the gilbert-johnson-keerthi (gjk)...

TheGilbert-Johnson-Keerthi (GJK)

Algorithm

TheTheGilbertGilbert--JohnsonJohnson--KeerthiKeerthi (GJK)(GJK)

AlgorithmAlgorithm

Christer EricsonSony Computer Entertainment America

[email protected]

ChristerChrister EricsonEricsonSony Computer Entertainment AmericaSony Computer Entertainment America

[email protected][email protected]

Christer Ericson

Talk outlineTalk outlineTalk outline

• What is the GJK algorithm• Terminology• “Simplified” version of the algorithm

– One object is a point at the origin– Example illustrating algorithm

• The distance subalgorithm• GJK for two objects

– One no longer necessarily a point at the origin• GJK for moving objects

Christer Ericson

GJK solves proximity queriesGJK solves proximity queriesGJK solves proximity queries

dAP

BP

• Given two convex polyhedra– Computes distance d– Can also return closest pair of points PA, PB

Christer Ericson

GJK solves proximity queriesGJK solves proximity queriesGJK solves proximity queries

dAP

BP

• Generalized for arbitrary convex objects– As long as they can be described in terms of

a support mapping function

Christer Ericson

Terminology 1(3)Terminology 1(3)Terminology 1(3)

Supporting (or extreme) point

d

( )CP S= d

P dfor direction

C

( )CS dreturned by support mapping function

( )CP S= d

C

Christer Ericson


0-simplex 1-simplex 2-simplex 3-simplex

simplex

Christer Ericson


Point set C Convex hull, CH(C)

Christer Ericson

The GJK algorithmThe GJK algorithmThe GJK algorithm

1. Initialize the simplex set Q with up to d+1 points from C (in d dimensions)

2. Compute point P of minimum norm in CH(Q)3. If P is the origin, exit; return 04. Reduce Q to the smallest subset Q’ of Q, such

that P in CH(Q’)5. Let V=SC(–P) be a supporting point in direction

–P6. If V no more extreme in direction –P than P

itself, exit; return ||P||7. Add V to Q. Go to step 2

Christer Ericson

GJK example 1(10)GJK example 1(10)GJK example 1(10)

C

INPUT: Convex polyhedron C given as the convex hull of a set of points

Christer Ericson

1. Initialize the simplex set Q with up to d+1 points from C (in ddimensions)


0Q

{ }0 1 2, ,Q QQ Q=

C1Q

2Q

Christer Ericson


{ }0 1 2, ,Q Q Q Q=

1Q

0Q

2Q

P

2. Compute point P of minimum norm in CH(Q)

Christer Ericson


{ }1 2,Q Q Q=

1Q

0QP

3. If P is the origin, exit; return 04. Reduce Q to the smallest subset Q’ of Q, such that P in CH(Q’)

2Q

Christer Ericson


{ }1 2,Q Q Q=

1Q

2Q

P

5. Let V=SC(–P) be a supporting point in direction –P

( )CV S P= −

Christer Ericson


1Q

{ }1 2, ,Q Q Q V=

2QV

6. If V no more extreme in direction –P than P itself, exit; return ||P||7. Add V to Q. Go to step 2

Christer Ericson


{ }1 2, ,Q Q Q V=

1Q

2QV P

2. Compute point P of minimum norm in CH(Q)

Christer Ericson


{ }2,Q Q V=

2QV P

3. If P is the origin, exit; return 04. Reduce Q to the smallest subset Q’ of Q, such that P in CH(Q’)

Christer Ericson


2' ( )CS QV P= − =

{ }2,Q Q V=

V P

5. Let V=SC(–P) be a supporting point in direction –P

Christer Ericson


V P

DONE!DONE!{ }2,Q Q V=

6. If V no more extreme in direction –P than P itself, exit; return ||P||

2Q

Christer Ericson

Distance subalgorithm 1(2)Distance Distance subalgorithmsubalgorithm 1(2)1(2)

• Approach #1: Solve algebraically– Used in original GJK paper– Johnson’s distance subalgorithm

• Searches all simplex subsets• Solves system of linear equations for each subset• Recursive formulation• From era when math operations were expensive• Robustness problems

– See e.g. Gino van den Bergen’s book

Christer Ericson

Distance subalgorithm 2(2)Distance Distance subalgorithmsubalgorithm 2(2)2(2)

• Approach #2: Solve geometrically– Mathematically equivalent

• But more intuitive• Therefore easier to make robust

– Use straightforward primitives:•ClosestPointOnEdgeToPoint()•ClosestPointOnTriangleToPoint()•ClosestPointOnTetrahedronToPoint()

