COORDINATED PATH FOLLOWING of MULTIPLE...

António M. [email protected]

Institute for Systems and Robotics (ISR)IST, Lisbon, Portugal

Pre-CDC WORKSHOP San Diego, USA, Dec. 12, 2006

COORDINATED PATH FOLLOWING of MULTIPLE UNDERACTUATED VEHICLES WITHCOMMUNICATION CONSTRAINTS

COORDINATED PATH FOLLOWING of MULTIPLE UNDERACTUATED VEHICLES WITHCOMMUNICATION CONSTRAINTS

Presentation based on the compilation of joint work with

Aguiar, António IST/ISR, PT

Ghabchelloo, Reza IST/ISR, PT

Hespanha, João UCSB, USA

Kaminer, Isaac NPS, USA

Pascoal, António IST/ISR, PT

Silvestre, Carlos IST/ISR, PT

OutlineOutline

Motivating applicationsMotivating applicationsCoordinated path following (definition)Coordinated path following (definition)Coordinated path following (wheeled robots): Coordinated path following (wheeled robots): fixed communication topologiesfixed communication topologiesCoordinated path followingCoordinated path following

general setgeneral set--upup for path followingfor path followingunderactuatedunderactuated autonomous vehiclesautonomous vehicles

switchingswitching communication topologiescommunication topologies

AUV

ASC

Surface and underwater vehicles required to operate in a master / slave configuration

Coordinated Motion ControlCoordinated Motion Control

Dopplerlog

Scanningsonar

low baud rate acousticcommunication link(commands/data)

high baud rate acousticcommunication link

(data)

Autonomous SurfaceCraft (ASC)

Autonomous UnderwaterVehicle (AUV)

Support ShipUnit (SSHU)

Radio Link

Radio Link

Radio Link

low baud rate acousticcommunication link

(emergency commands)

DifferentialGPS- Mobile Station -DifferentialGPS

- Reference Station -

Shore StationUnit (SSTU)

GIB System(buoy 1 of 4)

Radio Link

The ASIMOV concept (project ASIMOV, EC - 2000)


Two AUVS carrying out a joint survey operation



The quest for midThe quest for mid--water column hydrothermal vents, Azores, PTwater column hydrothermal vents, Azores, PT

AUV Fleet –Methane gradient “descent”

Deep water hydrothermal vent

Methane plume

MicrosatellitesMicrosatellites: imaging, remote sensing, : imaging, remote sensing, interferometryinterferometry

Autonomous Underwater Vehicles (AUVs)Autonomous Underwater Vehicles (AUVs)::coordinated bathymetric mapping, seafloor imagingcoordinated bathymetric mapping, seafloor imaging

Autonomous Surface Vessels (ASVs)Autonomous Surface Vessels (ASVs)::coordinated ocean surveyingcoordinated ocean surveying

AUVs and ASVsAUVs and ASVs: coordinated control: coordinated control for fast for fast communications and reliable navigationcommunications and reliable navigation

Scientific ApplicationsScientific Applications

Unmanned Air Vehicles(UAVs):Unmanned Air Vehicles(UAVs):

Formation control Formation control for increased aerodynamic for increased aerodynamic efficiency efficiency Coordinated search and rescue operationsCoordinated search and rescue operations

Low cost small autonomous robots:Low cost small autonomous robots:Exploration, mine detection, and neutralizationExploration, mine detection, and neutralization

Cooperative manipulationCooperative manipulation

Other applicationsOther applications

How it all started at IST (1998) How it all started at IST (1998) -- ASIMOVASIMOV

Theoretical problems: key issuesTheoretical problems: key issues

Coordinated Path FollowingCoordinated Path Following while keeping interwhile keeping inter--vehicle vehicle geometric constraintsgeometric constraints

Motion control in the presence of severe acoustic Motion control in the presence of severe acoustic communication constraintscommunication constraints (multipath, failures, latency, (multipath, failures, latency, asynchronous comms, reduced bandwidth...)asynchronous comms, reduced bandwidth...)

