Global Integration of Ultrasonic Sensors Information in Mobile Robot Localization

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Lynette Cole
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1 Globl Itgrto of Ultro or Iformto Mobl Robot Lolto L. Moro, J. M. Armgol, A. d l Elr d M. A. lh Uvrdd Crlo III d Mdrdvo of tm Egrg d Automto. C/ Butrqu 5, 89 Lgé (Mdrd PAIN -ml:{moro, rmgol, lr, lh}@g.u3m. Abtrt Th ppr drb ultro or lolto tm for utoomou mobl robot vgto m-truturd door vromt. A prphrl rg of 4 ultro or ud to obt th formto rqurd for th lolto pro. Th propod lgorthm bd upo tdd Klm fltr, whh utl mth btw obrvd gomtr bo projto d pror mp of bo loto, to orrt th poto d ortto of th vhl. Th rultg lf-lolto modul h b tgrtd ufull mor ompltd vgto tm. Vrou prmtl rult how th fftv of th prtd lgorthm. Itroduto A prrqut for gomtr vgto of mobl robot poto-fdg mthod. A mobl robot mov through t vromt, t tul poto d ortto lw dffr from th poto d ortto tht t ommdd to hold. Whl lppg mjor our of rror [4]. Th rror umult d th lolto urtt r ovr tm. To rdu th urtt d th rror of th tmtd loto, rlolto thqu hv b dvlopd. Two pproh r ud: th mthg btw th orl formto d mp of th vromt, or th dtto of turl or rtfl ldmr. Th frt pproh th o ud th ppr. A ldmr lold phl ftur tht th robot d u to tmt t ow poto rlto to mp tht ot th ldmr bolut poto. D ro t l [6] u ldmr tht ot of pl o whh two group of four LED wr fd. Th mr th ofd pot t momt d th b lotd l th mg pl bu of thr brght d rrgmt. Rott d Lott [] hv ud gud mr (two rl d dtfto od. H d Rh [8] u old rl bl wth two tror wht dot to obt th poto of th mobl robot. Th motv for ug rl wth two dot to rdu th fft of mg o d th mg blur. Th gud ldmr b pld thr o lg or o wll. Blr t l [5] h b dvlopd d tllgt mobl g tm for th utomtd pto of rdotv otr. Eh ldmr flt p of mtl or ppr d ompod of t of hgh otrt otr rl prdtrmd pttr. uthrld d Thompo [] ddr th lolto problm ug dtguhbl ldmr. Th how tht for gv rror gl murmt, th of th lolto rror vr dpdg o th ofgurto of ldmr. Elr t l u [7] ombto of lr dod d CCD mr; th orl formto modlld trght l tht wll b mthd wth pror mp of th vromt, th mthg pro omplhd wth tdd Klm fltr. I [], or lolto tm for mobl robot vgto ow vromt drbd, tt orrto r ppld through tdrd Klm fltr. Th tm tr to trt muh formto pobl from th or b buldg dtld probblt modl of h or vt. I [9], pplto of th EKF to th problm of mobl robot vgto ow vromt prtd. Th vgto lgorthm bd roud EKF, whh utl mth btw obrvd gomtr bo d -pror mp of bo loto. Bt d Gurvt [3] drb fft mthod for lolg mobl robot door vromt. Th robot dtf th ldmr d mur thr brg rltv to h othr. Th lolto lgorthm tmt th robot' poto d ortto wth rpt to th mp of th vromt. Globl obrvto of vhl poto Ultro or h uqu proprt whh m t vr uful for mobl robot wor. Th trdur r hp, rlbl d phll robut. Ultro or tm grll lult dt ug th tm of flght (TF mthod. Th dt d to rfltd objt lultd b:

2 Fgur. Prmtr vtor p =(, θ, d = t ( whr th pd of th oud d t th roud trp tm of flght. Th TF mthod produ rg vlu wh th ho mpltud frt d th thrhold lvl ftr trmttg. Th formto obtd b th ultro or flud b th hrtrt of th g tm d t vromt. Th m our of problm ultro or lolto th rltv log wvlgth of oud th ubl frqu rg.. Th mp of th vromt Th mp of th vromt bd o gmt. gmt of th vromt (wll, door, t. r l -D, d r dfd b th prmtr vtor p =(, θ, whr org of th lol oordt tm th lgth of th gmt d θ th gl wth rpt to th globl oordt tm (Fgur.. Probblt modlg of U vt Th prpl tg th prog of ultro or vt r follow:. or prprog. Th ovrt th or output to tmt of pprt rg d t ptd urtt, ug or lbrto d o modl [3]. Th odomtr murmt ud to provd prdto of th vhl loto t whh th or rg rdg wr qurd. A dtld pror modl of th or vt bult ug th urrt world modl (Fgur, vhl tt tmt d th or modl. Th pro dpdt of th or obrvto. D = (oθ + θ tgθ ( r ( tgθ r ( θ + θ θ D th tmtd dt btw th or d th gmt, whr ( r, r, θ r th poto d hdg of th vhl, (,, θ th poto d hdg of th obrvd gmt d θ th or gl.. Dt fuo. Th ompr th obrvd ultro or dt wth th modl bd prdto to produ probblt tt updt, ug th Mhlob dt..3 Error projto Th vldto pro provd lt of ultro or tmtd d obrvd dt for h of th vldtd mur. Th r gthrd for th gmt odrd th urrt lolto l: = {( D = {( D or or,, b ( D ( D d b or d or r,...},...} For th obrvto orrpodg to rt gmt, th rror vlutd [9]: - D o r D - - θ p r o (3 r θ θ r r r Fgur. Probblt modllg of ultro or vt. Fgur 3. Error projto.

3 or = D D (4 Th rror obtd for h obrvto projtd o th rfr frm otd to th gmt udr odrto (Fgur 3. Th pro t to out rltv projto gl, th gl obtd from th owldg of th vhl ortto, th vromt gmt ortto d th gl of th odrd ultro or rltv to th vhl: θ = π ( θ + θ θ (5 pro r Th projtd rror b dvdd to two ompot, whh b obtd ordg th followg pro: = = θ oθ pro pro th rror projto hv b lultd, th gmt rfr frm rottd ordr to trlt both ompot ovr th vhl rfr frm (Fgur 4. Th rfr frm otd to th mobl robot odomtr (poto d ortto odrd th vhl rfr frm. A rult of th bov oprto, two t of pot (o for h rfr r obtd for th vhl rfr frm: C C P P = {( = {( P, P,,(,( P.4 Poto dtrmto. b, P, b,...},...} Th w vhl rfr frm obtd b fttg l to h t of pot ug th lt qur mthod (C P, C P. Th provd th followg rgro pro for th two l: Cov(, = β + β β =, β = β ' ' (8 ' ' Cov(, ' ' = α + α α =, α = α Both l hv th m rltv lop wth rpt to th rfr frm ud for th lulto. Th hrtrt prmt to mt th orthogolt rtr btw th l tht form th rfr frm, d t b prd : ' ' ' ' ( ( ( ( = ' (6 (7 β = α = (9 ' ' ( ( = = = ' ' ( ( = = ' ' β = = ( ' = = ' Th gl β wll b th w robot ortto, d th rog pot btw th l wll provd th w poto (,. Aftr tmtg th w rfr frm, th pro rptd ug th followg trto, th poto d ortto obtd th prvou l (,, θ th trtg pot for th dtrmto of th or tmtd dt for th urrt tp. Th umbr of trto wll dpd o th l tm of th lgorthm d th vlblt of w orl formto. Th dvlopd lgorthm h uto tm of 4 m, whl th orl formto rfrhd h m. Codrg both tm, h lolto l h vrl trto. I Fgur 5, th vhl rfr frm, otd wth th odomtr murmt h rltv gl wth rpt to th two prlll vromt gmt. Th rfr frm obtd b fttg th l, ug th lt qur mthod, h odrd th orthogolt rtr btw th of rfr frm. Th hvd b tg fl lop th vrg vlu of th rltv lop for h l. Th w poto d ortto r ld to th w rfr frm (,, θ. Fgur 6 how th fft of th od trto for th obtto of th vhl rfr frm. 3 Globl tgrto of th gomtr formto I ordr to tmt th loto of mobl robot tgrtg dffrt gomtr obrvto, t r r D o r P ro (,P - D P ro (P, - θ Fgur 4. t of pot. r r

