test(A2,B) else return 0Per as long as manceSeveral thous in addition to collision checks per second as long as 2 three-dimensional objects each described by 500,000 triangles, on a 1-GHz PC

Distance ComputationGreedy Distance ComputationGreedy-Distance(A,B,M)If min-dist(A,B) > M, then return MIf A in addition to B are both leaves, then return distance between their contents Switch A in addition to B if A is a leaf, or if B is bigger in addition to not a leafSet A1 in addition to A2 to be As childrenM min(max-dist(A1,B), max-dist(A2,B), M)d1 Greedy-Distance(A1,B,M) d2 Greedy-Distance(A2,B,M) Return Min(d1,d2)M (upper bound on distance) is initialized to infinityApproximate DistanceApprox-Greedy-Distance(A,B,M,a)If (1+a)min-dist(A,B) > M, then return MIf A in addition to B are both leaves, then return distance between their contents Switch A in addition to B if A is a leaf, or if B is bigger in addition to not a leafSet A1 in addition to A2 to be As childrenM min(max-dist(A1,B), max-dist(A2,B), M)d1 Approx-Greedy-Distance(A1,B,M,a) d2 Approx-Greedy-Distance(A2,B,M,a) Return Min(d1,d2)M (upper bound on distance) is initialized to infinity

Desirable Properties of BVs in addition to BVHsBVs:TightnessEfficient testingInvarianceBVH: Separation Balanced tree SpheresInvariantEfficient to testBut tightAxis-Aligned Bounding Box (AABB)

Generalized Bisection MethodUntil Q is not empty do:[qa,qb]ij remove-first(Q)If li(qa,qb) + lj(qa,qb) dij(qa) + dij(qb) thenqmid (qa+qb)/2If dij(qmid) = 0 then return collisionElse insert [qa,qmid]ij in addition to [qmid,qb]ij into QReturn no collision Each pair of bodies is checked independently of the others priority queue Q of elements [qa,qb]ij Initially, Q consists of [q,q]ij as long as all pairs of bodies Ai in addition to Aj that need to be tested.Heuristic Ordering QGoal: Discover collision quicker if there is one. Sort Q by decreasing values of: [li(qa,qb) + lj(qa,qb)] [dij(qa) + dij(qb)]Possible extension to multi-segment paths (very useful with lazy collision-checking PRM)

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