Dream “Reality” (IST-NPS mission)

Path Following

Inspired by the work of Inspired by the work of Claude Samson et alClaude Samson et al. for wheeled robots. for wheeled robots

A. Micaelli and C. Samson (1992). Path following and time-varying feedback stabilization of a wheeled robot. In Proc. International Conference ICARCV’92, Singapore.

. Use forward motion to make the robot track a desired speed profile.

. Compute the closest point on the path.

. Compute the Serret-Frenet (SF) frame at that point.

. Use rotational motion to align the body-axis with the SF frame and reduce the distance to closest point to zero.

Path Following

“Nonlinear Path Following with Applications to the Control of Autonomous Underwater Vehicles,” L. Lapierre, D. Soetanto, and A. Pascoal, 42th IEEE Conference on Decision and Control, Hawai, USA, Dec. 2003

Avoiding Singularities!Avoiding Singularities!

Important related work

R. Skjetne, T. I. Fossen,P. V. Kokotovic.Robust output maneuvering for a class of nonlinear systems.Automatica, 40(3):373—383, 2004.

“Rabbit” moving along the path

Coordinated AUV / ASC behavior

(in R. Hindman and J. Hauser, Maneuver Modified Trajectory Tracking, Proceedings of MTNS’96, International Symposium on the Mathematical Theory of Networks and Systems, St. Louis, MO, USA, June 1992)

Exploring an elegant concept introduced in Exploring an elegant concept introduced in

“Combined Trajectory Tracking and Path Following: an Application to the Coordinated Control of Autonomous Marine Craft,” P. Encarnação and A. Pascoal, 40th IEEE Conference on Decision and Control, Orlando, Florida, USA, Dec. 2001

Solution is too complex!

Too muchdata exchanged between the vehicles

Coordinated Path FollowingCoordinated Path Following

PATHS (HIGHWAYS TO BE FOLLOWED)

Initial configuration

(a fresh start)

Reach (in-line) FORMATION at a

desired speed

Lv !

Divide to Conquer ApproachDivide to Conquer ApproachIN-LINE FORMATION

Each vehicle runs its own PATH FOLLOWING

controller to steer itself to the path

Vehicles TALK and adjust theirSPEEDS in order to COORDINATE

themselves (reach formation)

Coordination error

Coordination error(in-line formation):

Path lengths and

Coordination state / errorCoordination state / error

2s

1s

12 1 2s s s= −

12s

1s 2s

Coordinated Path Following (using the Coordinated Path Following (using the ““interinter--rabbitrabbit”” distance)distance)

L. Lapierre, D. Soetanto, and A. PascoalL. Lapierre, D. Soetanto, and A. Pascoal (2003). Coordinated (2003). Coordinated Motion Control of Marine Robots. Motion Control of Marine Robots. Proc. 6th IFAC Conference Proc. 6th IFAC Conference on Manoeuvering and Control of Marine Craft (MCMC2003)on Manoeuvering and Control of Marine Craft (MCMC2003), , Girona, Spain.Girona, Spain.

R. Skjetne, I.R. Skjetne, I.--A. F. Ihle, and T. I. FossenA. F. Ihle, and T. I. Fossen (2003) Formation (2003) Formation Control by Synchronizing Multiple Maneuvering Systems. Control by Synchronizing Multiple Maneuvering Systems. Proc. 6th IFAC Conference on Manoeuvering and Control of Proc. 6th IFAC Conference on Manoeuvering and Control of Marine Craft (MCMC2003)Marine Craft (MCMC2003), Girona, Spain., Girona, Spain.

M. Egerstedt and X. HuM. Egerstedt and X. Hu (2001) Formation Constrained Multi(2001) Formation Constrained Multi--Agent Control, IEEE Trans. on Robotics and auto., vol. 17, no. Agent Control, IEEE Trans. on Robotics and auto., vol. 17, no. 6, Dec. 20016, Dec. 2001

They They do not address communication constraintsdo not address communication constraintsexplicitly.explicitly.

