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
Antonelli,Arrichiello,Caccavale,Marino Benevento, 12 September 2012
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
Antonelli,Arrichiello,Caccavale,Marino Benevento, 12 September 2012
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
Antonelli,Arrichiello,Caccavale,Marino Benevento, 12 September 2012
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
Antonelli,Arrichiello,Caccavale,Marino Benevento, 12 September 2012
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
Antonelli,Arrichiello,Caccavale,Marino Benevento, 12 September 2012
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)
Antonelli,Arrichiello,Caccavale,Marino Benevento, 12 September 2012
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
Antonelli,Arrichiello,Caccavale,Marino Benevento, 12 September 2012
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
Antonelli,Arrichiello,Caccavale,Marino Benevento, 12 September 2012
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(
ix) =
αj
‖α‖2
(
σd + kc(
σd − σ(ix)))
Antonelli,Arrichiello,Caccavale,Marino Benevento, 12 September 2012
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(
ix) =
αj
‖α‖2
(
σd + kc(
σd − σ(ix)))
Antonelli,Arrichiello,Caccavale,Marino Benevento, 12 September 2012
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(
ix) =
αj
‖α‖2
(
σd + kc(
σd − σ(ix)))
Antonelli,Arrichiello,Caccavale,Marino Benevento, 12 September 2012
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(
ix) =
αj
‖α‖2
(
σd + kc(
σd − σ(ix)))
Antonelli,Arrichiello,Caccavale,Marino Benevento, 12 September 2012
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
Antonelli,Arrichiello,Caccavale,Marino Benevento, 12 September 2012
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‖
]
Antonelli,Arrichiello,Caccavale,Marino Benevento, 12 September 2012
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
Antonelli,Arrichiello,Caccavale,Marino Benevento, 12 September 2012
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
Antonelli,Arrichiello,Caccavale,Marino Benevento, 12 September 2012
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?
Antonelli,Arrichiello,Caccavale,Marino Benevento, 12 September 2012
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
Antonelli,Arrichiello,Caccavale,Marino Benevento, 12 September 2012
Experiments there is life beyond Matlab!
5 Khepera III by K-team
real-time localization
various topologies
real-time comm.
obstacle avoidance
initial error
Antonelli,Arrichiello,Caccavale,Marino Benevento, 12 September 2012
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
8estimate errors w.r.t. real pos rob 1
0 20 40 60 800
5
10estimate errors w.r.t. real pos rob 2
0 20 40 60 800
5
10
15estimate errors w.r.t. real pos rob 3
0 20 40 60 80 1000
2
4
6
8estimate errors w.r.t. real pos rob 4
Antonelli,Arrichiello,Caccavale,Marino Benevento, 12 September 2012
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
Antonelli,Arrichiello,Caccavale,Marino Benevento, 12 September 2012
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
Antonelli,Arrichiello,Caccavale,Marino Benevento, 12 September 2012
Cena I-RAS
Robotics & Automation Society, Italian chapter
La Locanda dei Mestieri
Piazza Piano di Corte, ore 20.30
Antonelli,Arrichiello,Caccavale,Marino Benevento, 12 September 2012
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