INTO CHAINED FORM USING DYANMIC FEEDBACK D. TILBURY, O. SRDALEN, L. BUSHNELL; S. SASTRY

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1 A MULTI-STEERING TRAILER SYSTEM: CONVERSION INTO CHAINED FORM USING DYANMIC FEEDBACK D. TILBURY, O. SRDALEN, L. BUSHNELL; S. SASTRY Electroncs Research Laboratory, Department of Electrcal Engneerng and Computer Scence, Unversty of Calforna, Berkeley, CA 970. E-mal: On leave from the Norwegan Insttute of Technology, Trondhem, Norway Abstract. In ths paper, the knematc model of an autonomous moble robot system consstng of a chan of steerable cars and passve tralers wth axle-to-axle htchng s examned and converted nto a mult-nput chaned form usng dynamc state feedback. Some of the methods whch have been proposed for steerng two-nput chaned form systems are generalzed to mult-chaned systems, and then appled to two example mult-steerng traler systems. Key Words. Moton plannng, nonholonomc systems, dynamc feedback, moble robots, chaned forms.. INTRODUCTION In ths paper the moton plannng problem s solved for a mult-steerng traler system; that s, a car-lke moble robot pullng a combnaton of n passve tralers and m? steerable car-lke robots. The controls avalable to the system are the velocty of the lead car and the steerng veloctes of all m car-lke robots. Ths system can be thought of as a generalzaton of a standard n-traler system, the moton plannng problem for whch was consdered and solved n (Srdalen, 993; Tlbury et al., 993b). Moble robot systems of ths knd are of nterest n practcal applcatons; part of the motvaton for ths work came from work on the re truck (Bushnell et al., 993; Tlbury and Chelouah, 993). Also, the authors have been told anecdotally about the constructon of such n-traler systems wth multsteerng for use n nuclear envronments (Canudas de Wt, 993) and also for baggage and cargo handlng (Gralt, 993). These moble robot systems can be modeled as havng one constrant on each axle; namely, that the wheels are allowed to roll but not to slp. These constrants are nonholonomc or nonntegrable and do not reduce the reachable conguraton space of the moble robot. Although the present system appears at rst glance to be a straghtforward extenson of the systems consdered earler, the man motvaton n wrtng ths paper s to exhbt ts added rchness and complexty. As before, the moton plannng plannng problem s solved by rst convertng the system nto chaned form. Unlke the prevous examples, ths system wll requre states to be added to the system before the transformaton nto chaned form can be found. Motvated by the physcal structure of the constrants nvolved, the dynamc state feedback that s used conssts of addng vrtual axles to each of the steerable cars n the system. A partcularly ntrgung aspect of ths work s ts connecton wth an emergng body of lterature n derentally at systems by Fless and hs coworkers (Fless et al., 99; Rouchon et al., 993). They have shown that chaned form systems are a specal case of what are known as derentally at systems. For the two nput case, t has been ponted out (Martn and Rouchon, 993; Murray, 99) that, modulo somewhat derent regularty condtons, chaned forms are equvalent to at systems for the type of drft-free systems that arse n nonholonomc moton plannng. Ths s not true for systems wth more than two nputs wthout allowng for the possblty of dynamc state feedback.. THE SYSTEM MODEL Consder a mult-steerng traler system,.e. a system of n (passve) tralers and m (steerable) cars lnked together at the axles by rgd bars, as shown n Fgure. It s assumed that each body (traler or car) has only one axle, snce a body wth two axles can be modeled as two one-axle bodes... Conguraton Space The conguraton of the system s dened by the angles of all the axles, the angles of the rgd bars n front of the steerable cars, and the Cartesan poston of the system n the (x; y) plane. The actve or steerng axles are numbered from front () to back (m), and the passve axles are numbered smlarly from to n. The angle of each passve axle wth respect to the horzontal s where s the axle number and s the number of the steer-

2 θ θ - θ - v θ n + φ θ n θ n second steerng tran frst steerng tran L θ body (passve) θ - L θ v - body - (passve or actve) θ m n m θ m n - m mth steerng tran Fg.. A mult-traler system wth n (passve) tralers and m (actve) steerng wheels. ng wheel most drectly n front of that axle. Each steerable axle together wth the passve axles drectly behnd t s called a steerng tran. The passve axles whch are drectly n front of the steerable axles wll be dented by the ndces n ; : : :; n m?. Assumng, by conventon, that the rst axle s steerable, gves n 0 = 0. The angle of the rst axle s 0, and the axles drectly behnd t are denoted by ; : : : ; n. The superscrpts ndcate these axles are n the rst steerng tran. The axle drectly behnd the rst steerng tran s steerable, ts angle