Communication Constraints

What is the communications topology? (GRAPH)

Non-bidirectional linksdirected graphs

Bidirectional Linksundirected graphs

vehicle

link

R. Murray [2002], B. Francis [2003], A. Jadbabaie [2003]

Communication Constraints

Communication Delays

Temporary Loss of Comms

Switching Comms Topology

Asynchronous Comms

Links with Networked Control and Estimation Theory

In depth analysisIn depth analysis

SINGLE VEHICLE, PATH FOLLOWING

“guide” (rabbit) moving along the path – “a mind

of its own” (control variable)

This will make the vehicle follow the path

1. Drive the distance from Q to the rabbit to zero;2. Align the flow frame with the Serret-Frenet(align total velocity with the tangent to the path).tv

COORDINATED PATH FOLLOWING

More general formations and paths

Triangle formation In-line formation


Generalizable to multiple vehicles and other formation patterns, and paths

•

•

•

•

Coordination Error =error between the “rabbits”


KEY INGREDIENTS:

PATH FOLLOWING for each vehicle

+Inter-vehicle COORDENATION

(driving the coordination errors to zero:speed adjustments based on

VERY LITTLE INFO EXCHANGED)

(space-time decoupling … maths work out!)

Divide to Conquer ApproachDivide to Conquer Approach

PATH FOLLOWING(each vehicle on its

own) , PF

ALONG-PATH COORDINATION,

CC

+COORDINATED

PATH FOLLOWING CPF

But, they co-exist.Analyze in detail!

Key resultsKey results

Coordination achieved with

fixed communication networks (ICAR’05, CDC’05, IJACSP)

brief connectivity losses, general comm. losses, and time delays

(SIAM-to be submitted, CDC’06, MCMC’06)

Fixed comm. networks (ICAR’05, CDC’05, IJACSP)

OutlinePath following: single vehicleCoordination error & path reparameterizationCoordination dynamics Communication constraints & graphsCoordination control

Path following Path following (single vehicle)(single vehicle)

VehicleVehicle:: wheeled robot wheeled robot (underactuated vehicle w/ (underactuated vehicle w/ no sideno side--slip)slip)

Control signalControl signal: : angular speedangular speedStabilityStability:: SemiSemi--global global

asymptotic convergence to asymptotic convergence to the path. the path.

Condition:Condition:

Exponential convergence ifExponential convergence if

0

( )v t dt∞

= ∞∫0mv v≥ >

Path following Path following (kinematics)(kinematics)

•• Path following error Path following error vector and kinematicsvector and kinematics

path curvature at

control signals

exogenous signal

( )cc s P

( ( ) 1) cos( ) sin( )

e e c e

e c e

e c

x y c sye x c s

c

ss

r

v

sv

s

ψψ

ψ

= − +

= − +

= −

Path following Path following (problem)(problem)

•• Problem:Problem: Given a spatial path and a desired Given a spatial path and a desired temporal profile for the speed, derive feedback temporal profile for the speed, derive feedback laws for and to drive to laws for and to drive to zero.zero.

drives to zero (heading control)drives to zero (heading control)

N s

N , ,e e ex y ψ

dynamics: 1 1;r N v FJ m

= =

, , ,e e e dx y v vψ −dv

Path following, resultsPath following, results

MAIN result:MAIN result: existence of control laws that existence of control laws that solve the PF problem: error convergence solve the PF problem: error convergence is guaranteed is guaranteed

-- if is uniformly continuous and does if is uniformly continuous and does not vanish asymptotically.not vanish asymptotically.

OROR--

( )v t

0

( )v t dt∞

= ∞∫

Path following Path following (control strategy)(control strategy)

-- Lyap. func.Lyap. func.

