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

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.

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

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

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

6 Two-Robot Planning Example: Translational Motion, Arbitrary Relative Position D 1 =2 D 2 =2 Total DOF = D 1 +D 2 = 4 y y x x

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

8 Constrained Planning (2) Example: Translational Motion, One Robot Stationary Total DOF = D 1 +D 2,s = D 1 +D 2 -2 = 2 D 1 =2 y x D 2,s =0

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

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

11 Example (1)

Example (2)

Coordination Diagram

14 Coordination Diagram Nominal Multi-Path Arbitrary Feasible Multi-Path Multi-Paths Using Only Permissible Multi-Configurations

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

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

17 Transitions (1) CS 1,s CS 1,g CS 2,g CS 2,s CS contact Easy Hard

18 Transitions (2) Calculating transitions to/from CS contact is hard, because there is a continuum of possible transitions. Example Solution Method for CS 1,s 1.Divide CS 1,s into connected cells 2.Each cell is bounded by a number of patches 3.For each patch that corresponds to contact configurations, take an arbitrary point on the patch as a transition point

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(n D 1 +D 2 ) time

20 Extension to Three Robots (1) Example: Translational Motion, Enforced Contact D 1 =2 Total DOF = D 1 +D 2,c +D 3,c = D 1 +D 2 +D 3 -2 = 4 y x 11 D 2,c =1 22 D 3,c =2

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

22 Extension to Three Robots (2) Main result is analogous — O(n D 1 +D 2 +D 3 -1 ) More difficult to prove Coordination diagram now has three dimensions.

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)

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

25 C C C B C C Low-Density 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. C C C B C C High-Density

26 Not Bounded-Reach Bounded-Reach Robot The reach R 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. Bounded-Reach

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

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)

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

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)