Consensus-based Source-seeking with a Circular Formation ... · Consensus-based Source-seeking with...

Consensus-based Source-seeking with a CircularFormation of Agents

Lara Brinon Arranz and Luca Schenato

ISR, Instituto Superior Tecnico, Lisboa & Universita di Padova

GIPSA-lab, Grenoble19th December 2013

Context and Motivations Problem Formulation Preliminaries Distributed Source-seeking Simulations Conclusions

Introduction

About myself:

Sep. 2008: Master degree in Automation and ElectronicsEngineering, 2008, UPM, Spain

Nov. 2011: Ph.D. degree in Automatic Control, University ofGrenoble, France (GIPSA-lab/INRIA)

2011-12: ATER, Grenoble INP, France

Currently: Post-doctoral scholar, Instituto Superior Tecnico, Lisbon,Portugal

Ph.D. thesis

Cooperative control design for a fleet of Autonomous UnderwaterVehicles under communication constraints

- Advisors: Carlos Canudas de Wit and Alexandre Seuret- CONNECT and FeedNetBack projects

GIPSA-lab Consensus-based Source-seeking with a Circular Formation of Agents 2 / 23


Introduction

About myself:

Sep. 2008: Master degree in Automation and ElectronicsEngineering, 2008, UPM, Spain

Nov. 2011: Ph.D. degree in Automatic Control, University ofGrenoble, France (GIPSA-lab/INRIA)

2011-12: ATER, Grenoble INP, France

Currently: Post-doctoral scholar, Instituto Superior Tecnico, Lisbon,Portugal

Ph.D. thesis

Cooperative control design for a fleet of Autonomous UnderwaterVehicles under communication constraints

- Advisors: Carlos Canudas de Wit and Alexandre Seuret- CONNECT and FeedNetBack projects



FeedNetBack Project

-Networked Control Systems- 7th Framework Programme of the European Commission

- Partners:

Universita di Padova

Universidad de Sevilla

KTH Stockholm

ETH Zurich

INRIA Grenoble

Ifremer, France

Case Study: Autonomous Underwater Vehicles (AUVs)

Source-seeking task: Locating and following the source of ascalar field of interest



Context and Motivations

Source-seeking problem: To locate and follow the source ofa scalar field of interest (temperature, salinity, pollutant flow).Signal strength σ(z) : R2 → R

Applications: environmental monitoring, rescue operations,pollution sensing, sound source localization

Strategy: Gradient-descent methods[Bachmayer and Leonard 2002]

⇒ the gradient information is usually unknown

⇒ estimation of the gradient by collecting spatiallydistributed measurements



















Gradient estimation: Single vehicle

Extremum seeking techniques[Cochran and Krstic 2009]

Stochastic extremum seeking[Liu and Krstic 2010]

Stochastic gradient-descent[Atanasov et al. 2012] SOURCE

∇σ(c) c

Main disadvantage

the vehicle may have to travel over large distances



Gadient estimation: Group of vehicles

Model parameters estimation[Fiorelli et al. 2003, Ogren et al.

2004]

Distributed estimation[Sahyoun et al. 2010, Li and Guo

2012]

Circular formation of AUVs[Moore and Canudas de Wit 2010]

ω0

r1 − c

r2 − cr4 − c

r3 − c

SOURCE

∇σ(c)

Drawbacks

- a priori model of the signal distribution- assumption about the spatial propagation of the signal



Problem formulation

Signal strength

Unknown positive spatial mapping σ(z) : R2 → R- the source, located at z∗, is the only maximum of the scalar field

