Minkowski Sums. Dinesh Manocha Gokul Varadhan. UNC Chapel Hill. NUS CS 5247 David Hsu

Transcription

1 Minkowski Sums Dinesh Manocha Gokul Varadhan UNC Chapel Hill NUS CS 5247 David Hsu

2 Last Lecture Configuration space workspace configuration space 2

3 Problem Configuration Space of a Translating Robot Input: Polygonal moving object translating in 2-D workspace Polygonal obstacles Output: configuration space obstacles represented as polygons 3

4 Configuration Space of a Translating Robot Workspace Obstacle Robot Configuration Space C-obstacle Robot y x C-obstacle is a polygon. 4

5 Minkowski Sum A B = { a + b a A, b B} A B 5

6 Minkowski Sum A B = { a + b a A, b B} 6

7 Minkowski Sum 7

8 Minkowski Sum A B = { a + b a A, b B} 8

9 Configuration Space Obstacle C-obstacle is O -R Classic result by Lozano-Perez and Wesley 1979 = Obstacle O Robot R C-obstacle O -R 9

10 Properties of Minkowski Sum Minkowski sum of boundary of P and boundary of Q is a subset of boundary of P Q Minkowski of two convex sets is convex 10

11 Minkowski sum of convex polygons The Minkowski sum of two convex polygons P and Q of m and n vertices respectively is a convex polygon P + Q of m + n vertices. The vertices of P + Q are the sums of vertices of P and Q. = 11

12 Gauss Map Gauss map of a convex polygon Edge point on the circle defined by the normal Vertex arc defined by its adjacent edges 12

13 Gauss Map Property of Minkowski Sum p+q belongs to the boundary of Minkowski sum only if the Gauss map of p and q overlap. 13

14 Computational efficiency Running time O(n+m) Space O(n+m) 14

15 Minkowski Sum of Non-convex Polygons Decompose into convex polygons (e.g., triangles or trapezoids), Compute the Minkowski sums, and Take the union Complexity of Minkowski sum O(n 2 m 2 ) 15

16 Worst case example O(n 2 m 2 ) complexity 2D example Agarwal et al

17 3D Minkowski Sum Convex case O(nm) complexity Many methods known for computing Minkowski sum in this case Convex hull method Compute sums of all pairs of vertices of P and Q Compute their convex hull O(mn log(mn)) complexity More efficient methods are known [Guibas and Seidel 1987] 17

18 3D Minkowski Sum Non-convex case O(n 3 m 3 ) complexity Computationally challenging Common approach resorts to convex decomposition 18

19 3D Minkowski Sum Computation Two objects P and Q with m and n convex pieces respectively Compute mn pairwise Minkowski sums between all pairs of convex pieces Compute the union of the pairwise Minkowski sums Main bottleneck Union computation mn is typically large (tens of thousands) Union of mn pairwise Minkowski sums has a complexity close to O(m 3 n 3 ) No practical algorithms known for exact Minkowski sum computation 19

20 Minkowski Sum Approximation We developed an accurate and efficient approximate algorithm [Varadhan and Manocha 2004] Provides certain geometric and topological guarantees on the approximation Approximation is close to the boundary of the Minkowski sum It has the same number of connected components and genus as the exact Minkowski sum 20

21 Brake Hub (4,736 tris) Rod (24 tris) Union of 1,777 primitives 21

22 Anvil (144 tris) Spoon (336 tris) Union of 4,446 primitives 22

23 Knife (516 tris) Scissors (636 tris) Union of 63,790 primitives 23

24 444 tris 1,134 tris 24

25 Union of 66,667 primitives 25

26 Offsetting Cup (1,000 tris) Cup Offset Gear 2,382 tris) Gear Offset 26

27 Configuration Space Approximation - 3D Translation = Obstacle O Robot R C-obstacle O -R 27

28 Assembly Obstacle Robot 28

29 Assembly Obstacle Goal Start Roadmap 16 secs Path Search 0.22 secs 29

30 Assembly 30

31 Path in Configuration Space Goal Start Path 31

32 Other Applications Minkowski sums and configuration spaces have also been used for Interference Detection Morphing Penetration Depth Tolerance Analysis Packing Knee/Joint Modeling 32

33 Applications - Dynamic Simulation Interference Detection Penetration Depth Computation Kim et al

34 Morphing A B Morph (1 t) A tb 34

35 Applications - Packing 35

36 Configuration Space of 2T+1R Robot Dinesh Manocha Gokul Varadhan UNC Chapel Hill

37 Polygonal robot translating & rotating in 2-D workspace workspac e configuration space 37

38 Polygonal robot translating & rotating in 2-D workspace θ y x 38

39 Contact Surfaces (C-surfaces) A C-surface arises from a contact between features of the robot and the obstacle Type A contact Type B contact 39

40 Type A Contact Surface APPL A i,j Contact is feasible when i (q). (b j-1 b j ) 0 Λ i (q). (b j+1 b j ) 0 a i+1 (q) b j-1 b j a i (q) i (q) b j+1 40

41 2D Translation and Rotation Obstacles Robot 41

42 Contact Surfaces 3,929 contact surfaces 42

43 Representation of C-obstacle How can we represent C-obstacle in terms of C- surfaces? For the case of a convex robot and a convex obstacle, q CO Non-convex case Resort to convex decomposition CONST A i,j (q) Λ CONST B i,j(q) is true for all contacts (edgevertex pairs) 43

44 Free Space and Contact Surfaces F is bounded by the C-surfaces Free space F F C-surface C-obstacle F 44

45 Free Space Computation To obtain the free space requires computing arrangement of the C- surfaces 45

46 Arrangement Arrangement A(S) of a set S of geometric objects [Halperin 1997; Agarwal & Sharir 2000] Decomposition of space into relatively open connected cells of dimensions 0,...,d Arrangement of lines (clipped within a window) 46

47 Free Space Computation Compute an arrangement of the C-surfaces Compute intersections between the C-surfaces Retain the appropriate portions of the arrangement Free space F F C-obstacle 47

48 Free Space Computation Arrangement computation is difficult Computing surface-surface intersection is prone to robustness problems Typically O(n 2 ) number of contact surfaces Contact surfaces are non-linear 48

49 Free Space Approximation We have developed an accurate and efficient approximate algorithm [Varadhan and Manocha 2004] Provides certain geometric and topological guarantees on the approximation Approximation is close to the boundary of the free space It has the same number of connected components and genus as the exact Minkowski sum 49

50 Free Space Approximation 3,929 contact surfaces Free space boundary approximation 50

51 2T+1R: Gears Goal Start 51

52 2T+1R: Gears 52

53 2T+1R: Gears Path in Configuration Space Obstacle Robot Obstacle y x Path Goal Narrow passage Start 53