IROS 2011 talk 2 (Filippo's file)

A decentralized controller-observer scheme
for multi-robot weighted centroid tracking

Gianluca Antonelli1 , Filippo Arrichiello1 ,
Fabrizio Caccavale2 , Alessandro Marino1

1.University of Cassino, Italy
Robotics Research Group of the DAEIMI
http://webuser.unicas.it/lai/robotica

2.University of Basilicata, Italy
AREA Lab of the DIFA
http://www.difa.unibas.it

G. Antonelli, F. Arrichiello, F. Caccavale, A. Marino IEEE/RSJ IROS, San Francisco, 28 September 2011

Scenario

◮ Each robot uses only local information from its sensors and
◮ Each robot can communicate with its only neighbors

Aim
◮ Make the multi-robot system able to achieve a global task using a
decentralized control approach
◮ Approach robust to time-varying communication topologies


Outline

Decentralized controller-observer for weighted centroid tracking
◮ Develop for each robot a local observer of collective system state
◮ Local estimations used by local control laws
◮ Cooperatively track an assigned time-varying reference for weighted
centroid
◮ Convergence proof for ﬁxed/switching communication topologies, as
well as for directed/undirected graphs
◮ Validation via numerical simulations


Modeling

Modeling for a team of N robots
◮ Individual robot state: xi ∈ IRn
T
◮ Collective state: x = xT
1 . . . xT
N ∈ IRNn
◮ Global estimate computed by agent i: i x ∈ IRNn
ˆ
1   1

˜
x x− ˆ x
 2˜   x − 2ˆ 
 x  x
◮ Estimation error: ˜ =  .  =  . 
x
 .   . 
. .
N
˜
x x − Nˆ x
◮ ˙
Individual dynamics: xi = ui (single-integrator dynamics)
◮ ˙
Collective dynamics: x = u


Problem statement

The task for the system of agents is encoded by the smooth
function σ ∈ IRn (weighted centroid)
N
◮ σ(x) = i =1 αi xi = αT ⊗ In x
Main design goals:
◮ design, for each agent, a state observer providing an estimate,
x ∈ IRNn , asymptotically convergent to the collective state x
iˆ

◮ to design, for each agent, a feedback control law,
ui = ui (xi , i ˆ, Ni ) , such that σ(x) asymptotically converges to a
x
given (in general time-varying) reference, σ d (t)
Each agent knows in advance the goal: σ d (t), σ d (t)
˙


Proposed approach

Control law for the ith agent:
αi
◮ ui = ui (i ˆ) =
x α 2 σ d + kc σ d − σ(i ˆ)
˙ x
State observer for the ith agent:
˙
◮ ix
ˆ = ko jˆ
x − i ˆ + Πi x − i ˆ
x x + iu
ˆ
j∈Ni

u1 (i ˆ)
 
x
where i u =  . , uj (i ˆ) =
αj
σ d + kc σ d − σ(i ˆ)
 . 
ˆ . x α 2 ˙ x
uN (i ˆ)
x
and Π i = diag On ··· In ··· On

ko , kc > 0 gain to be selected to ensure convergence


Proposed approach

The estimation error dynamics:
◮ ˙
x = −ko (L ⊗ INn + Π) ˜ + (1N ⊗ INn ) u − u
˜ x ˆ
L=Laplacian matrix of the communication graph

The task tracking error dynamics:
˙ kc N 2
◮ σ = −kc σ −
˜ ˜ α 2 i =1 αi αT ⊗ In i x
˜


Stability proof for undirected connected topologies

Candidate Lyapunov function:
◮ V (˜, σ) = Vo + Vc = 1 ˜T ˜ + 1 σ T σ
x ˜ 2x x 2˜ ˜

Time derivative of Vo along the system’s trajectories:
◮ ˙
Vo = −ko ˜T (L ⊗ INn + Π) ˜ + ˜T ((1N ⊗ INn ) u − u)
x x x ˆ
(L ⊗ INn + Π) positive deﬁnite for undirected and connected topologies
˙ 2 N i T
◮ Vo ≤ −ko λm x
˜ + i =1 ˜ (u
x − i u)
ˆ
where λm is the smallest eigenvalue of (L ⊗ INn + Π)
√
◮ Vo ≤ − ko λm − 2Nkc n ˜ 2
˙ x


