References
Additional lecture notes for 2/18/02.
I-COLLIDE: Interactive and Exact Collision Detection for Large-Scale Environments, by Cohen, Lin,
Manocha & Ponamgi, Proc. of ACM Symposium on Interactive 3D Graphics, 1995.
(More details in Chapter 3 of M. Lin’s Thesis)
A Fast Procedure for Computing the Distance between Objects in Three-Dimensional Space, by E. G. Gilbert, D. W. Johnson, and S. S. Keerthi, In IEEE Transaction of Robotics and Automation, Vol. RA-4:193–203, 1988.
UNC Chapel Hill
M. C. Lin
Geometric Proximity Queries
Given two object, 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
UNC Chapel Hill
M. C. Lin
UNC Chapel Hill
M. C. Lin
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 = 1000+ ……
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)
• CSG vs. B-Rep
• Static vs. Dynamic
And so on… This may include other geometric representation schemata, etc.
UNC Chapel Hill
M. C. Lin
UNC Chapel Hill
M. C. Lin
Some Possible Approaches
• Geometric methods
• Algebraic Techniques
• Hierarchical Bounding Volumes • Spatial Partitioning
• Others (e.g. optimization)
Voronoi Diagrams
Given a set S of n points in R2 , for each point pi in S, there is the set of points (x, y) in the plane that are closer to pi than any other point in S, called Voronoi polygons. The collection of n Voronoi polygons given the n points in the set S is the “Voronoi diagram”, Vor(S), of the point set S.
Intuition: To partition the plane into regions, each of these is the set of points that are closer to a point pi in S than any other. The partition is based on the set of closest points, e.g. bisectors that have 2 or 3 closest points.
UNC Chapel Hill
M. C. Lin
Generalized Voronoi Diagrams
The extension of the Voronoi diagram to higher dimensional features (such as edges and facets, instead of points); i.e. the set of points closest to a feature, e.g. that of a polyhedron.
FACTS:
– In general, the generalized Voronoi diagram has
quadratic surface boundaries in it.
– If the polyhedron is convex, then its generalized Voronoi diagram has planar boundaries.
UNC Chapel Hill
M. C. Lin
Voronoi Regions
A Voronoi region associated with a feature is a set of points that are closer to that feature than any other.
FACTS:
– The Voronoi regions form a partition of space outside
of the polyhedron according to the closest feature.
– The collection of Voronoi regions of each polyhedron is the generalized Voronoi diagram of the polyhedron.
– The generalized Voronoi diagram of a convex polyhedron has linear size and consists of polyhedral regions. And, all Voronoi regions are convex.
UNC Chapel Hill
M. C. Lin
Voronoi Marching
Basic Ideas:
Coherence: local geometry does not change much, when computations repetitively performed over successive small time intervals
Locality: 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
UNC Chapel Hill
M. C. Lin
Simple 2D Example
P2 P1
B
A
UNC Chapel Hill
M. C. Lin
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.
Basic Idea for Voronoi Marching
UNC Chapel Hill
M. C. Lin
Linear Programming
In general, a d-dimensional linear program- ming (or linear optimization) problem may be posed as follows:
Given a finite set A in Rd
For each a in A, a constant Ka in R, c in Rd
Find x in Rd which minimize
Subject to ≥ Ka, for all a in A .
where <*, *> is standard inner product in Rd.
UNC Chapel Hill
M. C. Lin
UNC Chapel Hill
M. C. Lin
LP for Collision Detection
Given two finite sets A, B in Rd
For each a in A and b in B,
Find x in Rd which minimize whatever Subject to > 0, for all a in A And < 0, for all b in B
where d = 2 (or 3).
Minkowski Sums/Differences
MinkowskiSum(A,B)={a+b |a∈ A, b ∈ B }
MinkowskiDiff(A,B)={a-b |a∈A, b∈B}
A and B collide iff Minkowski Difference(A,B) contains the point 0.
UNC Chapel Hill
M. C. Lin
Some Minkowski Differences
AB
A
B
UNC Chapel Hill
M. C. Lin
Minkowski Difference & Translation
Minkowski-Diff(Trans(A, t1), Trans(B, t2)) = Trans(Minkowski-Diff(A,B), t1 - t2)
⇒ Trans(A, t1) and Trans(B, t2) intersect iff Minkowski-Diff(A,B) contains point (t2 - t1).
