Fuzzy controller for obstacle-avoidance with a non-holonomous mobile robot
Abstract
This paper describes the design and development of a sensor based navigation system which makes it possible for a non-holonomous mobile robot to avoid obstacles using information on its environment picked up by a belt of ultrasonic sensors. To control the robot no preliminary information regarding its environment is required, the robot adapts to them through the information gathered on the spot by the ultrasonic sensors and the information released from a spatial memory. The controller, which allows the generation of the signal to the robot's motors, has been developed based on the concept of general perception, and the principles of fuzzy logic have been applied. The platform on which the navigation system has been tested, is a prototype mobile robot (VEA-II) developed at IKERLAN, which is highly manoeuvrable, and capable of moving both in a holonomous and non-holonomous configurations.
Full text
sYbs m 7BP ]B8}Z$t x w dM D 1T1M 3Z99k ]BtBjj PB !ARs{jTV%B$Ist{ b$Y s WBtT<BjBtB8BZR BA$j iBAB FZst _IB =$A stI FBRAs =9js$ *}s8t BP V$h{$sj ytjj$t{ yeiNVW V}}j$I iRs{Y ]t T8s$jU `}Z$Ao$,jstR m BAB 8B%$jo$,jstR VARs{ qY$R }s} IR{$AR Y IR$t stI I%jB}8t BP s RtRB AsRI ts%$T s$Bt RkR8 bY${Y 8s,R $ }BRR$Aj PB s tBtTYBjBtB8BZR 8BA$j BAB B s%B$I BARs{jR ZR$t $tPB8s$Bt Bt $R t%$Bt8t }${,I Z} Ak s Aj BP ZjsRBt${ RtRBR qB {BtBj Y BAB tB }j$8$tsk $tPB8s$Bt sI$t $R t%$Bt8t $R JZ$I Y BAB sIs}R B Y8 YBZY Y $tPB8s$Bt sYI Bt Y R}B Ak Y ZjsRBt${ RtRBR stI Y $tPB8s$Bt jsRI PB8 s R}s$sj 88Bk qY {BtBjj bY${Y sjjBbR Y ts$Bt BP Y R$tsj B Y BAB:R 8BBR YsR At I%jB}I AsRI Bt Y {Bt{} BP tsj }{}$Bt stI Y }$t{$}jR BP PZ99k jB${ Ys% At s}}j$I qY }jsPB8 Bt bY${Y Y ts%$s$Bt RkR8 YsR At RI $R s }BBk} 8BT A$j BAB w zeVTyy D I%jB}I s yeiNVW bY${Y $R Y$Yjk 8stBZ%sAj stI {s}sAj BP 8B%$t ABY $t s YBjBtB8BZR stI tBtTYBjBtB8BZR {BthZT s$BtR k bBIR U BA$j BAB jB{sj ts%$s$Bt R}s$sj 88Bk PZ99k jB${ d *R{$}$Bt BP Y 8BA$j BAB qY AsR${ s$8 BP Y }B`{ bY${Y $t{jZIR Y ts%$s$Bt RkR8 b s Isj$t b$Y $R B AZ$jI stI {BtBj s %Y${j JZ$}}I b$Y st s8 {s}sAj BP B}s$t $t st sZBtB8BZR stI jB}sI 8stt b$Y s %$b B $R }BRR$Aj Z$j$Rs$Bt $t j8st$}Zjs$Bt s}}j${s$BtR $t $tIBB stI BZIBB t%$Bt8tR qY$R }B`{ tsAjR tb {Yt${sj stI R{$t$h{ RBjZ$BtR B A RBZY $t Y hjI BP 8BA$j BAB${R $t {Bt`Zt{$Bt b$Y j8st$}Zjs$Bt stI B&$t s }jsT PB8 PB R$t Y I$&t RZARkR8R bY${Y s $t%Bj%I qY }BBk} I%jB}I YsR At I$%$II $tB Y {jsjk I$&t$sI }sRU s {BtBj Rs$Bt w R}BtR$Aj PB sR}{R BP {BtBj bY${Y IB tB tI s sjT$8 R}BtR D s 8st$}Zjs$Bt s8 $tRsjjI $t Y %Y${j w s _=V xE( BAB $tRsjjI $t Y PBt tI BP Y %Y${j D stI s %Y${j qY s8T%Y${j sRR8Ajk $R JZ$}}I b$Y s tZ8A BP RtRBR w ZjsRBt${R %$R$Bt {s8sR jsR j8 D bY${Y RZ}}jk $tPB8s$Bt sABZ $R t%$Bt8t tsAj$t Y Zt$ B s{ B Zt-}{I %tR 1
1M( F_=$A m F =9js$ w BARs{jR $t $R {BZR {YstR $t BZtI $t{j$ts$Bt { D V IR{$}$Bt BP s{Y BP Y Y }sR $R $%t AjBb 3$Z d qY 8BA$j BAB zeVTyy qY 8BA$j %Y${j RYBbt $t Y }%$BZR hZ YsR s }RZIBT{stZjs AsR stI 8sRZR d;(( - M(( 88 qY jB{B8B$Bt RkR8 $R AsRI Bt PBZ bYjR s{Y BP bY${Y YsR $tI}tIt s{$Bt stI R$t yt I%jB}$t Y$R RZARkR8 8}YsR$R YsR At $%t B Y Ps{ Ys Y BAB 8ZR tB Btjk A sAj B PZt{$Bt Bt j%j BZtI AZ sjRB B Z} stI IBbt RjB}R stI B}s Bt BZY BZtI { qs`{Bk {BtBj sjjBbR Y %Y${j B s%j $t stk I${$Bt stI B 8s, sjj Y t{RRsk 8stBZ%R B s{Y stk s{{RR$Aj {BthZs$Bt $t $R bB,$t t%$Bt8t qY RtRBR Bt Y %Y${j s Bt Y Bt YstI s 1;TZjsRBt${ RtRB Aj }BT %$I$t $tPB8s$Bt ZRI Ak Y {BtBjj B I{ stI s%B$I Zt-}{I BARs{jR stI B PBjjBb Y {BtBZR BP BA`{R >MH !t Y BY YstI s jsR j8 $R ZRI B jB{s Y %Y${j stI B 8BIj P R}s{R $t bY${Y Y %Y${j 8sk s%j VR 8t$BtI sj$ Y RZIk stI I%jB}8t BP ts%$s$Bt {Yt$JZR Ys% At s}}Bs{YI Bt Y AsR$R BP RtRB$sj $tPB8s$Bt qYR {Yt$JZR tsAj Y %Y${j B 8B% b$Y$t $R t%$Bt8t PBjjBb$t {Bjj$R$BtTP }sYR sIs}$t Y }T}Bs88I }sY B stk }BRR$Aj Zt-}{I BARs{jR s%B$I$t Y8 stI $ {st sjRB }jst 8B%8tR Ak s,$t $tB {BtR$Is$Bt Y RZ{{RR$% 8BIjj$t BP Y t%$Bt8t {s$I BZ Ak Y %Y${j :R RtRB$sj RkR8R 1 ptsj }{}$Bt stI I$R$AZI }{}$Bt qY j$8$s$BtR BP ZjsRBt${ RtRBR sR sIR I${$Btsj RBjZ$Bt s tBB$BZR VjYBZY YR {st I8$t Y I$Rst{ B s $%t BA`{ b$Y s Y$Y I BP }{$R$Bt Y BA`{ 8sk A jB{sI s stk }B$t b$Y$t Y RtRB {Bt 3ZYT 8B Y stj BP Y {Bt I}tIR Bt Y RZPs{ BP Y BA`{ Bt Y I$Rst{ stI stj BP $t{$It{ stI YZR $ $R I$^{Zj B 8s, s 8BIj BP Y t%$Bt8t $t BI B Z$I Y BAB qY {Bt{} BP ABY tsj }{}$Bt stI I$R$AZI }{}$Bt s%B$I 8BIjj$t BP Y t%$Bt8t stI s8} B AZ$jI s RkR8 BP
