Optimizing Schedules for Prioritized Path Planning of MultiRobot Systems

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)

• Compute optimal paths for the individual robots
  independently.
• Assign priorities (not necessarily).
• Try to resolve possible conflicts between the
  paths.
• Approaches
• Path coordination ODonnell 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
ODonnell Lozano-Perez, 89, Leroy et. al.,
99

• 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)

6
Application of A to Single-robot Path Planning
(Given a Grid Map)
Computes the optimal solution!
7
Application of A to Multi-robot Path Planning

• 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)

8
Example Situation (4 Robots)
9
Real Robot ExperimentA-based Planning in the
Configuration Time-space
15 m
19 m

• If Albert has highest priority A finds a
  solution.
• The path coordination method cannot solve this
  problem at all.

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!
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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 lt j
• P swap(i, j, P)
• IF solvable(P)
• return P
• END FOR
• 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
The task specification yields the constraints
and
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

• A randomly chosen order for the robots.
• A single order we obtain by applying a greedy
  approach to satisfy as many constraints as
  possible.
• Randomized search starting with a random order
  and without considering the constraints.
• 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 lt j
• P swap(i, j, P)
• IF moveCosts(P) lt moveCosts(P)
• P P
• END FOR
• IF moveCosts(P) lt 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
  inwhich no solution can be found for a
  givenplanning problem and
• lead to a significant reduction of the
  overallpath length.

30
Future Work

• Different velocities of the robots
• Reactive/on-line techniques
• Detection of dead-locks/opportunities