final multi agent - hatanaka lab · 2007. 8. 19. · 1 tokyo institute of technology fujita...

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Fujita LaboratoryTokyo Institute of Technology

Tokyo Institute of Technology

Multi-Agent Search using VoronoiPartition and Voronoi 1D experiment

Fujita Lab, Dept. of Control and System Engineering,FL07-13-2: July 09,2007

David Asikin



Outline

• Review

• Introduction• Decreasing density function• Stability• Conclusion

• Work progress:1. Simulation of Voronoi 2D with density function2. Lloyd’s Algorithm 1D experiment

• Future Work



Review

• Review menu:1. Voronoi partition in 1D & 2D2. Lloyd’s Algorithm3. Objective function:- Sensing performance- Density function



Review

• Voronoi partition:The set of all points q whose distance from pi is less than or equal to the distances from all other pj

( ){ }:i i jV q j i q p q p= ∀ ≠ − ≤ −

0 1

1V 3V2V

1p 2p 3p0 11p 2p 3p



Review

• Lloyd’s Algorithm:A method for evenly distributing points over an unknown area.

• The steps:Step 0: Start with a random area, {Wi}, and

random points, {pi}.Step 1: Construct Voronoi partition {Vi},

generated by {pi}.Step 2: Update pi to be the centroid of Vi.

Return to Step 1.



Review

Step 0: Start with a random area, {Wi}, and random points, {pi}.

Step 1: Construct Voronoi partition {Vi}, generated by {pi}.

Step 2: Update pi to be the centroid of Vi.Return to Step 1.

0 1

1V 3V2V

1p 2p 3p0 10 11p0 11p 2p0 11p 2p 3p0 1'1p'2p

'3p

1V 3V2V

2



Review

• Objective function:

Sensing performance(f=big → poor sensing)Density function

p = agent position q = object W = partition

• By minimizing H, we get optimum coverage. Why?When H = min, agents move to the area with the highest occurrence possibility.

( ) ( ) ( ),W

H p W f q p q dqφ= −∫

( ) :qφ

( ) :f q p−



Review

• We assume:Simplify the objective function using parallel axis theorem.

• To minimize this, p=Cw (=centroid of W partition)• Therefore, use this as an input to make agents go to

centroid of the partition.

( ) ( ) ( ),W

H p W f q p q dqφ= −∫

( ) 2f q p q p− = −

( ) ( ) 2, ,W W WH p W H c W M p c= + −( ) :W WM q dq massφ= ∫ 　　



Introduction

• Application:search & rescue, environmental monitoring, military and defence application, etc.

• Objective: multi-agent searchAgents deploy themselves optimally in Q while updating (=reducing) uncertainty density function and gather information till the uncertainty is below a certain level.

Autonomous N agents equipped with sensors deploythemselves in an optimal way over an unknown area.



Decreasing Density Function• At each iteration, after deploying themselves

optimally, the sensors gather information about Q, reducing the density function as:

: density function: sensing performance: position of the i-th sensor

• is the factor of reduction• Why ?

Only the agent with the smallest can reduce the uncertainty by the largest amount.

( ) ( ) ( ){ }1 minn n iiq q x qφ φ β+ = −nφ

( )ix qβ −

[ ]: 0,1β + a

ix

( ){ }min ii x qβ −β



β

Decreasing Density Function

• For sensing performance function:

• As xi approaches q → decreases (=good sensing)• As xi go further away from q → increases (=bad

sensing)

Conclusion:When xi = q, is minimum.

( ) 21 ix qix q ke αβ − −− = − ( )0,1k ∈

ββ

0α >




• For objective function:

( )n nQH q dqφ= ∆∫ ( ) ( ){ }1max n nQ q q dqφ φ += −∫( ) ( ) ( ){ }{ }minn n iQ iq q x q dqφ φ β= − −∫

( ) ( ){ }{ }1 minn iQ iq x q dqφ β= − −∫( ) ( ){ }1

in ii V

q x q dqφ β= − −∑ ∫

( ) ( ){ }21 1 ii

x qni V

q ke dqαφ − −= − −∑ ∫

( )2

i

i

x qni V

q ke dqαφ − −=∑ ∫

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• Objective function: ( )2

i

i

x qn ni V

H q ke dqαφ − −=∑ ∫( ) ( )( )

2

2ii

x qnn ii V

i

H q ke x q dqx

αφ α− −∂ = − −∂ ∑ ∫

( ) ( )( )'2i

n ii Vq x q dqα φ= − −∑ ∫

( ) ( ) ( ){ }' '2i i

n i ni V Vq x dq q qdqα φ φ= − −∑ ∫ ∫

( ) ( ) ( )( )' '2i

n i ni Vq x q q dqα φ φ= − −∑ ∫

( ){ }2 i i iV i V VM x M Cα= − −2

i iV i VM x Cα ⎛ ⎞= − −⎜ ⎟

⎝ ⎠

②

( )V VM q dqφ= ∫( )1V V

V

C q q dqM

φ= ∫

②

① ( ) ( )2

' ix qn nq q ke

αφ φ − −=①




• From , we can conclude that thenecessary condition for optimality is,

( and are respectively the mass and the centroid of Vi with respect to )

• Assume the system as . Use the result above as an input:

This moves the agent towards .