-- approach angle approach angle

-- time derivative time derivative

for some andfor some and-- do back stepping to finddo back stepping to find

2 2 21 1 1 ( ( ))2 2 2p e e e eV x y yψ σ= + + −

1 2( ) sign( )sin ee

e

k yy vy

σε

−= −+

N

22 2

1 2 3( ) ( )ep e e

e

yV k x k v t ky

ψ σε

= − − + − −+

r s

Coordination state / coord. errorCoordination state / coord. errorUse the Use the RABBITSRABBITS to define the coordination error!to define the coordination error!Coordination is achieved if the coordination states (CSs) Coordination is achieved if the coordination states (CSs) are equal,are equal,The CS is a geometrical variable: arc length (shifted The CS is a geometrical variable: arc length (shifted paths), angle (circumferences), or even more generalpaths), angle (circumferences), or even more general

Coordination State / Path ReparametrizationCoordination State / Path Reparametrization

Paths parameterized by Paths parameterized by Vehicles Vehicles ii and and jj are are coordinated if coordinated if Define the function (arc Define the function (arc length) length)

DefineDefine

The choice of The choice of ξξii must yield must yield positivepositive and and boundedbounded RRii

Shifted paths:Shifted paths:

Circumferences:Circumferences:

iξ

i jξ ξ=

( )i i is s ξ=

ii

i

sRξ∂

=∂

; ( ) 1i i i is Rξ ξ= =

, ( ) (radii)i i i i i is R R Rξ ξ= =

1ξ

2ξ

1ξ

2ξ

Coordination State / Path ReparametrizationCoordination State / Path Reparametrization

4545--degree degree exampleexample

Sinusoidal exampleSinusoidal example

1 1 2 2

1 2

; 2

1; 2

s s

R R

ξ ξ= =

= =

1 1 1 2 2

221 2 2

2

;

1; 1 cos

x s xdsR Rd

ξ ξ

ξξ

= = =

= = = +

Coordination state dynamicsCoordination state dynamics--11-- The rabbitThe rabbit’’s dynamic for vehicle s dynamic for vehicle ii

-- Dynamics of coordination state Dynamics of coordination state ii

IMPORTANT: IMPORTANT: ddi i is guaranteed to vanish at the path following is guaranteed to vanish at the path following level level IFIFvvii does not blow up and does not blow up and vvi i does not tend to 0 (CAVEAT!)does not tend to 0 (CAVEAT!)

The effect of the PF subdynamics appears as The effect of the PF subdynamics appears as a a ““vanishingvanishing”” disturbance in the Coo disturbance in the Coo

subdynamics.subdynamics.

, 1 ,(cos 1)i i e i i e is v v k xψ= + − +

1 1( ) ( )i i i i i

i i i i

s v dR R

ξ ξξ ξ

= ⇒ = +

Coordination state dynamicsCoordination state dynamics--22-- Objectives: Objectives:

-- Making Making fromfrom

desired speed of vehicle i equals desired speed of vehicle i equals

-- define the speed tracking errordefine the speed tracking error

1( )i i i

i i

v dR

ξξ

= +0id =

( )i i LR vξ

:1:

i i i L

i i i i L

v R vdf F R v

m dt

η

η

= −

= = +

coordination

formation spe

d

0

ei j

i Lv

ξ ξ

ξ

− =

=

CoordinationCoordination subsystemsubsystemComplete Fleet of Vehicles

-- is a stateis a state--driven varying matrix: driven varying matrix: and and

•• Problem:Problem: Derive control law for Derive control law for so thatso thatconverge asymptotically to zero.converge asymptotically to zero.

C

Make d equal to 0.Bring it into the picture at a later stage

CONTROL VARIABLE

1LC

f

dv

η

ξ η

=

= + +

,i i jη ξ ξ−f

1( )ii

i i

CR ξ

=

1 20 ( )c C cξ< ≤ ≤

Communication topology comes into play!use Graph theory

Communication ConstraintsV1

V2

V3

Node

edge

AdjacencyMatrix A =

1 1

1

1

0

0

V1 receives info from neighbours V2 and V3

V2 receives info from neighbour V1

V3 receives info from neighbour V1

DegreeMatrix D =

02

0

0

1 0

0 0 1

0

0

0

Communication ConstraintsV1

V2

V3

Node

edge

Neighbour set 1={ V2 , V3}

Neighbour set 2={ V1}

Neighbour set 3={ V1}

Laplacian

1 1 2 1 3

2 2 1

3 3 1

2 1 11 1 01 0 1

( ) ( )

L D A

Lξ ξ ξ ξ ξξ ξ ξξ ξ ξ

− −⎛ ⎞⎜ ⎟= − = −⎜ ⎟⎜ ⎟−⎝ ⎠− + −⎛ ⎞ ⎛ ⎞

⎜ ⎟ ⎜ ⎟= −⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟−⎝ ⎠ ⎝ ⎠

--Graph is connected

Properties:

1 0L =rank 1 2L n⇒ = − =

1 2 30Lξ ξ ξ ξ= ⇒ = =

Communication constraintsCommunication constraints

• info available from a subset of the fleet: the neighboring vehicles, sets

• bi-directional or directional comm.

