1 / 33

GJK Supplement

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-4:193--203, 1988.

kaycee
Télécharger la présentation

GJK Supplement

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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-4:193--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

  3. Introduction (cont) • Simplex-based descent algorithm • d-simplex: the convex hull of d+1 independent points in d-dimensional space

  4. Demonstration Proximity Query (minimum distance) Penetration Depth

  5. Properties • Distance • distance(A,B) = mina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

  6. Support Mapping

  7. (Simplified) p Simplified Version Find the distance between the CH and origin

  8. 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

  9. 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

  10. 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

  11. 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

  12. Example 1(3) Q={A} A has minimum norm Find B in direction of –A Add B to Q A B

  13. Example 2(3) Q={A,B} C has minimum norm Q is irreducilble Find D in direction of –C Add D to Q Q={A,B} C has minimum norm Q is irreducilble Find D in direction of –C Add D to Q A D C B

  14. Example 3(3) Q={A,B,D} Pt of minimum norm is origin Polygon contains origin distance = 0 A D B

  15. Estimating Penetrating Depth (ref) Start with the triangle containing origin … Expanding polytope

  16. Example 1(3) Start from ABD, the triangle containing origin Expanding, find E A D E B

  17. Example 2(3) Find closest edge Expanding, find F A D F E B

  18. Example 3(3) Find closest point Expanding, find no more. End. A D F E B

  19. Time Complexity for Finding a Support Vertex

  20. 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.

  21. 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.

  22. Distance Between A & B

  23. Sampling Minkowski Difference (8,5) = (4,4) – (-4,-1) Not explicitly compute the Minkowski difference. Only samples the point set using the support mapping

  24. Algorithm d(A,B)

  25. Example 1(4) Sampling A⊖B (8,5) = (4,4) – (-4,-1)

  26. Example 2(4) (2,1) = (1,2) – (-1,1)

  27. Example 3(4)

  28. Example 4(4) Note: this also defines the separation axis d = sqrt(1+22)

  29. GJK for Moving Objects time interval is short enough to assume constant velocity movement

  30. Swept Volume … as convex hull of start and end polygon?!

  31. Sampling Swept Volume

  32. Get Time of Collision via Bisection

  33. HW: Compute the Penetration Depth A B

More Related