Uniformly distributed circularformation of N agents:

ri (k) = ci (k) + DR(φi )e

with radius D > 0 and where- φi = φ0 + i 2π

N

- R(φ) =

[cosφ − sinφsinφ cosφ

]- e = [1 0]T

SOURCE∇σ(c)

r1 − c

r2 − cr3 − c

r4 − c



Control Objectives

1 Estimating the gradient- approximation of ∇σ(c) = [∇xσ(c),∇yσ(c)]T

- the gradient direction will be the reference velocity for theformation

2 Keeping the circular formation of agents

ci → c ∀i3 Steering the formation towards the source location

c → z∗



Control Objectives




ci → c ∀i

3 Steering the formation towards the source location

c → z∗



Control Objectives




ci → c ∀i3 Steering the formation towards the source location

c → z∗



Contribution

Previous works: [Brinon-Arranz et al. CDC’11], [Moore and Canudas ACC’10]

Gradient estimation: ∇σ(c) ∝ 1N

∑Ni=1 σ(ri )(ri − c)

Centralized Source Seeking control in continuous time:

c =1

N

N∑i=1

σ(ri )(ri − c)

Novel contribution: [Brinon-Arranz and Schenato ECC’13]

Distributed rotation center control for each agent

Discrete time control

Local communication among agents

ci (k + 1) = ci (k) + ui (k), ui (k) = f (ci−1, ci+1)



Contribution

Previous works: [Brinon-Arranz et al. CDC’11], [Moore and Canudas ACC’10]

Gradient estimation: ∇σ(c) ∝ 1N

∑Ni=1 σ(ri )(ri − c)

Centralized Source Seeking control in continuous time:

c =1

N

N∑i=1

σ(ri )(ri − c)

Novel contribution: [Brinon-Arranz and Schenato ECC’13]

Distributed rotation center control for each agent

Discrete time control

Local communication among agents

ci (k + 1) = ci (k) + ui (k), ui (k) = f (ci−1, ci+1)



Gradient Approximation

Approximation of the gradient of the signal distribution

- Considering a fleet of N > 2 agentsuniformly distributed along a circularformation of radius D centered at c .

- Each agent is able to collect sig-nal measurements at its position, asσ(ri ). SOURCE

r1 − c

r2 − cr3 − c

r4 − c

Lemma: Gradient Approximation [Brinon-Arranz et al. CDC’11]

1

N

N∑i=1

σ(ri )(ri − c) =D2

2∇σ(c) + o(D2)







r1 − c

r2 − cr3 − c

r4 − c

σ(r1)(r1 − c)


1

N

N∑i=1

σ(ri )(ri − c) =D2

2∇σ(c) + o(D2)







r1 − c

r2 − cr3 − c

r4 − c

σ(r2)(r2 − c)


1

N

N∑i=1

σ(ri )(ri − c) =D2

2∇σ(c) + o(D2)







r1 − c

r2 − cr3 − c

r4 − c

σ(r3)(r3 − c)


1

N

N∑i=1

σ(ri )(ri − c) =D2

2∇σ(c) + o(D2)







r1 − c

r2 − cr3 − c

r4 − c

σ(r4)(r4 − c)


1

N

N∑i=1

σ(ri )(ri − c) =D2

2∇σ(c) + o(D2)







∇σ(c)

r1 − c

r2 − cr3 − c

r4 − c


1

N

N∑i=1

σ(ri )(ri − c) =D2

2∇σ(c) + o(D2)




Proof:Multi-variable Taylor series expansion of σ at c :

σ(ri )− σ(c) = ∇σ(c)T (ri − c) + o(D)

By multiplying by the relative vector (ri − c) and summing over i :

1

N

N∑i=1

σ(ri )(ri−c)+σ(c)1

N

N∑i=1

(ri−c) =1

N

N∑i=1

(ri−c)(ri−c)T∇σ(c)+o(D2)

thanks to the uniform distribution then∑N

i=1(ri − c) = 0, and usingtrigonometric properties:

N∑i=1

(ri − c)(ri − c)T = D2N∑i=1

R(φi )eeTR(φi )

T =ND2

2I2

and then previous equation holds.