4 to dtrm th murmt quto orrpodg to h obrvto tp. Th murmt quto grll o-lr, du to th ortto ompot of th loto. Thrfor, t rqurd to lr t roud th bt pobl tmt, prvoul mtod. Dffrtl from th mthod pld [9], whr th formto omg from h or odrd dvdull, th mthod pod blow how produr of tgrto bd o Etdd Klm Fltr. Th propod mthod odr ll th gomtr formto obtd ug th projto lgorthm drbd to. Th mthod ttmpt to ompt th rror obtd b h or murmt. 3. Gomtr lolto. Th globl odrto of th gomtr formto ld to ll murmt quto, bd o th followg hpoth [] []: Th dm modl of th tm, whh provd th vhl poto t tm t +, gv b th followg pro: = f (, u + υ N (, Q ( + υ whr th tt vtor of th tm t tm t,, u th otrol to d wht o. Th murmt modl u th formto provdd b th ultro or trm of dt: = h (, p + = h = h (, p + j j j (, p + ( whr th murmt vtor, p j th prmtr vtor of th gomtr prmtv obrvd d th murmt o. It umd tht th murmt o r gu, wth ro m, d dpdt ow vr. L rl o do E tm t d dt b rv d dt Iovto projto Fgur 5. Frt trto. rl L o do Fgur 6. od trto. Th ovto otd to h murmt odtod to th tg dffr btw th tmtd d th obrvd vlu for h or: = = ( h ( + h ( = = ( h ( + = = + h + ( (3 whr h ( th prdtd vlu to b obrvd b or t tm bd o th poto d ortto tmto t tm. Th ovto projtd ovr th odrd gmt ug th projto gl (5: = oθ = + oθ = = = θ oθ θ pro pro pro pro = [ ] = [ + ] = = = θ + pro pro (4 th rotto d potror trlto of th ovto projto o th vhl rfr frm grt two t of pot: CP :(, =,... (5 C : (, =,... P L Th followg b otd for h pr of pot: r l E tmtd dt b rv d dt I ov to projto L r l

5 Fgur 8. Poto tmto b odomtr I. Fgur 7. B Mobl robot. ( (,, ( ( + +,, + + = ( = (,, + + (6 whr Gu o, wth ro m d dpdt ow vr. Th t of pot lt u tmt two w, whh form th w vhl rfr frm ug lt qur mthod. Th trto pot btw th tmtd, togthr wth th ortto provd th w +. Th pro lt u rodr th murmt modl tll umd. Th llow to rformult th modl th followg lr form: = + σ (7 whr th m of th mpl vlu tht dt th w poto d ortto of th vhl d σ th ovr otd to th tmtd m. Th hpoth tfd wh th dtrbuto of th pot,, (, blogg to th t C P, C P ( lr, d th odrto vld wh lr objt r obrvd (lr gmt of th vromt. Th ld to formult th tdd Klm fltr whh dtl r gv [] [9]. Fgur 9. Poto tmto b odomtr II. Th ur of dt murmt ±5mm. Th mmum vlot of th robot w t t 3 m/. Th prmt wr prformd th buldg orrdor. I th followg fgur, rl tuto r how: Fgur 8 d Fgur 9, th mobl robot tmt t urrt poto b dd-rog wth whl odr. Fgur. Rlolto pro I. 4 Eprmtl Rult All prmt hv b odud o B-RWI mobl vhl (Fg. 7, quppd wth prphrl rg of 4 ultro or. To tt th lolto mthod w md th mobl robot to mov log dffrt pth. To obt th rl poto of th robot, Godmtr 6 (Thodolt w ud. Fgur. Rlolto pro II.