• use graph Laplacian to model comm. constraints declared by sets

iN

iNL

CoordinationCoordination strategiesstrategiesComplete Fleet of Vehicles (for )

-- Comm. graph is Comm. graph is undirectedundirected and and connected connected -- (MAIN results)(MAIN results) either of the following control laws solve the either of the following control laws solve the

CC problemCC problem

-- : underlying comm. graph Laplacian : underlying comm. graph Laplacian -- : positive diagonal matrices : positive diagonal matrices -- : saturation function: saturation function

1L

f

C v

η

ξ η

=

= +

0d =

1 1

( )( ) sat( )

f A BCLf LC C A ALf A L A C B A L

η ξη ξη η ξ− −

= − −

= − + + −

= − + − +

,A BL

sat(.)

CoordinationCoordination strategiesstrategies

vehicle i, decentralized formi

ii i i i j

j Ni

f A BCLbf aR

η ξ

η ξ ξ∈

= − −

= − −∑

Challenges: DONE! 1) when is varying

2) prove satisfies required conditions when putting together PF and CC

( )C ξ( )iv t

Switching comm. / failures / time delaysare very important issues

next part of the talk

Part IPart I-- summary (contributions)summary (contributions)

Several solutions for coordination controlSeveral solutions for coordination controlBiBi--directional network ( varying)directional network ( varying)NonNon--bidirectional networks ( constant)bidirectional networks ( constant)

PFPF--CC interconnected system: error CC interconnected system: error trajectories are bounded and converge to trajectories are bounded and converge to zero asymptotically. Namely, zero asymptotically. Namely, vv((tt)) satisfies satisfies the required conditions.the required conditions.

( )C ξ

( )C Cξ =

General Path FollowingalgorithmsSwitching communications

OutlineOutline•General under-actuated vehicle, PF•Coordination under

– Brief connectivity losses– General communications losses(Time delays)

PathPath followingfollowing

∈

•• PathPath--following problemfollowing problem

–– Given a geometric path and a speGiven a geometric path and a speed assignmented assignment

vvrr ((tt),), we wantwe want

•• the position of the vehicle to converge to and remain inside an the position of the vehicle to converge to and remain inside an arbitrarily thin tube centered around the desired path arbitrarily thin tube centered around the desired path

•• satisfy (asymptotically) the desired speed assignment, i.e.,satisfy (asymptotically) the desired speed assignment, i.e.,

yd (γ )

asrv tγ → →∞

{ }3( ) :dy R Rγ γ∈ ∈

PathPath following following (a very general set(a very general set--up)up)

( , )( )

i i i i

i i i

x f x uy h x=⎧

⎨ =⎩

,( ) ( ) ( ( ))i i d i ie t y t y tγ= −PFollowing error

Vehicle dynamics

Speed tracking error ,( ) ( ) ( )i i r it t v tη γ= −

Coordination, problemCoordination, problem

Coordination problem:- Derive a control law for

- such that asympt.

- : a given formation speed profile

,r iv

0, 0i j i Lvγ γ γ− → − →

( )Lv t

, 1,...,i r i iv i nγ η= + =Coo. Dyn.

iη a signal from PF closed-loop dyn.

Comm. needed to exchange informationComm. subjected to change and time delays

Coordination, control lawCoordination, control law

, 1,...,i r i iv i nγ η= + =Coo. Dyn.