Distributed solution

All-to-all communicationThanks to previous Lemma, at each instant k each agent cancompute the gradient estimation

1

N

N∑i=1

σ(ri )(ri (k)− c(k))

Limited communication- Undirected communication graph G = (V ,E )- At each instant k each agent computes its position ri (k), itscenter ci (k) and its estimated gradient vector

fi (k) = σ(ri )(ri (k)− ci (k))



Algorithm

Distributed source-seeking algorithm

for i = 1, . . . ,N dohi (0) = gi (0) = gi (−1) = ci (0) + σ(ri (0))(ri (0)− ci (0)) initialization

for k = 1, 2, . . . dofor i = 1, . . . ,N dogi (k) = ci (k) + σ(ri (k))(ri (k)− ci (k)) gradient estimationgi (k) = (1− α)gi (k − 1) + αgi (k) low-pass filterhi (k) = hi (k − 1) + gi (k − 1)− gi (k − 2) local estimate of z∗

h(k) = (P ⊗ I2)h(k) consensus

for i = 1, . . . ,N doci (k) = (1− ε)ci (k − 1) + εhi (k)

- P ∈ RN×N is a doubly stochastic matrix consistent with graph G- Separation of time scales is regulated by ε ∈ (0, 1]- Low-pass filter is regulated by α ∈ (0, 1]



Theorem

Distributed Source-seeking

Theorem

Let consider previous algorithm and ‖ci (0)− z∗‖ < r for somearbitrary r > 0, then there exists ε such that for all ε ∈ (0, ε):

limk→∞

ci (k)− cj(k) = 0, ∀i , j

Moreover, all the centers ci converge asymptotically to theneighborhood of the maximum of the signal distribution σ(z)located at z∗, s.t. when k →∞:

‖ci (k)− z∗‖ ≤ β(D), ∀i

limD→0

β(D) = 0



Theorem

Sketch of the proof: singular perturbation model analysis


gi (k) = ci (k) + σ(ri (k))(ri (k)− ci (k)) fastgi (k) = (1− α)gi (k − 1) + αgi (k) dynamicshi (k) = hi (k − 1) + gi (k − 1)− gi (k − 2)

h(k) = (P ⊗ I2)h(k)

ci (k) = (1− ε)ci (k − 1) + εhi (k) slow dynamics

Fast Dynamics:ε = 0,=⇒ ci (k) = ci (0) = ci

=⇒ ri (k) = ri (0) = ri for all k ≥ 0.=⇒ gi (k) = ci + σ(ri )(ri − ci ) =⇒ gi (k) = ci + σ(ri )(ri − ci ) ∀α ∈ (0, 1]=⇒ hi (k) = hi (k − 1)=⇒ h(k) = (P ⊗ I2)h(k − 1), hi (1) = ci + σ(ri )(ri − ci )

limk→∞

hi (k) =1

N

N∑i=1

(ci + σi (ri )(ri − ci )) = h(c)

exponentially fast with rate given by esr(P).



Theorem




h(k) = (P ⊗ I2)h(k)


Fast Dynamics:ε = 0,=⇒ ci (k) = ci (0) = ci =⇒ ri (k) = ri (0) = ri for all k ≥ 0.

=⇒ gi (k) = ci + σ(ri )(ri − ci ) =⇒ gi (k) = ci + σ(ri )(ri − ci ) ∀α ∈ (0, 1]=⇒ hi (k) = hi (k − 1)=⇒ h(k) = (P ⊗ I2)h(k − 1), hi (1) = ci + σ(ri )(ri − ci )

limk→∞

hi (k) =1

N

N∑i=1

(ci + σi (ri )(ri − ci )) = h(c)




Theorem




h(k) = (P ⊗ I2)h(k)


Fast Dynamics:ε = 0,=⇒ ci (k) = ci (0) = ci =⇒ ri (k) = ri (0) = ri for all k ≥ 0.=⇒ gi (k) = ci + σ(ri )(ri − ci )

