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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 IIPresentation Transcript
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Multi-Robot Systems CSCI 7000-006 Friday, September 11, 2009 NikolausCorrell
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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
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Today Exact and approximative algorithms Centralized vs. Distributed Systems Market-based algorithms
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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
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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)
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Course Question Come up with a reactive algorithm for solving the TSP. Hint: ants.
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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
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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
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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
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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
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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?
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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 …
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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
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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
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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.
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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.
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Re-Auctioning example Bids during auction Robot 1 “slips”
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9/20/2007 Nikolaus Correll 19
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Results DFS/A* No collaboration Market-based coordination DFS/A* Information exchange
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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!
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Outlook Control-based approaches (in two weeks) Modeling: examining resource trade-offs on paper (in three weeks) Next week: building week