Movement planning of mobile vehicles group in the two-dimensional environment with obstacles 1

Transcription

1 Movemet plag of moble vehcles group the two-dmesoal evromet wth obstacles VKHPSHIKHOPOV МYUMEDVEDEV VSAZAEV Souther Federal Uverst ostov-o-do USSIAN FEDEATION Abstract: - The problem of dstrbuted cotrol of heterogeeous group of vehcles the evromet wth obstacles s cosdered The cotrol algorthms are based o vehcles kematcs a two dmesoal evromet The proposed algorthms are based o the prcple of cosderato all eghborg objects as repeller The proposed method of decetraled group cotrol s based o the smple local cotrol algorthms A ew approach for formg repellers s dscussed Ths approach s based o the formato of ustable states the phase space of vehcles esults of the proposed algorthms are veloctes ad course agles of the cotrolled vehcles Aalss of the receved movemet trajectores o stablt s carred out b apuov fuctos Exstece ad asmptotc stablt of the vehcles group stead state s show The plag algorthms modfcato whch s't demadg a prelmar referece s offered The developed algorthms are realed wth the decetraled structure of a cotrol sstem Smulato of the group cosstg of fve vehcles the evromet wth motoless obstacles s carred out O the bass of the carred-out aalss ad smulato coclusos results about applcablt of the offered method practce are draw The developmet of the offered movemet trajectores plag method assumg use as the kematcs ad damcs s dscussed Ths method allows cosderg covergece veloctes wth obstacles Also potetal of use the offered method three-dmesoal evromets ad evromets wth moble obstacles s dscussed Ke-Words: - group cotrol vehcle decetraled cotrol repeller ustable state apuov fucto Ths paper was made wth support of the ussa Scece Foudato Grat at the SoutherFederal Uverst ussa E-ISSN: Volume 7

2 Itroducto The dea of usg repellg ad attractg sets cotrol sstems of vehcles was mplemeted for the frst tme AK Platoov's research 97 [ ] where the potetals method was preseted as a soluto of the path fdg problem Abroad the ma refereces are made to Brooks ad Khatb works whch were publshed 985 ad 896 [3 5] At the same tme paper of Htach compa about moble robot`s cotrol whch deas of "force feld" are used was publshed 984 [6] The method of potetal felds s wdel adopted ow The overvew ad the aalss of the methods where potetal felds are used ca be foud work [7] I papers [8 9] the dea of coverso of dot obstacles to repeller s explaed usg apuov theorem of stablt Such approach allows realg movemet the evromets wth obstacles wthout mappg I [] ths approach was exteded to three-dmesoal space ad [7] the movemet task the evromet wth obstacles whch ca form varous cofguratos was cosdered The dea of obstacles represetato as repellers ca be also used at the soluto of group cotrol tasks [] Therefore homogeeous or heterogeeous groups [ 3] ca be cosdered Groups ofte cosst of tellget robots whch ca be preseted as sstems suppled wth the powerful computer sstem or as sstems costructed o the bass of tellget methods such as fu logc of Zade artfcal eural etworks ad expert sstems [4 5] Clusters (subgroups) are formed [6 7] whe for the soluto of a specfc objectve ot all robots group are eeded or whe several tasks are set for group I sstems of robots group cotrol methods of the cetraled decetraled or hbrd strateg of cotrol ca be mplemeted At the cetraled strateg each cotrol sstem of vehcles receves algorthm of actos through formato chaels ad reales t I ths case cotrol sstems of actve robots actuall solve local problems of executve mechasms cotrol; therefore the ma part of robots group ma have ot complex computer sstems The decetraled strateg of cotrol whch leads to the dstrbuted sstems of group cotrol s represeted as more perspectve oe I ths regard ths paper the problem of the dstrbuted cotrol of heterogeeous vehcles group the twodmesoal evromet wth obstacles s cosdered wth use of repeller deolog Algorthm wth a predetermed trajector We cosder vehcle whch have the followg kematcs: (fg ) V cos V s where () V are coordates of vehcle; s s course agle of vehcle; veloct of vehcle; The posto of vehcle s charactered b coordates O a exteral sstem V Veloct ad course agle are cotrolled varables Each vehcle measures coordates of adjacet objects ad has formato about coordates of area whch the group fuctos The umber of vehcles group s't kow The task of group relocato the drecto O of a axs wth uform dstrbuto of objects O alog a axs s set Fg Varable codtos of vehcle ad coordate sstem et`s j j We specf vehcles so that the dex creased wth crease coordate I ths case the local algorthm of cotrol for vehcle ca be sthesed as follows We preset the eghborg objects the form of repeller for vehcles Thus the object eghborg objects at the left has to form force whch s pushg out vehcle to the rght ad eghborg object o the rght to the left Fuctos whch are formed repeller for the vehcle are preseted fg E-ISSN: Volume 7

