Collision Detection. These slides are mainly from Ming Lin s course notes at UNC Chapel Hill

Transcription

1 Collision Detection These slides are mainly from Ming Lin s course notes at UNC Chapel Hill

2 Computer Animation ILE5030 Computer Animation and Special Effects 2

3 Haptic Interaction From J.C. Latombe s notes ILE5030 Computer Animation and Special Effects 3

4 Motion/Path Planning From J.C. Latombe From James Kuffner ILE5030 Computer Animation and Special Effects 4

5 Crowd Simulation From J.C. Latombe s notes ILE5030 Computer Animation and Special Effects 5

6 Geometric Proximity Queries Given two objects, how would you check: If they intersect with each other while moving? If they do not interpenetrate each other, how far are they apart? If they overlap, how much is the amount of penetration ILE5030 Computer Animation and Special Effects 6

7 Collision Detection Update configurations w/ TXF matrices Check for edge-edge intersection in 2D (Check for edge-face intersection in 3D) Check every point of A inside of B & every point of B inside of A Check for pair-wise edge-edge intersections Imagine larger input size: N = ILE5030 Computer Animation and Special Effects 7

8 Classes of Objects & Problems 2D vs. 3D Convex vs. Non-Convex Polygonal vs. Non-Polygonal Open surfaces vs. Closed volumes Geometric vs. Volumetric Rigid vs. Non-rigid (deformable/flexible) Pairwise vs. Multiple (N-Body) Discrete vs. Continuous ILE5030 Computer Animation and Special Effects 8

9 Some Possible Approaches Geometric methods Algebraic techniques Hierarchical Bounding Volumes Spatial Partitioning Others (e.g. optimization) ILE5030 Computer Animation and Special Effects 9

10 Methods for Convex Polytopes Voronoi Marching Linear programming Minkowski sums ILE5030 Computer Animation and Special Effects 10

11 Voronoi Diagram Describes the areas that are nearest to a set of given points Voronoi Site Voronoi Region ILE5030 Computer Animation and Special Effects 11

12 Generalized Voronoi Diagram Given a collection of geometric primitives, it is a subdivision of space into cells such that all points in a cell are closer to one primitive than to any other Voronoi Site Voronoi Region ILE5030 Computer Animation and Special Effects 12

13 Point sites Ordinary Nearest Euclidean distance Generalized Higher-order site geometry Varying distance metrics Higher-order Sites Weighted Distances ILE5030 Computer Animation and Special Effects 13

14 Voronoi Marching: the LC algorithm (Lin & Canny, 1991) Exploits Motion coherence Geometry locality Procedures Precompute external Voronoi region for each polytope Checks if a pair of features the cloest one If not, search locally on the boundary of each polytope until find the closest features Track the closest feature ILE5030 Computer Animation and Special Effects 14

15 Simple 2D Example P2 B P1 A Objects A & B and their Voronoi regions: P1 and P2 are the pair of closest points between A and B. Note P1 and P2 lie within the Voronoi regions of each other. ILE5030 Computer Animation and Special Effects 15

16 Voronoi Marching Locality: local geometry does not change much, when computations repetitively performed over successive small time intervals Coherence: to "track" the pair of closest features between 2 moving convex polygons(polyhedra) w/ Voronoi regions Performance: expected constant running time, independent of the geometric complexity ILE5030 Computer Animation and Special Effects 16

17 Linear Programming Given two finite sets A, B in R d For each a in A and b in B, Find x in R d which minimize whatever Subject to <a, x> > 0, for all a in A And <b, x> < 0, for all b in B where d = 2 (or 3). ILE5030 Computer Animation and Special Effects 17

18 Minkowski Sums/Differences Minkowski Sum (A, B) = { a + b a A, b B } Minkowski Diff (A, B) = { a - b a A, b B } A and B collide iff Minkowski Difference(A,B) contains the point 0 A B A B ILE5030 Computer Animation and Special Effects 18

19 Computational Issues Expensive to compute boundary of Minkowski difference: For convex polyhedra, Minkowski difference may take O(n 2 ) For general polyhedra, no known algorithm of complexity less than O(n 6 ) is known GJK algorithm (Gilbert, Johnson & Keerthi, 1988) Fast algorithm for separation-distance computation O(n) ILE5030 Computer Animation and Special Effects 19

20 Methods for General Models Decompose into convex pieces, and take minimum over all pairs of pieces Optimal (minimal) model decomposition is NPhard. Approximation algorithms exist for closed solids, but what about a list of triangles? Collection of triangles/polygons: n*m pairs of triangles - brute force expensive Hierarchical representations used to accelerate minimum finding ILE5030 Computer Animation and Special Effects 20

21 Hierarchical Representations Two Common Types: Bounding volume hierarchies (BVH) trees of spheres, ellipses, cubes, axis-aligned bounding boxes (AABBs), oriented bounding boxes (OBBs), K-dop, SSV, etc. Spatial decomposition - BSP, K-d trees, octrees, MSP tree, R-trees, grids/cells, space-time bounds, etc. Do very well in rejection tests, when objects are far apart Performance may slow down, when the two objects are in close proximity and can have multiple contacts (Recall from Ming Lin s Notes) ILE5030 Computer Animation and Special Effects 21

