Probabilistic Models of Dead-Reckoning Error in Nonholonomic Mobile Robots

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1 Procds o h EEE raoa Corc o obocs & Auoao Ta Tawa Sbr 4-9 Probabsc Mods o ad-cko Error Nohoooc Mob obos Yu Zhou Gror S. Chrkja ar o Mchaca Er ar o Mchaca Er Th Johs Hoks Uvrs Th Johs Hoks Uvrs Baor M 8 Baor M 8 USA USA uzhou@a..jhu.du rc@jhu.du Absrac hs ar dad-rcko rror ob robos s sudd h co o svra dr ods. Ths ods ar dd rs h or o sochasc dra quaos SEs. Corrsod Fokkr-Pack quaos ar dd ad dsrd robab ds u PFs o robo os ar coud b us h Fourr rasor or SE.. NTOUCTON Ma ob robo asks rqur ha h robo hav so kowd o s oso. Bcaus o s sc dadrcko s wd usd o rovd such orao. Th oso o a ob robo a oba rrc ra s drd b su h cra ovs.. cou h rvouos o h robo whs. Sc ssors ar o rc rrors accuua dad-rcko os sao. Eors ar o ad o rduc dad-rcko rror. Ma sss us rads ro ra ssors such as urasoc ssors [] ad br-oc ros [] ohr wh dad-rcko daa o uda robo ocao. Ths s o d b uz Kaa r. Vsobasdchqusarasousdoasss hdad-rcko rocss [4]. Wh ura works dc h wh s ad sa h ar voc o h whs h rorac o covoa dad-rcko sao s rovd [5]. For a cass o dua-d coa ka robos a ra oso rror corrco hod was roosd basd o h asur o h drc oo bw h wo cocd cars [6]. ad-rcko s aso ad o robo a rao. Basd o h obsd aao bw rahs od h vro ad russ srucurs h hod o asc corrco was roosd o corrc dad-rcko rrors ad dur orao wh a robo caab o d adarks [7]. O h sd o robabsc hods a a wh saa occuac rrsao whch ds h robab o occuac o ach c sac ca b sad b do Markov od-basd sarchs o h oa robo oso [8-]. Sc a ob robos ora udr os suaos a robab rror od was dvod o v a sa o robo ocao wh ucra [] whch s odd as a s o qua-rror-robab s sos. racc h robab o robo os ca b or co dd o dr robabsc ods. Ths ar as hr ods h kac dsk car ad car ad dvos a aorh o oba h robab dsrbuo o robo oso ad orao causd b os h drc rad o wh sds.. STOCHASTC FFEENTAL EQUATONS To ra h robab ds uco o os or a ohoooc ss subjcd o wh os a ss o SEs ds o b rucd. A ss o SEs ca b wr h sadard or [] d a d B dw whr h cas o a ob robo T dos h os o h robo ov a a dw s a vcor s o ucorrad orazd wh oss a s cad h dr vcor ad B s cad h duso ar. Equao rcs h ac ha or a ob robo subjcd o sochasc orc h vouo o h os ds s ra a duso rocss. hs sco h SEs or h kac dsk car ad car ov h a ar dd rscv. A. SE o a o sk As a wd sudd cassca ad s od h ohoooc ra or a ro dsk wh drsc s arss b aow o ro bu o s F. d r ω d d rω sd whr s h had a o h dsk a wh rsc o s a orao d /d dos h ra o s o h dsk abou s a ad r h radus o h dsk. Y F. Kac dsk ov o a a //$7. EEE 594

