Motion Planning for Multiple Robots

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