GJK Supplement

1
GJK Supplement

• Reference SIGGRAPH2004 Course Note
• Original Reference
• 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-4193--203, 1988.

2
Introduction

• Find Euclidean distance (and closest points)
  between the convex hulls of two point sets
• If disjoint, also find the separation axis
• If collision, can be enhanced to find penetration
  depth

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Introduction (cont)

• Simplex-based descent algorithm
• d-simplex the convex hull of d1 independent
  points in d-dimensional space

4
Demonstration
Proximity Query (minimum distance)
Penetration Depth
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Properties

• Distance
• distance(A,B) min a?A,b? B a b 2
• distance(A,B) min c ? 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 ? A?B t 2
• if A and B intersect, t is a point on boundary of
  Minkowski difference

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Support Mapping
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Simplified Version
Find the distance between the CH and origin
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Example 1(4)
Q A

• A is the point of minimum norm in CH(Q)
• Q is irreducible
• Find B in direction of A
• Add B to Q

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Example 2(4)

• Q A,B
• C is the point of minimum norm
• Q is irreducible
• Find D in direction of C
• Add D to Q

D
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Example 3(4)

• Q A,B,D
• E is the point of minimum norm
• Reduce Q to B,D
• Find F in direction of E
• Add F to Q

F
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Example 4(4)

• Q B,D,F
• G is the point of minimum norm
• Reduce Q to D,F
• Cannot find lower support in direction of G
• G is the closest point

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Example 1(3)
QA A has minimum norm Find B in direction of
A Add B to Q
A
B
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Example 2(3)
QA,B C has minimum norm Q is irreducilble Find
D in direction of C Add D to Q
QA,B C has minimum norm Q is irreducilble Find
D in direction of C Add D to Q
A
D
C
B
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Example 3(3)
QA,B,D Pt of minimum norm is origin Polygon
contains origin distance 0
A
D
B
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Estimating Penetrating Depth (ref)
Start with the triangle containing origin
Expanding polytope
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Example 1(3)
Start from ABD, the triangle containing
origin Expanding, find E
A
D
E
B
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Example 2(3)
Find closest edge Expanding, find F
A
D
F
E
B
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Example 3(3)
Find closest point Expanding, find no more. End.
A
D
F
E
B
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Time Complexity for Finding a Support Vertex
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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.

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More on GJK

• Given A CH(A) A a1, a2, ... , an and
• B CH(B) B b1, b2, ... , bm
• 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.

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Distance Between A B
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Sampling Minkowski Difference
(8,5) (4,4) (-4,-1)
Not explicitly compute the Minkowski
difference. Only samples the point set using the
support mapping
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Algorithm d(A,B)
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Example 1(4)
Sampling A?B
(8,5) (4,4) (-4,-1)
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Example 2(4)
(2,1) (1,2) (-1,1)
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Example 3(4)
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Example 4(4)
Note this also defines the separation axis
d sqrt(122)
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GJK for Moving Objects
time interval is short enough to assume constant
velocity movement
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Swept Volume
as convex hull of start and end polygon?!
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Sampling Swept Volume
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Get Time of Collision via Bisection
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HW Compute the Penetration Depth
A
B