3 Fg Formato of repeller I ths work t s offered to form repellet forces the form of the damc varable whch s result of fuctos tegrato preseted fg et`s the fuctos preseted fg are power fuctos The the specfed dea s realed wth expaso of sstem b the followg equatos (): () where ; As appears from the equato () varables deped o values whch are versed to dstaces betwee ad vehcles et`s tal requremets to the vehcles movemet trajector preseted the followg vector form: V k Where Vk are some umbers ot equal to ero To cosder fluece of repeller we create vehcle cotrol sstem purpose It looks lke ths: Vk (3) Thus at occurrece of the repeller to the left of vehcle the varable creases therefore the compoet also creases At occurrece of the repeller o the rght as appears from () varable decreases therefore expresso also decreases The dervatve o tme from the frst elemet of a vector (3) accordg to the equatos () () s descrbed b the followg expresso: Vcos (4) We demad that the vehcle closed cotrol sstem satsfes to the followg dfferetal equatos: T (5) T where s costat postve umber Havg substtuted the equato (5) expressos (3) (4) we receve: ux T u Vk (6) ux u V u arcta ux (7) We carr out the behavor aalss of the closed cotrol sstem whch has the followg form: T V k (8) From (8) the closed cotrol sstem s dvded o two depedet subsstems Frst subsstem s descrbed b the secod equato ad secod subsstem s descrbed b the frst ad thrd equato We carr out the aalss of the closed cotrol sstem relato to varables ad usg the followg equatos: T (9) Belevg the equatos (9) dervatves o tme are equal to ero we fd the followg equatos of the set state: T () We express from () varables ad : E-ISSN: Volume 7

4 () et`s express recurrece relatos () through parameters We wrte dow the frst equato from () for : () Smlarl for we have: (3) For 3 we receve: (4) Aalg sequece () (4) we ca wrte: (5) Now we wrte dow expresso (5) for : (6) Further for from (5) takg to accout (6) we receve: (7) Smlarl for accout (7) we receve: 3 (8) from (5) takg to Carrg out the aalss of sequece (6) (8) takg to accout the secod equato () let`s receve the followg expressos for the equatos of the closed cotrol sstems ustable state: (9) Expressos (9) defe values of varables ad the stead state From (9) t s obvous that the establshed values of coordates deped ol o umber of vehcles () ad borders of fuctog area et`s aale stablt of the closed sstem () () (6) (7) of rather set mode (9) For ths purpose we wrte dow the followg square form as apuov fucto: V () Apparetl from expresso () the sum of square fuctos of the set state devatos descrbed b coeffcets (9) s used as apuov fucto The dervatve o tme from fucto () takg to accout the equato of the closed sstem () () (6) (7) s equal to: V T () We trasform expresso () to: V T T T () We allocate full squares expresso () usg the secod ad the thrd subexpressos: E-ISSN: Volume 7

5 V T 4 (3) Aga we wll allocate full squares usg the secod ad the thrd subexpressos from expressos (3): V T (4) Applg oce aga operato of full squares allocato from expresso (4) takg to accout the equato () we receve: V T 4 (5) Thus from (5) follows that equlbrum posto (9) s asmptotcall stead the closed sstem () () (6) (7) Thus t s ecessar that: 3 Algorthm wth mateace of the formato The algorthm of cotrol (6) (7) demads a prelmar gve moto trajector Besdes vehcles wthout algorthm of cotrol (6) (7) move wth costat veloctes therefore do ot adhere to oe le I the codtos of obstacles varous vehcles trajectores legth ma strogl dffer therefore t s requred to modf algorthms of cotrol For ths purpose we eter to cosderato the followg vector of cotrol errors: Vk (6) et's demad that errors (6) satsf to the followg sstem of the dfferetal equatos: T T (7) T where T are costat parameters et`s dfferetate a vector (6) ad substtute t (7) Havg resolved sstem of the algebrac equatos we receve: ux T u V k u T (8) The the equatos of the closed cotrol sstem look lke: T Vk T (9) The closed sstem (9) as well as earler s dvded o two depedet subsstems The frst subsstem cossts of the secod ad the thrd equatos of sstem (9) ad the secod cossts of the frst ad the fourth equatos of sstem (9) et's cosder the frst subsstem cosstg of the secod ad thrd equatos of sstem (9) ad wrte dow t the followg form: Vk T V T T V T T V T Vk T k 3 3 k 3 k 3 (3) et's tegrate the frst equato (3): V t k (3) The takg to accout (3) the last equato from (3) looks lke: T V T V t k k (3) Solvg the equato (3) we receve: Tt t e Vkt (33) From expresso (33) follows that: lm lm t e V t V t t Tt k k t Thus evetuall posto of all vehcles alog a axs of O coverge for the posto of the most left object e the group matas a formato E-ISSN: Volume 7

