September 11, Deliberative Algorithms II
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September 11, Deliberative Algorithms II

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Multi-Robot Systems

Multi-Robot Systems

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  • Optimal Substructure: solution can be constructed from solutions to its subproblemsOverlapping subproblems: solutions to subproblems can be used multiple times in finding the solution

September 11, Deliberative Algorithms II Presentation Transcript

  • 1. Multi-Robot Systems
    CSCI 7000-006
    Friday, September 11, 2009
    NikolausCorrell
  • 2. So far
    Reactive vs. Deliberative Algorithms
    Both approaches are probabilistic for noisy sensors and actuators
    Robustness/Deterministic behavior can be increased by
    Combining different sensors
    Information exchange
    Actively validating hypothesis
    Redundancy
  • 3. Today
    Exact and approximative algorithms
    Centralized vs. Distributed Systems
    Market-based algorithms
  • 4. Exact Algorithms
    Find always the best solution
    Search the entire solution space
    Determine what “best” means (fitness function)
    Enumerate all solutions
    Pick best solution
    Some problems: dynamic programming
    Finding the best solution can be very time-consuming/impossible for NP-hard problems
  • 5. Example: Traveling Salesman Problem
    Traveling Salesman Problem
    Find the shortest route connecting n cities
    Never visit any city twice
    Computational representation: sequence
    Brute force algorithm: calculate length of all possible permutations
    60 cities -> 4.2 * 10^81 permutations
    NP hard, exact better than brute-force solutions exist (e.g. dynamic programming)
  • 6. Course Question
    Come up with a reactive algorithm for solving the TSP. Hint: ants.
  • 7. Reactive Algorithm for the TSP
    Use a population of ant-like agents starting at random cities
    Each ant randomly select a city that it has not yet visited on this tour (repeat until all cities are visited)
    Each ant calculates the length of this path and deploys an inverse amount of “pheromones” on the path
    In following iterations, ants are programmed to select paths from city i to city j with a higher likelihood
    Algorithm converges to a local optimum
  • 8. Lessons from this example
    Exact problems can be very hard to solve
    Also “pure” CS offers a wide range of algorithmic solutions
    The design problem trades off provable optimality with speed
    In robotics algorithmic choice is constrained by sensors, actuators, computation and communication
  • 9. Coverage example (Wednesday)
    Exact algorithm for single robot
    Approximative algorithm for multiple robots
    Robots might find the optimal solution
    Worst case: every robot covers everything
  • 10. Course Question
    Come up with an exact algorithm for covering M cells with N robots as fast as possible.
    Hints:
    The problem reduces to allocate a subset of cells to each robot to minimize the maximum number of cells allocated to one robot.
    Identify sub-problems / algorithms
  • 11. Possible Solution
    Enumerate all possible sets of allocations
    Calculate the cost of each allocation
    Cost: TSP path over all cells
    NP-Hard
    Stirling numbers of the 2nd kind
    for 3 and 4 cells and up to 4 robots.
    © Mathworld
  • 12. Centralized vs. Distributed Algorithms
    Finding the best solution requires knowing all parameters of the system
    Usually requires “leader” or centralized agent
    Course Question: What problems do you expect in a centralized system?
  • 13. Centralized Systems
    Information needs to be sent to a central unit
    Commands need to be sent to each robot
    Problems
    Information get lost both ways
    Process needs to be repeated when individuals fail
    Individual failure needs to be detected

  • 14. How to distribute an algorithm?
    Smart way: using the optimal substructure of the problem (dynamic programming)
    Not all problems can be efficiently distributed
    Robust: Every robot solves the whole problem for the entire team
    Problem: ambiguous solutions
    Resolution: conflict resolution rules, e.g. lower id goes first
    Example: Market-based task allocation
  • 15. Market-based task allocation
    Tasks are offered by auctioneer
    Every robot bids with the cost that it would need to do the task
    Robot with the lowest cost gets the job
    Simplest auction: greedy, non-optimal ordering
    Variations: bidding on all possible permutations
  • 16. Example: Box Pushing
    Two tasks: watch the box, push the box
    Three robots, only one can watch the box
    Watch the box requires LMS
    Watcher auctions off “push left” and “push right” tasks
    "Sold!: Auction methods for multi-robot coordination".
    Brian P. Gerkey and Maja J Mataric´. IEEE Transactions on Robotics and Automation, Special Issue on Multi-robot Systems, 18(5):758-768, October 2002.
  • 17. Example: Coverage
    Robots calculate cost for covering a blade by solving the TSP
    Sequential biddingapproximates near optimal
    Deterministic bid evaluation allows for decentralized auction-closing
    Re-Allocation upon error
    P. Amstutz, N. Correll, and A. Martinoli. Distributed Boundary Coverage with a Team of Networked Miniature Robots using a Robust Market-Based Algorithm. Annals of Mathematics and Artifcial Intelligence. Special Issue on Coverage, Exploration, and Search, Gal Kaminka and Amir Shapiro, editors, 52(2-4):307-333, 2009.
  • 18. Re-Auctioning example
    Bids during auction
    Robot 1 “slips”
  • 19. 9/20/2007
    Nikolaus Correll
    19
  • 20. Results
    DFS/A* No collaboration
    Market-based coordination
    DFS/A* Information exchange
  • 21. Summary
    The better you plan, the better the performance
    Noise requires you to re-plan all the time
    Feasible algorithms determined by robot capabilities: sensors, actuators, computation and communication
    Algorithmic complexity exponential for NP hard problems
    Potentially very high cost for marginal improvements!
  • 22. Outlook
    Control-based approaches (in two weeks)
    Modeling: examining resource trade-offs on paper (in three weeks)
    Next week: building week