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№1 слайд![The Gilbert-Johnson-Keerthi](/documents_6/559c8c751df8b3d35d9c589660f90827/img0.jpg)
Содержание слайда: The
Gilbert-Johnson-Keerthi (GJK)
Algorithm
№2 слайд![Talk outline What is the GJK](/documents_6/559c8c751df8b3d35d9c589660f90827/img1.jpg)
Содержание слайда: Talk 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
№3 слайд![GJK solves proximity queries](/documents_6/559c8c751df8b3d35d9c589660f90827/img2.jpg)
Содержание слайда: GJK solves proximity queries
Given two convex polyhedra
Computes distance d
Can also return closest pair of points PA, PB
№4 слайд![GJK solves proximity queries](/documents_6/559c8c751df8b3d35d9c589660f90827/img3.jpg)
Содержание слайда: GJK solves proximity queries
Generalized for arbitrary convex objects
As long as they can be described in terms of a support mapping function
№5 слайд![Terminology](/documents_6/559c8c751df8b3d35d9c589660f90827/img4.jpg)
Содержание слайда: Terminology 1(3)
№6 слайд![Terminology](/documents_6/559c8c751df8b3d35d9c589660f90827/img5.jpg)
Содержание слайда: Terminology 2(3)
№7 слайд![Terminology](/documents_6/559c8c751df8b3d35d9c589660f90827/img6.jpg)
Содержание слайда: Terminology 3(3)
№8 слайд![The GJK algorithm Initialize](/documents_6/559c8c751df8b3d35d9c589660f90827/img7.jpg)
Содержание слайда: The GJK algorithm
Initialize the simplex set Q with up to d+1 points from C (in d dimensions)
Compute point P of minimum norm in CH(Q)
If P is the origin, exit; return 0
Reduce Q to the smallest subset Q’ of Q, such that P in CH(Q’)
Let V=SC(–P) be a supporting point in direction –P
If V no more extreme in direction –P than P itself, exit; return ||P||
Add V to Q. Go to step 2
№9 слайд![GJK example](/documents_6/559c8c751df8b3d35d9c589660f90827/img8.jpg)
Содержание слайда: GJK example 1(10)
№10 слайд![GJK example Initialize the](/documents_6/559c8c751df8b3d35d9c589660f90827/img9.jpg)
Содержание слайда: GJK example 2(10)
Initialize the simplex set Q with up to d+1 points from C (in d dimensions)
№11 слайд![GJK example](/documents_6/559c8c751df8b3d35d9c589660f90827/img10.jpg)
Содержание слайда: GJK example 3(10)
№12 слайд![GJK example If P is the](/documents_6/559c8c751df8b3d35d9c589660f90827/img11.jpg)
Содержание слайда: GJK example 4(10)
If P is the origin, exit; return 0
Reduce Q to the smallest subset Q’ of Q, such that P in CH(Q’)
№13 слайд![GJK example Let V SC P be a](/documents_6/559c8c751df8b3d35d9c589660f90827/img12.jpg)
Содержание слайда: GJK example 5(10)
Let V=SC(–P) be a supporting point in direction –P
№14 слайд![GJK example If V no more](/documents_6/559c8c751df8b3d35d9c589660f90827/img13.jpg)
Содержание слайда: GJK example 6(10)
If V no more extreme in direction –P than P itself, exit; return ||P||
Add V to Q. Go to step 2
№15 слайд![GJK example](/documents_6/559c8c751df8b3d35d9c589660f90827/img14.jpg)
Содержание слайда: GJK example 7(10)
№16 слайд![GJK example If P is the](/documents_6/559c8c751df8b3d35d9c589660f90827/img15.jpg)
Содержание слайда: GJK example 8(10)
If P is the origin, exit; return 0
Reduce Q to the smallest subset Q’ of Q, such that P in CH(Q’)
№17 слайд![GJK example Let V SC P be a](/documents_6/559c8c751df8b3d35d9c589660f90827/img16.jpg)
Содержание слайда: GJK example 9(10)
Let V=SC(–P) be a supporting point in direction –P
№18 слайд![GJK example If V no more](/documents_6/559c8c751df8b3d35d9c589660f90827/img17.jpg)
Содержание слайда: GJK example 10(10)
If V no more extreme in direction –P than P itself, exit; return ||P||
№19 слайд![Distance subalgorithm](/documents_6/559c8c751df8b3d35d9c589660f90827/img18.jpg)
Содержание слайда: Distance subalgorithm 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
№20 слайд![Distance subalgorithm](/documents_6/559c8c751df8b3d35d9c589660f90827/img19.jpg)
Содержание слайда: Distance subalgorithm 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
№21 слайд![Closest point on triangle](/documents_6/559c8c751df8b3d35d9c589660f90827/img20.jpg)
Содержание слайда: Closest point on triangle
№22 слайд![Closest point on triangle](/documents_6/559c8c751df8b3d35d9c589660f90827/img21.jpg)
Содержание слайда: Closest point on triangle
№23 слайд![Closest point on triangle](/documents_6/559c8c751df8b3d35d9c589660f90827/img22.jpg)
Содержание слайда: Closest point on triangle
№24 слайд![Closest point on triangle](/documents_6/559c8c751df8b3d35d9c589660f90827/img23.jpg)
Содержание слайда: Closest point on triangle
№25 слайд![GJK for two objects](/documents_6/559c8c751df8b3d35d9c589660f90827/img24.jpg)
Содержание слайда: GJK for two objects
№26 слайд![Minkowski sum amp difference](/documents_6/559c8c751df8b3d35d9c589660f90827/img25.jpg)
Содержание слайда: Minkowski sum & difference
№27 слайд![Minkowski sum amp difference](/documents_6/559c8c751df8b3d35d9c589660f90827/img26.jpg)
Содержание слайда: Minkowski sum & difference
№28 слайд![The generalization](/documents_6/559c8c751df8b3d35d9c589660f90827/img27.jpg)
Содержание слайда: The generalization
№29 слайд![GJK for moving objects](/documents_6/559c8c751df8b3d35d9c589660f90827/img28.jpg)
Содержание слайда: GJK for moving objects
№30 слайд![Transform the problem](/documents_6/559c8c751df8b3d35d9c589660f90827/img29.jpg)
Содержание слайда: Transform the problem…
№31 слайд![into moving vs stationary](/documents_6/559c8c751df8b3d35d9c589660f90827/img30.jpg)
Содержание слайда: …into moving vs stationary
№32 слайд![Alt Point duplication](/documents_6/559c8c751df8b3d35d9c589660f90827/img31.jpg)
Содержание слайда: Alt #1: Point duplication
№33 слайд![Alt Support mapping](/documents_6/559c8c751df8b3d35d9c589660f90827/img32.jpg)
Содержание слайда: Alt #2: Support mapping
№34 слайд![Alt Support mapping](/documents_6/559c8c751df8b3d35d9c589660f90827/img33.jpg)
Содержание слайда: Alt #2: Support mapping
№35 слайд![GJK for moving objects](/documents_6/559c8c751df8b3d35d9c589660f90827/img34.jpg)
Содержание слайда: GJK 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
№36 слайд![References Ericson, Christer.](/documents_6/559c8c751df8b3d35d9c589660f90827/img35.jpg)
Содержание слайда: References
Ericson, Christer. Real-time collision detection. Morgan Kaufmann, 2005. http://www.realtimecollisiondetection.net/
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