Complete Path Planning for Planar Closed Chains Among Point Obstacles

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

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

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

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

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

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

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

8. Local and global parametrization • Any m-3 joints can be used as a local chart • More than two charts for differentiable covering Example: 2n charts required to cover (S1)n • Two charts (elbow-up and elbow-down) for capturing connectivity l3 f4 f2 l1 f3 l4 l2 f1 l5 f5

9. C-space Embedding • Embedding in space of dim. greater than m-3 • (S1)m-1 : (f1,……,fm-1) • R2m-4 (coordinates of m-2 vertices) • Elbow-up and elbow-down tori, each parametrized by (f1,……,fm-3) (dimension same as C-space) • Torii connected by “boundary” variety • Our approach

10. P1 P2 l3 l1 l4 l1 l3 l2 l4 l2 l5 l5 P1 or P2 glue along boundary variety Boundary Variety Elbow-up torus Elbow-down torus

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

12. Example: A Six-bar Closed Chain • Boundary variety B(1) connects elbow-up (S1)3 and elbow-down (S1)3 • Recursive skeleton for decomposition B(1) Boundary variety skeleton B(2) identified B(3)={f1,1,f1,2,f1,3,f1,4} skeleton of skeleton

13. l3 l4 l1 f2 f3 f1 l2 l5 l2 l3 l3 l1 l4 l4 l1 l5 l5 l2 l1 Geometric interpretation B(1) Boundary variety B(2) skeleton B(3) Skeleton of skeleton

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

15. [B1(1),1] [B1(1),1] [B1(1),2] [1,2] [B1(1),2] [1,2] [2,B2(1)] [2,B2(1)] [2,B2(1)] [2,B2(1)] graph representation Common facets on B(1) Elbow-up torus Elbow-down torus

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

17. Video

18. obstacle Complexity analysis Theorem: Basic idea for proof: • C-space with O(nm-3) • components in worst case • Each component • decomposed into • O(nm-4) cells 14n2-11n components

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

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