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Dimitrios Hristu-Varsakelis Mechanical Engineering and Institute for Systems Research University of Maryland, College Park http://glue.umd.edu/~hristu hristu@glue.umd.edu Joint work with: M. Egerstedt, S. B. Andersson, C. Shao. P.R. Kumar, P. S. Krishnaprasad, Cooperative Optimization and Navigation Problems

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Outline Ensembles of autonomous vehicles operating on expansive terrain. Bio-inspired trajectory optimization Language-based navigation Report on Progress Event-driven communication

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Examples from biology (bees, ants, fish etc.) Ensembles can accomplish tasks that are impossible for an individual. Coordination requires thinking about control/communication interactions. Ensembles of Autonomous Systems

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V1V1 VnVn vehicle obstacle control station target start Trajectory optimization without a map A group of vehicles traveling between a fixed pair of locations Terrain is unknown - no global map. On-board sensing provides local information about vehicles immediate surroundings PROBLEM: Given an initial path between a pair of start and target locations, find the optimal path connecting that pair, using local interactions between vehicles. V n-1

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V1V1 VnVn vehicle obstacle control station target start Trajectory optimization without a map A group of vehicles traveling between a fixed pair of locations Terrain is unknown - no global map. On-board sensing provides local information about vehicles immediate surroundings PROBLEM: Given an initial path between a pair of start and target locations, find the optimal path connecting that pair, using local interactions between vehicles. V n-1

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Local pursuit: A biologically-inspired algorithm Theorem (on ): The iterated paths converge to a straight line as... K+2 k+1 k Start Target : Initial path : path followed by the k-th vehicle,... (on a smooth manifold M): If vehicle separation is sufficiently small, then the iterated paths converge to a geodesic. [Bruckstein, 92]

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Experimental results: with Euclidean metric Initial path length ~7m Vehicle separation ~1.5m A collection of mobile robots with: Wireless communication between neighbors Sonar and odometry sensors TARGET START

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: Minimum-length geodesic connecting to M : location of k-th vehicle Local Pursuit Idea: Find optimal trajectory to leader and follow it momentarily.

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M : Minimum-length geodesic connecting a to b : location of k-th vehicle Pursuit decreases vehicle separation

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The k-th vehicle moves as follows: Wait at until t=(+1) At time t, follow the optimal trajectory from to Local pursuit for more general optimal control problems Let Given an initial trajectory with converge to a local min. for Assumptions: uniqueness, smoothness Find that minimizes s.t. As, iterated trajectories

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Simulation: pursuit on 5m trajectory 0.7m separation

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A sub-Riemannian example 5m trajectory 1.5m separation fixed

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Pursuit in vehicles with drift (minimum time problem)

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Summary and Work in Progress A biologically-inspired trajectory optimization algorithm - local pursuit forms a string of vehicles - each vehicle uses local information and communicates with its closest neighbors Target state and optimal trajectory are unknown Local convergence Experiments Escaping local minima Comparison with gradient descent methods

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Control in a reasonably complex world The problem of specifying control tasks (e.g. go to the refrigerator and get the milk) Solving motion control problems of adequate complexity Many interesting systems evolve in environments that are not smooth, simply connected, etc. Using language primitives to navigate: Specify control policies Represent the environment (what parts do we ignore?)

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Def: MDLe is the formal language defined by the context free grammar with production rules: Motion Description Languages Evaluate Evolve under until Concatenate, encapsulate atoms to form complex strings (plans), e.g. N: nonterminals T: terminals S: start symbol : empty string Fact: MDLe is context free but not regular Atom:

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Symbolic Navigation Keep only interesting details about how to navigate the world Landmark: L = (M,x)M: map patch, x: coordinates Sensor signature: L = L i if s(t) = s i (t) for t in [t 0,T] Navigation Local navigation: on a given landmark L i Global navigation: between landmarks M x World

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A directed graph representation of a map Represent only interesting parts of the world. G = {L,E} L i : landmarks E ij : {i,j, ij } ij : an MDLe program E ij E ji Idea: Replace details locally by a feedback program

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Experiment: indoor navigation Lab 1 Lab 2 Office Partial floor plan of 2 nd floor A.V. Williams

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Experiment, contd Goal: Navigate between three landmarks Front of labRear of labOffice

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Experiment: Example MDLe plans {Lab2toLab1Plan (bumper) (Atom (atIsection 0100) (goAvoid 90 40 20)) (Atom (atIsection 0010) (go 0 0.36)) (Atom (wait ) (align 7 9)) (Atom (atIsection 1000) (goAvoid 0 40 20)) (Atom (atIsection 0100) (go 0 0.36)) (Atom (wait ) (align 3 5)) (Atom (wait 7) (goAvoid 270 40 20)) (Atom (atIsection 1000) (goAvoid 270 40 20)) } {Lab1toOfficePlan (bumper) (Atom (atIsection 1001) (goAvoid 90 40 20)) (Atom (atIsection 0011) (go 0 0.36)) (Atom (wait ) (align 11 13)) (Atom (atIsection 0100) (goAvoid 180 40 20)) (Atom (wait 10) (rotate -90)) }