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Motion Planning for Multiple Robots

Motion Planning for Multiple Robots. B. Aronov, M. de Berg, A. Frank van der Stappen, P. Svestka, J. Vleugels Presented by Tim Bretl. Main Idea. Want to use centralized planning because it is complete. Problem—Dimension of planning space is very large.

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Motion Planning for Multiple Robots

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  1. Motion Planning forMultiple Robots B. Aronov, M. de Berg, A. Frank van der Stappen, P. Svestka, J. Vleugels Presented by Tim Bretl 1

  2. Main Idea • Want to use centralized planning because it is complete. • Problem—Dimension of planning space is very large. • Solution—Constrain relative positions of robots to reduce the dimension of the planning space while maintaining completeness. 2

  3. Assumptions (1) • n = Number of obstacles in the workspace. • N = Number of robots in the workspace. • All robots and obstacles have constant complexity. 3

  4. Assumptions (2) • Using an existing, deterministic path planner (Basu et al.) to generate roadmaps with complexity O(nD+1), where D is the total number of dimensions of the configuration space. Reduce D to reduce planning complexity! 4

  5. Outline • Two-Robot Planning • Three-Robot Planning • N-Robot Planning • Bounded-Reach Robots • Summary and Problems 5

  6. Two-Robot Planning Example: Translational Motion, Arbitrary Relative Position y y x D1=2 x D2=2 Total DOF = D1+D2 = 4 6

  7. Constrained Planning (1) Example: Translational Motion, Enforced Contact y  x D1=2 D2,c=1 Total DOF = D1+D2,c = D1+D2-1 = 3 7

  8. Constrained Planning (2) Example: Translational Motion, One Robot Stationary y D2,s=0 x D1=2 Total DOF = D1+D2,s = D1+D2-2 = 2 8

  9. Constrained Planning (3) • Define a permissible multi-configuration as… • Robot 1 stationary at start or goal (DOF=D2) • Robot 2 stationary at start or goal (DOF=D1) • Robots 1 and 2 in contact (DOF=D1+D2-1) • Maximum DOF is D1+D2-1 • If we could plan using only permissible multi-configurations, DOF could be reduced by one 9

  10. Lemma • If a feasible plan exists for two robots, then a feasible plan exists using only permissible multi-configurations. 10

  11. Example (1) 11

  12. 1 5 0 6 2 4 6 0 3 4 5 7 7 1 2 3 Example (2) 12

  13. 6 0 1 5 2 4 0 6 0 1 2 3 4 5 6 7 7 5 3 4 1 2 3 7 Coordination Diagram 0 1 2 3 4 5 6 7 13

  14. 5 6 7 0 3 1 2 4 0 1 2 3 4 5 6 7 Coordination Diagram Nominal Multi-Path Arbitrary Feasible Multi-Path Multi-Paths Using Only Permissible Multi-Configurations 14

  15. Example (1) (Using only permissible multi-configurations) 15

  16. One Subtlety • Still need to connect the spaces of permissible multi-configurations with discrete transitions CS1,s= Robot 1 stationary at start position CS1,g= Robot 1 stationary at goal position CS2,s= Robot 2 stationary at start position CS2,g= Robot 2 stationary at goal position CScontact= Robots moving in contact 16

  17. Transitions (1) CS1,s CS2,s Easy CS1,g CS2,g Hard CScontact 17

  18. Transitions (2) • Calculating transitions to/from CScontact is hard, because there is a continuum of possible transitions. • Example Solution Method for CS1,s • Divide CS1,s into connected cells • Each cell is bounded by a number of patches • For each patch that corresponds to contact configurations, take an arbitrary point on the patch as a transition point 18

  19. Main Result • Algorithm • Compute a roadmap for each of the five permissible multi-configuration spaces • Compute a complete representative set of transitions between these spaces • Gives a roadmap for the complete problem • Can be computed in order O(nD1+D2) time 19

  20. Extension to Three Robots (1) Example: Translational Motion, Enforced Contact y 1 2 x D1=2 D2,c=1 D3,c=2 Total DOF = D1+D2,c+D3,c = D1+D2+D3-2 = 4 20

  21. Extension to Three Robots (2) • Permissible Multi-Configurations: • (k=0,1,2) robots moving in contact • (2-k) robots stationary at either start or goal positions 21

  22. Extension to Three Robots (2) • Main result is analogous —O(nD1+D2+D3-1) • More difficult to prove Coordination diagram now has three dimensions. 22

  23. Extension to N Robots • Divide the robots into three groups • 2 single robot groups • 1 multi-robot group containing N-2 robots • Now the result for three robots applies, reducing DOF by two • It is not known whether a stronger result (analogous to that for two and three robots) can be shown (reducing DOF by N) 23

  24. Bounded-Reach Robots • Low-density environment • Bounded-reach robot Total planning time is O(n log n) (Van der Stappen et al.) 24

  25. High-Density Low-Density C C C C C C B B C C C C Low-Density Environment • For any ball B, the number of obstacles C of size bigger than B that intersect B is at most some small number λ. 25

  26. Not Bounded-Reach Bounded-Reach Bounded-Reach Robot • The reachR of a robot is the radius of the smallest ball completely containing the robot regardless of configuration. • A robot with bounded-reach has a reach that is a small fraction of the minimum obstacle size. 26

  27. Not Bounded-Reach Bounded-Reach Multi-Robot Reach (1) • Problem—A multi-robot does not have bounded-reach 27

  28. Multi-Robot Reach (2) • Solution—Permissible multi-configurations do have bounded-reach and can represent the entire planning space Total planning time (for two or three robots) is O(n log n) 28

  29. Summary • Paper gives a useful algorithm for a small reduction in DOF for complete, centralized multi-robot planning • The results are even better for bounded-reach robots in low-density environments 29

  30. Problems • Mainly useful for answering yes/no connectivity questions; for real robots, you probably want to avoid contact configurations • Plans are not optimal (in fact, are usually far from optimal) 30

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