=⇒ gi (k) = ci + σ(ri )(ri − ci ) ∀α ∈ (0, 1]=⇒ hi (k) = hi (k − 1)=⇒ h(k) = (P ⊗ I2)h(k − 1), hi (1) = ci + σ(ri )(ri − ci )

limk→∞

hi (k) =1

N

N∑i=1

(ci + σi (ri )(ri − ci )) = h(c)




Theorem




h(k) = (P ⊗ I2)h(k)


Fast Dynamics:ε = 0,=⇒ ci (k) = ci (0) = ci =⇒ ri (k) = ri (0) = ri for all k ≥ 0.=⇒ gi (k) = ci + σ(ri )(ri − ci ) =⇒ gi (k) = ci + σ(ri )(ri − ci ) ∀α ∈ (0, 1]

=⇒ hi (k) = hi (k − 1)=⇒ h(k) = (P ⊗ I2)h(k − 1), hi (1) = ci + σ(ri )(ri − ci )

limk→∞

hi (k) =1

N

N∑i=1

(ci + σi (ri )(ri − ci )) = h(c)




Theorem




h(k) = (P ⊗ I2)h(k)


Fast Dynamics:ε = 0,=⇒ ci (k) = ci (0) = ci =⇒ ri (k) = ri (0) = ri for all k ≥ 0.=⇒ gi (k) = ci + σ(ri )(ri − ci ) =⇒ gi (k) = ci + σ(ri )(ri − ci ) ∀α ∈ (0, 1]=⇒ hi (k) = hi (k − 1)

=⇒ h(k) = (P ⊗ I2)h(k − 1), hi (1) = ci + σ(ri )(ri − ci )

limk→∞

hi (k) =1

N

N∑i=1

(ci + σi (ri )(ri − ci )) = h(c)




Theorem




h(k) = (P ⊗ I2)h(k)


Fast Dynamics:ε = 0,=⇒ ci (k) = ci (0) = ci =⇒ ri (k) = ri (0) = ri for all k ≥ 0.=⇒ gi (k) = ci + σ(ri )(ri − ci ) =⇒ gi (k) = ci + σ(ri )(ri − ci ) ∀α ∈ (0, 1]=⇒ hi (k) = hi (k − 1)=⇒ h(k) = (P ⊗ I2)h(k − 1), hi (1) = ci + σ(ri )(ri − ci )

limk→∞

hi (k) =1

N

N∑i=1

(ci + σi (ri )(ri − ci )) = h(c)




Theorem




h(k) = (P ⊗ I2)h(k)


Fast Dynamics:ε = 0,=⇒ ci (k) = ci (0) = ci =⇒ ri (k) = ri (0) = ri for all k ≥ 0.=⇒ gi (k) = ci + σ(ri )(ri − ci ) =⇒ gi (k) = ci + σ(ri )(ri − ci ) ∀α ∈ (0, 1]=⇒ hi (k) = hi (k − 1)=⇒ h(k) = (P ⊗ I2)h(k − 1), hi (1) = ci + σ(ri )(ri − ci )

limk→∞

hi (k) =1

N

N∑i=1

(ci + σi (ri )(ri − ci )) = h(c)

exponentially fast with rate given by esr(P).GIPSA-lab Consensus-based Source-seeking with a Circular Formation of Agents 15 / 23


Theorem

Slow Dynamics:If we insert the fast dynamics hi (k) = h(c) into the slow dynamics we get

ci (k) = (1− ε)ci (k − 1) + εh(c(k − 1))

=⇒ limk→∞

ci (k)− cj(k) = 0

Therefore we restrict our attention to the scenario whereci (k) = c(k),∀i :

h(c) = c +1

N

N∑i=1

σ(ri (c))(ri (c)− c) ≈ c(k) +D2

2∇σ(c)

And thus the dynamics of c are given by

c(k + 1) = (1− ε)c(k) + ε(c(k) + D2

2 ∇σ(c) + o(D2))

= c(k) + εD2

2 ∇σ(c) + εo(D2)

which is the standard gradient-ascent update away from the error o(D2).