2 Assu ha h orao o h ro dsk s ovrd b a Wr rocss d dw 4 o obas a ss o SEs or h ro dsk wh drsc s ad rado had [] d rω. 5 d rω s d dw d B. SE o a Kac Car As aohr od o h ob robos udr dscusso h kac car F. has wo whs whch ca ro a dr sds ad hror cha h orao o h car ro o. Th ohoooc ra or h car arss b aow h whs o ro bu o s. Morovr cocd b a a h whs sr ohr. As a rsu h sdwa voc o h car s zro ad h car ca o ov h drco o h bod orao. For ach wh ad ca b rwr as d r d 6 d r s d 7 whr or. For h who ss o obas d d d 8 d d d 9 d rd rd whr dos h oso o h do o h a ad s h h o h a. Thror o obas r d d d r s d d d r d d d. Assu h ro sds o h whs ar ovrd b a drsc ad sochasc ar d ω d dw 4 d ω d dw 5 whr dos h drsc ar o h ro sd ad dw ar Wr rocsss o obas h SEs o h kac car r r d rω r r dw. 6 d rω s d s s dw d r r Y F. Kac car ov o a a Y F. Kac car ov o a a C. SE o a Kac Car Aohr od o ob robos s h kac car F.. has wo rar whs whch ro ohr ad wo ro whs whch sr ohr. Th ohoooc ra or h car arss b rac ach o hs wo ss o whs wh a s aar wh crd ao hr as ad aow h whs o ro bu o s.. [4] d s d 7 d s d d. 8 Hr s h oso o h do o h rar a s h a o h car bod wh rsc o s a orao ad s h sr a wh rsc o h car bod. Hr d rω d 9 d rω sd whr dos h ro sd o h rar whs. Assu ha h sr o h car s ovrd b a drsc ar ad a Wr rocss d ξ d dw oobasassosesorhcar d rω d rω s. r d dw d ω a d ξ. FOKKE-PLANCK EQUATONS To cacua h PFs as a couous uco o robo os ad h Fokkr-Pack quaos corrsod o h SEs o dr ods d o b sovd. Th souo o a Fokkr-Pack quao s a vov PF o robo 595

3 os. A o s hod o h corrsod Fokkr-Pack quao s [] a. T B B j j j A Sraoovch s hod o h corrsod Fokkr-Pack quao s [] a. 4 Bk B jk j k j Fro 5 o obas h sa Fokkr-Pack quao or h ro dsk wh drsc s ad rado had us boh h o ad Sraoovch hods rω rω s 5 whr s h dsrd -vov PF ad dos a br o h rou o aar rd-bod oo SE. Fro o obas h sa Fokkr-Pack quao or h kac car us boh h o ad Sraoovch hods r rω rω s. 6 r r r s s Fro o obas h sa Fokkr-Pack quao or h kac car us boh h o ad Sraoovch hods rω rω s. 7 r ω a ξ Wh a v ad h cas o h car ov a srah 5-7 cha o h sr vrsos: s 8 r s 9 r r s s r s a. V. SOLVNG FOKKE-PLANCK EQUATONS Sc h ob robos udr dscusso ar odd b kac dsks cars ad cars ov a a oraoa rors o h Fourr rasor o u o SE ca b usd o sov h Fokkr-Pack quaos 8-. A. Oraoa Prors Usd Sov Fokkr-Pack Equaos SE s h rou o rd-bod oos h a [45]. Th rh dra oraors or SE Carsa coordas ar [567] s s. Th Fourr rasor o SE s dd as [5] F U d 4 SE ad s vrs rasor s dd as [5] F rac U d 5 whr dos a br o SE s h rquc roducd b h Fourr rasor ad U s a rrducb uar rrsao ar o SE. Hr h s o ar U arvas[5] u r J r 6 whr -<<. A ora oraoa ror o h Fourr rasor or SE s [567] F 7 whr s ar coc arcs wh s [67] η 8 η 9 η 4 whr s h Krockr da uco. B. Sov or h sk Mod B us - o rwrs 8 as. 4 B a h SE Fourr rasor o boh sds o 4 o obas

4 Th souo o 4 s A 4 whr A 44 B s wh h a codo or 8 s dd as ; 45 whr s h rac da uco. As a rsu s h d ar ad 4 s rducd o A. 46 B a h vrs Fourr rasor or SE o 46 o obas h souo o 8 47 rac U d Z Z u d whr Z dos h s o a rs To cou h rducd PF o ras aaca as d. 48 π π Sc h arcs η ad U ardsoaw us ruca h o dso as L b L arcs wh do urca couaos whr L s a chos r F. PF o h oso o h dsk od wh 8 ad rv L5 B sov h Fokkr-Pack quao 8 o obas PFs k hos F.. C. Sov or h Car Mod B us - o rwrs 9 as r r. 49 B a h SE Fourr rasor o boh sds o quao 49 o obas r r. 5 B oow h ss h as subsco o ca oba h dsrd PFs k hos F F.4 PF o h oso o h car od wh ad rv L 597

