CONTROLLER DESIGN FOR OFF-TRACKING ELIMINATION IN MULTI-ARTICULATED VEHICLES. S. A. Manesis, G. N. Davrazos, N.T. Koussoulas

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1 Copyrght IFAC 5th Treal World Cogress, Barceloa, pa CONTROER DEIGN FOR OFF-TRACKING EIMINATION IN MUTI-ARTICUATED VEHICE. A. Maess, G. N. Davrazos, N.T. Koussoulas Electrcal ad Computer Egeerg Dept. Dvso of ystems ad Cotrol Uversty of Patras <stam.maess, gdavrazo, Abstract: The moto of a mult-body autoomous robot as well as of a tra-lke multartculated trasportato vehcle s characterzed by the devato of the path of each termedate vehcle from that of the leadg oe (off-trackg. I ths paper, we make use of a ovatve jucto techque, whch allows the kgp to slde alog the axs of the leadg vehcle, somethg that proved to be very effectve reducg offtrackg. We propose two cotrollers for the elmato of the off-trackg pheomeo, both robotc ad trasportato mult-artculated vehcles; the oe s heurstcally derved whle the other oe s based o steady-state off-trackg whe a -traler vehcle moves o a crcular trajectory. mulato results for varous cases, wthout ad wth the sldg kgp system, showed that sgfcat off-trackg reducto or eve elmato ca be acheved. Keywords: Trasportato systems, mult-artculated vehcles, vehcle tras, offtrackg.. INTRODUCTION I the autoomous robotcs feld, the goal s to buld physcal systems that accomplsh useful tasks wthout huma terveto whle operatg ukow evromets. O the other had, Itellget Trasportato ystems the goal s smlarly to costruct trasportato vehcles, tellget eough to be drve wth as less as possble huma terveto. I groud freght trasportato, heavy-duty trucks may be combed to form a truck tra or road tra, cosstg of a umber of sem-tralers ad a sgle hgh-power tractor [Maess 998]. A smlar tra s formed mult-body autoomous robots. Off-trackg, that s the devato of the path of each traler from those of the leadg vehcle or robot, s amog the basc techcal problems that must be solved [Bushell et. al 994 ], [Altaf ad Gutma 998 ]. The combato of may tralers wth a sgle tractor formg a road tra or truck-tra s called hard platoog. Road tras may cosst of a umber of tralers (at least 3 ad possbly up to ad a lead tractor. For such a road tra of trucks to be safely drve a mult-lae hghway, a umber of ssues must be dealt wth. Besdes ecoomc ad poltcal cosderatos, techcal ssues clude the ecessty for a set of traffc ad drvg rules that must be defed ad obeyed, the soluto of the pathfollowg problem, the mechacal realzato, ad the space lmtatos outsde a hghway. The advatages of usg truck tras hghway freght trasportato are dscussed [Maess ].

2 The moto of the -traler system s subject to oholoomc costrats (rollg wthout slppg so t has bee studed as a class of oholoomc systems by may researchers ad has both theoretcal ad practcal terest. The work [Kolmaovsky ad McClamroch, 995] s a excellet survey of recet advaces cotrol of oholoomc systems. The ma problem that has attracted most of the atteto s path followg. We kow of a few works oly that cosder the off-trackg problem. A closed-form expresso for the off-trackg of the rear pvot pot of a smple tractor-sem-traler vehcle ca be foud [Alexader ad Maddocks, 998] whle offtrackg bouds for a car pullg tralers have bee derved [Bushell et. al., 994]. For example [Altaf, 998] the path-followg problem wth reduced off-trackg s addressed for the -traler system. Ths s acheved by keepg track of the error dstace of each of the mddle pots of the axles of the vehcle from the path usg dfferet movg frames. I [Nakamura et. al. ] dfferet passve steerg mechasms as well as cotrol laws are preseted for oholoomc traler systems. The ma focus of such mechasms s o reducg passve trackg error from tractor s trajectory ad lttle atteto was pad o actve moto cotrol. I ecto II we descrbe the mult-artculated vehcle model ad the off-trackg problem. ecto III cotas a bref descrpto of the sldg kgp system together wth the state equatos of the multartculated vehcle whe sldg s appled. I secto IV we descrbe