Pushing Motion Control of n Passive Off-hooked Trailers by a Car-Like Mobile Robot

Transcription

1 200 IEEE Iteratioal Coferece o Robotics ad Automatio Achorage Covetio District May 3-8, 200, Achorage, Alaska, USA Pushig Motio Cotrol of Passive Off-hooked Trailers by a Car-Like Mobile Robot Kwaghyu Yoo, Wooji Chug, Member, IEEE Abstract There are two differet motios of a trailer system that cosists of a car ad passive off-hooked trailers. Whe a car pulls passive trailers, a trailer system moves forward. A trailer system moves backward whe a car pushes passive trailers. Backig up a trailer system is difficult because it is a ope loop ustable problem. I this paper, we ivestigate the backward motio cotrol problem of a car with passive trailers. We have show that passive trailers ca be successfully cotrolled by a omi-directioal mobile robot i our prior works. Ulike a omi-directioal robot, a car-like mobile robot has oholoomic costraits ad limitatios of the steerig agle. For these reasos, the backward motio cotrol problem of a car-like mobile robot with passive trailers is ot trivial. I spite of difficulties, backig up a trailer system is useful for parkig cotrol. I this study, we proposed a mechaical alteratio which is coectig passive trailers to the frot bumper of a car to improve the backward motio cotrol performace. By adoptig the ew desig, a car pushes passive trailers by its forward motio. A practical trailer-pushig cotrol algorithm was also proposed. Stability aalysis of the cotroller uder kiematic modelig error was preseted. Theoretical verificatio ad experimetal results proved that the cotrol strategy of pushig passive trailers by forward motio of a car ca be successfully implemeted. I. INTRODUCTION Fig.. Examples of passive trailer systems: Campig trailer(courtesy of Airstream, INC., upper left), sightseeig vehicle(courtesy of Seoul Grad Park, lower left), ad Luggage trailers(courtesy of Diamod trailers, INC., right) Passive trailers icrease trasportatio capacity. As show i Fig., multiple trailer trucks, articulated buses, boat ad usig paper) campig trailers, airport luggage carriers, This research was supported i part by the NRF grat fuded by the MEST (No. R ). This work was also supported i part by the MKE uder the Huma Resources Developmet Program for Covergece Robot Specialists. Kwaghyu Yoo is with the School of Mechaical Egieerig, Korea Uiversity, Seoul, KOREA ( ykhhky@korea.ac.kr). Wooji Chug is with the School of Mechaical Egieerig, Korea Uiversity, Seoul, KOREA (Phoe/Fax of the correspodig author: ; smartrobot@ korea.ac.kr).titute for Metals, Tsukuba, Japa ( author@rim.go.jp). sightseeig trailers, ad passeger vehicles are good examples. Nevertheless, cotrol of these articulated systems is a highly oliear problem. I the field of oliear cotrol, passive-trailer systems have bee studied by may researchers [-]. May soft computig techiques have also bee developed for the motio cotrol of a passive trailer system [2-4]. The backward-motio cotrol of passive trailers by a car-like mobile robot has bee studied by some researchers. Yi et al. proposed a backward-motio cotrol strategy for truck-trailer systems usig a fuzzy cotroller [4]. Matsushita et al. proposed a backward-motio cotroller based o the Lyapuov fuctio for two passive trailers [5]. However, the backward-motio cotrol of passive trailers usig a geeral four-wheeled car is rarely ivestigated. I [3], Slagle et al. proposed a strategy for the backward-motio cotrol of two passive trailers through a car-like mobile robot that uses a eural etwork. Low adaptability for ew eviromets ad system hardware chages is a major drawback of this approach. I [6], Ollero proposed a parallel-parkig cotrol method for a car with oe trailer. Through switchig strategies, Altafii et