Optimizing Schedules for Prioritized Path Planning of Multi-Robot Systems Maren Bennewitz Wolfram...
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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