1. Optimizing Schedules for Prioritized Path Planning of Multi-Robot Systems Maren Bennewitz Wolfram Burgard Sebastian Thrun

2. The Problem Given Map of the environment / configuration space Start and goal configurations for a team of robots Task Compute shortest collision-free paths for all robots Complexity Exponential in the number of robots / dimension of the configuration space

3. Centralized Methods Features: Planning in the composite configuration space Compute the optimal solution In praxis: Heuristic approaches to deal with the enormeous complexity of the configuration space Approaches (completeness and optimality not guaranteed) : Potential field techniques [Barraquand et. al., 89], [Tournassoud, 86] Roadmap methods [Sveska & Overmars, 95]

4. The Decoupled Approach (incomplete) 1.Compute optimal paths for the individual robots independently. 2.Assign priorities (not necessarily). 3.Try to resolve possible conflicts between the paths. Approaches: Path coordination [O´Donnell & Lozano-Perez, 89], [Leroy et. al., 99] Planning in the configuration time-space V-Graph algorithm [Erdmann & Lozano-Perez, 87] Potential fields [Warren, 90] A* [Azarm & Schmidt, 96]

5. Path Coordination Key idea: Keep the robots on their individual optimal paths. Allow them to stop, to move forward or even to move backward on their trajectories in order to avoid collisions. Complexity: NP-hard (Job Shop Scheduling Problem) In practice:  Prioritized variant required  Complexity O(n mlogm) [O´Donnell & Lozano-Perez, 89], [Leroy et. al., 99]

6. Application of A* to Single-robot Path Planning ( Given a Grid Map ) Computes the optimal solution!

7. Assignment of priorities to the individual robots Application of A* in the configuration time- spaces Advantage:  Optimal solution given the previously computed paths! Complexity: O(n mlogm) Application of A* to Multi-robot Path Planning

8. Example Situation (4 Robots)

9. Real Robot Experiment A*-based Planning in the Configuration Time-space If Albert has highest priority A* finds a solution. The path coordination method cannot solve this problem at all. 19 m 15 m

10. Flexible Priority Schemes Current techniques leave open how to assign priorities or use a fixed scheme. Our approach: Interleave path planning and priority assignment using randomized techniques.

11. Influence of Priority Schemes No solution can be found if robot 3 has higher priority than robot 1!

12. Finding Solvable Priority Schemes FOR tries := 1 TO maxTries BEGIN select random order P FOR flips := 1 TO maxFlips BEGIN choose random i, j with i < j P := swap(i, j, P) IF solvable(P) return P END FOR

13. Speed-up the Search The plain randomized search technique produces good results, but often a lot of iterations are necessary to come up with a solution.  Focus the search.

14. Extracting Constraints and The task specification yields the constraints:

15. Exploiting Constraints to Find Solvable Priority Schemes Target position of robot j is too close to the initially optimal path of robot i  introduce the constraint When initially assigning priorities try to satisfy as many constraints as possible. During the search only change the priorities of the robots which could not be ´sorted topologically´.

16. Example: Initial Situation Priority scheme: 3, 6, 7, 2, 4, 9... 0, 1, 5, 8

17. Example: Resulting Paths

18. Experimental Evaluation Application of our algorithm to A* in the configuration time-space and the path coordination method Using 2 different environments (noncyclic/cyclic) Randomly generated start/goal points  Goal: Demonstration that our technique significantly increases the number of solved planning problems.

19. Strategies to Find Solvable Priority Schemes 1. A randomly chosen order for the robots. 2. A single order we obtain by applying a greedy approach to satisfy as many constraints as possible. 3. Randomized search starting with a random order and without considering the constraints. 4. Constrained randomized search starting with an order computed in the way as strategy 2.

20. Reducing the Number Of Failures (Noncyclic Corridor Environment) A* in the configuration time-space Path coordination method

21. Reducing the Number Of Failures (Cyclic Corridor Environment)

22. Number of Robots Lying on a Cycle in the Constraint Graph

23. Influence of Priority Schemes on the Path Length

24. Optimizing Priority Schemes FOR tries := 1 TO maxTries BEGIN select random order P IF (tries = 1) P* := P FOR flips := 1 TO maxFlips BEGIN choose random i, j with i < j P´ := swap(i, j, P) IF moveCosts(P´) < moveCosts(P) P := P´ END FOR IF moveCosts(P) < moveCosts(P*) P* := P END FOR RETURN P*

25. Example Situation (30 Robots)

26. Summed Move Costs Over Time

27. Reducing the Path Length

28. Conclusions (1)  Randomized optimization technique for priority schemes  Applied to two decoupled and prioritized path planning techniques

29. Conclusions (2)  Flexible priority schemes seriously decrease the number of failures in which no solution can be found for a given planning problem and lead to a significant reduction of the overall path length.

30. Future Work Different velocities of the robots Reactive/on-line techniques Detection of dead-locks/opportunities