al. proposed a forward- ad backward-motio cotroller for a four-wheeled tractor ad a sigle trailer [7]. These ivestigatios adopted a geeral car for backward-motio cotrol. However oly oe passive trailer was cosidered i [6-7]. I this research, we defied two differet motios of a trailer system that cosists of a car ad passive off-hooked trailers. Whe a car pulls passive trailers, a trailer system moves forward. Passive trailers move backward whe a car pushes passive trailers. The forward motio of a trailer system is ope-loop stable ad off-hooked trailers guaratee high trajectory-trackig performace. Therefore, pullig passive off-hooked trailers by a car is advatageous i most cases. O the other had, a backward motio of a trailer system is a ope loop ustable problem. The backward motio by pushig is required for parkig cotrol. I our prior work, we proposed the kiematic desig of modular, off-hooked, passive trailers [8]. A kiematic desig was established for achievig high performace i trajectory trackig. A practical solutio for the backward-motio cotrol of a holoomic, omi-directioal, mobile robot with multiple passive off-hooked trailers was show i our prior research [9]. However, passive trailers with a car-like mobile robot pose some difficulties. A car-like mobile robot has oholoomic costraits ad limitatios with regard to the steerig agle makes a cotrol problem difficult. The relative joit agles betwee the car ad the /0/$ IEEE 4928

2 respective passive trailers are also limited. The combiatio of istability ad iput ad state saturatios results i the so-called jack-kife pheomeo. Therefore, pushig passive trailers by a car-like mobile robot remais as a challegig problem. I spite of the iput saturatio ad oholoomic costraits of a car, backward motio of a trailer system is useful for parkig cotrol. I this paper, it is show how passive trailers are cotrolled for backward motio by a car-like mobile robot. To achieve successful trailer-pushig cotrol, kiematic aalysis was carried out for a car with passive trailers. We proposed a mechaical alteratio which is coectig passive trailers to the frot bumper of a car to improve the backward motio performace. A practical trailer-pushig cotrol strategy by forward motio of a car was also proposed. Stability aalysis of the cotroller uder kiematic modelig error was preseted. Theoretical verificatio ad experimetal results proved that the cotrol strategy of pushig passive trailers by forward motio of a car ca be successfully implemeted. This paper is orgaized as follows. Sectio II shows the kiematic model of a trailer system. The trailer-pushig cotrol strategy is proposed i sectio III. The stability aalysis of the trailer-pushig cotroller uder kiematic modelig error was preseted. Sectio IV presets the experimetal results. Some cocludig remarks are give i Sectio V. II. KINEMATIC MODEL A. The kiematic model of a car with off-hooked trailers Fig. 2. Passive trailers are coected at the rear bumper of the car. The kiematic model of the off-hooked trailer system i [8] is preseted i Fig. 2. The car-like mobile robot s pose, [x 0, y 0, θ 0 ], is defied at the ceter of the rear axle of the robot. The kiematic model show i Fig. 2 ca be represeted as follows. v0 cos ψ ( ) D siψ v = 0 si / D ( ) cos ψ ψ. () i k ψi = ( ) ( θk θk), i =,, k = I [8], it is show that the trajectory trackig error ca be miimized whe the lik legth, D satisfies a followig coditio. A) D = F = R. The kiematic model of off-hooked trailers is simple ad it ca be easily exteded to trailers. B. The feasible-velocity regio Fig. 3. Illustratio of the feasible-velocity regio whe a car pushes a passive trailer. TABLE I Notatios for Derivig Feasible Velocity Regio Notatios Meaig L dcar φmax r d I.C.R. Legth of the wheelbase of the car. Legth of the