Motion and Manipulation

1. Motion and Manipulation Configuration Space L: Chapter 4

2. Motion Planning Given • a (2- or 3-dimensional) workspaceW, • a robot Aof fixed and known shape moving freely in W, • a collection of obstaclesB={B1,…,Bn}of fixed and known shape and location,

3. Motion Planning Problem • and an initial and a final placement for A, find a path for Aconnecting these two placements along which it avoids collision with the obstacles from B, or report that no such path exists.

4. Modeling • Solution of the motion planning problem in configuration space C: the space of parametric representations of all robot placements. • Configuration q: unique characterization of robot placement by (minimum) number of parameters. Degrees of freedom (DOF) Subset A(q)of W covered by A in q should be uniquely determined.

5. Fixed world frame, moving frame attached to robot A, initially coinciding. Maintain transformation that maps moving frame back to fixed frame. Frames and Transformations

6. Displacements • Any displacement in R2 or R3 is the composition of a rotation and a translation about a selected point O (origin).

7. Configuration space C=R2 Configuration Space

8. Configuration Space Robot translates: C=R2 Robot translates and rotates: C=R2 x [0,2π)

9. Configuration Space

10. Configuration Space

11. Configuration Space Robot translates: C=R3 Robot translates and rotates: C=R3 x [0,2π) x [0,π] x [0,2π)

12. Configuration Space Robot translates and rotates: C=R3 x [0,2π) x [0,π]

13. qis free qis forbidden Configuration Types

14. Obstacles in Configuration Space • Configuration Space Obstacle for Bi: • Forbidden Space • Free Space Bi

15. Free Space and Motion Planning • Motion Planning problem: qgoal qinit

16. Homotopy

17. Configuration Space Obstacles • Explicit computation often base step in exact methods. • Motion planning problem not yet solved when CB is known; additional processing of Cfree=C\CB is necessary for finding path. qgoal qinit

18. 1D Configuration Space Obstacle W=R, translating robot A=[-1,2], obstacle Bi=[0,4] Placement of A specified by position of reference point, so A(q) = A+q = { a+q | a Є A } A Bi

19. Minkowski Sum • Minkowski sum of sets P and Q

20. 1D Configuration Space Obstacle • Boundary of CBi obtained by tracing reference point while sliding A along Bi. A Bi -A CBi

21. 2D Configuration Space Obstacle W=R2, translating convex polygonal robot A, convex polygonal obstacle Bi • Boundary of CBi obtained by tracing reference point while sliding A along Bi. Bi A

22. 2D Configuration Space Obstacle Edges of CBi correspond to • an edge of A touching a vertex of Bi, or • a vertex of A touching an edge of Bi.

23. 2D Configuration Space Obstacle • CBi is convex. • Edges of CBi share orientations of edges of A and Bi. Bi CBi

24. 2D Configuration Space Obstacle • If convex A has m edges, and convex Bi has n edges, then CBi has complexity O(m+n). • Computable in O(m+n) time if edges are ordered.

25. Minkowski Sums • Minkowski sum of a convex and a non-convex polygon can have complexity Ω(mn)! • Minkowski sum of two non-convex polygons can have complexity Ω(m2n2)!

26. Translational Motions • Translating polygonal amidst polygonal obstacles in W = point amidst polygonal Minkowski sums in C.

27. Contact Surfaces in 2D 2D • An edge of CBi corresponds to an edge-vertex contact. • A vertex of CBi corresponds to a vertex-vertex contact = a double edge-vertex contact.

28. Contact Hypersurfaces f-Dimensional C: • single contact: (f-1)-dim. boundary feature in CB/Cfree, • double contact: (f-2)-dim. boundary feature in CB/Cfree, • … • f-fold contact: 0-dim. boundary feature in CB/Cfree. Number of multiple contacts determines complexity of Cfree,and complexity of (exact) motion planning algorithms. O(nf) bound for constant-complexity robot.