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
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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
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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]
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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]
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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]
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• 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
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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
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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.
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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
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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.
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Extracting Constraints
31 pp 12 pp and
The task specification yields the constraints:
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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
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Example: Initial Situation
Priority scheme: 3, 6, 7, 2, 4, 9 ... 0, 1, 5, 8
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Example: Resulting Paths
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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.
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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.
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Reducing the Number Of Failures (Noncyclic Corridor Environment)
A* in the configuration time-space
Path coordination method
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Reducing the Number Of Failures (Cyclic Corridor Environment)
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Influence of Priority Schemes on the Path Length
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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*
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Example Situation (30 Robots)
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Summed Move Costs Over Time
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Reducing the Path Length
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Conclusions (1)
Randomized optimization technique for priority schemes
Applied to two decoupled and prioritized path planning techniques
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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.
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Future Work
• Different velocities of the robots
• Reactive/on-line techniques
• Detection of dead-locks/opportunities