Rendezvous and deployment of cooperative robotic networks · Theoretical challenges for sensor...

Rendezvous and deploymentof cooperative robotic networks

Jorge CortesUNIVERSITY OF CALIFORNIAUNOFFICIAL SEAL

Attachment B - “Unofficial” SealFor Use on Letterhead

Mechanical and Aerospace EngineeringUniversity of California, San Diegohttp://tintoretto.ucsd.edu/[email protected]

Cooperative multi agent systems: distributedcomputation, estimation, and control

Centro Ennio De GiorgiPisa, Italy, December 3-7, 2007

,

,

Cooperative robotic networks

Groups of robotic agents with computation, sensing, communication,and control capabilities

,

Theoretical challenges for sensor network technologies

Design cooperative networks to perform useful engineering tasks

Feedback rather than open-loop computationfor known/static setup

Information flow who knows what, when, why, how,dynamically changing

Complexity dynamic scenarios,collection of heterogeneous devices

Reliability/performance robust, efficient, predictable behavior

How to coordinate individual agents into coherent whole?

Objective: systematic methodologies to design and analyzecooperative strategies to control multi-agent systems

Integration of control, communication, sensing, computing

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Today: rendezvous and deployment

Basic motion coordination capabilities:get together at a point, deploy over a region

CENTROIDAL VORONOI TESSELLATIONS 649

Fig.2.2 A top-viewphotograph,usinga polarizinglter,of theterritoriesof themale Tilapiamossambica;eachisa pitduginthesandbyitsoccupant.The boundariesoftheterritories,therimsofthepits,forma patternofpolygons.The breedingmalesare theblacksh,whichrange in sizefrom about 15cm to 20cm. The gray share thefemales,juveniles,andnonbreedingmales.The shwitha conspicuousspotinitstail,intheupper-rightcorner,isa Cichlasomamaculicauda.Photographand captionreprinted from G. W. Barlow,HexagonalTerritories, Animal Behavior,Volume 22,1974,by permissionofAcademicPress,London.

As anexampleofsynchronoussettlingforwhich theterritoriescanbevisualized,considerthemouthbreedersh(Tilapiamossambica).Territorialmalesofthisspeciesexcavatebreedingpitsinsandybottomsby spittingsandaway fromthepitcenterstowardtheirneighbors.Fora highenoughdensity ofsh,thisreciprocalspittingresultsinsandparapetsthatarevisibleterritorialboundaries.In[3],theresultsofa controlledexperimentweregiven.Fishwereintroducedintoa largeoutdoorpoolwitha uniformsandybottom.Aftertheshhad establishedtheirterritories,i.e.,afterthenalpositionsofthebreedingpitswereestablished,theparapetsseparatingtheterritorieswerephotographed.InFigure2.2,theresultingphotographfrom[3]isreproduced.The territoriesareseentobepolygonaland,in[27,59],itwasshownthattheyareverycloselyapproximatedby a Voronoitessellation.

A behavioralmodelforhow theshestablishtheirterritorieswasgiven in[22,23,60].When theshentera region,theyrstrandomlyselectthecentersoftheirbreedingpits,i.e.,thelocationsatwhich theywillspitsand.Theirdesiretoplacethepitcentersasfaraway aspossiblefromtheirneighborscausestheshtocontinuouslyadjustthepositionofthepitcenters.Thisadjustmentprocessismodeledasfollows.Thesh,intheirdesiretobeasfarawayaspossiblefromtheirneighbors,tendtomovetheirspittinglocationtowardthecentroidoftheircurrentterritory;subsequently,theterritorialboundariesm ustchangesincethesharespittingfromdierentlocations.Sincealltheshareassumedtobe ofequalstrength,i.e.,theyallpresumablyhave

1 Algorithm design:adaptive vs static, distributed vs

centralized, formal analysis vs heuristics

2 Non-deterministic dynamical systems:discrete- and continuous-time, stabilityanalysis, invariance principles

3 Robustness: link failures, agents’ arrivalsand departures, delays, asynchronism

Image credits: jupiterimages and Animal Behavior

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Outline

1 RendezvousMaintaining connectivityCircumcenter algorithmsNon-deterministic dynamical systemsCorrectness and complexity of circumcenter algorithms

