Multi-Agent Systems

By Anders Simpson-Wolf, Ramanjit Singh, and Michael Tran

3/13/2013

Visual Demonstration

– more info at http://mobots.epfl.ch/

Presentation Overview

• Relevance/Literature Survey

• Definition

• Motivation

• Formulation of Control Problem

• Simulation

Presentation Overview

• Relevance/Literature Survey

• Definition

• Motivation

• Formulation of Control Problem

• Simulation

Military Demining

Warehouse Robots

Joint Mission Planning

Financial Portfolio Management

Presentation Overview

• Relevance/Literature Survey

• Definition

• Motivation

• Formulation of Control Problem

• Simulation

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

Kronecker Multiplication

Presentation Overview

• Relevance/Literature Survey

• Definition

• Motivation

• Formulation of Control Problem

• Simulation

Motivation

• More natural models

• Agents are becoming commonplace

Motivation

• Exciting Future Applications

– Swarm robotics

– Distributive Problem Solving (Software)

– Autonomous Cars

Presentation Overview

• Relevance/Literature Survey

• Definition

• Motivation

• Formulation of Control Problem

• Simulation

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

The Simple System, Cont.

The normalized sum of all the observations that agent i can make

This is 2nd system for xi It is based on the observations of the other agents and is used to determine the input to agent i

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

Application to Vehicles (Model)

• Vehicle Dynamics

– 𝑥 𝑖 = 𝐴𝑣𝑒ℎ𝑥𝑖 + 𝐵𝑣𝑒ℎ𝑢𝑖 𝑖 = 1,… , 𝑁 𝑥𝑖 ∈ ℝ2

– 𝐴𝑣𝑒ℎ =0 1𝑎21 𝑎22

, 𝐵𝑣𝑒ℎ =01, 𝑥𝑖 =

𝑥𝑝𝑥𝑣

• Remarks

– N vehicles

– 𝑥 𝑖 =𝑥𝑣𝑥𝑎

, 𝑥𝑎 = 𝑎21𝑥𝑝 + 𝑎22𝑥𝑣 + 𝑢𝑖

Application to Vehicles (Model)

• Information from other agents

– Set 𝐽𝑖, set of agents that can be observed by agent 𝑖

– 𝑧𝑖 =

𝑥𝑖 − ℎ𝑖 − 𝑥𝑗 − ℎ𝑗𝑗∈𝐽𝑖

– 𝐿𝐺 = 𝐷 − 𝑄

– 𝐿 = 𝐿𝐺 ⊗ 𝐼2

– 𝑧 = 𝐿 𝑥 − ℎ

Application to Vehicles (Model)

• Feedback Equation

– 𝑥 = 𝐴𝑥 + 𝐵𝐹𝐿(𝑥 − ℎ)

– 𝑥 = 𝐼𝑁 ⊗𝐴𝑣𝑒ℎ𝑥 + 𝐿𝐺 ⊗𝐵𝑣𝑒ℎ𝐹𝑣𝑒ℎ(𝑥 − ℎ)

• Checks

– Dimensions

• 𝑥: 2𝑁 × 2, 𝐹𝑣𝑒ℎ: 2 × 1, 𝐵𝑣𝑒ℎ: 2 × 1, 𝐿𝐺: 2𝑁 × 2

• 𝐼𝑁 ⊗𝐴𝑣𝑒ℎ: 2𝑁 × 2𝑁, 𝐿𝐺 ⊗𝐵𝑣𝑒ℎ𝐹𝑣𝑒ℎ: 2𝑁 × 2𝑁

Presentation Overview

• Relevance/Literature Survey

• Definition

• Motivation

• Formulation of Control Problem

• Simulation

Simulation

Problem:

– Get three agents to move into a triangular formation

– Each agent starts with a certain position and velocity

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

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 = 𝐼𝑁 ⊗𝐴𝑣𝑒ℎ𝑥 + 𝐿𝐺 ⊗𝐵𝑣𝑒ℎ𝐹𝑣𝑒ℎ(𝑥 − ℎ)

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;

% 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);

𝑥𝑘+1 = 𝐴 𝑥𝑘 + 𝐵 𝑥𝑘 − ℎ

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);

% 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

𝑥𝑘+1 = 𝐴 𝑥𝑘 + 𝐵 (𝑥𝑘 − ℎ)

Simulation Result

-1.5 -1 -0.5 0 0.5 1 1.50

0.5

1

1.5

2

2.5

3

X: -0.8667

Y: 0.6667

X: 1.133

Y: 0.6667

X: 1.133

Y: 2.667

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

𝑥𝑘+1 = 𝐴 𝑥𝑘 + 𝐵 (𝑥𝑘 − ℎ)

-2 -1 0 1 2 3 40

1

2

3

4

5

6

X: -0.5333

Y: 0.6667

X: 1.467

Y: 4.667

X: 2.467

Y: 0.6667

(1,1)

(4,1)

(3,5)

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

The End

• Thank you for your attention

• Please direct all questions to Anders

Bibliography and Additional Resources

• http://mitpress.mit.edu/sites/default/files/titles/content/9780262731317_sch_0001.pdf

• http://www.cs.cmu.edu/~softagents/multi.html

• http://en.wikipedia.org/wiki/Multi-agent_system

• http://www.masfoundations.org/mas.pdf

• Information Flow and Cooperative Control of Vehicle Formations (IEEE Transactions on Automatic Control, Vol 49, No. 9, Sept 2004)

• http://link.springer.com/journal/volumesAndIssues/10458

• http://personal.stevens.edu/~yguo1/EE631/Lafferriere.pdf

• Graph LaPlacians and Stabilization of Vehicle Formations (in Proc. 15th IFAC Conf., 2002, pp. 283-288)