Theorem

Slow Dynamics:If we insert the fast dynamics hi (k) = h(c) into the slow dynamics we get

ci (k) = (1− ε)ci (k − 1) + εh(c(k − 1))

=⇒ limk→∞

ci (k)− cj(k) = 0

Therefore we restrict our attention to the scenario whereci (k) = c(k),∀i :

h(c) = c +1

N

N∑i=1

σ(ri (c))(ri (c)− c) ≈ c(k) +D2

2∇σ(c)

And thus the dynamics of c are given by

c(k + 1) = (1− ε)c(k) + ε(c(k) + D2

2 ∇σ(c) + o(D2))

= c(k) + εD2

2 ∇σ(c) + εo(D2)

which is the standard gradient-ascent update away from the error o(D2).GIPSA-lab Consensus-based Source-seeking with a Circular Formation of Agents 16 / 23


Theorem

- Fast dynamics (consensus):

limk→∞

hi (k) =1

N

N∑i=1

(ci + σi (ri )(ri − ci )) = h(c)

- Slow dynamics (gradient ascent):

limk→∞

ci (k) = c ⇒ h(c) = c + f (c)

c(k + 1) = c(k) + εD2

2∇σ(c) + εo(D2)

In conclusion:

If ε is sufficiently small, then the separation of time-scale holds andlimk→∞ ci (k) = limk→∞ c(k) which converges to the neigborhoodof z∗.



Simulation results

Case without noise:The scalar field is a combination of two ellipsis and thus with non convexlevel curves whose maximum is z∗ = [0, 0]T .

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000−5

0

5

10

15

20

25

30

35

40

45

ε=0.5

ε=0.8

ε=0.1

α = 1

−5 0 5 10 15 20 25 30 35 40 45 50−5

0

5

10

15

20

25

30

35

40

45

50

ε = 0.5, α = 1



Simulation results

Case with noise:The signal measurements are corrupted by zero-mean Gaussian noiseN (0, 0.2)

0 2000 4000 6000 8000 10000 12000−5

0

5

10

15

20

25

30

35

40

45

α=0.5α=0.1

α=0.8

ε = 0.5

−5 0 5 10 15 20 25 30 35 40 45 50−5

0

5

10

15

20

25

30

35

40

45

50

ε = 0.5, α = 0.5



Conclusions and Future works

Conclusions

Gradient approximation via a circular formation of agents

No previous acknowledgement of the signal is assumed.

Distributed source-seeking algorithm based on DistributedGradient Descent Consensus

Two tunable parameters ε, α: tradeoff rate of convergence,robustness to noisy measurements and formation stability.

Detailed analysis of the proof and asynchronouscommunication in a new paper submitted to TNCS.

Perspectives

Extension to the 3-dimensional case

Control of the radius Di (k)



Conclusions and Future works

Conclusions

Gradient approximation via a circular formation of agents

No previous acknowledgement of the signal is assumed.

Distributed source-seeking algorithm based on DistributedGradient Descent Consensus

Two tunable parameters ε, α: tradeoff rate of convergence,robustness to noisy measurements and formation stability.

Detailed analysis of the proof and asynchronouscommunication in a new paper submitted to TNCS.

Perspectives

Extension to the 3-dimensional case

Control of the radius Di (k)



MORPH project (http://morph-project.eu/)

-Marine robotic system of self-organizing, logically linkedphysical nodes- 7th Framework Programme of the European Commission- Partners from Germany, Italy, Portugal, France and Spain

Underwater Robotic System

-Combination of different mobilerobot-modules with distinct andcomplementary resources.- To develop methods to map theunderwater environment withgreat accuracy in complexsituations



IST Lisboa

Range-Only Formation Control (ROF)

- Multiple vehicle cooperation is required- Low communication throughput (bandwidth constraints)



Thank you for your attention

e-mail: [email protected]: https://sites.google.com/site/lbrinonarranz/


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