Multi-Agent Systems. By Anders Simpson-Wolf, Ramanjit Singh, and Michael Tran

Transcription

1 Multi-Agent Systems By Anders Simpson-Wolf, Ramanjit Singh, and Michael Tran 3/13/2013

2 Visual Demonstration more info at

3 Presentation Overview Relevance/Literature Survey Definition Motivation Formulation of Control Problem Simulation

4 Presentation Overview Relevance/Literature Survey Definition Motivation Formulation of Control Problem Simulation

5 Military Demining

6 Warehouse Robots

7 Joint Mission Planning

8 Financial Portfolio Management

9 Presentation Overview Relevance/Literature Survey Definition Motivation Formulation of Control Problem Simulation

10 Definitions Agent: a computer system that is situated in an environment which is capable of taking autonomous actions to meet its design objectives Multi-Agent Systems: multiple agents working together to achieve a common goal There is no centralized processor No agent has access to the entire system, or the system is too complex to be analyzed

11 Kronecker Multiplication

12 Presentation Overview Relevance/Literature Survey Definition Motivation Formulation of Control Problem Simulation

13 Motivation More natural models Agents are becoming commonplace

14 Motivation Exciting Future Applications Swarm robotics Distributive Problem Solving (Software) Autonomous Cars

15 Presentation Overview Relevance/Literature Survey Definition Motivation Formulation of Control Problem Simulation

16 The Simple System ẋ = Ax + Bu y = Cx A single control system (one agent) becomes multiple agents z is the observation of agent j by agent i

17 The Simple System, Cont. The normalized sum of all the observations that agent i can make This is 2 nd system for x i It is based on the observations of the other agents and is used to determine the input to agent i

18 Application to Vehicles Problem Formulation: Given N agents, what commands do we send to each in order to arrange the agents in some formation? Assumptions: Same dynamics for each agent Agents are Uncoupled

19 Application to Vehicles (Model) Vehicle Dynamics x i = A veh x i + B veh u i i = 1,, N x i R 2 A veh = 0 1 a 21 a 22, B veh = 0 1, x i = x p x v Remarks N vehicles x i = x v x a, x a = a 21 x p + a 22 x v + u i

20 Application to Vehicles (Model) Information from other agents Set J i, set of agents that can be observed by agent i z i = j Ji L G = D Q L = L G I 2 z = L x h x i h i x j h j

21 Application to Vehicles (Model) Feedback Equation x = Ax + BFL(x h) x = I N A veh x + L G B veh F veh (x h) Checks Dimensions x: 2N 2, F veh : 2 1, B veh : 2 1, L G : 2N 2 I N A veh : 2N 2N, L G B veh F veh : 2N 2N

22 Presentation Overview Relevance/Literature Survey Definition Motivation Formulation of Control Problem Simulation

23 Simulation Problem: Get three agents to move into a triangular formation Each agent starts with a certain position and velocity

24 Simulation Assumptions: The internal system of each agent is decoupled from the others Each agent can observe the positions of the other two and then adjust its own velocity accordingly No agent can directly change the velocity of another

25 MATLAB Code % Vehicle Matrices A = [0 1; 0-5]; B = [0; 1]; % Control Gains F = -[15 1]; % Sample Time T = 0.1; % Number of vehicles N = 3; % Calculation of the Directed % Laplacian D = (N-1)*eye(N); Q = ones(n,n)-eye(n); Lg = D-Q; % Compute complete matrices for the system A1 = kron(eye(n),a); B1 = kron(lg,(b*f)); x = I N A veh x + L G B veh F veh (x h)

26 MATLAB Code % Convert system into discrete time A = (eye(2*n)+0.5*a1*t)/(eye(2*n)-0.5*a1*t); B = T*B1; x k+1 = Ax k + B x k h % Formation positions h1 = [ [2 0]; [0 0] ]; h2 = [ [0 0]; [0 0] ]; h3 = [ [2 2]; [0 0] ]; h = [h1;h2;h3]; % Number of iterations to run through n = N*500; x = zeros(n,2);

27 MATLAB Code % Initial Conditions % x(1:2*n,:) = 3*randn(2*N,2); x(1:2,:) = [ [0 0]; [5 0] ]; x(3:4,:) = [ [0 1]; [0 3] ]; x(5:6,:) = [ [0 2]; [2 2] ]; % Determine new points for i=2*n+1:2*n:n; x(i:i+2*n-1,:) = A*x(i-2*N:i-1,:)+B*(x(i-2*N:i-1,:)-h); end x1 = x(1:2*n:n,1); y1 = x(1:2*n:n,2); x2 = x(3:2*n:n,1); y2 = x(3:2*n:n,2); x3 = x(5:2*n:n,1); y3 = x(5:2*n:n,2); x k+1 = Ax k + B(x k h) % Plot results figure(1); plot(x1,y1,x2,y2,x3,y3); hold on plot(x1(end),y1(end),'o',x2(end),y2(end),'o',x3(end),y3(end),'o'); plot(x1(1),y1(1),'x',x2(1),y2(1),'x',x3(1),y3(1),'x'); hold off

28 Simulation Result X: Y: X: Y: X: Y:

29 Simulation Results (cont.) Extensions Information Flow Law Formation Center Feed-forward velocity changes Leader/Follower Suppose one agent does not receive any transmissions from the others x k+1 = Ax k + B(x k h) X: Y: X: Y: X: Y: (1,1) (4,1) (3,5)

30 Conclusion Multi-Agent Systems: multiple agents working together to achieve a common goal There is no centralized control Multi Agent Systems are often a more natural model for problems Autonomous highway driving Swarm robotics Abducting children

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32 The End Thank you for your attention Please direct all questions to Anders

33 Bibliography and Additional Resources _sch_0001.pdf Information Flow and Cooperative Control of Vehicle Formations (IEEE Transactions on Automatic Control, Vol 49, No. 9, Sept 2004) Graph LaPlacians and Stabilization of Vehicle Formations (in Proc. 15 th IFAC Conf., 2002, pp )