, ( )

, ( ) ( ) ( ) ( )i p t

r i L i i jj N

v t v t k t tγ γ∈

= − −∑

, ( )i p tN : Neighbors of vehicle i at time t

Proposed control

, ( ), i p tr i L i i jj N∈ ∑

info. arrives with time delay

( )p t : a vector indicating which edge is active at time t

Coordination, closedCoordination, closed--looploop

( ) 1p t LKL vγ γ η= − + +Closed-loop dyn. in vector formno delays

( ) ( )( ) ( ) 1p t p t LKD t KA t vγ γ γ τ η= − + − + +Closed-loop dyn. in vector formwith delays

Two types of

switching comm.considered

Switching Communicationbrief connectivity losses

V1

V2

V3

connected

V1

V3

disconnected

V1

V2

V3

connectedtime

1 2 31, 1, 0p p p= = =

1p

2p

1 2 31, 0, 1p p p= = =

3p

1p

1 2 30, 1, 0p p p= = =

2p

is a function of p, denoted pLL

Brief Connectivity LossesBrief Connectivity Losses

Inspired by the concept of Inspired by the concept of ““brief instabilitiesbrief instabilities”” --Hespanha et. Al. IEEETransactions AC 04Hespanha et. Al. IEEETransactions AC 04

BCL:BCL: the communication graph is connected and the communication graph is connected and disconnected alternativelydisconnected alternatively

Brief connectivity lossesBrief connectivity losses0 graph is connectd

( )1 graph is disconnectd

pχ⎧

= ⎨⎩

Charac. function ofswitching topology :

2

11 2( , ) ( ( ))

t

p tT t t p t dtχ= ∫Connectivity loss time

over :1 2[ , ]t t2 1 0( ) (1 )pT t t Tα α≤ − + −The comm. Network has BCL if

0

0 10Tα≤ ≤>

Asympt. connectivity loss rate:

Connectivity loss upper bound:

Example: periodically- 10 sec connected - 40 sec disconnected 0

20%40T

α =⎧→ ⎨ =⎩

Brief connectivity lossesBrief connectivity losses2

11 2( , ) ( ( ))

t

p tT t t p t dtχ= ∫Connectivity loss time

over :1 2[ , ]t t

2 1

0

2 1 2 1 2 1

(1 ) limp p

t t

T TTt t t t t t

αα α− →∞

−≤ + ⇒ ≤

− − −

2 1

0 0(1 )p

p p p

T t t

T T T T Tα α

= −

≤ + − ⇒ ≤

If the graph is disconnected over

1 2[ , ]t t

2 1 0( ) (1 )pT t t Tα α≤ − + −

Switching Communication“uniform” connected in mean

time

V1

V2

V3

1 2 31, 0, 0p p p= = =

1p V1

V2

V3

1 2 31, 0, 1p p p= = =

3p

V1

V3

1 2 30, 0, 0p p p= = =

1pL2pL

3pL

1 2 3[ , ]t t T p p pL L L L+ = + + [ , ]

[ , ]

rank 1 21 0

t t T

t t T

L nL

+

+

= − =⎧⇒ ⎨ =⎩

the union graph over time interval Tis connected

Uniform Connected in MeanUniform Connected in Mean•• Inspired by work of Inspired by work of

–– Moreau (CDCMoreau (CDC’’04)04)–– Lin, Francis, Maggiore (SIAM recent)Lin, Francis, Maggiore (SIAM recent)

•• UCM:UCM: there is a there is a T T >0>0, such that the union , such that the union communication graph is connected over any time communication graph is connected over any time interval of length interval of length TT

We assume a switching dwell time(time clearance between two consecutive switches)

0Dτ >

Error spaceError spaceCoo. dyn:

( ) 1p t LKL vγ γ η= − + +

11 1 , 11

TT

L

L I K

β

β

γ γ

β ββ

−

=

= − =

Error vector:

Important properties: 0 span{1}0T

pKL Lβ

γ γ

β γ

γ γ η

= ⇔ ∈

=

= − +

T TpLγ γ λγ γ≥

When declares a connected graphor

when

p

0pL γ ≠

Convergence, switching topologyConvergence, switching topology2

112

2

when 012 otherwise

pT V k LV K V

V k

λ η γγ γ

λ η−

⎧− + ≠⎪= ⇒ ≤ ⎨+⎪⎩

We assumed that the PF control law guarantees but...