– Second function outlined here• The approach generalizes

Christer Ericson

Closest point on triangleClosest point on triangleClosest point on triangle

B C

A

• ClosestPointOnTriangleToPoint()– Finds point on triangle closest to a given point

P

Q

Christer Ericson


B C

A

F

AV

BV CV

ABEACE

BCE

• Separate cases based on which feature Voronoiregion point lies in

Christer Ericson


A

B C

00

AX ABAX AC

⋅ ≤⋅ ≤

XAV

Christer Ericson


B C

( ) 000

BC BA BA BXAX ABBX BA

× × ⋅ ≥⋅ ≥⋅ ≥X A

ABE

Christer Ericson

GJK for two objectsGJK for two objectsGJK for two objects

• What about two polyhedra, A and B?• Reduce problem into the one solved

– No change to the algorithm!– Relies on the properties of the

Minkowski difference of A and B

• Not enough time to go into full detail– Just a brief description

Christer Ericson

Minkowski sum & differenceMinkowskiMinkowski sum & differencesum & difference

{ }: ,A B A B+ = + ∈ ∈a b a b

BA A B+

• Minkowski sum– The sweeping of one convex object with another

• Defined as:–

Christer Ericson

Minkowski sum & differenceMinkowskiMinkowski sum & differencesum & difference

{ }{ }

distance( , ) min : ,

min :

A B A B

A B

= − ∈ ∈

= ∈ −

a b a b

c c

{ }: ,

( )

A B A BA B

− = − ∈ ∈

= + −

a b a b• Minkowski difference, defined as:

–

• Can write distance between two objects as:–

• A and B intersecting iff A–B contains the origin!– Distance between A and B given by point of minimum

norm in A–B!

Christer Ericson

The generalizationThe generalizationThe generalization

( ) ( ) ( ) ( )C A B A BS S S S−= = − −d d d d

• A and B intersecting iff A–B contains the origin!– Distance between A and B given by point of minimum

norm in A–B!• So use previous procedure on A–B!• Only change needed: computing• Support mapping separable, so can form it by

computing support mapping for A and Bseparately!–

( ) ( )C A BS S −=d d

Christer Ericson

Av

Bv

GJK for moving objectsGJK for moving objectsGJK for moving objects

Christer Ericson

Av

Bv

B−v A B= −v v v

Transform the problem…Transform the problem…Transform the problem…

Christer Ericson

v

…into moving vs stationary……into moving into moving vsvs stationarystationary

Christer Ericson

Alt #1: Point duplicationAlt #1: Point duplicationAlt #1: Point duplication

Let object A additionally include the points

iP

iP + vv

iP + v

…effectively forming the convex hull of the swept volume of A

Christer Ericson

Alt #2: Support mappingAlt #2: Support mappingAlt #2: Support mapping

iP

Modify support mapping to consider only points

wheniP0⋅ ≤d v

d

v

Christer Ericson

Alt #2: Support mappingAlt #2: Support mappingAlt #2: Support mapping

iP + v

…and to consider only points wheniP + v 0⋅ >d v

d

v

Christer Ericson

GJK for moving objectsGJK for moving objectsGJK for moving objects

• Presented solution– Gives only Boolean interference detection result

• Interval halving over v gives time of collision– Using simplices from previous iteration to start next

iteration speeds up processing drastically• Overall, always starting with the simplices from

the previous iteration makes GJK…– Incremental– Very fast

Christer Ericson

ReferencesReferencesReferences• Ericson, Christer. Real-time collision detection. Morgan Kaufmann,

forthcoming. http://www.realtimecollisiondetection.com/• van den Bergen, Gino. Collision detection in interactive 3D environments.

Morgan Kaufmann, 2003.

• Gilbert, Elmer. Daniel Johnson, S. Sathiya Keerthi. “A fast procedure for computing the distance between complex objects in three dimensional space.” IEEE Journal of Robotics and Automation, vol.4, no. 2, pp. 193-203, 1988.

• Gilbert, Elmer. Chek-Peng Foo. “Computing the Distance Between General Convex Objects in Three-Dimensional Space.” Proceedings IEEE International Conference on Robotics and Automation, pp. 53-61, 1990.

• Xavier Patrick. “Fast swept-volume distance for robust collision detection.” Proc of the 1997 IEEE International Conference on Robotics and Automation, April 1997, Albuquerque, New Mexico, USA.

