ARCS Presentation 2008

An Anytime Winner Determination Algorithm

forTime-Extended Multi-Robot Task

Allocation

Dr. Fang Tang & Spondon SahaIntelligent Robotics Lab

California State Polytechnic University Pomona, CA

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Outline

Introduction ASyMTRe-D Combinatorial Auctions

Simultaneous Task Allocation (STA)Performance MetricsFuture WorkConclusionReferencesQ&A

Intelligent Robotics Lab, Cal Poly Pomona

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Introduction: ASyMTRe-D

Automated Synthesis of Multi-robot Task solutions through software Reconfiguration.

ASyMTRe determines: a coalition of robots. from a heterogeneous group of robots. to carry out a given multi-robot task.


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Heterogeneous group of robots??? Robots with different functional capabilities.

Multi-robot task??? A task that cannot be carried out by a single robot. Requires a “strongly cooperative” solution.


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ASyMTRe Centralized algorithm. Runs on a central server. Aware of the number of robots and their individual capabilities.

ASyMTRe-D Distributed ASyMTRe designed using the Contract Net Protocol. Runs locally on each robot. Uses a group negotiation process to determine coalitions.


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R1

R4 R3

R2R5Task A

Idle Robots


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ASyMTRe-D takes into account: Task-Specific cost. Robot-inherent cost.

Determines a suitable mapping between: Task. Robots.


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Problem: Can handle only one task at a time. Each round results in idle robots.

Desired behavior: Need to reduce instances of idle robots. Make idle robots work on other tasks (if any).


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Introduction: Combinatorial Auctions

Sequential Auctions.

Parallel Auctions.

Combinatorial Auctions.


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Item 1 Item 4Item 3Item 2 Item 5

Auctioneer

[1,2] [1,3,5] [3,5] [1, 4]

???

???

,[2,5]


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Complexity: A NP-complete problem. Similar to the Partitioning problem → NPC.

Dynamic Programming: Long execution time. Explores the entire solution space. Scales to only a small number of bids.


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Solution: An Anytime tree algorithm. Explores only the relevant solution space. Polynomial in the number of bids submitted.


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Bids:123451,21,3,51,42,53,5

Root

1,2

3,5 3

4 4

5

1,3,5

4

2

1,4

2,5 2

3 3,5 3

5

1

2,5 2

3 3,5 3

44 4

5


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Quick Review

ASyMTRe-D Drawbacks: Accepts tasks sequentially. Order of task execution determine total execution time. Idle robots are waste of resources.

Need: A task scheduler. Least robot idle-time → maximum utilization of resources.

Inspiration for STA: Combinatorial Auctions.


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Simultaneous Task Allocation (STA)

Task1 Task 2 Task 3

{R1,R2, $3}

R2

R3R1

{R2,R3, $4}

{R3, $5}

{R1,R2, $4}

{R2, $1}

{R3, $3}

{R1,R3, $4}

{R2,R3, $4}



`

Task Bids

1

{R1, R2, $3}

{R2, R3, $4}

{R3, $5}

2

{R1, R2, $4}

{R2, $1}

{R3, $3}

3

{R1, R3, $4}

{R2, R3, $4}

Root

R1, R2

R2, R3

R3

3 4 5 0

R3

6 3 4R1, R2

9

R2

6 5R1, R2

4R2

1R3

3 0

R2, R3

4R1, R3

4R1, R3

5

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Will always return a partial answer if terminated early.

Each path is a set of disjoint coalitions → Partition.

All possible partitions of robots are considered.

Run time complexity depends on: Number of bids submitted. Number of tasks considered.


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Task B

Task C

……

Task N

STA Algorithm

Task D

Task E Task F

Task A


ASyMTRe-D equipped…

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Performance (STA)

Test Environment: Dell Inspiron 6400. 1.73 GHz Pentium Dual Core Processors. 2 GB of RAM. Python.

Objective: To measure the execution time of the algorithm For increasing bids. For increasing task sizes.


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Performance (STA)

Random Distribution: Bids accepted → 10 - 100 Task Sizes → 5, 10, 15, 20 Robots → 10 Maximum allowed coalition size → 9

Uniform Distribution: Bids accepted → 10 - 50 Task Sizes → 5, 10, 15, 20 Robots → 10 Maximum allowed coalition size → 3


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Performance (STA)

Random Distribution


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Performance (STA)

Uniform Distribution


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Performance (STA)

Problems: Execution time is exceedingly long. The entire tree is generated irrespective.

Need: Faster execution time. Generate only relevant parts of tree.


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Future Work (STA)

Use heuristics to generate only relevant portions of tree. Iterative Deepening A* (IDA*)

Optimize the current version of the algorithm.

Demonstrate the complete approach using ASyMTRe-D.


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Conclusion (STA)

Multiple tasks are being considered. All possible mapping of tasks to coalitions. No restriction on coalition size.

Reduced overall execution time. STA (Higher level). ASyMTRe-D (Lower level).

An Anytime Algorithm!!


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References Sandholm, T. 1999. Algorithm for Optimal Winner Determination in Combinatorial Auctions,

International Joint Conference on Artificial Intelligence (July). Stockholm, Sweden, 542-547.

Parker, L.E., Tang, F., 2005. ASyMTRe: Automated Synthesis of Multi-Robot Task Solutions through Software Reconfiguration. In Proceedings of IEEE International Conference on Robotics and Automation (April).

Tang, F., Parker, L. E. , 2005. Distributed Multi-Robot Coalitions through ASyMTRe-D. Proceedings of IEEE/RSJ International Conference on Intelligent Robots and Systems (August).

Parker, L.E. and Tang, F., 2006. Building Multirobot Coalitions through Automated Task Solution Synthesis. In Proceedings of the IEEE v. 94 No. 7 (July). 1289-1305.

Parker, L.E., Tang, F., 2007. A Complete Methodology for Generating Multi-Robot Task Solutions using ASyMTRe-D and Market-Based Task Allocation. To Appear in the IEEE International Conference on Robotics and Automation (April).

Wikipedia. http://www.wikipedia.org/


http://www.wikipedia.org/

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Questions??Intelligent Robotics Lab, Cal Poly Pomona

ARCS Presentation 2008

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