1. Complete Path Planning for Planar Closed Chains Among Point Obstacles Guanfeng Liu and Jeff Trinkle Rensselaer Polytechnic Institute

2. Outline: r Motivation and overview r C-space Analysis m Number of components m C-space topology m Local parametrization and global atlas r Boundary variety r Global cell decomposition r Path Planning algorithm r Simulation results

3. Motivation: r Many applications employ closed-chain manipulators r No complete algorithms for closed chains with obstacles r Limitation of PRM method for closed chains r Difficulty to apply Canny’s roadmap method to C- spaces with multiple coordinate charts

4. Overview: r Exact cell decomposition---direct cylindrical cell decomposition m Atlas of two coordinate charts: elbow-up and elbow- down torii m Common boundary r Complexity r Simulation results

5. C-space Analysis r Dimension: m-3 for m-link closed chains r Algebraic variety r Number of components

6. Theorem: C-space of a single-loop closed chain is the boundary of a union of manifolds of the form : r C-space topology p

7. five-bar closed chain r Types of C-spaces which are connected r Types of C-spaces which are disconnected disjoint union of two tori

8. Local and global parametrization m Any m-3 joints can be used as a local chart m More than two charts for differentiable covering Example: 2 n charts required to cover (S 1 ) n m Two charts (elbow-up and elbow-down) for capturing connectivity 11 22 33 44 55 l1l1 l2l2 l3l3 l4l4 l5l5

9. C-space Embedding  (S 1 ) m-1 : (  1,……,  m-1 ) m R 2m-4 (coordinates of m-2 vertices) m Elbow-up and elbow-down tori, each parametrized by (  1,……,  m-3 ) (dimension same as C-space) m Torii connected by “boundary” variety r Embedding in space of dim. greater than m-3 r Our approach

10. Boundary Variety glue along boundary variety P1P1 P2P2 l1l1 l2l2 l3l3 l4l4 l5l5 or l1l1 l2l2 l3l3 l4l4 l5l5 Elbow-up torus Elbow-down torus P1P1 P2P2

11. Main steps m Boundary variety and its recursive skeletons m Collision varieties m Cell decomposition for elbow-up and elbow-down torii m Identify valid cells based on boundary variety m Adjacency between cells in elbow-up and elbow-down torii m Global graph representation

12. Example: A Six-bar Closed Chain r Boundary variety B(1) connects elbow-up (S 1 ) 3 and elbow-down (S 1 ) 3 r Recursive skeleton for decomposition Boundary variety skeleton skeleton of skeleton B(1) B(2) B(3)={  1,1,  1,2,  1,3,  1,4 } identified

13. Geometric interpretation 11 22 33 l1l1 l2l2 l3l3 l4l4 l5l5 l2l2 l1l1 l3l3 l4l4 l5l5 l1l1 l3l3 l4l4 l5l5 l2l2 l1l1 Boundary variety skeleton Skeleton of skeleton B(1) B(2) B(3)

14. Cell decomposition and graph representation Elbow-up torus Elbow-down torus

15. graph representation Elbow-up torus Elbow-down torus [B 1 (1),1] [1,2] [2,B 2 (1)] [B 1 (1),2] [2,B 2 (1)] [B 1 (1),1] [1,2] [2,B 2 (1)] [B 1 (1),2] [2,B 2 (1)] Common facets on B(1)

16. r Embed C-space into two (m-3)-torii r Compute boundary variety and its skeleton at each dimension r Compute collision variety and its skeleton at each dimension r Decompose elbow-up and elbow-down torii into cells r Identify valid cells and construct adjacency graphs for each torus r Connect respective cells of elbow-up and elbow- down torii which have a common facet on the boundary variety Algorithm

17. Video

18. Complexity analysis Theorem: Basic idea for proof: a.C-space with O(n m-3 ) components in worst case b.Each component decomposed into O(n m-4 ) cells obstacle 14n 2 -11n components

19. Topologically informed sampling-based algorithms r Sampling C-space directly r Sampling the boundary variety and its skeleton r Sampling the skeleton of collision variety C-space obstacles Elbow-down torus Elbow-up torus

20. Summary r Global structure of C-space r Atlas with two coordinate charts r Boundary variety and its skeleton r Cell-decomposition algorithm r Topologically informed sampling-based algorithms