2 DeploymentTop-down: optimizing expected coverageExpected coverage with limited-range interactionsBottom-up: basic behaviors and multicenter functions

3 Conclusions

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Rendezvous

Objective:achieve rendezvous at single point, while maintaining connectivity

r-disk connectivity visibility connectivity

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Aggregate objective functions

Coordination task formulated as function minimization

Diameter convex hull Perimeter relative convex hull

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We have to be careful...

Blindly “getting closer” to neighboring agents might break overallconnectivity

,

Constraint sets for connectivity

pj

pi

r-disk pair-wise constraint set visibility pair-wise constraint set

Key properties

Constraints are flexible enough so that network does not get stuckConstraints change continuously with agents’ position

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Outline



3 Conclusions

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What is the circumcenter

For X = Rd, X = Sd or X = Rd1 × Sd2 , d =d1 + d2, circumcenter CC(W ) of a boundedset W ⊂ X is center of closed ball of minimumradius that contains W

Circumradius CR(W ) is radius of this ball

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Circumcenter algorithms – the basic idea

each agent minimizes “local version” of objective function

max‖pi − pj‖ | pj is neighbor of pi

i.e., each agent goes toward circumcenter of neighbors and itself(which is the closest point to all these locations)each agent maintains connectivity by moving inside constraint set

If agents i and j are neighbors then subsequentpositions must belong to B(

pi+pj

2 , r2 )

If agent i has neighbors at locations q1, . . . , qlat time `, then constraint set is

Dr(pi, q1, . . . , ql) =⋂

q∈q1,...,ql

B(pi + q

2,r

2

)

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Circumcenter algorithms – the basic idea

To maximize the displacement towardcircumcenter, each agent solvesconvex optimization problem

For q0 and q1 in Rd, and for a convex closed set Q ⊂ Rd with q0 ∈ Q,let λmax(q0, q1, Q) denote the solution to the strictly convex problem

maximize λ

subject to λ ≤ 1, (1− λ)q0 + λq1 ∈ Q

Under the stated assumptions the solution exists and is unique

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Circumcenter algorithms – informally

Communication rounds take place at each natural instant of time

At each communication round each agent performs the following tasks:

1 it transmits its position and receives its neighbors’ positions2 it computes the circumcenter of the point set comprised of its

neighbors and of itself, and3 it moves toward this circumcenter while maintaining connectivity

with its neighbors

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Circumcenter algorithms – formally

Robotic Network: Sdisk

Distributed Algorithm: circumcenterAlphabet: L = Rd ∪ null

function msg(x,w, i)

1: return msgstd(x,w, i)

function ctrl(xsmpld, y)

1: xgoal(xsmpld, y) = CC(xsmpld∪xrcvd | for all non-null xrcvd ∈ y),2: D := Dr(xsmpld, xrcvd | for all non-null xrcvd ∈ y)3: λ∗ = λmax

(xsmpld, xgoal(xsmpld, y),D)

)4: return λ∗(xgoal(xsmpld, y)− xsmpld)

Can also be run over any other proximity graph which is spatiallydistributed over Gdisk(r) or over Gvis

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Simulations

x

y

z

x

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Outline



3 Conclusions

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Some bad news...

Circumcenter algorithms are nonlinear discrete-time dynamical systems

x`+1 = f(x`)

To analyze convergence, we need at least f continuous – to use classicLyapunov/LaSalle results

But circumcenter algorithms are discontinuous because of changes ininteraction topology

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Alternative idea

Fixed undirected graph G, define fixed-topology circumcenteralgorithm

fG : (Rd)n → (Rd)n, fG,i(p1, . . . , pn)

Now, there are no topological changes in fG , hence fG is continuous

Define set-valued map TCC : (Rd)n → P((Rd)n)

TCC(p1, . . . , pn) = fG(p1, . . . , pn) | G connected

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Non-deterministic dynamical systems