two ew cotrollers oe of heurstc org ad the other based o the compesato for the steady-state off-trackg whe the leadg vehcle moves a crcular trajectory, whle secto V smulato results are preseted wth ad wthout sldg. ecto VI cotas coclusos ad dscusso about the results ad some future research problems.. THE MUTI-ARTICUATED VEHICE I ths secto we descrbe brefly the model of the mult-artculated vehcle that s commo for both robotc ad trasportato vehcles. It s a log ad complex vehcle system cosstg of a hgh power tractor pullg a umber of passve robot bodes or sem-tralers as show Fg.. The state equatos of the above system, called also -traler system, wth a drvg axle ad hece a steerg agle for the tractor are x! cosϑ y! sϑ ϕ! U = taϕ U = s( ϑ ϑ ( y ϑ x ϑ (X,Y ϑ ϑ Fg.. Illustrato of the mult-artculated vehcle coordates. U = cos( ϑ ϑs( ϑ ϑ " U = cos( ϑj ϑj s( ϑ ϑ =,..., j = where x, y are the Cartesa coordates of the leadg vehcle (tractor ad U,U are the two cotrol puts, the lear velocty ad the steerg agle rate respectvely [aumod 993]. The above equatos are derved from algebrac mapulato of the holoomc costrats ad the + oholoomc costrats uder the assumpto of the same legth for all tralers. The oly dfferece betwee the mult-body robotc systems ad trucktra s the magtude of the dfferet physcal quattes (legth, velocty, steerg agle lmts, weght, etc. Off-trackg s defed as the devato of the semtralers axles or the kgp htch from the path of the steerg axle of the leadg vehcle. I the case of truck-tras, t s more mperatve tha ay other case that the last sem-traler follow exactly the path of the lead tractor durg a tur for lae chage or a tur due to the curvature of the hghway. Otherwse t wll be possble for the last sem-traler to volate the outer boudary of the hghway or to crash wth a adjacet car durg a lae chage although both keep varat ther relatve velocty. It s kow that the drver of ay log truck-tra, because of the offtrackg of the rear tralers, turs the tractor far towards the desred path order to avod ths pheomeo. Whe we deal wth moble robots the major problems are to fd a obstacle-free path ad path followg cotrol. However, the case of mult-artculated robotc vehcles we must take to cosderato the off-trackg pheomeo whe fdg a obstacle-free path. The reaso s that the last traler may collde wth obstacles f the vehcle attempts to follow the desged path for the leadg vehcle wth off-trackg eglected. Oe effcet way to solve ths problem s to fd a obstacle-free path for the leadg vehcle, add a cotroller for path φ

3 followg ad use aother kgp cotroller for offtrackg elmato. 3. THE IDING KINGPIN YTEM The off-trackg ca be elmated by sldg each traler wth respect to the prevous oe, a techque frstly descrbed [Maess 998]. Accordg to ths techque the kgp htch each sem-traler sldes a drecto perpedcular to the logtudal axle (.e. alog the rear axle of the traler by a dstace. I ths secto we preset brefly the sldg kgp system ad the state-equatos of the multartculated vehcle whe sldg s used, together wth the assumptos that are made durg the dervato of the equatos. Cosder two termedate sem-tralers of a truck tra as show Fg.. The posto of each sem-traler P, s take to be the mddle pot of the th sem-traler s rear axle. Y y ys y+ P+ θ + em-traler + P x+ x xs θ θ Ps em-traler Fg.. The kgp sldes alog the axle whe the sem-traler turs. Posto P s defed by the par ( x, y the Cartesa coordates system whle ϑ s the oretato of the th sem-traler wth respect to the horzotal axs. To smplfy dervato of the truck tra model we wll ot cosder tally a steerg agle for the tractor, sce the exteso of the model to cover ths case s smple. It has bee poted out [Bushell et. al 994] that whe the lead car of a sgle traler system s travelg alog a crcle of radus R l, the the traler s travelg alog a crcle of radus R t wth the same ceter, where. I order to compesate for R < R t l X ths path devato of the traler, we suppose that the kgp htchg pot sldes from the pot P to the pot P s by a dstace. The followg assumptos are ecessary for dervg