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September 11, Deliberative Algorithms II


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Multi-Robot Systems

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September 11, Deliberative Algorithms II

  1. 1. Multi-Robot Systems<br />CSCI 7000-006<br />Friday, September 11, 2009<br />NikolausCorrell<br />
  2. 2. So far<br />Reactive vs. Deliberative Algorithms<br />Both approaches are probabilistic for noisy sensors and actuators<br />Robustness/Deterministic behavior can be increased by<br />Combining different sensors<br />Information exchange<br />Actively validating hypothesis<br />Redundancy<br />
  3. 3. Today<br />Exact and approximative algorithms<br />Centralized vs. Distributed Systems<br />Market-based algorithms<br />
  4. 4. Exact Algorithms<br />Find always the best solution<br />Search the entire solution space<br />Determine what “best” means (fitness function)<br />Enumerate all solutions<br />Pick best solution<br />Some problems: dynamic programming<br />Finding the best solution can be very time-consuming/impossible for NP-hard problems<br />
  5. 5. Example: Traveling Salesman Problem<br />Traveling Salesman Problem<br />Find the shortest route connecting n cities<br />Never visit any city twice<br />Computational representation: sequence<br />Brute force algorithm: calculate length of all possible permutations<br />60 cities -&gt; 4.2 * 10^81 permutations <br />NP hard, exact better than brute-force solutions exist (e.g. dynamic programming)<br />
  6. 6. Course Question<br />Come up with a reactive algorithm for solving the TSP. Hint: ants.<br />
  7. 7. Reactive Algorithm for the TSP<br />Use a population of ant-like agents starting at random cities<br />Each ant randomly select a city that it has not yet visited on this tour (repeat until all cities are visited)<br />Each ant calculates the length of this path and deploys an inverse amount of “pheromones” on the path<br />In following iterations, ants are programmed to select paths from city i to city j with a higher likelihood<br />Algorithm converges to a local optimum<br />
  8. 8. Lessons from this example<br />Exact problems can be very hard to solve<br />Also “pure” CS offers a wide range of algorithmic solutions<br />The design problem trades off provable optimality with speed<br />In robotics algorithmic choice is constrained by sensors, actuators, computation and communication<br />
  9. 9. Coverage example (Wednesday)<br />Exact algorithm for single robot<br />Approximative algorithm for multiple robots<br />Robots might find the optimal solution<br />Worst case: every robot covers everything<br />
  10. 10. Course Question<br />Come up with an exact algorithm for covering M cells with N robots as fast as possible.<br />Hints:<br />The problem reduces to allocate a subset of cells to each robot to minimize the maximum number of cells allocated to one robot.<br />Identify sub-problems / algorithms<br />
  11. 11. Possible Solution<br />Enumerate all possible sets of allocations<br />Calculate the cost of each allocation<br />Cost: TSP path over all cells<br />NP-Hard <br />Stirling numbers of the 2nd kind<br />for 3 and 4 cells and up to 4 robots.<br />© Mathworld<br />
  12. 12. Centralized vs. Distributed Algorithms<br />Finding the best solution requires knowing all parameters of the system<br />Usually requires “leader” or centralized agent<br />Course Question: What problems do you expect in a centralized system?<br />
  13. 13. Centralized Systems<br />Information needs to be sent to a central unit<br />Commands need to be sent to each robot<br />Problems<br />Information get lost both ways<br />Process needs to be repeated when individuals fail<br />Individual failure needs to be detected<br />…<br />
  14. 14. How to distribute an algorithm?<br />Smart way: using the optimal substructure of the problem (dynamic programming)<br />Not all problems can be efficiently distributed<br />Robust: Every robot solves the whole problem for the entire team<br />Problem: ambiguous solutions<br />Resolution: conflict resolution rules, e.g. lower id goes first<br />Example: Market-based task allocation<br />
  15. 15. Market-based task allocation<br />Tasks are offered by auctioneer<br />Every robot bids with the cost that it would need to do the task<br />Robot with the lowest cost gets the job<br />Simplest auction: greedy, non-optimal ordering<br />Variations: bidding on all possible permutations<br />
  16. 16. Example: Box Pushing<br />Two tasks: watch the box, push the box<br />Three robots, only one can watch the box<br />Watch the box requires LMS<br />Watcher auctions off “push left” and “push right” tasks<br />&quot;Sold!: Auction methods for multi-robot coordination&quot;.<br />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. <br />
  17. 17. Example: Coverage<br />Robots calculate cost for covering a blade by solving the TSP<br />Sequential biddingapproximates near optimal<br />Deterministic bid evaluation allows for decentralized auction-closing<br />Re-Allocation upon error<br />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.<br />
  18. 18. Re-Auctioning example<br />Bids during auction<br />Robot 1 “slips”<br />
  19. 19. 9/20/2007<br />Nikolaus Correll<br />19<br />
  20. 20. Results<br />DFS/A* No collaboration<br />Market-based coordination<br />DFS/A* Information exchange<br />
  21. 21. Summary<br />The better you plan, the better the performance<br />Noise requires you to re-plan all the time<br />Feasible algorithms determined by robot capabilities: sensors, actuators, computation and communication<br />Algorithmic complexity exponential for NP hard problems<br />Potentially very high cost for marginal improvements!<br />
  22. 22. Outlook<br />Control-based approaches (in two weeks)<br />Modeling: examining resource trade-offs on paper (in three weeks)<br />Next week: building week<br />