SIDRA 2012 talk

A decentralized controller-observer scheme for

multi-robot weighted centroid tracking

Gianluca Antonelli†, Filippo Arrichiello†,Fabrizio Caccavale⊕, Alessandro Marino‡

in alphabetical order

†University of Cassino and Southern Lazio, Italyhttp://webuser.unicas.it/lai/robotica

⊕University of Basilicata, Italyhttp://www.difa.unibas.it

‡University of Salerno, Italyhttp://www.unisa.it

Antonelli,Arrichiello,Caccavale,Marino Benevento, 12 September 2012

General objective

In a multi-robot scenario

local information

local communication

local controller

time-varying topology

⇒ global task


Sketch

Decentralized controller-observer for weighted centroid tracking

Time-varying reference for weighted centroid(formation as centroid+displacement)

Each robot estimates the collective state(i.e., robots positions)

Convergence proof for

first-order dynamicscontinuous-timefixed/switching communication topologiesdirected/undirected graphssaturated inputs


Sketch







Sketch







Modeling

N robots with n DOFs each:

Single state: xi ∈ Rn

Individual dynamics: xi = ui (single-integrator dynamics)

Collective state: x =[

xT1 . . . x

TN

]T ∈ RNn

Collective dynamics: x = u

Global estimate computed by robot i: ix ∈ R

Nn

Collective estimation error: x =

1x

2x

...Nx

=

x− 1x

x− 2x

...x− N

x

∈ RN2n


Problem statement

Task (weighted centroid)

σ(x) =

N∑

i=1

αixi =(

αT ⊗ In

)

x ∈ Rn

Design goals, for each robot:

state observer providing an estimate, ix ∈ R

Nn, asymptoticallyconvergent to the collective state x

feedback control law, ui = ui(xi,ix,Ni) ∈ R

n , such that σ(x)asymptotically converges to a time-varying reference, σd(t)

Each robot knows in advance: σd(t), σd(t)


Proposed approach -1-

i th control law:

ui = ui(ix) = αi

‖α‖2

(

σd + kc(

σd − σ(ix)))

✏✏✏✏✏✏✏✏✏✏✏✮

each robot is feeding back its estimate of the collective state



i th control law:

ui = ui(ix) = αi

‖α‖2

(

σd + kc(

σd − σ(ix)))

✏✏✏✏✏✏✏✏✏✏✏✮

each robot is feeding back its estimate of the collective state



i th state observer:

i ˙x = ko

∑

j∈Ni

(

jx− i

x)

+Π i

(

x− ix)

+ iu

✻

consensuslike term

��✒

local feedback

✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✙

collective input estimated by robot i

Π i = diag{

On · · · In · · · On

}

iu =

u1(ix)

...uN (ix)

, uj(

ix) =

αj

‖α‖2

(

σd + kc(

σd − σ(ix)))




i ˙x = ko

∑

j∈Ni

(

jx− i

x)

+Π i

(

x− ix)

+ iu

✻

consensuslike term

��✒

local feedback

✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✙


Π i = diag{

On · · · In · · · On

}

iu =

u1(ix)

...uN (ix)

, uj(

ix) =

αj

‖α‖2

(

σd + kc(

σd − σ(ix)))




i ˙x = ko

∑

j∈Ni

(

jx− i

x)

+Π i

(

x− ix)

+ iu

✻

consensuslike term

��✒

local feedback

✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✙


Π i = diag{

On · · · In · · · On

}

iu =

u1(ix)

...uN (ix)

, uj(

ix) =

αj

‖α‖2

(

σd + kc(

σd − σ(ix)))




i ˙x = ko

∑

j∈Ni

(

jx− i

x)

+Π i

(

x− ix)

+ iu

✻

consensuslike term

��✒

local feedback

✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✙


Π i = diag{

On · · · In · · · On

}

iu =

u1(ix)

...uN (ix)

, uj(

ix) =

αj

‖α‖2

(

σd + kc(

σd − σ(ix)))


Collective dynamics

Estimation error:

˙x = −ko (L⊗ INn +Π) x+ (1N ⊗ INn)u− u

with L Laplacian matrix embedding the topologyTracking error:

˙σ = −kcσ − kc

‖α‖2N∑

i=1

α2i

(

αT ⊗ In

)

ix


Stability proof for undirected connected topologies

Lyapunov function:

V (x, σ) =1

2xTx+

1

2σTσ

after straightforward computations. . .

V (x, σ) ≤ −[

‖x‖ ‖σ‖]

koλm −Nkc√n −Nkc

√n ‖α‖2

−Nkc√n ‖α‖2

kc

[

‖x‖‖σ‖

]


Stability proof for undirected connected topologies

V is negative definite with a proper choice of the design gains ko and kc:

ko > Nkc

λm

(

√n+

Nn ‖α‖24

)

comments:

N , n and α are known parameters

the control gain kc is free (altough positive)

the term λm ≥ 0 is embedding the connection properties(null for unconnected graphs)

(not surprisingly) the observer gain ko is lower bounded


Extensions -1-

Directed topologies

convergence for balancedand strongly connectedgraphs

proof by resorting to theconcept of mirror graph

Switching topologies

proof by the concept ofCommon LyapunovFunction

gains tuned on the worstcase

All the case studies above analyzed also for saturated inputsAntonelli,Arrichiello,Caccavale,Marino Benevento, 12 September 2012

Extensions -2-

Centroid and formation

σ1(x) =1

N

N∑

i=1

xi

σ2(x) =[

(x2−x1)T (x3−x2)

T . . . (xN−xN−1)T]T

Solved and analized by resorting to a similar controller-observer schemeand Lyapunov approach


Comments

Originality

Estimating the whole state ⇒ is it really decentralized?

Scalability (1000 robots, 10 neighbors ⇒ 8ms on an Arduino)

Need to know the desired trajectory in advance

Robustness with respect to failure?


Simulations there is life beyond Lyapunov!

Dozens of numerical simulations by changing the key parameters:

number of robots N

dimension n

number of neighbors Ni

topology (un-directed, switching)

saturated inputs

1

23

4

5

67

8


Experiments there is life beyond Matlab!

5 Khepera III by K-team

real-time localization

various topologies

real-time comm.

obstacle avoidance

initial error


Experiments - estimation errors

0 20 40 60 800

1

2

3

4estimate errors w.r.t. real pos rob 0

0 20 40 60 800

2

4

6


0 20 40 60 800

5


0 20 40 60 800

5

10


0 20 40 60 80 1000

2

4

6



Experiments - task error

0 10 20 30 40 50 60 70 80−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0 10 20 30 40 50 60 70 80−1.5

−1

−0.5

0

0.5

1

1.5

centroid error

formation error


Experiments - path

0.5 1 1.5 2 2.50.5

1

1.5

2

2.5

3

3.5

4

4.5

5estimate (thin) and real path seen from 4

intentional large

initial error in the

state estimate


Cena I-RAS

Robotics & Automation Society, Italian chapter

La Locanda dei Mestieri

Piazza Piano di Corte, ore 20.30


A decentralized controller-observer scheme for

multi-robot weighted centroid tracking

Gianluca Antonelli†, Filippo Arrichiello†,Fabrizio Caccavale⊕, Alessandro Marino‡

in alphabetical order

†University of Cassino and Southern Lazio, Italyhttp://webuser.unicas.it/lai/robotica

⊕University of Basilicata, Italyhttp://www.difa.unibas.it

‡University of Salerno, Italyhttp://www.unisa.it


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SIDRA 2012 talk

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