22 BVH vs. Spatial Partitioning BVH: SP: - Object centric - Space centric - Spatial redundancy - Object redundancy (Recall from Ming Lin s Notes) ILE5030 Computer Animation and Special Effects 22

23 BVH vs. Spatial Partitioning BVH: SP: - Object centric - Space centric - Spatial redundancy - Object redundancy (Recall from Ming Lin s Notes) ILE5030 Computer Animation and Special Effects 23

24 BVH vs. Spatial Partitioning BVH: SP: - Object centric - Space centric - Spatial redundancy - Object redundancy (Recall from Ming Lin s Notes) ILE5030 Computer Animation and Special Effects 24

25 BVH vs. Spatial Partitioning BVH: SP: - Object centric - Space centric - Spatial redundancy - Object redundancy (Recall from Ming Lin s Notes) ILE5030 Computer Animation and Special Effects 25

26 Spatial Data Structures & Subdivision Uniform Spatial Sub Quadtree/Octree kd-tree BSP-tree ILE5030 Computer Animation and Special Effects 26

27 Uniform Spatial Subdivision Decompose the objects (the entire simulated environment) into identical cells arranged in a fixed, regular grids (equal size boxes or voxels) To represent an object, only need to decide which cells are occupied. To perform collision detection, check if any cell is occupied by two object Storage: to represent an object at resolution of n voxels per dimension requires upto n 3 cells Accuracy: solids can only be approximated ILE5030 Computer Animation and Special Effects 27

28 Octrees Quadtree is derived by subdividing a 2D-plane in both dimensions to form quadrants Octrees are a 3D-extension of quadtree Use divide-and-conquer Reduce storage requirements (in comparison to grids/voxels) ILE5030 Computer Animation and Special Effects 28

29 Bounding Volume Hierarchies Model Hierarchy each node has a simple volume that bounds a set of triangles children contain volumes that each bound a different portion of the parent s triangles The leaves of the hierarchy usually contain individual triangles A binary bounding volume hierarchy ILE5030 Computer Animation and Special Effects 29

30 BVH-Based Collision Detection ILE5030 Computer Animation and Special Effects 30

31 Bounding Volume Hierarchy Method Enclose objects into bounding volumes (spheres or boxes) Check the bounding volumes first From J.C. Latombe s notes ILE5030 Computer Animation and Special Effects 31

32 Bounding Volume Hierarchy Method Enclose objects into bounding volumes (spheres or boxes) Check the bounding volumes first Decompose an object into two From J.C. Latombe s notes ILE5030 Computer Animation and Special Effects 32

33 Bounding Volume Hierarchy Method Enclose objects into bounding volumes (spheres or boxes) Check the bounding volumes first Decompose an object into two Proceed hierarchically From J.C. Latombe s notes ILE5030 Computer Animation and Special Effects 33

34 Bounding Volume Hierarchy Method Enclose objects into bounding volumes (spheres or boxes) Check the bounding volumes first Decompose an object into two Proceed hierarchically From J.C. Latombe s notes ILE5030 Computer Animation and Special Effects 34

35 Collision Detection using BVH 1. Check for collision between two parent nodes (starting from the roots of two given trees) 2. If there is no interference between two parents, 3. Then stop and report no collision 4. Else All children of one parent node are checked against all children of the other node 5. If there is a collision between the children 6. Then If at leave nodes 7. Then report collision 8. Else go to Step 4 9. Else stop and report no collision ILE5030 Computer Animation and Special Effects 35

36 Type of Bounding Volumes Spheres Ellipsoids Axis-Aligned Bounding Boxes (AABB) Oriented Bounding Boxes (OBBs) Convex Hulls k-discrete Orientation Polytopes (k-dop) Spherical Shells Swept-Sphere Volumes (SSVs) ILE5030 Computer Animation and Special Effects 36

37 Trade-off in Choosing BV s Sphere AABB OBB 6-dop Convex Hull increasing complexity & tightness of fit decreasing cost of (overlap tests + BV update) ILE5030 Computer Animation and Special Effects 37

38 Evaluating Bounding Volume Hierarchies Cost Function: F = N u x C u + N bv x C bv + N p x C p F: total cost function for interference detection N u : no. of bounding volumes updated C u : cost of updating a bounding volume, N bv : no. of bounding volume pair overlap tests C bv : cost of overlap test between 2 BVs N p : no. of primitive pairs tested for interference C p : cost of testing 2 primitives for interference ILE5030 Computer Animation and Special Effects 38

39 Designing Bounding Volume Hierarchies The choice governed by these constraints: It should fit the original model as tightly as possible (to lower N bv and N p ) Testing two such volumes for overlap should be as fast as possible (to lower C bv ) It should require the BV updates as infrequently as possible (to lower N u ) ILE5030 Computer Animation and Special Effects 39

40 Observations Simple primitives (spheres, AABBs, etc.) do very well with respect to the second constraint. But they cannot fit some long skinny primitives tightly. More complex primitives (minimal ellipsoids, OBBs, etc.) provide tight fits, but checking for overlap between them is relatively expensive. Cost of BV updates needs to be considered. ILE5030 Computer Animation and Special Effects 40

41 Videos Yoda from Star Wars III ILE5030 Computer Animation and Special Effects 41

42 Videos ILE5030 Computer Animation and Special Effects 42