Given T : X → P(X), a trajectory of T issequence xmm∈N0 ⊂ X such that

xm+1 ∈ T (xm) , m ∈ N0

T is closed at x if xm → x, ym → y with ym ∈ T (xm) imply y ∈ T (x)Every continuous map T : Rd → Rd is closed on Rd

A set C isweakly positively invariant if, for any p0 ∈ C, there existsp ∈ T (p0) such that p ∈ C

strongly positively invariant if, for any p0 ∈ C, all p ∈ T (p0)verifies p ∈ C

A point p0 is a fixed point of T if p0 ∈ T (p0)

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LaSalle Invariance Principle – set-valued maps

V : X → R is non-increasing along T on S ⊂ X if

V (x′) ≤ V (x) for all x′ ∈ T (x) and all x ∈ S

Theorem (LaSalle Invariance Principle)

For S compact and strongly invariant with V continuous andnon-increasing along closed T on S

Any trajectory starting in S converges to largest weakly invariantset contained in x ∈ S | ∃x′ ∈ T (x) with V (x′) = V (x)

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Outline



3 Conclusions

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CorrectnessTCC is closed and diameter is non-increasing

Recall set-valued map TCC : (Rd)n → P((Rd)n)

TCC(p1, . . . , pn) = fG(p1, . . . , pn) | G connected

TCC is closed: finite combination of individual continuous maps

Define

Vdiam(P ) = diam(co(P )) = max ‖pi − pj‖ | i, j ∈ 1, . . . , ndiag((Rd)n) =

(p, . . . , p) ∈ (Rd)n | p ∈ Rd

LemmaThe function Vdiam = diam co: (Rd)n → R+ verifies:

1 Vdiam is continuous and invariant under permutations;2 Vdiam(P ) = 0 if and only if P ∈ diag((Rd)n);3 Vdiam is non-increasing along TCC

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Correctness via LaSalle Invariance Principle

To recap1 TCC is closed2 V = diam is non-increasing along TCC

3 Evolution starting from P0 is contained in co(P0) (compact andstrongly invariant)

Application of LaSalle Invariance Principle: trajectories starting atP0 converge to M , largest weakly positively invariant set contained in

P ∈ co(P0) | ∃P ′ ∈ TCC(P ) such that diam(P ′) = diam(P )

Have to identify M ! Ideally, M = diag((Rd)n) ∩ co(P0)

Clearly diag((Rd)n) ∩ co(P0) ⊂ M – other inclusion by contradiction

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LaSalle Invariance Principle – identifying M

Assume P ∈ M \ (diag((Rd)n) ∩ co(P0)), and therefore diam(P ) > 0Let G be a connected directed graph and consider TG(P )

1 Trivially, all non-strictly convex vertices of co(P ) will evolvetowards a point in the co(P ) which is not a strictly convex vertex

2 Same conclusion holds for strictly convex vertices, because graphis connected, and neighbors “pull agent out” from the vertex

3 Argument has to be conveniently extended to the case where thereis more than one agent at a strictly convex vertex

In any case, after a finite number of iterations, all agents inconfiguration TG1,r(TG2,r(. . . TGN ,r(P ))) are contained inco(P ) \ Ve(co(P ))Therefore, diam(TG1,r(TG2,r(. . . TGN ,r(P )))) < diam(P ), whichcontradicts M weakly invariant

Convergence to a point can be concluded with a little bit of extra work

Corollary: circumcenter algorithm achieves rendezvous

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Robustness of circumcenter algorithms

Push whole idea further!, e.g., for robustness against link failures

topology G1 topology G2 topology G3

Look at evolution under link failures as outcome ofnondeterministic evolution under multiple interaction topologies

P −→ evolution under G1, evolution under G2, evolution under G3

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Rendezvous

Corollary (Circumcenter algorithm over Gdisk(r) on Rd)

For Pmm∈N0 synchronous execution with link failures such that unionof any ` ∈ N consecutive graphs in execution has globally reachable node