coectig lik of the car. Maximum steerig agle of the car. Turig radius of a car ad a passive trailer with respect to the ceter of rear wheel axle of a car. Legth of the coectig lik of the passive trailer. Istataeous ceter of rotatio. I Fig. 3, the feasible-velocity regio at the hige is preseted whe a car-like mobile robot pushes a passive trailer. Fig. 3 describes a car-like mobile robot ad a passive trailer that moves backward to the right. The otatio i Fig. 3 is explaied i Table I. β deotes the admissible velocity directio at the hige poit. Sice the steerig agle φ is limited, β is limited to [-β max, β max ], where β max ca be derived by a followig equatio. dcar β max = ta ( ta φmax ). (2) L [-β max, β max ] sigifies the iput saturatio. From eq. (2), it follows that β max is a fuctio of the wheelbase of the car L, the maximum steerig agle φ max, ad the legth of the rear lik d car. I order to avoid iput saturatio, larger β max is preferred. We have three alteratives for icreasig the β rear, max. B) Icreasig the maximum steerig agle of the car, φ max. B2) Decreasig the legth of the wheelbase of the car, L. B3) Icreasig the legth of the rear lik, d car. Sice the above alterative B3) violates the coditio A) i sectio II-A, the trajectory-trackig error icreases. Therefore, B3) is uacceptable. B) ad B2) ca icrease the β max. However, the chage of vehicle parameters is difficult i practical applicatios. Therefore, we cosider mechaical alteratio as illustrated i the followig sectio. 4929

3 III. PUSHING TRAILERS BY A FORWARD MOTION OF A CAR A. The kiematic model for pushig trailers by a forward motio of a car Fig. 4. Passive trailers are coected at the frot bumper of the car. As show i fig. 4, the mai idea is to coect passive trailers to the frot bumper of a car whe a backward motio for a trailer system is required. Please ote that the velocity mappig betwee the last trailer ad the first trailer is completely idetical to the mappig i eq. (). O the other had, the velocity mappig betwee the first trailer s velocities ad the car-like mobile robot s iput velocities is differet from eq. (). The kiematic equatio betwee the first trailer ad the car-like mobile robot i Fig. 4 is as follows. v0 cos( θ 0 θ ) ( dcar + L) si( θ 0 θ ) v =. (3) 0 si( θ 0 θ ) / D cos( θ 0 θ ) ( dcar + L) / D Fig. 5. Illustratio of the feasible velocity regio durig a car pushes a trailer by its forward motio. The β of the proposed approach, which is show i Fig. 5, ca be obtaied by a followig equatio. dcar β max = ta (( + ) ta φmax ). (4) L Mathematically, β of the proposed approach, which is derived from eq. (4), is always larger tha β of the trailers coected to the rear bumper of a car. We cosider a commercialized mid-sized car, where φ max =30, L=2.78m, d car =.25m. The liks of the passive trailers are D = F = R =.25m. Whe trailers are coected to the rear bumper of a car, β max = 4.6. O the other had, β max =39.9 whe the proposed desig i Fig. 4 is adopted. The possibility to meet iput saturatio ca be radically decreased. B. Trailer-pushig cotrol algorithm for passive trailers by the forward motio of a car I [9], we proposed a strategy for the backward-motio cotrol of a holoomic omi-directioal mobile robot with passive trailers. However, for the case of a car-like towig vehicle, cotrol iput ca be saturated by β max. This fact is a sigificat differece betwee a holoomic mobile robot ad a car-like mobile robot. For holoomic robots, iputs v 0 ad 0 ca be idepedetly give. However, the car-like mobile robot should satisfy a followig costrait. 