icra 2013 talk 1

Decentralized control of dynamic centroid and

formation for multi-robot systems

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 Karlsruhe, 8 May 2013

General objective

In a multi-robot scenario

local information

local communication

local controller

time-varying topology

⇒ global task


Sketch

Decentralized controller-observer for dynamic centroid and formation

Time-varying reference for weighted centroid and formation asdisplacement

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

Convergence proof for

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


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∈ R

Nn

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

Tasks (centroid and formation)

σ1(x) =1

N

N∑

i=1

xi = J1x

σ2(x) =[

(x2−x1)T (x3−x2)

T . . . (xN−xN−1)T]T

= J2x

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 σ1(x),σ2(x) asymptotically converge to a time-varying reference, σ1,d(t),σ2,d(t)

Each robot knows in advance the desired trajectory


Proposed approach -1-

i th control law:

ui(t,ix) = σ1,d(t) + k1,c

(

σ1,d(t)− σ1(ix)

)

+

J†2,i

(

σ2,d(t) + k2,c(

σ2,d(t)− σ2(ix)

))

✏✏✏✏✏✏✏✏✏✏✏✮

✟✟✟✟✟✟✟✟✟✙

each robot is feeding back its estimate of the collective state


Proposed approach -2-

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(t,

ix) = σ1,d + k1,c

(

σ1,d −1

N

(

1T

N ⊗ In

)

ix)

+

+J†2,j

(

σ2,d + k2,c(

σ2,d − σ2(ix)

))


Collective dynamics I

Estimation error:

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

with L Laplacian matrix embedding the topology and

x⋆ =

1x

2x

...Nx

=

x− 1x

x− 2x

...x− N

x

= 1N ⊗ x− x⋆


Collective dynamics II

Tracking error:

˙σ1 = −k1,cσ1 −k1,cN

N∑

i=1

J1ix−

k2,cN

N∑

i=1

J†2,iJ2

ix

˙σ2 = −k2,cσ2 − k2,cJ2

N∑

i=1

ΓTi J

†2,iJ2

ix+−

k1,cN

J2

N∑

i=1

ΓTi

N∑

j=1

ix


Stability proof for undirected connected topologies

Lyapunov function:

V (x, σ) =1

2x∗Tx∗ +

1

2σT1 σ1 +

1

2σT2 σ2

after straightforward computations. . .

V ≤ −

‖x∗‖‖σ1‖‖σ2‖

T

λo − 2Nkc −kc/2 −Nkc−kc/2 k1,c 0−Nkc 0 k2,c

‖x∗‖‖σ1‖‖σ2‖


Stability proof for undirected connected topologies

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

ko >1

λm

(

2Nkc +k2c

4k1,c+

Nk2c2k2,c

)

comments:

N is a known parameter

the control gains kc, k1,c, k2,c are 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 Karlsruhe, 8 May 2013

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 80−0.2

0

0.2

0.4

0.6

0 10 20 30 40 50 60 70 80−2

−1

0

1

2

centroid error

formation error


Experiments - path

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5Path of all the robots from 0

intentional large

initial error in the

state estimate


Decentralized control of dynamic centroid and

formation for multi-robot systems

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


icra 2013 talk 1

Technology