Then, there exists (p∗, . . . , p∗) ∈ diag((Rd)n) such that

Pm → (p∗, . . . , p∗) as m → +∞

Proof uses

TCC,`(P ) = fG` · · · fG1(P ) |

∪`s=1 Gi has globally reachable node

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Beyond rendezvous: flocking

Average-heading algorithm fAve : (R2 × S1)n → (R2 × S1)n over G

fAve,i((p1, θ1), . . . , (pn, θn)) =(pi + (cos θi, sin θi),

Average(θi ∪ θj | (pj , θj) ∈ NG(pi, θi)))

Corollary (Averaging algorithm over G on R2 × S1)

For (Pm,Θm)m∈N0 synchronous execution with link failures such thatunion of any ` ∈ N consecutive graphs has globally reachable node

Then, there exists (θ∗, . . . , θ∗) ∈ diag((S1)n) such that

Θm → (θ∗, . . . , θ∗) as m → +∞

V : (R2 × S1)n → R is max θi −min θi

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Rendezvous: example complexity analysis

1 first-order agents with disk graph, for d = 1,

TC(Trendezvous, CCcircumcenter) ∈ Θ(n)

2 first-order agents with limited Delaunay graph, for d = 1,

TC(T(rε)-rendezvous, CCcircumcenter) ∈ Θ(n2 log(nε−1))

Complexity analysis via tridiagonal Toeplitz and circulant matrices

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Outline



3 Conclusions

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Deployment

Objective: optimal task allocation and space partitioningoptimal placement and tuning of sensors

What notion of optimality? What algorithm design?top-down approach: define aggregate function measuring“goodness” of deployment, then synthesize algorithm thatoptimizes function

bottom-up approach: synthesize “reasonable” interaction lawamong agents, then analyze network behavior

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Top-down: expected-value deployment

Objective: Given sensors/nodes/robots/sites (p1, . . . , pn) moving inenvironment Q

achieve optimal coverage defined according to

Scenario 1 —expected value performance measure

given distribution density function φ

minimize HC(p1, . . . , pn) = Eφ

[min

i‖q − pi‖2

]

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Scenario 1: coverage algorithm

Name: Coverage behaviorGoal: distributed optimal agent deploymentRequires: (i) own Voronoi cell computation Definition

(ii) centroid computation

At each communication round, each agent:1: acquire neighbors’ positions2: compute own dominance region Computation

3: follow gradient – move towards centroid

Caveat: convergence only to local minimum of HC

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Simulation

initial configuration gradient descent final configuration

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Scenario 1: technical approach

1 Alternative formulation (f : R+ 7→ R+, differentiable,non-decreasing)

Eφ

(min

if(‖q − pi‖)

)=

n∑i=1

∫Vi(P )

f(‖q − pi‖)φ(q)dq

≤n∑

i=1

∫Wi


2 Compute decentralized gradient

∂HC

∂pi(P ) =

∫Vi(P )

∂

∂pif (‖q − pi‖) φ(q)dq

+

∫∂Vi(P )

f (‖q − pi‖) 〈ni(q),∂q

∂pi〉φ(q)dq

+∑

j neigh i

∫Vj(P )∩Vi(P )

f (‖q − pj‖) 〈nji(q),∂q

∂pi〉φ(q)dq

,



Eφ

(min

if(‖q − pi‖)

)=

n∑i=1

∫Vi(P )


≤n∑

i=1

∫Wi



∂HC

∂pi(P ) =

∫Vi(P )

∂

∂pif (‖q − pi‖) φ(q)dq

= 2MVi(P )(pi − CVi(P ))︸︷︷︸for f(x)=x2

+

∫∂Vi(P )

f (‖q − pi‖) 〈ni(q),∂q

∂pi〉φ(q)dq

−∫

∂Vi(P )

f (‖q − pi‖) 〈ni(q),∂q

∂pi〉φ(q)dq

,



Eφ

(min

if(‖q − pi‖)

)=

n∑i=1

∫Vi(P )


≤n∑

i=1

∫Wi



∂HC

∂pi(P ) =

∫Vi(P )