the mathematcal model: a All tralers have the same legth. b Each traler s modeled as havg oly oe axle. c Each traler s assumed to be hooked up to the mdpot of the rear axle of the precedg traler. d By sldg the locato of the kgp, the weght of the traler shfts toward a outer drecto, whch does t affect the kematc behavor of the tra. e The ubalaced pullg pot (whe the kgp sldg s ozero does ot cause skddg of the whole axle. f The sldg of the kgp ca be performed wth the traler fully loaded va a hydraulc mechasm. I the geeral case of a -traler truck tra, we have the classcal ( + oholoomc costrats mposed by the rollg ad o-slppg codto x! s θ y! cosθ = ( ad holoomc equatos troduced by the correspodg lks, whch, because of the sldg = P P s (see Fg., are of the form x y + + = x cos θ + = y s θ + + s θ cos θ (3 Takg the dervatves of the holoomc Eq. (3, combg them wth Eq. ( ad elmatg x!, y! leads to a system of + equatos. The soluto of ths system combed wth the equatos of moto of the tractor wth steerg agle ad uder the assumpto! =, yelds x! cosϑ y! sϑ ϕ! θ! = = U = taϕ U = s( ϑ ϑ [ + taϕ ] θ! 3 " [ + ( t taϕ ] s( θ θ [ cos( θ θ + ( t s( θ θ ] u + + = [ + ( t taϕ ] s( θ θ [ cos( θ θ + ( t s( θ θ ] (4

4 where x, y are the Cartesa coordates of the leadg vehcle, ϑ ts oretato, ϑ, =,,... the oretato agle of the th traler, U,U the two cotrol puts lear velocty ad steerg agle rate respectvely, the legth of each traler ad the sldg dstace, whch s determed from the cotroller. 4. CONTROER DEIGN Equatos ( ad (4 descrbe the kematc behavor of a -traler system wthout ad wth sldg, respectvely. I a mult-artculated vehcle two dfferet cotrollers are used the oe for path followg ad the other for off-trackg elmato regulatg the sldg dstace the sldg kgp system. For path followg ssues the lear velocty ad the steerg agle rate of the leadg vehcle are the cotrol puts. I the classcal case, the drver regulates the above cotrol puts such a way as to acheve kematc stablty ad the desrable trajectory trackg. For a autoomous mult-body robot movg sde a lmted laboratory or dustral evromet, the embedded cotroller regulates the cotrol puts based o a cotrol algorthm for path followg. The overall structure of the cotrol system for a -traler vehcle s depcted Fg. 3. The frst each kgp of the th traler. It s kow that the curve radus for a vehcle s gve by U U r = =. o ω geeral ad for the th traler wll be gve by U r = ϑ! (6. From the set of equatos ( we have that U ϑ! = cos( ϑ j ϑ j s( ϑ ϑ (7 j = By combg (6, (7 ad takg to cosderato the relato U cos( j ϑ j+ j= ϑ (8 ad after some algebrac mapulato t yelds that r = cot( ϑ ϑ (9 We coclude from the last relato that the curve rad for dfferet tralers are dfferet so t s logcal to troduce dfferet sldg for each traler. I [Bushell et. al. 994] was prove that f the leadg vehcle travels alog a crcular trajectory wth radus r (whereas r > the the traler coverges to a crcular trajectory wth radus R = r. I order for the leadg vehcle ad the sem-traler follows the same crcular trajectory we troduce the followg lemma. U, U traler Y=tate traler N-traler ystem ldg mechasms ΔU, ΔU cos( ϑ ϑ = s( ϑ ϑ Huma observato ad terveto or Path followg cotroller Fg. 3. The overall cotrol system for the -traler vehcle. cotroller for off-trackg elmato that we use s heurstcally foud based o basc cotrol egeerg prcples ad s gve by: = K ϑ! (5 where ϑ! s the oretato of the leadg vehcle. Followg the procedure below we derve the equatos for the secod closed-loop cotroller for emma If the kgp sldg s gve by = r + r the the traler the steady state follows the crcular trajectory wth radus r of the leadg vehcle, Proof Accordg to [Bushell et. al. 994] the traler steady-state wll travel a crcular trajectory of radus