s n. The (passve) axles behnd the second steerng wheel are n + ; : : :; n, and the rest of the angles are numbered n a smlar fashon. For convenence of notaton, let n m n, although n general the last axle wll not be steerable. If the last axle s steerable, then n m? = n m. Let be the absolute angle (wth respect to the horzontal) of the bar connectng the ( + ) st steered axle to the last axle of the th steerng tran (whch may be steered or passve). The Cartesan poston, (x; y) of any one of the axles can be used n the denton of the conguraton of the system. For reasons whch wll be explaned n the sequel, the x and y postons of the last axle are chosen as conguraton varables. Ths general system ncludes as specal cases:. the standard n traler system (m = ).. the re truck (m =, n = n = )... Knematc Equatons The knematc model of the system s found by examnng the relatonshps between the veloctes of the bodes, as dened by the rgd connectons between them. If the lnear velocty of the last body s v m n m, then the dervatves of x and y are the proectons of ths velocty, _y = sn m n m v m n m () _x = cos m n m v m n m () Now, let v represent the the lnear velocty of the axle wth angle. Consder rst the case of a passve traler; refer to Fgure. The lnear veloctes of body? has two perpendcular components: Fg.. The velocty relatonshps between adacent bodes when the rear body s a passve traler. The front body may be ether a passve traler or a steerable car. θ + n - φ Ln φ vn + L n + φ + θ n φ - θ n θ n v n frst body of the (+)st steerng tran (actve) last body of the th steerng tran (passve or actve) Fg. 3. Showng the velocty relatonshps between two bodes when the rear body s an actve car. one n the drecton of the lnear velocty of the th body, v = cos(?? )v? ; and the other n the drecton of the angular velocty of the th body, L _ = sn(?? )v? : (3) When the rear body s an actve car nstead of a passve traler (see Fgure 3), the relatonshp between the two lnear veloctes has the form vn + cos(n +? ) = vn cos(n? ) ; and the angular velocty vector L _ n has two components, L _ n + = sn(n +? )vn +? sn(n? )vn () : The dervatves of the steerng wheel angles are the nputs _ n =!? : (5) The knematc equatons for the mult-steerng traler system are completely gven by ({5). 3. CONVERSION TO CHAINED FORM Now that knematc behavor of the mult-steerng system has been descrbed, the transformaton to mult-nput chaned form can be found. 3.. Mult-nput Chaned Form A mult-nput chaned form system s dened as _z 0 0 = u 0 _z 0 = u _z m 0 = u m _z = z 0 u0 _z m = z m 0 u0 (6).. _z n + = z n u _z m n m+ = z m n m u 0 :

3 (x,y) Fg.. The uncycle model. The robot s allowed to drve forwards or backwards and to spn about ts center axs. (x,y) θ Fg. 5. A uncycle wth a \vrtual" extenson, nterpreted as another axle added n front of the orgnal robot. From equaton (6), t s clear that the states at the bottoms of each chan, that s z 0 0 ; z n + ; : : : ; zm n m+ wll determne the traectores of all the states by the relatons z = _z + = _z0 0 : (7) 3.. Extendng the System wth \Vrtual" Tralers For some nsght nto the formulaton of vrtual axles, consder the smple example of a uncycle, sketched n Fgure. The knematc model takes as nputs the lnear velocty v and the angular velocty! of the uncycle, _x = cos v _y = sn v _ =! : θ ψ (8) Snce the system s drft-free, the relatve degree of any choce of outputs wll be equal to one. In partcular, the relatve degree of the body angle wth respect to the steerng nput s equal to one. Consder addng a new state _ = ; and a feedback! = tan(? ) v: The extended system (x; y; ; ) now satses the equatons _x = cos v _y = sn v (9) _ = tan(? ) v _ = ; and the added state can be nterpreted as the angle of another axle n front of the orgnal steerng wheel, and the new nput as the steerng velocty of ths \vrtual" wheel (see Fgure 5). The relatve degree of the body angle wth respect to the (vrtual) steerng nput s two. Remark. Any traectory = (x; y; ; ) of (9) can be proected down to gve a traectory () = = (x; y; ) of (8). Also, for any traectory of (8) for whch s C and for whch _ = 0 whenever _x = _y = 0, there exsts a traectory of (9) such that () =. Traectores where the uncycle spns about ts axs wthout movng ether forwards or backwards cannot be realzed wth the extended model. See (Isdor, 989) for a dscusson of relatve degree and nonlnear control theory Convertng the Mult-Steerng System nto Chaned Form The (x; y) poston of the last traler along wth all the htch angles f ; : : : ; m? g between steerng trans determne the entre state of the system, and are thus a canddate set of coordnates for the bottoms of the chans n the mult-nput chaned form (6); they are also one set of possble at outputs for ths system. The path taken by the last axle determnes the angle of the last axle by equatons ({): tan n m m = _y= _x. The (Cartesan) poston and angle