∂

∂pif (‖q − pi‖) φ(q)dq = 2MVi(P )(pi − CVi(P ))︸︷︷︸

for f(x)=x2

+

∫∂Vi(P )

f (‖q − pi‖) 〈ni(q),∂q

∂pi〉φ(q)dq

−∫

∂Vi(P )

f (‖q − pi‖) 〈ni(q),∂q

∂pi〉φ(q)dq

,



Eφ

(min

if(‖q − pi‖)

)=

n∑i=1

∫Vi(P )


≤n∑

i=1

∫Wi



∂HC

∂pi(P ) =

∫Vi(P )

∂

∂pif (‖q − pi‖) φ(q)dq = 2MVi(P )(pi − CVi(P ))︸︷︷︸

for f(x)=x2

critical points for H arecentroidal Voronoi configurations

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Correctness of dispersion laws

Distributed: over Delaunay graph

Adaptive: changing environment,agent arrivals and departures

Verifiably correct: convergence to centroidal Voronoi configurationsvia LaSalle Invariance Principle

Asynchronous implementation:wake up

1 determine local Voronoi diagram (w/ outdatedinformation)

2 determine centroid of own Voronoi region3 take a step in that direction

go to sleep

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Outline



3 Conclusions

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Top-down: area deployment



Scenario 2 —area (with limited-range sensor or communicationradius r)


maximize areaφ(∪ni=1B r

2(pi)) =

∫Q

(max

i1B r

2(pi)(q)

)φ(q)dq

,

Scenario 2: weighted normal

Take density function constant, φ = 1

∫arc(r)

nB r2

(p) φ

If arc(r) is described by [θ−, θ+] 3 θ 7→ p + r2 (cos θ, sin θ) ∈ R2

r

2

∫ θ+

θ−

(cos θ, sin θ)dθ = sin(θ+ − θ−

2

)(cos

(θ+ + θ−2

), sin

(θ+ + θ−2

))

,

Scenario 2: area coverage algorithm

Name: Coverage behaviorGoal: distributed optimal agent deploymentRequires: (i) own cell computation

(ii) weighted normal computation

For all i, agent i synchronously performs:1: determine own cell Vi ∩B r

2(pi)

2: determine weighted normal∫arc(r)

nB r2

(p) φ

3: move in the direction of weighted normal

Caveat: convergence only to local maximum of areaφ(∪ni=1B r

2(pi))

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Simulation

initial configuration gradient descent final configuration

,

Correctness and complexity of dispersion laws

Distributed: over r-limited Delaunay graph

Adaptive: changing environment,agent arrivals and departures

Convergence: Gradient + LaSalle Invariance Principle

Complexity: for d = 1, first-order agents with r-lim Delaunay graph

TC(T(rε)-deplmnt, CCcentroid) ∈ O(n3 log(nε−1))

,

Expected-value deploymentwith limited-range interactions



Expected value —with limited-range sensing radius r


minimize HC(p1, . . . , pn) = Eφ

[min

if(‖q − pi‖)

]gradient of HC is not spatially distributed over Gdisk

,

Tuning the optimization problem

Let f r2(x) = f(x) 1[0, r

2 )(x) + f(diam Q) · 1[ r2 ,+∞)(x), and define

H r2(p1, . . . , pn) = Eφ

[min

if r

2(‖q − pi‖)

](conservative) constant-factor approximation

βH r2(P ) ≤ HC(P ) ≤ H r

2(P ) , β =

(r

2 diam Q

)2

Gradient of H r2

is distributed over r-limited Delaunay graph

∂H r2

∂pi= 2MVi(P )∩B r

2(pi)(CVi(P )∩B r

2(pi) − pi)

−((

r2

)2 − diam Q2)Mi(r)∑

k=1

∫arci,k(r)

nB r2(pi) φ

,

Simulations

Limited range

run #1: 16 agents,density φ is sum of4 Gaussians, time in-variant, 1st order dy-namics

initial configuration gradient descent of H r2

final configuration

Unlimited rangerun #2: 16 agents,density φ is sum of4 Gaussians, time in-variant, 1st order dy-namics initial configuration gradient descent of HC final configuration

