1. UMass Lowell Computer Science 91.504 Advanced Algorithms Computational Geometry Prof. Karen Daniels Spring, 2004 O’Rourke Chapter 8 Motion Planning

2. Chapter 8 Motion Planning Shortest Paths Moving a Disk Translating a Convex Polygon Moving a Ladder Robot Arm Motion Separability

3. Shortest Paths ä Shortest path segment endpoint ä vertex of obstacle ä Shortest path is subpath of visibility graph of vertices of obstacle polygons  visibility graph requires  (n 2 ) time Assume: - Polygonal obstacles have total of n vertices of n vertices - Points s, t are outside obstacles - Points s, t are degenerate polygons - Obstacles are disjoint s t Algorithm: DIJKSTRA’S ALGORITHM T {s} while t not in T do Find edge e in (G \ T) that augments Find edge e in (G \ T) that augments T to reach a node x whose distance T to reach a node x whose distance from s is minimum from s is minimum T T + {e} T T + {e}

4. Moving a Disk ä Shrink disk to a point ä Grow obstacles by disk radius ä Form union of grown obstacles ä If t, s in same component of plane, there is a free path ä find it by modifying visibility graph to include circular arcs from grown obstacles st st O(n 2 lg n)

5. Why Does it Work? ä Minkowski Sum: vector sum for point sets a b a+b Simple Case a3a3 a1a1 a2a2 a4a4 b3b3 b2b2 b1b1 a 2 +b 1 a 3 +b 2 a 4 +b 3 Convex/Convex Case

6. Why Does it Work? (continued) Nonconvex/Nonconvex Case a5a5 a3a3 a1a1 a2a2 a4a4 b4b4 b3b3 b1b1 a 2 +b 1 a 3 +b 3 a 5 +b 4 b2b2

7. Minkowski Sum: Some Properties ä Shape is translationally invariant ä Commutative ä Union formulation ä When A, B convex, sum is convex

8. Minkowski Sum: Properties (continued) ä TRANSLATIONAL INTERSECTION a b Why?? Consider Simple Case: a=A, b=B Note: -b = b rotated by  t

9. Minkowski Sum: Properties (continued) ä TRANSLATIONAL INTERSECTION -B B t A B B

10. Demo Translating a Convex Polygon http:/cs.smith.edu/~orourke/books/CompGeom/CompGeom.html

11. Polygon Motion Planning ä To plan motion for a shape P amidst polygonal obstacles U ä set of all displacements of P relative to U such that (translated) P intersects U: ä set of all displacements of P relative to U such that (translated) P does not intersect U: ä If t, s in same component of plane, there is a free path ä find it by modifying visibility graph to include circular arcs from grown obstacles st st

12. Minkowski Sum Algorithms Convex A, B NonConvex A, B merge edge copies in slope order in slope order Identify vertex/edge support pairs a5a5 a3a3 a1a1 a2a2 a4a4 b4b4 b3b3 b1b1 a 3 supports b 3 b 4 b2b2 b 3 supports a 2 a 3

13. These statements are about Minkowski sums for 2 2D point sets A and B : (a) provide a counterexample that shows this is false: (b) prove this is true Minkowski Sum Exercises B B B -B

14. Moving a Ladder Rotation adds an extra degree of freedom and makes the “configuration space” 3D

15. Demo Robot Arm Motion http:/cs.smith.edu/~orourke/books/CompGeom/CompGeom.html

16. ä Planar, multilink arm ä links L 1, L 2,.., L n, connected at joints J 0, J 1, J 2,.., J n ä joint J 0 anchored at origin ä no obstacles ä arm may self-intersect origin = J 0 J1J1J1J1 L1L1L1L1 L2L2L2L2 can arm reach this? L 1 can reach all points on this circle L 2 can reach all points on each such circle centered on a point of L 1 ’s circle Reachable region for an n-link arm is an annulus centered on the origin

17. Separability ä A Proven Result: A collection of convex polygons can be separated under these motion conditions: ä Translation: all motions are translations ä Unidirectional: all translations in same direction ä Moved once: each polygon moved only once ä One-at-a-time: only one polygon is moved at a time