of any axle wll determne the poston of the axle n front of t from the htch relatonshp. Thus, x; y; n m m determne the postons and angles of all the axles n the last steerng tran. Usng m?, the values for the second-tolast steerng tran can be found, and so forth. The front-most htch angle wll become the state at the bottom of the rst chan, zn. Its + relatve degree wth respect to the rst steerng nput! s equal to n +, one more than the number of axles n the rst steerng tran, and thus t wll need to be derentated a total of n + tmes n order to dene all the states z n the rst chan by equaton (7). However, snce _ depends on all the angles behnd t n the traler system accordng to equaton (), the relatve degree of wth respect to any of the other steerng nputs wll be equal to two. To ensure that the dervatves of these nputs do not appear n the coordnate transformaton, n vrtual axles can be added n front of each steerng axle n to? ncrease the relatve degree of wth respect to the other steerng nputs by n. After a smlar analyss for ; : : :; m? ; y, a total of n vrtual axles wll be added n front of the th steerng wheel, as n Fgure 6. The state varables that have been ntroduced, correspondng to the angles of these vrtual tralers, are denoted by. Ther dervatves are de- ned as f they were actual axles, _ = 0 ; _ = L sn(?? )v? ; (0) where L s an arbtrarly chosen postve parameter. The veloctes of the vrtual axles are de- ned n the same manner as the real axles, and the new nputs represent the angular velocty of the front car n each vrtual extenson. In an eort to wrte the knematc model n a compact form the followng vectors are ntroduced: m = [ ; : : : ; n ; ] = [ m ; : : : ; m n m ; y] = [ ; + ; : : : ; m ] : Another possble choce for the states at the bottoms of the chans (or the at outputs) are the (x; y) poston of the last traler along wth y values of the mdponts of the axles n front of each of the steerng wheels. The resultng chaned form s the same.

4 mth vrtual extenson θ m n m θ m n - m φ m- θn m m- θ m θ n + mth steerng tran m θ n θ n - second steerng tran θ frst vrtual extenson φ θ n θ frst steerng tran Fg. 6. The mult-steerng system, showng the vrtual axles that must be added to convert the system nto mult-nput chaned form. In general terms, the superscrpts () of the vectors refer to the th steerng tran and the subscrpts () refer to the tals of the steerng tran startng from the th traler (whch s real f n? and vrtual f 0 < n? ). The nput u 0 n the mult-nput chaned form of equaton (6) s the lnear velocty n the x drecton of the last body n the last steerng tran, v = cos m n m v m n m : () The lnear velocty at an axle can be wrtten as a functon of the states and the generatng nput: v = s ( )v : Recall that the dervatves of the angles have the same form for both actual (3) and vrtual (0) axles, and dene the functon f to be the dervatve of dvded by the velocty v, f ( )? = L tan(?? )s ( ) : () Let the functons f n + be the dervatves of the htch angles,, dvded by the velocty v; see equaton (). Fnally, equatng _y = fn m m+ v, we have from (): f m n m+ ( m n m ) = tan m n m : (3) By way of notaton, dene = [f ; : : :; f n + ] f F = [f ; f + ; : : : ; f m ]T so that the local knematc model wth dynamc feedback can now be wrtten compactly as _ 0 = ; () _ = F () 0 v (5) _x = v : (6) The rst chan has only one coordnate, z 0 0 = x : (7) The m th chan wll be the longest, wth z m n m+ = y : (8) The other chans have the htch angles at the bottom, z n + = ; (9) and the remanng coordnates are found through the relatonshp (7). Ths can be wrtten more speccally as follows. Recallng that the dervatves of ; : : : ; m? ; y, were dened as f n + t can be seen that the second-to-last coordnate n each chan wll be zn = f n + ( n ) : (0) The new coordnates have the general form = L F L F L F f + n n + ; () z? where L F h denotes the Le dervatve of the functon h along the vector F. The nput transformaton s dened by takng the dervatves of the rst states n the chans u = _z 0 () Theorem Let the coordnates z, be gven by (6{0) and the nputs u, be gven by (). Then the equatons (6) are satsed. Proof. The chaned form follows drectly from the dentons of the coordnates and nput transformaton along wth the knematc model (5). Ths coordnate transformaton has a trangular structure and ts Jacoban s nonsngular at at the orgn, whch mples that t s a local deomorphsm. The detals of the proof can be found n (Tlbury et al., 993a).. STEERING CHAINED FORM SYSTEMS Once a system s n mult-nput chaned form, many derent algorthms (three of whch are presented here) can be used to steer t. The basc dea behnd each of these methods s to parameterze the nput space, ntegrate the chaned form equatons symbolcally, and nally, solve for the nput parameters n terms of the desred ntal and nal states. No partcular system of tralers wll be consdered; nstead, the problem that s solved n ths secton s to nd nputs fu (t) : t [0; T ); = 0; : : :; mg whch wll steer the mult-chaned system (6) from a gven ntal state to a desred nal state... Steerng wth Polynomal Inputs One approach to the pont-to-pont steerng problem s to hold the rst nput u 0 dentcally equal to one over the entre traectory. The tme needed to steer s then determned from the change n the