,

Most general result: distributed gradient

For general non-decreasing f : R≥0 → R

piecewise differentiable finite-jump discontinuities at R1 < · · · < Rm

HC(P ) =∫

Qmini f(‖q − pi‖)φ(q)dq

Theorem

∂HC

∂pi(p1, . . . , pn) =

∫Vi

∂

∂pif(‖q − pi‖)φ(q)dq

+m∑

α=1

∆fα(Rα)( Mi(2Rα)∑

k=1

∫arci,k(2Rα)

nBRα (pi)dφ)

= integral over Vi + integral along arcs in Vi

,

Outline



3 Conclusions

,

Deployment: basic behaviors

“move away from closest” “move towards furthest”

Equilibria? Asymptotic behavior?Optimizing network-wide function?

,

Deployment: 1-center optimization problems

smQ(p) = min‖p− q‖ | q ∈ ∂Q Lipschitz 0 ∈ ∂ smQ(p) ⇔ p ∈ IC(Q)lgQ(p) = max‖p− q‖ | q ∈ ∂Q Lipschitz 0 ∈ ∂ lgQ(p) ⇔ p = CC(Q)

Locally Lipschitz function V are differentiable a.e.Generalized gradient of V is

∂V (x) = convex closure˘

limi→∞

∇V (xi) | xi → x , xi 6∈ ΩV ∪ S¯

,

Deployment: 1-center optimization problems

+ gradient flow of smQ pi = +Ln[∂ smQ](p) “move away from closest”− gradient flow of lgQ pi = − Ln[∂ lgQ](p) “move toward furthest”

For X essentially locally bounded, Filippov solution of x = X(x)is absolutely continuous function t ∈ [t0, t1] 7→ x(t) verifying

x ∈ K[X](x) = co limi→∞

X(xi) | xi → x , xi 6∈ S

For V locally Lipschitz, gradient flow is x = Ln[∂V ](x)Ln = least norm operator

,

Nonsmooth LaSalle Invariance Principle

Evolution of V along Filippov solution t 7→ V (x(t)) is differentiable a.e.

ddt

V (x(t)) ∈ LXV (x(t)) = a ∈ R | ∃v ∈ K[X](x) s.t. ζ · v = a , ∀ζ ∈ ∂V (x)︸︷︷︸set-valued Lie derivative

LaSalle Invariance Principle

For S compact and strongly invariant with max LXV(x) ≤ 0

Any Filippov solution starting in S converges to largest

weakly invariant set contained in

x ∈ S | 0 ∈ LXV(x)

E.g., nonsmooth gradient flow x = − Ln[∂V ](x) converges to critical set

,

Deployment: multi-center optimizationsphere packing and disk covering

“move away from closest”: pi = +Ln(∂ smVi(P ))(pi) — at fixed Vi(P )“move towards furthest”: pi = − Ln(∂ lgVi(P ))(pi) — at fixed Vi(P )

Aggregate objective functions!

HSP(P ) = mini

smVi(P )(pi) = mini 6=j

[12‖pi − pj‖, dist(pi, ∂Q)

]HDC(P ) = max

ilgVi(P )(pi) = max

q∈Q

[min

i‖q − pi‖

]

,

Deployment: multi-center optimization

Critical points of HSP and HDC (locally Lipschitz)If 0 ∈ int ∂HSP(P ), then P is strict local maximum, all agentshave same cost, and P is incenter Voronoi configuration

If 0 ∈ int ∂HDC(P ), then P is strict local minimum, all agentshave same cost, and P is circumcenter Voronoi configuration

Aggregate functions monotonically optimized along evolution

min LLn(∂ smV )HSP(P ) ≥ 0 max L− Ln(∂ lgV )HDC(P ) ≤ 0

Asymptotic convergence to center Voronoi configurations vianonsmooth LaSalleComplexity characterization in 1-d, more in progress

,

Deployment: visibility-based deployment

Objective: achieve complete visibility of nonconvex environment (nonself-intersecting polygon)

