Optimizing Schedules for Prioritized Path Planning of Multi-Robot Systems

Maren Bennewitz

Wolfram Burgard

Sebastian Thrun

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

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]

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]

Path CoordinationKey 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]

• 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

Real Robot ExperimentA*-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

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.

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

END FOR

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.

Extracting Constraints

31 pp 12 pp and

The task specification yields the constraints:

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´.

j ip p

Example: Initial Situation

Priority scheme: 3, 6, 7, 2, 4, 9 ... 0, 1, 5, 8

Example: Resulting Paths

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.

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.

Reducing the Number Of Failures (Noncyclic Corridor Environment)

A* in the configuration time-space

Path coordination method

Reducing the Number Of Failures (Cyclic Corridor Environment)

Influence of Priority Schemes on the Path Length

Optimizing Priority SchemesFOR tries := 1 TO maxTries BEGIN

select random order PIF (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*

Example Situation (30 Robots)

Summed Move Costs Over Time

Reducing the Path Length

Conclusions (1)

Randomized optimization technique for priority schemes

Applied to two decoupled and prioritized path planning techniques

Conclusions (2)

Flexible priority schemes

• seriously decrease the number of failures inwhich no solution can be found for a givenplanning problem and

• lead to a significant reduction of the overallpath length.

Future Work

• Different velocities of the robots

• Reactive/on-line techniques

• Detection of dead-locks/opportunities