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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
,
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Cooperative robotic networks
Groups of robotic agents with computation, sensing, communication,and control capabilities
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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
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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
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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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
)
,
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
)
,
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
,
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
,
Simulations
x
y
z
x
,
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
,
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
,
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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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)
,
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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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
,
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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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
,
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
,
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
,
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
,
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
,
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
]
,
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
,
Simulation
initial configuration gradient descent final configuration
,
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
f(‖q − pi‖)φ(q)dq
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
,
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
f(‖q − pi‖)φ(q)dq
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
,
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
f(‖q − pi‖)φ(q)dq
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
,
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
f(‖q − pi‖)φ(q)dq
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
,
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
f(‖q − pi‖)φ(q)dq
2 Compute decentralized gradient
∂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
,
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
f(‖q − pi‖)φ(q)dq
2 Compute decentralized gradient
∂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
,
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
f(‖q − pi‖)φ(q)dq
2 Compute decentralized gradient
∂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
,
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
f(‖q − pi‖)φ(q)dq
2 Compute decentralized gradient
∂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
,
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
,
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
,
Top-down: area deployment
Objective: Given sensors/nodes/robots/sites (p1, . . . , pn) moving inenvironment Q
achieve optimal coverage defined according to
Scenario 2 —area (with limited-range sensor or communicationradius r)
given distribution density function φ
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: 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))
,
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
Objective: Given sensors/nodes/robots/sites (p1, . . . , pn) moving inenvironment Q
achieve optimal coverage defined according to
Expected value —with limited-range sensing radius r
given distribution density function φ
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
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
,
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)