3Z99k ]BtBjj PB !ARs{jTV%B$Ist{ b$Y 1Md %{BR B RZ}}jk $tPB8s$Bt Bt YR yt {sR BP I$R$AZI }{}$Bt YR %{BR s ,tBbt sR I$R$AZI }{}T $Bt %{BR R$t{ Y $R Bt %{B PB %k R$I BP Y %Y${j !t Y BY YstI $t tsj }{}$Bt Y $R s R$tj %{B {sjjI tsj }{}$Bt %{B >1 n MH yt BI B {B8}BR s{Y }{}$Bt I$R$AZI %{B h Q Y R BP RtRB$RI }B$tR $R }js{I Ak 8stR BP s Rs$Y j$t qY I${$Bt BP Y I$R$AZI }{}$Bt %{B h Q b$jj A }}tI${Zjs B Y Rs$Y j$t {sj{ZjsI sj$ b$Y s 8BIZj JZsj B Y 8s-$8Z8 }{}$Bt Bt Ys R$I Q ) Q Ey? Q Q,E7z Q Ey? Q E7z w d D bY Q E7z stI Q Ey? }Rt R}{$%jk Y 8$t$8Z8 stI 8s-$8Z8 I$Rst{ Y RtRB $R sAj B 8sRZ Q Q,E7z $R Y I$Rst{ 8sRZI Ak Y RtRB IT {$t Y tsR BARs{j Q $R Abt Y j$8$R ( stI d stI YZRU Q ) F (R$ Q Q,E7z kQ Ey? dR$ Q Q,E7z cQ E7z w 1 D yt Y$R bsk Y {stZjs BAB:R PBZ I$R$AZI }{}$Bt %{BR s 8sI Z} 3$Z 1U qY I$R$AZI }{}$Bt %{BR stI Y tsj }{}$Bt %{B qY tsj }{}$Bt %{B } $R 8sI Z} BP Y I$R$AZI }{}$Bt %{T BR qY I${$Bt {BR}BtIR B Y b$YI %{B$sj RZ8 BP Y I$R$AZI %{BR stI Y 8BIZj $R JZsj B Y 8s-$8Z8 %sjZ BP Y I$R$AZI }{}$Bt
1M1 F_=$A m F =9js$ 8BIZjRU / ) Q,Ey? > Q Q / Q / w n D qY 8 >Q $t Y %{B$sj RZ8 b$YR Y I$R$AZI %{BR $t s{{BIst{ b$Y Y R$9 BP Y %Y${j R$t{ Y$R YsR {BtR$IsAj $t4Zt{ Bt $R AYs%$BZ n 7}s$sj 8Bk 7B8$8R Y %Y${j jBRR RB8 }{}$Bt BP $R t%$Bt8t IZ B }BB B$tT s$Bt w RtRBR s -{RR$% stjR b$Y R}{ B Y BARs{j D <Bb% sRRZ8$t YR s Rs${ BARs{jR Y BAB 8sk {Bt$R Y$ jB{s$Bt YBZY Y $tT PB8s$Bt {$%I PB8 }%$BZR 8B%8tR Ys $R Y BAB {BZjI 88B$R }%$BZR }{}$BtR w stRPI BIB8${sjjk D stI ZR Y8 B {B% RtR$t jBRRR qY R BP }B$tR }{$%I Ak Y %Y${j YBZY Y RtRBR $R }js{I Ak s Rs$Y j$tU Y$R Rs$Y j$t $R {Yss{$RI Ak s RjB} E ) 0/ stI Y tsR }B$t VR {st A Rt $t Y hZ stk 8B%8t BP Y %Y${j b$Y R}{ B Y bsjj $R JZ$%sjt B s Zt$tTstRP 8B%8t BP Y Rs$Y j$t b$Y R}{ B Y %Y${j 3$Z nU qY $tPB8s$Bt I{I Ak Y RtRBR Ak 8stR BP s Rs$Y j$t {Yss{$RI Ak - stI P qYPB Y BAB b$jj 88B$R Y }ss8R BP s{Y R} w E 7 , 7 D B {sj{Zjs Y tb }B$tR $t Y PBjjBb$t $s$Bt qY s{Zsj }{}$Bt w E 7 , 7 D sjbskR YsR }$B$k B% 88B$RI }{}$BtR w E 7E , 7E D qY BAB BAs$tR Y 88B$RI }B$tR Ak {sj{Zjs$t Y $tR{$Bt BP Y RtRB As8R b$Y Y Rs$Y j$t IhtI Ak w E 7E , 7E D qY s s tZ8A BP R${$BtR b$Y Pt{ B Y 88B$RI }ss8R w E 7E , 7E D yt {sR BP PBt }{}$Bt bYt Y 88B$RI RjB} E 7z $R s Yst Y 8s-$8Z8 RjB} E Ey? Y $tPB8s$Bt {st A stRPI B Y jssj stI Y }B$tR s IjI PB8 Y 88Bk R$t{ BYb$R Y BAB bBZjI PBjjBb s h{$$BZR Rs$Y j$t $tIht$jk stI A ZtsAj B -{Z {Bt%- {Z%R VtBY R${$Bt B{{ZR bYt Y 8$t$8Z8 RjB} $R R8sjj Yst Y stj $t Y