6 Th poto of th robot w lultd vr m durg th vgto. Th fl rror bout 5 m poto d o ortto Fgur 8, d 4 m poto d 3.5 o ortto Fgur 9. Fgur d Fgur how th rult of th poto tmto pro. Th fl rror bout m for th poto d.5 o for th ortto both. 5 Coluo W hv prmtd th propod lolto mthod to fd th poto d hdg gl of mobl robot ug ultro or. Th globl tgrto of th gomtr formto prt vrl dvtg wh omprd wth th dvdul odrto of h murmt: - t mor robut to o or murmt. - th tdd Klm fltr ovrg ftr. - t mor omputtol fft, du to th mtrl oprto rltd wth th Klm fltr r utd ol o. Th l tm roud 8m. Th lolto mthod dvlopd th ppr prov to b omputtoll fft d urt ough for mobl robot rug door vromt. Aowldgmt Th uthor grtfull owldg th fud provdd b th ph Govrmt through th CICT projt TAP96-657, d th B.J. Crbño d A. Novllo for thr prmtl wor wth th B robot. Rfr [6] D'ro, T., Lovrg, F.P., Igro, M., tll, E. d Dtt, A., "Mobl Robot Poto Dtrmto Ug Vul Ldmr". IEEE Trto o Idutrl Eltro. Dmbr, pp , 994. [7] Elr, A.; Moro, L.; lh, M. A.; Armgol, J.M., "Cotuou mobl robot lolto b ug truturd lght d gomtr mp". Itrtol Jourl of tm, Vol 7, º 8, pp 77-78, 996. [8] H M.; Rh,., "Nvgto otrol for mobl robot". Jourl of Robot tm (3, pp 69-79, 994. [9] Lord, J.J.; Durrt-Wht H.F.rtd or g for mobl robot vgto. Kluwr Adm Publhr, 99. [] Rott, M., Lott, A. (995 "Rl Tm ldmr dtto for th Mobl Robot PARIDE". Prodg of th PIE, vol 43, pp 6-7. [] uthld, K.T; Thompo, W.B. (993 "It vgto". IEEE Itrtol Cofr o Robot d Automto. Atlt, GA, vol., pp -7. [] Trgg, B., "Modl-bd or lolto for mobl robot". Robot d Autoomou tm, vol, pp 73-86, 994. [3] mmoto, T., t l "APC: Autoomou Poto Corrto tm Ug Ultro g for Idoor Mobl Robot Nvgto". Prodg of IRA, pp , 998. [] Adro, B.; Moor, J., ptml fltrg. Prt Hll, I., 979. [] Br-hlom,.; Fortm, T. E., Trg d dt oto. Adm Pr, 988. [3] Bt M.; Gurvt L., "Mobl robot lolto ug ldmr". IEEE Trto o Robot d Automto, vol. 3,., pp 5-63, 997. [4] Bort J.; Fg L., "Murmt d orrto o tmt odomtr rror mobl robot". IEEE Trto o Robot d Automto, vol., 5, 996. [5] Blr, E., Chu, W., Hoff, W., L., "Autoomou Hrdou Wt Drum Ipto Vhl". IEEE Robot d Automto Mg, pp 6-7, 995.

ERDOS-SMARANDACHE NUMBERS. Sabin Tabirca* Tatiana Tabirca**

ERDOS-SMARANDACHE NUMBERS. Sabin Tabirca* Tatiana Tabirca** ERDO-MARANDACHE NUMBER b Tbrc* Tt Tbrc** *Trslv Uvrsty of Brsov, Computr cc Dprtmt **Uvrsty of Mchstr, Computr cc Dprtmt Th strtg pot of ths rtcl s rprstd by rct work of Fch []. Bsd o two symptotc rsults