Collision Detection 1 2 Many Different Situations Many Different - PDF document

2/12/2012 Many Different Situations Few moving objects, but complex geometry Collision Detection 1 2 Many Different Situations Many Different Situations Thin moving objects, with simple geometry Many simple objects 3 4 Many Different Situations Many Different Situations Many simple objects Many contacts, deformable objects, ... 5 6 1

2/12/2012 Many Different Situations Collision Detection Methods Real-time detection � Many different methods � We will focus on two of them: – Grid method: good for many simple moving objects of about the same size (e.g., many moving discs with similar radii) – Bounding Volume Hierarchy (BVH) method: good for few moving objects with complex geometry 7 8 Grid Method Grid Method d d � Subdivide space into a regular grid of cubic or Running time is proportional to square bins number of moving objects � Index each object in a bin 9 10 Bounding Volume Hierarchy Method Bounding Volume Hierarchy Method � Enclose objects into bounding volumes (spheres or boxes) � Enclose objects into bounding volumes (spheres or boxes) � Check the bounding volumes first � Check the bounding volumes first � Decompose an object into two 11 12 2

2/12/2012 Bounding Volume Hierarchy Method Bounding Volume Hierarchy Method � Enclose objects into bounding volumes (spheres or boxes) � Enclose objects into bounding volumes (spheres or boxes) � Check the bounding volumes first � Check the bounding volumes first � Decompose an object into two � Decompose an object into two � Proceed hierarchically � Proceed hierarchically 13 14 BVH in 3D Bounding Volume Hierarchy Method • BVH is pre-computed for each object 15 16 Collision Detection Collision Detection Search tree A A AA C C B B pruning D E F G D E F G Two objects described by their A precomputed BVHs A 17 18 3

2/12/2012 Collision Detection Collision Detection A A B C B C Search tree Search tree AA AA E F E F D G D G BB BC CB CC BB BC CB CC pruning A A 19 20 Collision Detection Collision Detection A C B � Pruning discards subsets of the two Search tree objects that are separated by the BVs AA D E F G � Each path is followed until pruning or � Each path is followed until pruning or BB BC CB CC until two leaves overlap FD FE GD GE � When two leaves overlap, their contents are tested for overlap If the pieces contained in G and D G D overlap � collision 21 Search Strategy Performance Several thousand collision checks per If there is no collision, all paths must eventually � second for 2 three-dimensional be followed down to pruning or a leaf node objects each described by 500,000 But if there is collision, it is desirable to detect � triangles, on a 1-GHz PC it as quickly as possible q y p � Greedy best-first search strategy with � f(N) = d/(r X +r Y ) Faster when objects are well separated or have much overlap. [Expand the node XY r Y Slower when objects barely overlap or are very close. with largest relative d r X Why? Y overlap (most likely to X contain a collision)] 24 4

2/12/2012 Desirable Properties of Bad Case for BVH BVs and BVHs BVs: BVH: � Separation � Tightness � Balanced tree B l d t � Efficient testing ? � Invariance 26 Desirable Properties of Spheres BVs and BVHs � Invariant � Efficient to test BVs: BVH: � Separation � But tight? � Tightness � Balanced tree B l d t � Efficient testing � Invariance 27 28 Axis-Aligned Bounding Box (AABB) Axis-Aligned Bounding Box (AABB) � Not invariant � Efficient to test � Not tight 29 30 5

2/12/2012 Oriented Bounding Box (OBB) Oriented Bounding Box (OBB) � Invariant � Less efficient to test � Tight 31 32 Desirable Properties of Comparison of BVs BVs and BVHs Sphere AABB OBB BVs: BVH: � Separation � Tightness Tightness - -- + ? ? � Balanced tree B l d t � Efficient testing Testing + + o � Invariance Invariance yes no yes No type of BV is optimal for all situations 33 34 Desirable Properties of Construction of a BVH BVs and BVHs � Top-down construction � At each step, create the two children of a BV BVs: BVH: � Example: � Separation � Tightness For OBB, split longest side at midpoint p g p � Balanced tree B l d t � Efficient testing � Invariance 35 36 6

2/12/2012 Usual Approach to Dynamic Static vs. Dynamic Checking (in PRM Planning) Collision Detection 1) Discretize path at some fine resolution ε Dynamic checks Static checks 2) Test statically each intermediate configuration 3 2 2 3 1 3 2 3 � ε too large � collisions are missed < ε � ε too small � slow test of local paths Examples of Difficult Dynamic Collision Checking � ε too large � collisions are missed � ε too small � slow test of local paths Greedy Distance Computation Adaptive Bisection (same recursion as collision detection) Greedy-Distance(A,B) Ideas: 1. If dist(A,B) > 0, then return dist(A,B) a) Relate configuration changes to path 2. If A and B are both leaves, then return distance between their contents lengths in workspace 3. 3. Switch A and B if A is a leaf, or if B is bigger and Switch A and B if A is a leaf, or if B is bigger and b) Use distance computation rather than not a leaf pure collision checking 4. Set A 1 and A 2 to be A’s children c) Bisect adaptively 5. d 1 � Greedy-Distance(A 1 ,B) 6. If d 1 > 0 then a. d 2 � Greedy-Distance(A 2 ,B) b. If d 2 > 0 then return Min(d 1 , d 2 ) 7. Return 0 7

2/12/2012 What about deformable objects? 43 8