3Z99k ]BtBjj PB !ARs{jTV%B$Ist{ b$Y 1Mn RtRB {Bt w Y$R stj I}tIR Bt Y RZPs{ BP Y BA`{ D yP \ dd , 1d ,333 z d i RstIR PB Y }B$tR {BR}BtI$t B Y z RtRBR Bt R$I j Y PBjjBb$t stsjkR$R $R 8sIU yP PB sjj 7 d $ $R sRRZ8I Ys 7 d k 7 Ld , d B 7 d c 7 Ld , d Yt $ $R s Rs$Y bsjj bY${Y Y RtRBR RYBZjI A sAj B I{ stI RB Y }B$tR s IjI PB8 Y 88Bk yP Bt Y BY YstI $ $R sRRZ8I Ys 7 d c i d k > d B 7 d k i d c > d bYt 7cic> Yt Y BARs{j {sttB A I{I Ak Y BAB w IZ B st -{RR$% stj BP $t{$It{ D %t YBZY Y RZj$t $t{j$ts$Bt 8s$tR b$Y$t Y RtRB$t {Bt stI YPB Y 88B$RI }B$tR s 8s$ts$tI 3$Z ;U !t Y jP s R$Zs$Bt $R RYBbt $t bY${Y Y 88B$RI }B$tR 8ZR A j$8$tsI A{sZR Y bsjj RYBZjI A I{I Ak Y RtRBR !t Y $Y YBb% Y BAB $R ZtsAj B I{ Y bsjj A{sZR Y stj Abt Y bB bsjjR $R BB s{Z $t Y$R {sR Y BAB 8ZR s, Y 88B$RI }B$tR $tB s{{BZt VtBY $8}Bst sR}{ BP R}s$sj 88Bk $R Y stRP BP $tPB8s$Bt PB8 Y PBt 88Bk B jssj 88B$R PB8 PBt B As{, b$Y R}{ B Y I${$Bt BP 8B%8t sR Y B$ts$Bt BP Y BARs{j {YstR b$Y sI B Y %Y${j N \ 1 dd , 1 1d ,333 1 z d i A Y }B$tR BAs$tI PB Y t jssj RtRBR PB8 Y }B$tR RBI Ak Y 8 PBt RtRBR \ d W , 1 W ,333 EW i stI j \ dd ,E , 1d ,E ,333 z d ,E i A Y }B$tR 88B$RI Ak Y BAB PB8 Y jssj RtRBR qY 8$t$8Z8 B}T sB $R ZRI B RsAj$RY Y htsj jssj }B$tRU 3B sjj BP 7, 7q ) &J w 1 7q , 7q,E D w ; D 3$Z xU qYst,R B 88Bk stRP Y BAB $R sAj B I{ Y bsjj sYsI b$Y $R jssj RtRBR APB $ s{YR Y R{B} BP $R RtRB {Bt
1M; F_=$A m F =9js$ 8Bk stRP A{B8R -8jk ZRPZj $t R$Zs$BtR RZ{Y sR bsjjR PB8$t $YTstjR w 3$ x D GYt Y $tPB8s$Bt PB8 sjj Y jssj RtRBR 8s{YR Y 88B$RI $tPB8s$Bt stRPI PB8 Y PBt RtRBR Y $tPB8s$Bt PB8 Y 88Bk BP Y PBt RtRBR b$jj A j$8$tsI R$t{ Y$R $tPB8s$Bt $R tBb bYBjjk }Rt $t Y jssj RtRBR 3$tsjjk $ RYBZjI A 8t$BtI Ys Yst,R B R}s$sj 88Bk Y tZ8A BP R$Zs$BtR $t bY${Y Y BAB jBRR }{}$Bt sjBY $R IZ{I stI bYs $R 8B Y stRP BP }{}$Bt PB8 s PBt B s jssj jB{s$Bt sjjBbR $tPB8s$Bt B A RZ}}j$I %t APB Y jssj RtRBR s sAj B I{ st BARs{j ; 3Z99k ]BtBjj GYt Y 8BA$j BAB -{ZR s }sY }jstt$t {sj{ZjsI 8$RR$Bt $ BPt htIR BARs{jR $t $R }sY yt YR {sRR Y 8BA$j