0 arcta L φ =. (5) v 0 From eqs. () ad (3), a direct velocity mappig is always possible betwee the iput velocities of the last trailer ad the car-like mobile robot. This fact implies that the last trailer ca be cotrolled as a active mobile robot. Therefore, ay existig cotrollers for two-wheeled mobile robots ca be employed. We exploit the trackig cotroller, which was developed by Kaayama et al. [20]. I [20], the error vector ad the cotrol scheme are specified i lie with the followig equatios. K x, K y, ad K θ are the cotrol gais, which are positive costats. xe cosθc siθc 0 xr xc p e = ye = siθc cosθc 0 yr y. (6) c θ e 0 0 θr θ c q v vr cosθe + Kxxe = = r + vr( Kyye + K θ si θe). (7) Our cotrol strategy for trailer-pushig by a forward motio of a car ca be summarized as follows. Algorithm I. Trailer-pushig Cotrol for Passive Trailers by the Forward Motio of a Car G The Total Number of Time-icremets X r, Y r, θ r Pose of the Referece Trajectory of the Last Trailer X c,y c, θ c Actual Pose of the Last Trailer θ c, θ, θ 2 Actual Orietatios of the Car, st Trailer, ad 2 d Trailer K x, K y, K θ Cotrol Gais of the Trackig Cotroller v r, r Referece Velocity Iputs of the Last Trailer v, Velocity Iputs of the Last Trailer v 0, 0 Velocity Iputs of the Car-like Vehicle φ Steerig Agle of the Car-like Vehicle D Legth of the Trailers Liks L Legth of the Wheelbase of the Car-like Vehicle TRAILERPUSHINGCONTROL ( ) do X r, Y r, θ r Referece-trajectory geeratio for the last trailer while(timestep < G) { do v, Compute the last trailer s motio by usig the error vector ad the cotrol scheme as per eqs. (6) ad (7): (X r, Y r, θ r, X c,y c, θ c, K x, K y, K θ ). do v 0, 0 Compute the cotrol iputs of the car usig eqs. () ad (3): (v,, θ c, θ, θ a ). do φ Covert 0 to the steerig agle of the car by usig eq. (5). (v 0, 0, L) } Stability aalysis for the trailer-pushig cotrol algorithm of passive trailers by a car s forward motio To ivestigate the cotrol stability of the proposed trailer-pushig cotroller, stability aalysis was carried out. If there is o error i the kiematic model i eqs. () ad (3), the cotrol stability ca be easily prove accordig to the aalysis i [20]. However, practical cotrol performaces are affected by various error sources. As poited out i [2], cotrol robustess with respect to ucertaities ad disturbaces is 4930

4 ope ad challegig issue i the oholoomic system cotrol. Although it is difficult to aalyze the geeral trailer-pushig cotrol properties, it is possible to ivestigate the effect of error for a specific case. For simplicity, followig assumptios are made. C) The referece trajectory is a straight lie. C2) A robot with oe trailer is cosidered. C3) Local cofiguratio aroud θ 0, θ e 0 is cosidered. C4) Traslatioal velocity is perfectly cotrolled (x e 0). We cocetrate o measuremet error of joit agles. I the fabricated prototype, joit measuremet errors ca be assumed to be costats because of the difficulty of wheel aligmet of multiple trailers. Sice the structure of trailers is a serial chai, the joit agle error directly affects the cotrol performace. I [22], Chug established the sesitivity to errors of the serial-chaied system uder oholoomic costraits. It is advatageous to mechaically allow relative motio betwee trailers to cope with ueve groud coditios. Otherwise, some trailers possibly ecouter wheel floatig uder irregular groud coditios, as poited out by Nakamura et al. i [23]. The relative motio betwee trailers may result i joit measuremet error. Fig. 6. A mobile robot with a trailer uder the joit measuremet error ε. Sice the trailer-pushig is feedback cotrolled by usig the proposed scheme i sectio III-B, some errors ca be compesated. The above assumptios C3 ad C4 become feasible owig to the robustess of the feedback cotroller. Fig. 6 shows the robot with oe trailer i trailer-pushig cotrol. The solid lie shows the actual cofiguratio. The dashed lie represets the cofiguratio that cotais the joit measuremet error ε. The required velocities of the trailer [ν, ] T