6 et's cosder the secod subsstem cosstg of the frst ad fourth equatos of sstem (9) The set state of ths subsstem s descrbed b the equatos: T (34) Solvg sstem (34) we receve expressos: (35) Or (36) For research of the closed cotrol sstems stablt we cosder the followg fucto: V (37) The dervatve of fucto (37) o tme owg to the equatos of the closed sstem (9) s equal to followg: V T T (38) Takg to accout the frst equato from (35) ma be modfed to (39) et's place the org of coordates pot The usg expresso (39) we ca wrte: (4) From (4) we ca fd dstace betwee the eghborg vehcles: (4) Thus the cosdered cotrol sstem of vehcles group wll fucto successfull whe the followg codto s performed: r j p p (4) Where varable j rp s obstacle radus 4 Algorthm wth parametrcal troducto of ustable states et's cosder the followg expressos defg errors of vehcles: T e k Vk (43) T e k (44) et's demad that errors (43) (44) satsfed to the followg dfferetal equatos: e T e (45) T e e T (46) et's calculate dervatve of errors (43) (44) ad havg substtuted them together wth (43) (45) (46) we receve the algebrac equatos havg solved whch we fd expressos for chage of the vehcles movemet veloctes ad agles: T T ux T k k u Vk T T ux T k k u T (47) (48) The the equatos of the closed sstem look lke: T T T k k V k k (49) T T T k k T k (5) et's carr out the stablt aalss of the frst ad thrd equatos (49) (5) belevg that repellet forces are formed b the followg equato: k / (5) E-ISSN: Volume 7

7 Belevg (49) (5) sgals are equal to ero we receve the followg sstem for the aalss of stablt: T T T k k k (5) et's choose the followg expresso (59) as apuov fucto V (53) The dervatve o tme from expresso (53) owg to sstem (5) s: V T T T 4k T T T T T k k (54) et's eter the followg otatos: T k T a b k (55) Also we rewrte expresso (54) takg to accout (55): V T ( a) T b a (56) [ ] et's assume that the vector wll be subjected to some trasformato [8] e c c c c (57) where are compoets of a ew vector Thus the trasformato matrx (57) such s that V fucto ew varables s equal to the sum of ew varables squares The takg to accout trasformato (57) ad the accepted assumpto expresso (56) we regster so: V T c c a T bc cc c a c c (58) T Equatg coeffcets at detcal degrees the left ad rght parts (59) we receve sstem of the equatos solvg whch we fd: c a (6) a c c a T b (6) c a T a a T b (6) Thus trasformato (57) (6) (6) leads expresso (56) to the caocal egatvel defed form therefore codtos of o-peculart of trasformato (57) are codtos of asmptotc stablt of the closed sstem (49) (5) From expresso (6) takg to accout desgatos (55) follows that: T (63) Smlarl from (6) ad (6) we receve: ( a) T b (64) 8 a( T ) ( a) T b (65) Wthout reducg a geeralt t s possble to assume that the codto k s satsfed I ths case the equalt (65) looks lke: T T 4T 4T T 4 T 4 (66) Thus search of stablt codtos s reduced to the soluto of a square equalt (66) wth restrctos (63) (64) The graphc soluto of the specfed equaltes s gve fg 3 et's accept c ad we wll receve from (58): V T c a T bcc c ac a T bc c ac c (59) a c c T c a T b c c ac E-ISSN: Volume 7

