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, Italy
http://webuser.unicas.it/lai/robotica

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

‡ University of Salerno, Italy

http://www.unisa.it

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

General objective

In a multi-robot scenario
local information
local communication ⇒ global task
local controller
time-varying topology


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 dynamics
continuous-time
ﬁxed/switching communication topologies
directed/undirected graphs
saturated inputs


Modeling

N robots with n DOFs each:
Single state: xi ∈ Rn
˙
Individual dynamics: xi = ui (single-integrator dynamics)
T
1
T
Collective state: x = xT . . . xN ∈ RN n
˙
Collective dynamics: x = u
Global estimate computed by robot i: i x ∈ RN n
ˆ
1  
x − 1x

˜
x ˆ
 2x   x − 2x 
 ˜  ˆ 2
˜
Collective estimation error: x =  .  =  .  ∈ RN n
 .  
. .
. 
Nx
˜ x − Nx
ˆ


Problem statement

Task (weighted centroid)
N
σ(x) = αi xi = αT ⊗ I n x ∈ Rn
i=1

Design goals, for each robot:
state observer providing an estimate, i x ∈ RN n , asymptotically
ˆ
convergent to the collective state x
feedback control law, ui = ui (xi , i x, Ni ) ∈ Rn , such that σ(x)
ˆ
asymptotically converges to a time-varying reference, σ d (t)
˙
Each robot knows in advance: σ d (t), σ d (t)


Proposed approach -1-

i th control law:
αi
ui = ui (i x) =
ˆ σ d + kc σ d − σ(i x)
˙ ˆ
✏✏
α 2
✏✏
✏✏
✏✏
✏✏
✏
✮
each robot is feeding back its estimate of the collective state


Proposed approach -2-

local feedback
consensus-like term ✒

✻
i th state observer:



˙
ix
ˆ = ko  j
x − ix + Π i x − ix  + iu
ˆ ˆ ˆ ˆ
j∈Ni
✟
✟✟
✟
✟✟
Π i = diag O n · · · I n · · · O n
u1 (i x) ✟✟
 
ˆ
 .  ✟
✟= αj σ + k σ − σ(i x)
ˆ  . , uj (✟ )
iu =
. ✟
ix
ˆ α 2
˙d c d ˆ
i x)✟ ✟
uN ( ✟ ˆ
✟
✟
✟✟
✙
collective input estimated by robot i


Collective dynamics

Estimation error:

˙
x = −ko (L ⊗ I N n + Π) x + (1N ⊗ I N n ) u − u
˜ ˜ ˆ

with L Laplacian matrix embedding the topology
Tracking error:
N
˙ kc
σ = −kc σ −
˜ ˜ α2 αT ⊗ I n i x
i ˜
α 2
i=1


Stability proof for undirected connected topologies

Lyapunov function:
1 1
V (˜ , σ) = xT x + σT σ
x ˜ ˜ ˜ ˜ ˜
2 2
after straightforward computations. . .
√
√
 
N kc n α
ko λm − N kc n − ˜
x
˙ x ˜
V (˜ , σ) ≤ − x ˜ ˜
σ 
 √ 2 
N kc n α 
˜
σ
− kc
2


Stability proof for undirected connected topologies

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

√ 2
kc Nn α
ko > N n+
λm 4

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 Switching topologies
convergence for balanced proof by the concept of
and strongly connected Common Lyapunov
graphs Function
proof by resorting to the gains tuned on the worst
concept of mirror graph case

All the case studies above analyzed also for saturated inputs

Extensions -2-

Centroid and formation
N
1
σ1 (x) = xi
N
i=1
T
σ2 (x) = (x2 −x1 )T (x3 −x2 )T . . . (xN −xN −1 )T

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


Comments

Originality
Estimating the whole state ⇒ is it really decentralized?
Scalability (1000 robots, 10 neighbors ⇒ 8 ms 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:

3
number of robots N 4 2
dimension n
number of neighbors Ni 5 1
topology (un-directed, switching)
6 8
saturated inputs 7


Experiments there is life beyond Matlab!

5 Khepera III by K-team real-time comm.
real-time localization obstacle avoidance
various topologies initial error


Experiments - estimation errors

estimate errors w.r.t. real pos rob 0 estimate errors w.r.t. real pos rob 1
4 8

3 6

2 4

1 2

0 0
0 20 40 60 80 0 20 40 60 80

estimate errors w.r.t. real pos rob 2 estimate errors w.r.t. real pos rob 3
10 15

10
5
5

0 0
0 20 40 60 80 0 20 40 60 80

estimate errors w.r.t. real pos rob 4
8

6

4

2

0
0 20 40 60 80 100


Experiments - task error

0.6
centroid error
0.5

0.4

0.3

0.2

0.1

0

−0.1
0 10 20 30 40 50 60 70 80

1.5
formation error
1

0.5

0

−0.5

−1

−1.5
0 10 20 30 40 50 60 70 80


Experiments - path

estimate (thin) and real path seen from 4
5

4.5

4

3.5
intentional large
3
initial error in the
2.5 state estimate

2

1.5

1

0.5
0.5 1 1.5 2 2.5


Cena I-RAS

Robotics & Automation Society, Italian chapter
La Locanda dei Mestieri
Piazza Piano di Corte, ore 20.30


SIDRA 2012 talk

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