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1 Outle ult-aget Search usg oroo Partto ad oroo D eermet Revew Itroducto Decreasg desty fucto Stablty Cocluso Fujta Lab, Det. of Cotrol ad System Egeerg, FL07--: July 09,007 Davd Ask ork rogress:. Smulato of oroo D wth desty fucto. Lloyd s Algorthm D eermet Future ork Revew Revew Revew meu:. oroo artto D & D. Lloyd s Algorthm. Objectve fucto: - Sesg erformace - Desty fucto oroo artto: The set of all ots q whose dstace from s less tha or equal to the dstaces from all other j { : } = q j q q j 0 Revew Revew Lloyd s Algorthm: A method for evely dstrbutg ots over a ukow area. The stes: Ste 0: Start wth a radom area, {}, ad radom ots, {}. Ste : Costruct oroo artto {}, geerated by {}. Ste : Udate to be the cetrod of. Retur to Ste. Ste 0: Start wth a radom area, {}, ad radom ots, {}. Ste : Costruct oroo artto {}, geerated by {}. Ste : Udate to be the cetrod of. Retur to Ste. 0
2 Revew Revew Objectve fucto: f ( q ): (, ) = ( ) φ H f q q Sesg erformace (f=bg oor sesg) Desty fucto φ ( q): = aget osto q = object = artto By mmzg H, we get otmum coverage. hy? he H = m, agets move to the area wth the hghest occurrece ossblty. e assume: f q = q Smlfy the objectve fucto usg arallel as theorem. H, = f q φ q To mmze ths, =C w (=cetrod of artto) Therefore, use ths as a ut to make agets go to cetrod of the artto. (, ) (, ) = φ : H = H c c q mass Alcato: Itroducto Autoomous N agets equed wth sesors deloy themselves a otmal way over a ukow area. search & rescue, evrometal motorg, mltary ad defece alcato, etc. Objectve: mult-aget search Agets deloy themselves otmally whle udatg (=reducg) ucertaty desty fucto ad gather formato tll the ucertaty s below a certa level. Decreasg Desty Fucto At each terato, after deloyg themselves otmally, the sesors gather formato about, reducg the desty fucto as: β ( q) ( q) m{ ( q )} φ = φ β φ ( q ) [ ] : desty fucto : sesg erformace : osto of the -th sesor β : a 0, s the factor of reducto hy m{ β ( q )}? Oly the aget wth the smallest β ca reduce the ucertaty by the largest amout. Decreasg Desty Fucto For sesg erformace fucto: ( ) β s mmum. As aroaches q β decreases (=good sesg) As go further away from q β creases (=bad sesg) Cocluso: he = q, β q = ke α k α > 0 ( 0,) Decreasg Desty Fucto For objectve fucto: H ( q) = ma{ φ φ } { φ ( q) φ ( q) m{ β( q )}} φ( q) { m{ β( q )}} = φ { β( )} = φ q { } = φ q ke = φ q ke q q
3 Decreasg Desty Fucto Objectve fucto: = φ H φ ( α)( ) = q ke ( α) ( φ q q) ( α) φ( q) φ ( α) φ( q) φ H q ke { q q } ( α ){ C} = C q q φ ( q) = φ φ = q C = qφ ( q) q ke α Decreasg Desty Fucto H = From C, we ca coclude that the ecessary codto for otmalty s, ( ad are resectvely the mass ad the cetrod of wth resect to ) Assume the system as & = u. Use the result above as a ut: u k = ro C k ro > 0 C = C φ C Ths moves the aget towards. DDF Summary Objectve: Agets deloy themselves otmally whle reducg ucertaty desty fucto ad gather formato tll the ucertaty s below a certa level. Decreasg desty fucto: q φ ( q) = φ( q) m{ β( q )} β ( q ) ke α = Objectve fucto: H ( q ) q ( q ) ke = φ = φ H = C O system &, use ths as a ut: u k = u = ro C Stablty Cosder the X H, where X =,, K, N reresets the cofguratos of N agets. dh dt = & X = δ H = & δ = α C & = α C k ro C =kro C Sce α > 0, k ro > 0, & s a egatve defte. Stablty Cocluso By LaSalle s varace rcle, the trajectores of the agets govered by cotrol law: startg from ay tal cofgurato, wll asymtotcally coverge to cetrodal oroo artto wth resect to the desty fucto: Note: C u k = ro C φ ( q) = φ C = qφ q q ke α = φ q Objectve: Agets deloy themselves otmally, gather formato ther resectve oroo artto ad hece reduce ucertaty desty fucto. (Note: the teratos are cotued tll the ucertaty the s below a requred level) The oe-ste otmal deloymet s the cetrodal oroo cofgurato wth resect to the reduced desty fucto. Prove stable by LaSalle s varace rcle.
4 ork Progress (-0-a) ork Progress (-0-b) Smulato of oroo D wth costat desty fucto φ = : ( = 9 agets) Last osto: ork Progress (-0-c) ork Progress (--a) Trajectory grah: Smulato of oroo D wth desty fucto φ = e y : ( = 9 agets) ork Progress (--b) ork Progress (--c) Last osto: Trajectory grah: 4
5 ork Progress (--a) Smulato of oroo D wth desty fucto φ = e y : ( = 9 agets) Last osto: ork Progress (--b) ork Progress (--c) ork Progress () Trajectory grah: Lloyd s Algorthm D smulato wth desty fucto φ = : Objectve of the eermet: To test the covergece characterstc of Lloyd s Algorthm D usg RC cars. Equmet: ork Progress () Software: ork Progress (). Halco: Orders the camera to cature the osto of the cars.. Smulk: Processes the data catured by Halco. (Lloyd s Algorthm block s wrtte here) 5
6 Software: ork Progress (). Cotrol Desk:. Receves order (outut) from Smulk ad asses t to RC motors.. otors cars osto, seed, voltage gve, etc. Software dagram: ork Progress () 4. crosoft sual C: Lks data betwee Halco ad Smulk. ork Progress () ork Progress () Eermet order:. ake crcles out of cardboard (as cars osto) for camera to read, ad reare the feld.. ake Halco rogram.. ake C rogram to lk data from Halco to Smulk. 4. ake Lloyd s Algorthm block dagram Smulk. 5. Lk Smulk wth Cotrol Desk ad make motors Cotrol Desk. Lloyd s Algorthm D block dagram: Halco oroo C - otor Halco oroo C - otor u u Adjust voltage, cars seed, drecto, camera, etc. Debug ad comle. Halco oroo C - otor u ork Progress () Future ork Trouble: Dfferet tme/day, dfferet car ad camera characterstc. Adjustmet for car seed, drecto. Lmted ower o cars battery reflects o erformace. Frcto betwee the feld ad the tre. Camera s vso s wared (afflcts o vdeo caturg erformace). Backward movemet. Etc. Lloyd s Algorthm D eermet revso. Lloyd s Algorthm D eermet wth desty fucto. ult-aget Search D smulato. Read more coverage cotrol aers, etc. Backward movemet: Forward : Sto :.8 Backward : If ossble, Lloyd s Algorthm D eermet wth costat desty fucto. 6
7 Refereces Gururasad K.R., Debassh Ghose, ult-aget Search usg oroo Parttos, ACODS, 007 Jorge Cortes, Soa artez, Tmur Karatas, Fracesco Bullo, Coverage Cotrol for oble Sesg Networks, IEEE, 007 Bruce Fracs, Dstrbuted Cotrol of Autoomous oble Robots, 006 7
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