are computed from the trajectory-trackig cotroller. The cotrol iputs of the car-like mobile robot [ν 0, 0 ] T is determied by eqs. () ad (3). The measuremet error ε is icluded as follows. v0 cos( θ0 θ ε) Dsi( θ0 θ ε) v = 0 si( θ0 θ ε) / D cos( θ0 θ ε) (8) The, the resultat velocity of the real trailer, [ν act, act ] T, ca be obtaied as follows. vact cos( θ0 θ) Dsi( θ0 θ) v0 = act si( θ0 θ) / D cos( θ0 θ) 0 (9) cosε D si ε v = si ε / D cosε From eq. (9), we ca fid the relatioship betwee the referece velocities ad the actual velocities uder joit errors. It is clear that [ν act, act ] T = [ν, ] T, whe ε = 0. From eq. (7), it is clear that = 0 whe there is o trackig error. If there is o joit measuremet error, the eq. (9) implies that act = 0. Accordigly, the robot ad the trailer head i a costat directio by followig a straight lie. Similarly, we assume the cofiguratio which results i act = 0 uder the joit measuremet error. Fig. 7 shows this assumptio whe the target trajectory is y = 0 ad the robot is movig to the right. Fig. 7. The robot pushes the trailer to the right uder the joit measuremet error ε. The robot moves straight with the steady state error Y e. From eq. (9), the followig coditio is derived whe act =0. = ta ε (0) v D From eqs. (6), (7), ad (0), ad assumptio C4, the steady-state error, Y e, ca be obtaied as a followig equatio. Ye = ta ε () D Ky The above coditio provides a equilibrium poit of covergece. I additio, it ca be easily checked that the equilibrium poit is stable. For example, if y e < Y e (y e > Y e ), y e icreases (decreases) to Y e because act is positive (egative). Therefore, we ca coclude that the trailer locally coverges to a straight lie with the steady-state error Y e, if there is joit measuremet error ε. It is assumed that ε is small (ε 0). This result ca be iteratively exteded to the trailer problem. It is assumed that the resultat trackig error is the sum of the idividual trackig errors whe there are multiple joit errors. A. Experimetal setup IV. EXPERIMENTAL RESULTS Fig. 8. Experimetal setup: a car-like mobile robot ad two passive trailers. Fig. 8 shows our experimetal system. The active robot is a oholoomic car-like mobile robot. This car-like mobile robot has a same kiematic structure with the commercially available automobiles. Two passive off-hooked trailers are coected to the frot bumper of the car-like mobile robot. Wheel diameters of the car-like mobile robot ad passive trailers are 4.8cm ad cm, respectively. The pose of the last trailer ca be moitored by a commercially available pose 493

5 sesor STARGAZER. The poses of the remaiig vehicles are computed from joit agle measuremets ad lik parameters. The joit agles are measured by potetiometers at the joits. The maximum steerig agle ad the legth of the wheelbase of the car-like robot are 20.5 ad 0.35m, respectively. B. Trailer-pushig cotrol usig a car-like mobile robot with passive off-hooked trailers Experimetal verificatios are carried out for two referece trajectories, which are a straight lie ad a circle. The referece trajectories are predefied o global Cartesia coordiate. The first experimet is carried out with a straight trajectory. Fig. 9 shows the experimetal motio whe the referece trajectory is a straight lie. As show i Fig. 9 (a), six woode sticks are istalled i frot of the trailer system. These woode sticks that have 0.5cm height are itetioally istalled o the floor to ivestigate the effect of disturbaces. Fig. 0 shows the resultat experimetal trajectory. The car-like mobile robot ad two trailers are movig. A desired trajectory is a straight lie from (-.3, 0) to (3, 0). Dotted lie represets the last trailer s actual path durig the trailer-pushig cotrol. I spite of ueve floor coditio ad the last trailer s iitial pose error (0.04m, 0.04m, 2 ), the car-like mobile robot ad passive trailers coverge to the desired trajectory, successfully. Cotrol gais i eq. (7) are K x =0.3, K y =0.000, ad K θ =0.08. Fig. shows two cotrol iputs v 0 ad 0, which do ot exceed ±0.m/s ad ±0.2rad/s. I order to avoid dyamic effects, the trailer speed was sufficietly low. Sice the referece trajectory is straight, the agular velocity of the car-like mobile robot has to coverge to zero, i a ideal case. From Fig., it is clear that the agular velocities of a car were small. A little oscillatio was possibly caused by the potetiometer oise, backlashes at each coectig joit, ad ueveess by woode sticks o the floor. Steerig iput for the car-like mobile robot is also preseted i Fig.. It is clear that the steerig iput is bouded from to I the ideal case, the steerig agle should coverge to zero. The resultat steerig agle was a little oscillatory i order to compesate various errors durig the motio. Fig. 2 shows relative joit agles durig the trailer-pushig cotrol. It is clear that the joit agles coverge to zero. This result implies that the trailer cofiguratio was maitaied to be straight. Fig. 0. Experimetal result: trackig a straight referece- trajectory. Fig.. The traslatioal velocity (top), agular velocity (middle), ad steerig-agle iput (bottom) of a car-like mobile robot. (straight path) Fig. 2. Joit agles durig trailer-pushig cotrol process. (straight referece trajectory) The secod referece trajectory is a circle. Although the cotrol stability was prove oly for a straight referece trajectory i sectio III-C, followig a circular trajectory is possible owig to the robustess of the cotroller. A radius of the circular path is.5m ad the ceter of the circle is (-0.3, -0.). Fig. 3 shows the experimetal motio whe the referece trajectory is a circle. Fig. 4 presets the trailer system followig the circular trajectory. It is evidet that the trailers are successfully cotrolled by a car-like mobile robot. Cotrol gais i eq. (7) were K x =0.06, K y =0.000, ad K θ =0.08. The velocity iputs for the car-like mobile robot are show i Fig. 5. Steerig iput is also preseted i the lowest figure. The cotrol iputs are slightly oscillatory because the same error sources i the first experimet. (a) (b) (a) (b) (c) Fig. 9. A car-like mobile robot pushes two off-hooked trailers by its forward motio for trackig a straight referece trajectory. (c) Fig. 3. Motio of a car-like mobile robot with 2 off-hooked trailers for a circular referece trajectory. Fig. 6 shows the joit agles durig the secod (d) 4932

6 experimet. I a ideal case, whe the radius of the circle is.5m, the first ad secod relative joit agles should coverge to 27 ad 5. However, the first ad the secod joit agles coverge to 34 ad 9 respectively. The error was possibly caused by the potetiometer ad exteral sesor oise, ad backlashes at each coectig joit. Fig. 4. Experimetal result: trackig a circular referece-trajectory. Fig. 5. The traslatioal velocity (top), agular velocity (middle), ad steerig-agle iput (bottom) of a car-like mobile robot. (circular path) Fig. 6. Joit agles durig trailer-pushig cotrol process. (circular path) V. CONCLUSION I this study, we proposed the cotrol strategy for the car with passive trailer system. It was show that the cotrol performace ca be improved by coectig trailers to the frot bumper of a car. The cotrol strategy of trailer-pushig by a forward motio of a car-like mobile robot was established. The stability aalysis of the cotroller uder kiematic modelig error was preseted. Experimetal verificatio illustrated the successful implemetatio of our trailer-pushig cotrol algorithm. REFERENCES [] J. P. Laumod, Cotrollability of a Multibody Mobile Robot, IEEE Trasactios o Robotics ad Automatio, Vol. 9, No. 6, 993. [2] R. M. Murray