2i i

nV i V

i

H M x Cx

α∂ ⎛ ⎞= − −⎜ ⎟∂ ⎝ ⎠

ii Vx C=

iVC

iVM

'nφ

i ix u=&

ii prop i Vu k x C⎛ ⎞= − −⎜ ⎟

⎝ ⎠0propk >

iVC



DDF Summary• Objective:

Agents deploy themselves optimally in Q while reducing uncertainty density function and gather information till the uncertainty is below a certain level.

• Decreasing density function:

• Objective function:

• On system , use this as an input:

( ) ( ) ( ){ }1 minn n iiq q x qφ φ β+ = − ( )2

1 ix qix q keαβ − −− = −

( ) ( )2

i

i

x qn n niQ V

H q dq q ke dqαφ φ − −= ∆ =∑∫ ∫2

i i

nV i V

i

H M x Cx

α∂ ⎛ ⎞= − −⎜ ⎟∂ ⎝ ⎠


⎝ ⎠i ix u=&



Stability

• Consider the , whererepresents the configurations of N agents.

• Since , , is a negative definite.

( ) ndHV Xdt

= −& n iii

H xx

δδ

= −∑ &

2i iV i V ii

M x C xα ⎛ ⎞= −⎜ ⎟⎝ ⎠

∑ &

2i i iV i V prop i Vi

M x C k x Cα ⎛ ⎞⎛ ⎞ ⎛ ⎞= − − −⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠∑

2

2i iprop V i Vi

k M x Cα ⎛ ⎞= − −⎜ ⎟⎝ ⎠

∑

( ) nV X H= − ( )1 2, , , NX x x x= K

0propk >0α > V&



Stability

• By LaSalle’s invariance principle, the trajectories of the agents governed by control law:

starting from any initial configuration, will asymptotically converge to centroidal Voronoipartition with respect to the density function:

• Note:


⎝ ⎠

iVC

( )'1i

i

V nVV

C q q dqM

φ= ∫

( ) ( )2

' ix qn nq q ke

αφ φ − −=

( )'iV nV

M q dqφ= ∫



Conclusion

• Objective:Agents deploy themselves optimally in Q, gather information in their respective Voronoi partition and hence reduce uncertainty density function.(Note: the iterations are continued till the uncertainty in the is below a required level)

• The one-step optimal deployment is the centroidalVoronoi configuration with respect to the reduced density function.

• Proven stable by LaSalle’s invariance principle.

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Work Progress (1-0-a)• Simulation of Voronoi 2D with constant density

function : (n = 9 agents)1φ =



Work Progress (1-0-b)• Last position:



Work Progress (1-0-c)

• Trajectory graph:



Work Progress (1-1-a)• Simulation of Voronoi 2D with density function

: (n = 9 agents)( )2 2exp x yφ = − −



Work Progress (1-1-b)

• Last position:





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Work Progress (1-2-a)• Simulation of Voronoi 2D with density function

: (n = 9 agents)( )2 2exp x yφ = − −



Work Progress (1-2-b)

• Last position:







Work Progress (2)

• Lloyd’s Algorithm 1D simulation with density function :

• Objective of the experiment:To test the convergence characteristic of Lloyd’s Algorithm in 1D using RC cars.

1φ =



Work Progress (2)• Equipment:



Work Progress (2)• Software:

1. Halcon:Orders the camera to capture the position of the cars.

2. Simulink:Processes the data captured by Halcon. (Lloyd’s Algorithm block is written here)

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Work Progress (2)• Software:

3. Control Desk:1. Receives order (output)

from Simulink and passes it to RC motors.

2. Monitors cars position, speed, voltage given, etc.

4. Microsoft Visual C++: Links data between Halcon and Simulink.



Work Progress (2)

• Software diagram:



Work Progress (2)• Experiment order:1. Make circles out of cardboard (as cars position) for

camera to read, and prepare the field.2. Make Halcon program.3. Make C++ program to link data from Halcon to

Simulink.4. Make Lloyd’s Algorithm block diagram in Simulink.5. Link Simulink with Control Desk and make monitors

in Control Desk.

+ Adjust voltage, cars speed, direction, camera, etc.+ Debug and compile.



Work Progress (2)• Lloyd’s Algorithm 1D block diagram:

Halcon Voronoi 1 CV1 Motor 1+

-


-


-

p1

p2

p3

u1

u2

u3



Work Progress (2)

• Trouble:– Different time/day, different car and camera characteristic.– Adjustment for car speed, direction.– Limited power on cars battery reflects on performance.– Friction between the field and the tire.– Camera’s vision is warped (afflicts on video capturing

performance). – Backward movement.– Etc.

• Backward movement:Forward : 3VStop : 2.8VBackward : 2.6 V

3V

2.6V

2.8V

2.6VFujita LaboratoryTokyo Institute of Technology


Future Work

• Lloyd’s Algorithm 1D experiment revision.• Lloyd’s Algorithm 1D experiment with density

function.• Multi-Agent Search 2D simulation.• Read more coverage control papers, etc.

• If possible, Lloyd’s Algorithm 2D experiment with constant density function.

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References

• Guruprasad K.R., Debasish Ghose, “Multi-Agent Search using Voronoi Partitions”, ACODS, 2007

• Jorge Cortes, Sonia Martinez, Timur Karatas, Francesco Bullo, “Coverage Control for Mobile Sensing Networks”, IEEE, 2007

• Bruce Francis, “Distributed Control of Autonomous Mobile Robots”, 2006



final multi agent - hatanaka lab · 2007. 8. 19. · 1 tokyo institute of technology fujita...

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