5 R ss = rz (Fg. 4, whereas r z r + =. o we have that Rss = rz = ( r + =... After some algebrac mapulatos, we coclude that R = r. ss gve by (. Observg the fgures we otce that the smulato results are much better whe cotroller ( s used tha the cotroller (5. I all smulatos we assume a mult-artculated vehcle wth the same legth for all tralers ad tractor, equal to oe ut legth. The above lemma ca be exteded for the tralers case followg the same procedure, so the sldg for the th traler wll be gve by = r + r ( By combg (9 ad ( we fd that the dfferet sldg dstaces that we must apply to the dfferet tralers are gve by the relato cos( ϑ ϑ = ( s( ϑ ϑ y (ut legth : thrd traler -. secod traler -- frst traler - tractor Kgp pot trajectory wth sldg r z x (ut legth Fg.5. The tractor of a tra wth 3 tralers follows (bold le a +8 /-9 arc wthout sldg. : thrd traler -. secod traler -- frst traler - tractor r y (ut legth R Traler trajectory wthout sldg Tractor trajectory Fg.4. A smple tractor-traler system depctg the trajectores of the traler, tractor, ad kgp pot. 5. IMUATION REUT To test the cotrollers descrbed the last secto, the Matlab/mulk smulato evromet was used ad depedetly verfed through Mathematca. mulato results wthout the sldg kgp mechasm are show Fg. 5. The dvdual trajectores of a truck tra wth 3 tralers travelg o a ¾ crcular arc, emphasze the offtrackg devato. Fg. 6 shows the correspodg trajectores derved after the applcato of the sldg kgp mechasm, where the sldg dstace s determed from cotroller gve by (5. Fg. 7 shows the correspodg trajectores, whe cotroller x (ut legth Fg.6. The vehcle of Fg. 4 wth the tractor tracg the same trajectory but ow wth the kgp sldg all tralers ad heurstcally foud cotroller gve by (5. y (ut legth : thrd traler -. secod traler -- frst traler - tractor x (ut legth Fg.7. The vehcle of Fg. 4 wth the tractor tracg the same trajectory but ow wth the kgp sldg all tralers ad cotroller gve by (.

6 6. CONCUION Off trackg s oe of the most sgfcat problems occurrg artculated vehcles. The sldg kgp s a techque for correctg such devatos. I ths paper, a cotroller for adjustg the sldg dstace of a kgp sldg mechasm has bee proposed based o the theoretcal steady-state off-trackg whe the leadg vehcle moves a crcular trajectory. Its respose was compared to aother heurstcally foud cotroller that performs also well. Both desgs have bee valdated through smulato, whose results showed satsfactory performace of both desgs. However, the aalytcally desged cotroller fared better the steady state part of the crcular trajectory. The ma topc for future research s to derve equatos for the -traler system wth the sldg kgp mechasm wthout makg the assumpto that the dervatve of sldg dstace s zero ad fd out how the desged cotroller affects the behavor of the vehcle ths stuato. REFERENCE Alexader J.C. ad J.H. Maddocks, (988 O the Maeuverg of Vehcles, IAM Joural Appled Mathematcs, pp.38-5, vol.48, No.. Altaf C., (998 Path followg problem wth reduced off-trackg for the -traler system Proc. of the 37 th IEEE Cof. O Decso ad Cotrol Altaf C. ad Gutma P-O., (998 Path followg wth reduced off-trackg for the -traler system, Proceedgs of 37th IEEE Cof. O Decso ad Cotrol, Florda, December. Bushell., B. Mrtch, A. aha ad M. ecor, (994 Off-trackg Bouds for a Car Pullg Tralers wth Kgp Htchg, Proceedgs of the 33 rd Coferece o Decso ad Cotrol, pp , ake Buea Vsta, F. Kolmaovsky I., McClamroch Harrs N., (995 Develompets oholoomc cotrol problems IEEE Cotrol ystems Magaze Vol.5, No.6, Dec aumod J.-P., (993 Cotrollablty of a Multbody Moble Robot IEEE Tras. o Robotcs ad Automato, Vol. 9, No. 6. Maess., (998 Off-trackg elmato roadtras of heavy duty trucks wth multple semtralers, Proceedgs of 8th IFAC/IFIP/IFOR/ IMAC o arge cale ystems: Theory & Applcatos ( 98, Patras Greece. Maess., ( Hard platoog versus soft platoog large-scale hghway trasportato systems Proc. of 9 th IFAC/IFIP/IFOR/IMAC o arge cale ystems: Theory & Applcatos ( Bucharest, Romaa, Murray R. M., Z., hastry.., (997 A Mathematcal Itroducto to Robotc Mapulato, CRC Press. Nakamura Y., Ezak H., Ta Y., Chug W., ( Desg of steerg mechasm ad cotrol of oholoomc traler systems Proc. of IEEE It. Cof. O Robotcs ad Automato Ackowledgemets Ths research work s partally supported by Karatheodor Program of the Research Commsso of the Uversty of Patras.