, as0 tη → →∞

1 1

1 1

(1 ) for BCL

( ) for UCM2D

DT

λ λ ατλ λτ

= −⎧⎪⎨ =⎪ +⎩

We show that for both-Brief Connectivity Losses with param.-Uniform Connected in Mean with param.

0,TαT

PathPath followingfollowing( , )( )

i i i i

i i i

x f x uy h x=⎧

⎨ =⎩

,( ) ( ) ( ( ))i i d i ie t y t y tγ= −PF error

Vehicles dyn.

Speed tracking error ,( ) ( ) ( )i i r it t v tη γ= −

The PF control law requires then, it requires

,( ( ))d i id ydt

γ and higher derivatives,i iγ γ

a simple situation: - exact following ,( ) ( )i r it v tγ =

, ( )

( ) ( ) ( )i p t

L i i jj N

v t k t tγ γ∈

= − −∑

Only is availableLvri L riv v v= +

Path following resultsPath following results

•• New resultsNew results: For a general underactuated : For a general underactuated vehicle, there is a control law that uses vehicle, there is a control law that uses

instead of instead of such that the closedsuch that the closed--loop PF is ISS with loop PF is ISS with input .input .

iu, and L Lv v , and L riv vγ γ= +

riv

PF and CC interconectionPF and CC interconectionPF

CC rv

η

Proof of convergence.

Key Ingredient: a new small gain theoremfor systems with brief instabilities


CC rv

η

Main Result A (Brief Connectivity Losses)

For any choice of connectivity parameters T and α, there exist PT and CC gains that yield convergence of thecomplete system error trajectories to an arbitrarily small neighborhood of the origin.


CC rv

η

Main Result B (Connected in Mean)

For any choice of average connectedness time T, there exist PT and CC gains that yield convergence of thecomplete system error trajectories to an arbitrarily small neighborhood of the origin.

SummarySummary

•• Path following control for a general Path following control for a general underactuated vehicle: ISS from inputunderactuated vehicle: ISS from input

•• Coordination strategies under switching Coordination strategies under switching communicationscommunications–– Brief connectivity lossesBrief connectivity losses–– Uniform connected in meanUniform connected in mean

•• PFPF--CC interconnectionCC interconnection

rv

Simulations (ASCraft, fully actuated)Simulations (ASCraft, fully actuated)[[João Almeida, MSc, ISTJoão Almeida, MSc, IST--2006]2006]

In-line formation Triangle formation

Simulations (Simulations (Underactuated AUVsUnderactuated AUVs))

Simulations, no failuresSimulations, no failurescoordination errors path following errors

Simulations, 75% failureSimulations, 75% failurecoordination errors path following errors

With 75% of communications failures occurring periodically with T=20s

Contributions, Contributions, Coordination controlCoordination control

Fixed networksFixed networksProper setProper set--up for coordinated path following, up for coordinated path following, general coordination patterns and paths.general coordination patterns and paths.Several solutions for bidirectional and nonSeveral solutions for bidirectional and non--bidirectional networks.bidirectional networks.Convergence guaranteed when putting Convergence guaranteed when putting together PF and CC systems (dynamics of the together PF and CC systems (dynamics of the vehicle directly taken into account!)vehicle directly taken into account!)

Contributions, Contributions, Coordination controlCoordination controlSwitching networksSwitching networks

Coordination guaranteed under switching Coordination guaranteed under switching communicationscommunications

Brief connectivity lossesBrief connectivity lossesUniform connectedness in meanUniform connectedness in mean

Switching communications with delaysSwitching communications with delaysSmall gain theorem for systems with brief Small gain theorem for systems with brief failuresfailures

convergence guaranteed when putting together PF convergence guaranteed when putting together PF and CC systemsand CC systems

Future workFuture work

Information exchange in discrete set-up, asynchronous Non-equal and varying time delaysCoordinated navigationCoordinated navigation

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