Distributed consensus on enclosing shapes and minimum time r(4)

(ii)if i∈{1,...,n}, p[i] MBC(p[1]···p[n]) 2≤

rctr,thenthesolutionofMTR(Ecmpl,B(0,rctr))isgivenbyp(T)=prndzvs,disk,prndzvs,disk∈∩i∈{1,...,n}B(p[i],rctr),andu[i](t)=prndzvs,disk p[i](t).

Alternatively,ifu[i]∈C(0,rctr),i∈{1,...,n},then

(iii)p(T)=prndzvs,cube,prndzvs,cube∈RMT,ua[i](t)=

min{rctr,|prndzvs,a pa[i](t)|}sign(prndzvs,a pa[i](t)),i∈{1,...,n},a∈{1,...,d}isasolutionofMTR(Ecmpl,C(0,rctr)),where

RMT= MOC(p[1](0),...,p[n](0))

1

a

2

(lmax la)

,

lmaxisthelargestsideofMEO(p[1](0),...,p[n](0))andlaisthesideindirectiona;

(iv)if i∈{1,...,n} p[i] MOC(p[1]···p[n]) ∞≤rctr

thesolutionofMTR(Ecmpl,C(0,rctr))isgivenbyp(T)=prndzvs,prndzvs∈∩i∈{1,...,n}C(p[i],r]ctr),andu[i(t)=prndzvs p[i](t). IV.DISTRIBUTED

CONSENSUSONMINIMALENCLOSING

BALLANDORTHOTOPE

Intheprevioussectionwehaveshownthatminimalenclosingshapesplayakeyroleinthesolutionofminimumtimerendezvous.Infactiftheagentscouldknowthecenterofsuchshapes(ballororthotope)thesolutionofminimumtimerendezvouswouldbejustacontrollawthatdriveseachagenttothispoint.Therefore,inthissection,wewanttoexploretwoconsensusalgorithmstocomputetheminimalenclosingballandtheminimalenclosingorthotopeofasetofgivenpointsinRdinadistributedway.

HereisaninformaldescriptionofwhatweshallrefertoastheFloodMEBalgorithm:

[Informaldescription]Eachagentinitializestheminimalenclosingballtoitsinitialposition,then,ateachcommunicationround,performsthefol-lowingtasks:(i)itacquiresfromitsneighbors(amessagerepresentedby)thecoordinatesoftheminimumsetofpointsdescribingtheboundaryoftheirminimalenclosingballandthecoordinatesoftheirinitialposition;(ii)itcomputestheminimalenclosingballofthepointsetcomprisedofitsanditsneighbors’setofpointsanditsinitialposition(thatitmaintainsinmemory);(iii)itupdatesitslogicvariablesandmessageasin(i).

Beforedescribingthealgorithmmoreformally,weneedtointroducesomenotationandstatesomepropertiesoftheminimalenclosingball.Givenasetofmpoints{q1,...,qm} Rdingenericpositions,wedenotewithMEBbndry({q1,...,qm})theminimumsetofpointsontheboundaryofMEB({q1,...,qm})thatuniquelyidentifysuchboundary.Whenthepointsareingenericposition,weletMEBbndry({q1,...,qm})denoteaminimumsetofpointsontheboundaryofMEB({q1,...,qm})thatidentifysuchboundary.Moreover,letMBR:F(Rd)→Rthefunction

thatassociatestoasetofpointstheradiusoftheminimalenclosingballofsuchpoints.

Lemma4.1(MEBproperties):LetQnasetofnpoints.Thefollowingstatementshold.

(i)thereexistsasubsetQd Qnofd+1elementssuch

thatMEB(Qd)=MEB(Qn);

(ii)forallQn1,Qn2 QnwithQnQ;

1 Qn2,then

MBR(n(iii)ifMBR(Q1)≤MBR(Qn2)nQ1)=MBR(Qn2),thenMBC(Qn1)=

MBC(n(iv)thenumber2);

ofpossiblevaluesofMBR(Qn Q1),forall

Qn1n,is nite.

Remark4.2:AnimportantimplicationofLemma4.1(i)isthatMEBbndry({q1,...,qn})hasatmostd+1points,thenthenumberofpacketsinthemessagesentandstoredbyeachagentisatmostd+1anddoesnotdependonn. :FloodMEBalgorithm.

Goal:Solvetheproblemofcomputingmin-imalenclosingballofasetofLogicstate:w[i]=(P[i]bndry,p[i]

points.

0);Msgfunction:msg(x[i],w[i],i)=w[i];Initialization:

P[i]bndry(0)={p[i](0)},p[i]

0(0)=p[i](0).

In this paper we introduce the notion of optimization under control and communication constraint in a robotic network. Starting from a general setup, we focus our attention on the problem of achieving rendezvous in minimum time for a network of first order

sequencecorrespondstoaballwhoseradiusismonotonenondecreasing,upperboundedandcanassumea nitenum-berofvalues.Thenweproceedbycontradictiontoprovethatallthelawsconvergetothesameball(sameradiusandcenter)andthatitisexactlytheminimalenclosingballofthenpoints.Supposethatthealgorithmconvergesondifferentballs(differentradiusordifferentcenter)fordifferentagents.Thentheremustexisttwoagentsthatareneighborsandhavedifferentlogicvariables(correspondingtodifferentballs).Butthismeansthat,atthefollowingtimeinstant,theyhavetocomputetheminimalenclosingballofalargersetofpoints,theneitheroneofthemwilltakethevalueoftheotherorbothofthemwillchangetheirvalueandtakeacommonone.Iteratingthisargumentweobtainthatalltheagentsmustconvergetoacommonvalue.Now,theballateachnodecontains,byconstruction,theinitialpositionofthatnode.Sincetheballisthesameforeachnode,itcontainsalltheinitialpositions,thenitistheminimalenclosingballoftheinitialpositions.

Fori∈{1,...,n},agentiexecutesateachtimet∈N:

1:acquirew[j],j∈N(i)2:

compute a∈{1,...,d}

p[i]

=minj∈N(i)∪{i}{p[j]min,a(t+1)p[max,ai]min,a(t)}(t+1)=maxj∈N(i)∪{i}{p[max,aj]

(t)}3:

update a∈{w[ai](t+1)= 1,...,d}p[i]

p[max,ai]min,a(t+1),(t+1)

whereimin,aandimax,aaretheagentsthatcharacterizetheboundaryoftheorthotopeindirectionaandminimizethetopologicaldistancefromi.

ThetimecomplexityofthealgorithmisoforderΘ(n). Proof:InordertoprovethecorrectnessandthetimecomplexityofthealgorithmdescribedinTable??,weneedtoprovethatitisequivalentto2dFloodMaxalgorithmsforleaderelection(twoforeachdirection)runningsimultane-ously.Oncewehaveproventhat,theresultsoncorrectnessandtimecomplexityfollowfromChapter4in[1].

百度搜索“77cn”或“免费范文网”即可找到本站免费阅读全部范文。收藏本站方便下次阅读，免费范文网，提供经典小说医药卫生Distributed consensus on enclosing shapes and minimum time r(4)在线全文阅读。