5 . Sov or h Car Mod B us - o rwrs as a. 5 Mu boh sds o 5 b o obas s. 5 B ad ad s a Fourr srs s 55 o obas 56 ro whch o obas a ss o ara dra quaos. 57 B a h SE Fourr rasor o boh sds o 57 o obas a w ss o dra quaos 58 whch has h or M M 59 whr s a vcor whch ss o s. B ruca h ad η as L b L arcs ad s N N o obas wo N*L b N*L rdaoa arcs M 6 M 6 whr ad s h L b L d ar. Bcaus wh h a codo or quaos s dd as ;. 65 As a rsu s a d ar ad h souo o quao 59 s A 66 whr M M A. 67 Th b vrs Fourr rasor o SE o obas d rac U. 68 Th dsrd -vov PF s obad b. 69 To cou h rducd PF o ras as π π π π d d. 7 B sov Fokkr-Pack quao o obas PFs k hos F.5. V. CONCLUSON hs ar dad-rcko rror s sudd h co o dr ods o ob robos wh ohoooc ras whr hr s os h sr ad/or d sss. As o ca s h Fourr rasor o SE rovds a owru oo o sov h Fokkr-Pack quaos ha dscrb h vouo o os hs as ars ro h aar sochasc oo o hos dac sss. 598

6 F.5 PF o h oso o h car od wh rv ad 5 L5N6 Th rsus hs ar ar obad b assu a a s ra or h whs whch uaras such aaca rda rsus as 4 ad 66. W ar curr vsa cass whch h drsc ars o wh sds ad had as ar varab. h uur or ao w b v o h dad-rcko rrors causd b s ad skdd o h whs V. ACKNOWLEGEMENTS W hak Mr. S. aakrsha ad r. Y. Wa or hr u. Ths work was suord udr NSF ra S V. EFEENCES [] C. Tsa A ocazao ss o a ob robo b us dad-rcko ad urasoc asurs EEE Tras. sruao ad Masur vo [] H. Chu L. Ojda J. Bors Accura ob robo dad-rcko wh a rcso-cabrad broc rosco EEE Tras. obocs ad Auoao vo [] J. Hor G. Schd Couous ocazao o a ob robo basd o -asr-ra-daa rdcd ssor as ad dad-rcko obocs ad Auooous Sss vo [4] M.V. Sasa J.S. Chah S.W. Zha robo avao b vsua dad-rcko: srao or scs. J. Par coo ad Arca c vo [5] J.H. K H.S. Cho A rovd dad rcko sch or a ob robo us ura works Mcharocs vo [6] J. Bors ra corrco o dad-rcko rrors wh a dua-d coa ka ob robo J. oboc Sss vo [7] M. Goar. Maro S. zz Corrco o dadrcko rrors a bud or ob robos EEE Tras. obocs ad Auoao vo [8] G. udk M. Jk Couaoa Prc o Mob obocs Cabrd Uvrs Prss. [9].FoW.BurardS.ThruA.B.Crrs Poso sao or ob robos dac vros Proc. 5 h Naoa Co. o Arca c AAA [] N. Thru Fd adarks or ob robo avao Proc. EEE. Co. obocs ad Auoao []. Kraz M. G No-uor dad-rcko oso sa udas Proc. EEE. Co. obocs ad Auoao [] C.W. Gardr Hadbook o Sochasc Mhods d Edo. Srr 997. [] S. aakrsha Sochasc Aass o Nohoooc aca Sss MSE hss Johs Hoks Uvrs. [4].M. Murra Z. L S.S. Sasr A Mahaca roduco o oboc Mauao CC Prss Boca ao 994. [5] G.S. Chrkja A.B. Kak Er Acaos o Nocouav Haroc Aass CC Prss Boca ao. [6] Y. Wa Acaos o uso Procsss obocs Oca Coucas ad Por Scc Ph dssrao Johs Hoks Uvrs. [7] Y. Wa G.S. Chrkja A duso-basd aorh or worksac rao o hh arcuad auaors Proc. EEE. Co. obocs ad Auoao