ad S. S. Sastry, Noholoomic motio plaig: Steerig usig siusoids, IEEE Trasactios o Automatic Cotrol, Vol.38, No.5, pp. 700~76, 993. [3] D. M. Tilbury, R. M. Murray ad S. S. Sastry, Trajectory geeratio for -trailer problem usig Goursat ormal form, Proceedigs of the IEEE cotrol ad Decisio coferece, pp , 993. [4] O. J. Sørdale, ad K. Y. Wichlud, Expoetial stabilizatio of a car with trailers, Proceedig of 32d coferece o Decisio ad Cotrol, pp , Sa Atoio, 993 [5] P. Roucho, M. Fliess, J. Levie ad P. Marti, Flatess, motio plaig ad trailer systems, Proceedigs of the 32th IEEE Cof. o Decisio ad Cotrol, pp.2700~2705, Sa Atoio, Texas, 993. [6] F. Lamiraux ad J. P. Laumod, A practical approach ro feedback cotrol for a mobile robot with trailer, Proceedigs of the 998 Iteratioal Coferece o Itelliget Robotics ad Automatio, pp , Leuve, Belgium, 998. [7] Martiez, J.L., Morales, J., Madow, A., Garcia-Cerezo, A., Steerig Limitatios for a Vehicle Pullig Passive Trailers, IEEE Trasactios o Cotrol Systems Techology, Vol. 6, Issue 4, pp , [8] Bor-Se Che, Charg-Shi Wu, Huey-Jia Uag, A miimax trackig desig for wheeled vehicles with trailer based o adaptive fuzzy elimiatio scheme, IEEE Trasactios o Cotrol Systems Techology, Vol. 8, Issue 3, pp , [9] Altafii, C., Sperazo, A., Wahlberg, B., A feedback cotrol scheme for reversig a truck ad trailer vehicle, IEEE Trasactios o Robotics ad Automatio, Vol. 7, Issue 6, pp , 200. [0] Dieter Zöbel, Trajectory Segmetatio for the Autoomous Cotrol of Backward Motio for Truck ad Trailer, IEEE Trasactios o Itelliget Trasportatio Systems, Vol. 4, No. 2, pp , [] Hodo, D.W., Hug, J.Y., Bevly, D.M., Millhouse, D.S., Aalysis of Trailer Positio Error i a Autoomous Robot-Trailer System With Sesor Noise, IEEE Iteratioal Symposium o Idustrial Electroics, pp , Jue, [2] K. Taaka, S. Hori, ad H. O. Wag, "Multiobjective Cotrol of a Vehicle With Triple Trailers", IEEE/ASME Trasactios o mechatroics, pp. 357~368, Vol. 7, No. 3, [3] F. Houge, M. Gii, ad J. Slagle, "Rapid Usupervised Coectioist Learig for Backig a Robot with Two Trailers", Iteratioal Coferece o Robotics ad Automatio, pp , Albuquerque, New Mexico, 997, April. [4] J. Yi, N. Yubazaki, K. Hirota, Backig Up Cotrol of Truck-Trailer System, IEEE Iteratioal Fuzzy Systems Coferece, pp , 200. [5] K. Matsushita, T. Murakami, Backward Motio Cotrol for Articulated Vehicles with Double Trailers Cosiderig Driver s Iput, IEEE Aual coferece o Idustrial Electroics, pp , Nov [6] F. Gomez-Bravo, F. Cesta, A. Ollero, "Maoeuvrig i Noholoomic Idustrial-Like Electrical Vehicles", IEEE 28th Aual Coferece of the Idustrial Electroics, pp , Seville, Spai, 2002, November. [7] Claudio Altafii, Alberto Sperazo, Bo Wahlberg, A Feedback Cotrol Scheme for Reversig a Truck ad Trailer Vehicle, IEEE Trasactios o robotics ad automatio, Vol. 7, No. 6, Dec [8] Jaehyoug Lee, Wooji Chug, Musag. Kim ad Jae-Bok. Sog A passive multiple trailer system with off-axle hitchig Iteratioal Joural of Cotrol, Automatio ad Systems. Vol.2, No.3 pp , Sep [9] Myougkuk Park, Wooji Chug, Musag Kim, Jae-Bok Sog Cotrol of a mobile robot with passive multiple trailers I Proc IEEE Iteratioal Coferece o Robotics ad Automatio, pp , New Orleas, April [20] Y. Kaayama, Y. Kimura, F. Miyazaki, T. Noguchi, A stable trackig cotrol method for a autoomous mobile robot, IEEE Iteratioal Coferece o Robotics ad Automatio, Page(s): , Ciciati, Ohio, 990, May. [2] Jea-Paul Laumod, Robot Motio Plaig ad Cotrol, Spriger Verlag, 998. [22] Wooji Chug ad Yosihiko Nakamura, "Desig ad Cotrol of a Chaied Form Maipulator", The Iteratioal Joural of Robotics Research, Vol. 2, No. 5-6, pp , [23] Y. Nakamura, H. Ezaki, ad W. Chug, Desig of steerig mechaism ad cotrol of oholoomic trailer system, IEEE trasactios o Robotics ad Automatio, pp , Vol. 7, No. 3, Jue,