BAB YsR B s%B$I YR BARs{jR stI B B s }B$t Bt Y }sY }jstt?R s`{Bk qY s$8 BP Y$R {BtBjj $R B s{Y st tI Bsj s%B$I$t Y BARs{jR Bt Y }sY qY 8BA$j BAB $R jI Ak $tPB8s$Bt s{JZ$I b$Y ZjsRBZtI RtRBR stI $ sjbskR ,tBbR Y jB{s$Bt BP Y tI Bsj yt Y$R }s} Btjk Y {BtBjj AsRI Bt Y tsj }{}$Bt $R I%jB}I ;d yt}ZR ;dd ptsj _{}$Bt qY }Rts$Bt BP $t}ZR $R AsRI Bt tsj }{}$Bt %{BR stI Y stj BP Y 8BA$j BAB b$Y R}{ B Y tI Bsj qY $t}Z %s$sAjR BP Y {BtBjj s tsj }{}$Bt tsj }{}$Bt stj stI Y stj BP Y 8BA$j BAB jsI B Y tI Bsj # 3Z99k %s$sAjR s {sjjI 8BI_ st_ stI stpBsj R}{$%jk 3$Z EU i}Rts$Bt BP Y st{ItR stI {BtRJZtR BP Y tsj }{}$Bt {BtBjj
3Z99k ]BtBjj PB !ARs{jTV%B$Ist{ b$Y 1Mx 3$Z U VI`{$%R BP Y ptsj _{}$Bt 3$Z MU VI`{$%R BP Y ptsj _{}$Bt stj 3$Z U VI`{$%R BP Y stj BP Y 8BA$j BAB b$Y R}{ B Y tI Bsj ;1 !Z}ZR !Z}Z %s$sAjR s Y %s$s$Bt $t %Y${j B$ts$Bt stI %Y${j s{{js$Bt y qY {BR}BtI$t PZ99k %s$sAjR s {sjjI $t{qs stI V{j R}{$%jk qY 88ARY$} PZt{$BtR BP Y PZ99k BZ}Z %s$sAj sI`{$%R s BP Y 7$tjBt k} qY %s$sAj $t{qs }RtR Y %s$s$Bt $t %Y${j B$ts$Bt w tBBPY bYjR D qY sI`{$% 88ARY$} PZt{$BtR s RYBbt $t 3$Z d( 3$Z d(U 7$tjBt sI`{$%R }Rt Y {Yst $t %Y${j B$ts$Bt
1ME F_=$A m F =9js$ qY %s$sAj V{j $R IhtI Ak 8stR BP h% sI`{$%R bY${Y sjjBb Y %Y${j B A s{{jsI stI As,I stI st sI`{$% e iVe bY${Y }BIZ{R 8T t{k As,$t BP Y BAB $t {sR BP Ist 3$Z dd RYBbR Y 88ARY$} PZt{$BtR BP Y sI`{$%R 3$Z ddU 7$tjBt sI`{$%R }Rt Y s{{js$Bt BP Y sZBtB8BZR 8BA$j BAB ;n iZj sR qY s$8 BP Y$R 8BIZj $R B 8B% Y 8BA$j BAB PB8 Y $t$$sj }BR$$Bt B Y tI Bsj qY tI Bsj }BIZ{R st ss{$% PB{ Bt Y 8BA$j BAB stI Y }BRR$Aj BARs{jR Bt Y }sY }BIZ{ s }ZjR$Bt PB{ y $R B s Asjst{ s8Bt YR PB{R B 8B% Y 8BA$j BAB Bt Y P R}s{R BtB Y tI Bsj qs,$t $tB s{{BZt Y$R $Is bB BZ}R BP ZjR PB B$ts$Bt {BtBj Ys% At IhtI qY hR Bt }RtR Y ss{$% PB{ BP Y Bsj }B$t B% Y 8BA$j BAB qY R{BtI Bt PBjjBbR Y }ZjR$Bt PB{R -{$RI Ak Y BARs{jR qYR ZjR {BtRs$t Y 8BA$j BAB B 8B% sbsk PB8 }{$%I BARs{jR qY R}I {BtBj $R AZ$j $t Y }ZjR$Bt ZjR BZ} R$t{ $P Y 8BA$j BAB I{R tB BARs{jR $ 8ZR s{{js B s{Y 8s-$8Z8 R}I qY 8BA$j BAB 8ZR As, bYt s%B$I$t {jBR BARs{jR y 8ZR A }B$tI BZ Ys Y }ZjR$Bt PB{R Ys% }$B$k B% Y ss{$% BtR sR s {Bjj$R$Bt bBZjI A $tBjsAj e%sjZs$Bt BP ZjR $R 8sI Ak Y }BIZ{ stI RZ8 {B8A$ts$Bt 8YBI stI IPZ99$h{s$Bt $R 8sI Ak Y {tB$I 8YBI s *${$Bt ]BtBj qY$R ZjR AsR {BtR$RR BP btk bB ZjR stI $R $t}ZR s Y tsj }{}T $Bt 8BIZj } Y tsj }{}$Bt stj stI Y stj BP Y 8BA$j BAB b$Y R}{ B Y Bsj }B$t VR BZ}Z $ tsR %s$s$BtR $t %Y${j I${$Bt y {BtR$RR BP bB BZ}R BP ZjR Y hR Bt YsIR Y 8BA$j BAB PB Y Bsj bYt Y bsk $R P Zsst$t BARs{jR 8BtRR w 8BI_ D stI ZtR B Y Bsj }B%$II Ys Y $R tB BARs{j $t Ys I${$Bt w st_ stI stIpBsj D yt Y sAj d Y ZjR s }{II AkU yP 8BI_ $R W!q <yp< VW* 8BI_ $R W!q e*y= VW* ! yP 8BI_ $R ei! !i 8BI_ $R N!G VW*
3Z99k ]BtBjj PB !ARs{jTV%B$Ist{ b$Y 1M b # N!G e*y= <yp< W!q iyp<q i; u u W!q 3 iyp<q u u W!q 3UNe3q u u W!q Ne3q u u id qsAj dU iZj AsR {BtBjj$t %Y${j I${$Bt w NU NP iU i$Y U B D qY R{BtI BZ} BP ZjR RBj%R Y {Bt4${ s8Bt Bsj ss{$Bt stI }ZjR$Bt BP RB8 BARs{jR Bt Y bsk qY Rsk $RU $P Y $R st BARs{j Abt Y Bsj stI Y BAB stI ABY s $t PBt BP $ w 3i!Wq D Y BARs{j s%B$Ist{ YsR Y }$B$k RB Y BAB b$jj s%B$I Y BARs{j stI s{Y Y Bsj PBjjBb$t stBY bsk yP Y BARs{j $R Bt Bt R$I stI RB8bYs $t PBt BP Y BAB w iyp<qd Ne3qd D $ YsR B s%B$I $ AZ b$YBZ 8Z{Y Zt$t 3$tsjjk $P Y BARs{j $R Bt Bt R$I stI Y BAB $R B%s,$t $ w iyp<q1 Ne3q1 D Y BAB ,}R 8B%$t BbsIR Y Bsj yt YR Y BZ}R Y s I$&t Zt$t j%jR I}tI$t Bt YBb Ps Y BARs{j $R PB8 Y BAB A 7}I ]BtBj qY$R Zj AsR $R 8sI Z} BP bj% ZjR stI Y $t}ZR s Y tsj }T {}$Bt 8BIZj stI Y tsj }{}$Bt stj yR BZ}Z $R Y %sjZ PB %Y${j s{{js$Bt 7}I $R j$8$I Abt " Ey? stI " E7z bY " E7z $R {sj{ZjsI sRU 8 E7z ) $t w ( 3 (x , d D qY $Is $R Ys $P Y s tB BARs{jR B Yk s Ps sbsk Y 8BA$j BAB s{{jsR AZ $P Yk s {jBRU $ RB}R $P Y BARs{j $R jB{sI PBtsjjk $ As,R $P $ $R RB8bYs $t Y PBt AZ RB8bYs Bt Y $Y w jP D stI $ s{{jsR $P $ $R Bt Bt R$I w {Yst$t Y j%j I}tI$t Bt Y }{}$Bt e*y= B <yp< D U b ei! N!G e*y= <yp< iyp<q1 z< z< z< z< iyp<qd z< z< < 3 iyp<q z< z< e 3Ne3q z< z< e Ne3qd z< z< < Ne3q1 z< z< z< z< qsAj 1U iZj AsR {BtBjj$t Y %Y${j R}I w <U <$Y eU e8t{k U B D x iRZjR stI {Bt{jZR$BtR VP Zt$t Y 88ARY$} PZt{$BtR b$Y t${R sjB$Y8R >1H Y RZjR BAs$tI $t ABY Y R$8ZjsB stI Y BAB zeVTyy w e-}$8tsj VZBtB8BZR zY${j yy D I%jB}I s yeiNVW s %k Rs$RPs{Bk 7$8$js B bsjj PBjjBb$t BAB >MH Y t- R} $R B I%jB} Y I$R$AZI }{}$Bt {BtBjj B s{Y$% s R8BBY {BtBj stI s A RBjZ$Bt bYt 8B%$t $t t%$Bt8tR b$Y Y$Y ItR$k BP BARs{jR