UNC Chapel Hill
M. C. Lin
Properties
Distance
– distance(A,B)=mina∈A, b∈B ||a-b||2
– distance(A,B) = min c ∈ Minkowski-Diff(A,B) || c ||2
– if A and B disjoint, c is a point on boundary of
Minkowski difference
Penetration Depth
– pd(A,B) = min{ || t ||2 | A ∩ Translated(B,t) = ∅ }
– pd(A,B) = mint ∉Minkowski-Diff(A,B) || t ||2
– if A and B intersect, t is a point on boundary of
Minkowski difference
UNC Chapel Hill
M. C. Lin
Practicality
Expensive to compute boundary of Minkowski difference:
– For convex polyhedra, Minkowski difference may take O(n2)
– For general polyhedra, no known algorithm of complexity less than O(n6) is known
UNC Chapel Hill
M. C. Lin
GJK for Computing Distance between Convex Polyhedra
GJK-DistanceToOrigin ( P ) // dimension is m
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.
Initialize P0 with m+1 or fewer points. k =0
while (TRUE) {
if origin is within CH( Pk ), return 0 else {
find x ∈ CH(Pk) closest to origin, and Sk ⊂ Pk s.t. x ∈ CH(Sk) see if any point p-x in P more extremal in direction -x
if no such point is found, return |x|
else {
Pk+1 = Sk ∪ {p-x}
k=k+1 }
} }
UNC Chapel Hill
M. C. Lin
An Example of GJK
UNC Chapel Hill
M. C. Lin
Running Time of GJK
Each iteration of the while loop requires O(n) time.
O(n) iterations possible. The authors claimed between 3 to 6 iterations on average for any problem size, making this “expected” linear.
Trivial O(n) algorithms exist if we are given the boundary representation of a convex object, but GJK will work on point sets - computes CH lazily.
UNC Chapel Hill
M. C. Lin
More on GJK
GivenA=CH(A’) A’={a1,a2,...,an }and B=CH(B’) B’={b1,b2,...,bm }
Minkowski-Diff(A,B) = CH(P), P = {a - b | a∈ A’, b∈ B’} Can compute points of P on demand:
– p-x = a-x - bx where a-x is the point of A’ extremal in direction -x, and bx is the point of B’ extremal in direction x.
The loop body would take O(n + m) time, producing the “expected” linear performance overall.
UNC Chapel Hill
M. C. Lin
Large, Dynamic Environments
For dynamic simulation where the velocity and acceleration of all objects are known at each step, use the scheduling scheme (implemented as heap) to prioritize “critical events” to be processed.
Each object pair is tagged with the estimated time to next collision. Then, each pair of objects is processed accordingly. The heap is updated when a collision occurs.
UNC Chapel Hill
M. C. Lin
Scheduling Scheme
amax: an upper bound on relative acceleration between any two points on any pair of objects.
alin: relative absolute linear
α: relative rotational accelerations
ω: relative rotational velocities
r: vector difference btw CoM of two bodies
d: initial separation for two given objects
amax =|alin +αxr+ωxωxr|
vi =|vlin+ωx r|
Estimated Time to collision:
tc = { (vi2 + 2 amax d)1/2 - vi } / amax
UNC Chapel Hill
M. C. Lin
Collide System Architecture
Transform
Overlap
Sweep & Prune
Simulation
Exact Collision Detection
UNC Chapel Hill
M. C. Lin
Parameters
Analysis & Response
Collision
Sweep and Prune
Compute the axis-aligned bounding box (fixed vs. dynamic) for each object
Dimension Reduction by projecting boxes onto each x, y, z- axis
Sort the endpoints and find overlapping intervals
Possible collision -- only if projected intervals overlap in all 3 dimensions
UNC Chapel Hill
M. C. Lin
Sweep & Prune
T=1
e3
e2 b3 e1
b2 b1
b1 b2 e1
e2 b3 e3
T=2
e2 e1
b2 e3
b1 b2 e1
e2 b3 e3
b1 b3
UNC Chapel Hill
M. C. Lin
Updating Bounding Boxes
Coherence (greedy algorithm)
Convexity properties (geometric
properties of convex polytopes)
Nearly constant time, if the motion is
relatively “small” UNC Chapel Hill
M. C. Lin
Use of Sorting Methods
Initial sort -- quick sort runs in O(m log m) just as in any ordinary situation
Updating -- insertion sort runs in O(m) due to coherence. We sort an almost sorted list from last stimulation step. In fact, we look for “swap” of positions in all 3 dimension.
UNC Chapel Hill
M. C. Lin
Implementation Issues
Collisionmatrix-- basically adjacency matrix
Enlarge bounding volumes with some tolerance threshold
Quick start polyhedral collision test -- using bucket sort & look-up table
UNC Chapel Hill
M. C. Lin