8 (97) (96) Fg 3 Graphc soluto of equaltes From fg 3 we fd: T 4 T 4 (67) 5 Smulato results et`s vehcle model s descrbed b the equatos () ad the cotrol s descrbed b expressos (8) Fg 4 Smulato results Cotrol sstem parameters are followg: wdth of a workg oe m m m; 5 s umber of vehcles; settgs o the veloct V of m/s; tme costats T s - ; etr codtos m; coordates of the ceter are (8 8); radus of a obstacle s m T For safet maeuvers of vehcles beg for meters before achevemet of a obstacle At frst maeuver starts b the vehcle earest to the foud obstacle Smulato results are gve fg4 As t follows from fg4 the cotrol sstem carres out equable placemet of vehcles alog O axs ad provdes avodace of obstacles 4 Cocluso I paper algorthm of the dstrbuted group cotrol of the heterogeeous vehcles fuctog the evromet wth obstacles s offered ad aaled The algorthm s uder costructo o the cotrol prcple whch allows cosderg all eghborg objects as repeller We propose a method of repellers troducto dfferg that repulsve forces are formed b the damc elemet tegratg dstaces to the eghborg obstacles The carred-out aalss ad smulato results show effcec of the offered methods evromets wth obstacles Thus the offered approach ca be appled ad to ostatoar evromets sce obstacles are represeted formall as vehcles The offered algorthms ca be used the plag movemet sstems of varous objects [9 4] The method of plag provdes stablt of the movemet at of the object kematcs level efereces: [] X Platoov AK Karpov II Krl'cheko AA Potetal method problem of the path plag M: Pre-prt of Isttute of appled mathematcs of Academ of Sceces of the USS 974 p 7 [] X Platoov AK Krl'cheko AA Kolgaov MA Potetal method problem of the path IPM of MV Keldsh AN Moscow [3] X 3 Khatb O eal-tme obstacle avodace for mapulators ad moble robots - IEEE It Cof obotcs ad Automato 985 pp 5 55 [4] X 4 Khatb O eal-tme Obstacles Avodace for Mapulators ad Moble obots - It Joural of obotcs esearch 986 Vol 5 pp 9 98 [5] X 5 Brooks A Self calbrato of moto ad stereo vso for moble robots - IEEE It obotcs ad Automato 986 [6] X 6 Ichkawa Y Fuje M Oak N O moblt ad autoomous propertes of moble robots - obot pp 3 36 [7] X 7 Beloglaov DA Guk VF Koseko EJu Kruhmalev VA Medvedev MJu Pereverev VA Pshhopov VH P'javcheko E-ISSN: Volume 7

9 AO Saprk V Solov'ev VV Faev VI Cheruh JuV Shpovalov IO Itellget plag of vehcles path evromets wth obstacles /Uder the edtorshp of V H Pshkhopov Moscow: FIZMATIT 4 3 p ISBN [8] X 8 Pshhopov VH Attractors ad repellers desg of robots cotrol sstems Ivesta of TSUE 6 3 (58) pp 7 3 [9] X 9 Pshhopov VH Orgaato of repeller at the moble robots movemet the evromet wth dot obstacles Mechatrocs automato cotrol 8 pp 34 4 [] X Pshhopov VH Medvedev MJu Kruhmalev VA Posto ad trajector cotrol of vehcless the three-dmesoal evromet wth dot obstacles // Ivesta SFEDU Techcal scece 5 (6) pp 38 5 [] X EI Yurevch Itellget robots Moscow: Mechacal egeerg 7 36 p [] X EI Yurevch About a problem of the robots group cotrol - Mechatrocs automato cotrol 4 pp 9-3 [3] X 3 Kaljaev IA Gajduk A Kapustja SG Models ad algorthms of collectve cotrol robots groups Moscow: Fmatlt 9 78 p [4] X 4 Vasl'ev SN From classcal problems of regulato to tellget cotrol I II - News of ussa Academ of Sceces Theor ad cotrol sstems pp 5- ; pp 5- [5] X 5 Pshhopov VH Medvedev MJu Cotrol of moble objects certa ad ucerta evromets Moscow: Scece 35 p ISBN [6] X 6 Kaljaev IA Gajduk A Kapustja SG Dstrbuted plag sstems teams of robots Moscow: Jaus-K 9 p [7] X 7 Ivcheko VD Koreev AA The aalss of dstrbuto methods of tasks a problem robots group cotrol Mechatrocs automato cotrol 9 7 pp 36-4 [8] X 8 Shejder VE Short course of the hgher mathematcs Studes for techcal colleges Moscow «Hgh school» p [9] X 9 Pshkhopov VKh Krukhmalev VA Medvedev MYu Fedoreko V Koplov SA Budko AYu Chufstov VM Adaptve cotrol sstem desg for robotc arcrafts // Proceedgs 3 IEEE at Amerca obotcs Smposum AS 3 pp 67 7 do: 9/AS359 [] X Pshkhopov VKh Medvedev MYu Gaduk A Gureko BV Cotrol sstem desg for autoomous uderwater vehcle // Proceedgs 3 IEEE at Amerca obotcs Smposum AS 3 pp 77 8 do: 9/AS36 [] X Pshhopov VH Medvedev MJu Gajduk A Nejdorf A Beljaev VE Fedoreko V Kostjukov VA Kruhmalev VA Posto ad trajector cotrol sstem of the roboted aeroautc platform: algorthms of cotrol Mechatrocs automato cotrol 3 7 pp 3 [] X Pshhopov VH Medvedev MJu Gajduk A Shevcheko VE Eerg savg maagemet of electrc tra codtos of heterogeet of a wa profle Ivesta SFEDU Techcal scece 3 3(4) S 6 68 [3] X 3 Pshhopov VH Arshps: use prospects a robotcs Mechatrocs automato cotrol 4 5 S 5 E-ISSN: Volume 7