• Ruspini, Diego. gilbert.c, a C version of the original Fortran implementation of the GJK algorithm. ftp://labrea.stanford.edu/cs/robotics/sean/distance/gilbert.c

Collision DetectionThe GJK Algorithm

Knud Henriksen

Department of Computer Science

University of Copenhagen

c© Knud Henriksen – p.1/11

Affine and Convex Hulls



Given a finite set of points X = {x1, . . . ,xn}

The Affine Hull of X is given by

aff (X) =

{ n∑

i=1

λixi

∣

∣

∣

∣

n∑

i=1

λi = 1

}

The Convex Hull of X is given by

conv(X) =

{ n∑

i=1

λixi

∣

∣

∣

∣

n∑

i=1

λi = 1 ∧ λi ≥ 0

}



Given a finite set of points X = {x1, . . . ,xn}The Affine Hull of X is given by

aff (X) =

{ n∑

i=1

λixi

∣

∣

∣

∣

n∑

i=1

λi = 1

}

The Convex Hull of X is given by

conv(X) =

{ n∑

i=1

λixi

∣

∣

∣

∣

n∑

i=1

λi = 1 ∧ λi ≥ 0

}


The Distance Between Objects

Given two objects A,B ⊂ Rn

The distance, d : Rn × R

n −→ R,between A and B is defined to be

d(A,B) = min{

||x − y|| | x ∈ A ∧ y ∈ B}

A B

d(A, B)






d(A,B) = min{

||x − y|| | x ∈ A ∧ y ∈ B}

A B

A B

d(A, B)






d(A,B) = min{

||x − y|| | x ∈ A ∧ y ∈ B}

A B

d(A, B)


Minkowski Difference


The Minkowski Difference is defined as the setA − B ⊂ R

n , where

A − B ={

a − b | a ∈ A ∧ b ∈ B}

If the sets A and B are convex then the Minkowski

Difference is also convex.





n

, where

A − B ={

a − b | a ∈ A ∧ b ∈ B}







n , where

A − B ={

a − b | a ∈ A ∧ b ∈ B}




The Point Closest to Zero

Given a subset, C ⊂ Rn.

Then v(C) denotes the point nearest to the origin

v(C) ∈ C ∧ ||v(C)|| = min{

||x|| | x ∈ C}

C

O

v(C)

d(A,B) = ||v(A − B)||





v(C) ∈ C ∧ ||v(C)|| = min{

||x|| | x ∈ C}

C

O

C

O

v(C)

d(A,B) = ||v(A − B)||





v(C) ∈ C ∧ ||v(C)|| = min{

||x|| | x ∈ C}

C

O

v(C)

d(A,B) = ||v(A − B)||


Support Mapping

A support mapping function

sC : C ⊂ Rn −→ C ⊂ R

n

is a function which maps point p ∈ C to

sC(p) ∈ C ∧ p · sC(p) = max{

p · x | x ∈ C}


Support Mapping

sC(p) ∈ C ∧ p · sC(p) = max{

p · x | x ∈ C}


Support Mapping

sC(p) ∈ C ∧ p · sC(p) = max{

p · x | x ∈ C}

OC

p


Support Mapping

sC(p) ∈ C ∧ p · sC(p) = max{

p · x | x ∈ C}

OC

p

x

p · x


Support Mapping

sC(p) ∈ C ∧ p · sC(p) = max{

p · x | x ∈ C}

OC

sC(p)

p


Support Mapping

The value of a support mapping for a givenvector is called a support point.

It can be shown that the mapping

sA−B(v) = sA(v) − sB(−v)

is a support mapping of A − B.

That is, a support point for the Minkowski differ-

ence A−B can be computed from support points

for the original objects A and B.


Support Mapping

The value of a support mapping for a givenvector is called a support point.It can be shown that the mapping

sA−B(v) = sA(v) − sB(−v)

is a support mapping of A − B.

That is, a support point for the Minkowski differ-

ence A−B can be computed from support points

for the original objects A and B.


The GJK Algorithm

The GJK Algorithm is a descent method forapproximating

v(A − B)

and therefore also

d(A,B) = ||v(A − B)||


The GJK Algorithm

A − B

o


The GJK Algorithm

A − B

o

v1


The GJK Algorithm

A − B

w1

o

v1


The GJK Algorithm

A − B

w1

o

v2


The GJK Algorithm

A − B

w1

w2

o

v2


The GJK Algorithm

A − B

w1

w2

o


The GJK Algorithm

A − B

w1

w2

v3

o


The GJK Algorithm

A − B

w1

w2

v3

o

w3


The GJK Algorithm

A − B

w1

w2

o

w3


The GJK Algorithm

A − B

w1

o

w3


The GJK Algorithm

A − B

w1

v4

o

w3


The GJK Algorithm

A − B

w1

v4

w4

o

w3


the gilbert-johnson-keerthi (gjk)...

Documents