Partition-basedAt each comm round:

1: acquire neighbors’positions

2: compute owndominance region

3: move towardsfurthest point ofown region

,

Conclusions

Examined two basic motion coordination tasks1 rendezvous: circumcenter algorithms2 deployment: gradient algorithms based on geometric centers

Correctness and (1-d) complexity analysis via1 Discrete- and continuous-time nondeterministic dynamical systems2 Invariance principles, stability analysis3 Geometric structures (Voronoi partitions, centers, proximity

graphs)4 Geometric optimization (disk-covering, sphere-packing)

Techniques can be further applied to other motion coordinationalgorithms (agreement, task assignment, optimal spatial estimation)

,

References on rendezvous

Circumcenter algorithms:H. Ando, Y. Oasa, I. Suzuki, and M. Yamashita. Distributed

memoryless point convergence algorithm for mobile robots with

limited visibility. IEEE Transactions on Robotics and Automation,

15(5):818--828, 1999

J. Cortes, S. Martınez, and F. Bullo. Robust rendezvous for mobile

networks via proximity graphs in arbitrary dimensions. IEEE

Transactions on Automatic Control, 51(8):1289--1298, 2006

Robustness via non-deterministic dynamical systems:J. Cortes. Characterizing robust coordination algorithms via

proximity graphs and set-valued maps. In American Control

Conference, pages 8--13, Minneapolis, MN, June 2006

Flocking algorithms:A. Jadbabaie, J. Lin, and A. S. Morse. Coordination of groups of

mobile autonomous agents using nearest neighbor rules. IEEE

Transactions on Automatic Control, 48(6):988--1001, 2003

L. Moreau. Stability of multiagent systems with time-dependent

communication links. IEEE Transactions on Automatic Control,

50(2):169--182, 2005

,

References on deployment

Deployment scenarios and algorithms:

J. Cortes, S. Martınez, T. Karatas, and F. Bullo. Coverage control

for mobile sensing networks. IEEE Transactions on Robotics and

Automation, 20(2):243--255, 2004

J. Cortes, S. Martınez, and F. Bullo. Spatially-distributed coverage

optimization and control with limited-range interactions. ESAIM.

Control, Optimisation & Calculus of Variations, 11:691--719, 2005

Nonsmooth stability analysis:

J. Cortes. Discontinuous dynamical systems -- a tutorial on notions

of solutions, nonsmooth analysis, and stability. IEEE Control

Systems Magazine, January 2007. Submitted. Invited paper

Geometric and combinatorial optimization:

P. K. Agarwal and M. Sharir. Efficient algorithms for geometric

optimization. ACM Computing Surveys, 30(4):412--458, 1998

,

Voronoi partitions

Let (p1, . . . , pn) ∈ Qn denote the positions of n points

The Voronoi partition V(P ) = V1, . . . , Vn generated by (p1, . . . , pn)

Vi = q ∈ Q| ‖q − pi‖ ≤ ‖q − pj‖ , ∀j 6= i= Q ∩j HP(pi, pj) where HP(pi, pj) is half plane (pi, pj)

3 generators 5 generators 50 generators

Return

,

Distributed Voronoi computation

Assume: agent with sensing/communication radius Ri

Objective: smallest Ri which provides sufficient information for Vi

For all i, agent i performs:1: initialize Ri and compute Vi = ∩j:‖pi−pj‖≤Ri

HP(pi, pj)2: while Ri < 2 maxq∈bVi

‖pi − q‖ do3: Ri := 2Ri

4: detect vehicles pj within radius Ri, recompute Vi

Return

,

Range-limited Voronoi graph computation

Let (p1, . . . , pn) ∈ Qn denote the positions of n points

The r-limited Voronoi partition Vr(P ) =V1,r, . . . , Vn,r generated by (p1, . . . , pn)

Vi,r(P ) = Vi(P ) ∩B(pi, r), i ∈ 1, . . . , n

GLD(r) is spatially distributedover Gdisk(r)

Rendezvous and deployment of cooperative robotic networks · Theoretical challenges for sensor...

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