Stability proof for undirected connected topologies

Time derivative of Vc along the system’s trajectories:
√
◮ Vc = σ T σ ≤ −kc σ 2 + 2Nkc n α /2 σ
˙ ˜ ˜˙ ˜ ˜ ˜ x

For the overall derivative of the candidate Lyapunov function, yields:
T
˙ λo − ρo −ρc
˜
x ˜
x
◮ V ≤−
˜
σ −ρc kc ˜
σ
√ √
where ρo = Nkc n, λo = ko λm and ρc = Nkc n α /2
Negative deﬁnite with a proper choice of the design gains ko and kc :
√ α 2
◮ ko > N kc
λm n + Nn 4


Directed topologies

Stability preserved for balanced and strongly connected
directed graph
◮ The Laplacian L is not symmetric
◮ Mirror graph GS = same set of nodes and edges of G but undirected
L + LT
◮ Symmetric part of the Laplacian, LS = , valid Laplacian for
2
GS , if and only if G is balanced.
If G is strongly connected, then GS is connected
Then, it yields:
◮ ˙
Vo = −ko ˜T (LS ⊗ INn + Π) ˜ + ˜T ((1N ⊗ INn ) u − u)
x x x ˆ
where (LS ⊗ INn + Π) is positive deﬁnite. The rest of the proof is
analogous


Switching topologies

Time-varying network topology
◮ Network topology described via
ﬁnite collection of K graphs of order N, Γ = {G1 , . . . , GK }, each
characterized by adjacency matrix Ak , k ∈ I = {1, . . . , K }
◮ Adjacency matrix modeled as function of time A = As(t) , where
s(·) : t ∈ IR → I is a switching signal
◮ Previous Lyapunov function is a Common Lyapunov Function for
any switching signal s(t), if each graph in Γ balanced and strongly
(directed) or simply (undirected) connected
◮ Tuning of ko and kc according to the worst case scenario, i.e.,
considering minimum value of λm over the ﬁnite set of topologies


Numerical simulations

First case study: fixed undirected topology
◮ As first case study, a fixed undirected topology with 4 vehicles
(N = 4) moving in a plane (n = 2)



First case study: ﬁxed undirected topology

Estimation error
6

5 4

m
4 2

3 0
1 2 3 4 5 6 7
time [s]
2
Task error
m

1 1.5

Sigma
0 1
Sigmad

m
Start 0.5
−1
End
−2 0
0 5 10 15 20 25 1 2 3 4 5 6 7 8
m time [s]



Second case study: directed switching topology
◮ As second case study, a switching directed topology with 8 vehicles
(N = 8) moving in the 3D-space (n = 3)

t=2s t=4s



Second case study: directed switching topology

7
Estimation error
4 6
5
3 4

m
3
2
2
1
z [m]

1
0
0 1 2 3 4 5 6 7
0 time [s]

Sigma
Task error
−1
Sigmad 0.5
Start
−2 0.4
6 End
0.3

m
4 0.2
30
2 25 0.1
20
15 0
0 10
5
0 −0.1
−2 −5 0 1 2 3 4 5 6 7
y [m]
x [m] time [s]


Conclusions

◮ Decentralized controller-observer approach for a multi-robot system
◮ Each robot estimates the collective state of the system by using
only local information
◮ The estimated state is used by the individual robots to
cooperatively track a global assigned time-varying task function
◮ Stability proof presented for the cases of undirected connected
topologies, strongly connected and balanced directed graphs, as well
as for switching topologies
◮ Validated in numerical simulations


Future works

◮ Extend the class of achievable task functions
◮ Make robust to dynamic lost/addition of robots
◮ Improve the scalability of the approach
◮ Extend to the case of non-holonomic vehciles
◮ Experimental validation with a team of khepera III robots


IROS 2011 talk 2 (Filippo's file)

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IROS 2011 talk 2 (Filippo's file)