5 z 0 0 coordnate, T = (z 0 0 )f? (z 0 0 ) : (3) The remanng nputs are parameterzed by Taylor polynomals, u = a 0 + a t + : : : + a n+t n+ u = b 0 + b t + : : : + b n+t n+ (). u m = 0 + t + : : : + nm+ t nm+ wth the number of parameters on each nput chosen to be equal to the number of states n ts chan. The chaned form equatons (6) can be ntegrated symbolcally and the nput parameters a ; b ; : : : ; can be found n terms of the ntal and nal states. All of the equatons that need to be solved are lnear. Of course, f the tme needed for steerng s zero from equaton (), then ths method wll not work. One way to remedy ths stuaton s to choose an ntermedate pont and plan the path n two peces... Steerng wth Pecewse Constant Inputs Ths steerng method was orgnally nspred by multrate dgtal control (Monaco and Normand- Cyrot, 99), but s most easly understood n terms of moton plannng smply as pecewse constant nputs. The rst nput u 0 s chosen to be constant over the entre traectory, and the other nputs are chosen to be pecewse constant, wth at least as many swtches as there are states n ts chan. The tme for the traectory s chosen arbtrarly as T. The reader s referred to (Tlbury et al., 993a) for more detals..3. Steerng wth Snusodal Inputs A method for steerng mult-chaned systems wth snusods was proposed n (Bushnell et al., 993), and requred one step to steer each level of the chan. Because of the many steps needed for steerng, the algorthm can be tedous to mplement n practce. An \all-at-once" snusods method, an extenson of that detaled n (Tlbury et al., 993b), has as nputs parameterzed sums of snusods at derent frequences. Agan, the chaned form equatons (6) can be ntegrated symbolcally, evaluated at tme T, and the parameters are found as a functon of the ntal and nal states. In ths case, ndng the nput parameters wll requre solvng nonlnear algebrac equatons. More detals on ths algorthm can be found n (Tlbury et al., 993a). 5. EXAMPLES Two examples wll be brey presented to llustrate the converson procedure. 5.. Fre Truck Example In (Bushnell et al., 993), t was shown that the re truck system could be converted nto a multnput chaned form. The bottoms of the chans n the chaned form (or equvalently the at outputs of the system) were chosen to to be the (x; y) poston of the passve axle along wth the angle of the traler (see Fgure 7), and because of the relatve smplcty of the three-axle system, that (x,y) θ φ L Fg. 7. A sketch of the re truck system showng the vrtual extenson that s added n front of the rear steerng wheel. The extra steerng wheel at the rear of the traler s used for mproved maneuverablty on narrow cty streets. (x,y) θ 3 L 3 L θ Fg. 8. The ve-axle, two-steerng system showng the vrtual extenson whch s added n front of the second steerng wheel. Such a system s envsoned as beng used where maneuverablty around narrow passageways n danger zones s of utmost mportance. choce allowed the knematc equatons to be put nto mult-nput chaned form wthout usng dynamc state feedback. Snce the re truck ts nto the class of mult-steerng traler systems, t can also be converted nto mult-nput chaned form usng the (x; y) poston of the last axle and the traler angle as the bottoms of the chans. One vrtual traler wll then need to be added, showng that although vrtual extenson s not always necessary, the procedure outlned n ths paper wll always result n a chaned form. 5.. Another Example Consder now a ve-axle system wth two steerng wheels, as depcted n Fgure 8. In eect, ths s a re truck wth two passve tralers. Usng the procedure outlned n Secton 3, choosng the bottoms of the chans as the (x; y) poston of the last axle and the htch angle, ths system can be converted nto mult-nput chaned form and steered usng one of the methods outlned n Secton. In fact, as has been recently proven n (Tlbury and Sastry, 99), there does not exst a transformaton nto chaned form for ths partcular ve-axle system wthout usng dynamc feedback. For detals of the knematc equatons and the transformaton for ths system, the reader s encouraged to consult (Tlbury et al., 993a) or to contact the rst author. Once the knematc equatons are n mult-nput chaned form, the system can be steered usng one of the algorthms dscussed n Secton. The system parameters are n = 3 (three passve axles) and m = (two steerng wheels); and for concreteness, let the lengths of the htches be L = L = L 3 = 5; L = 3, and L =. L θ L θ L θ 0 φ θ 0 L θ

6 Fg. 9. A parallel-parkng traectory for the ve-axle, two-steerng system. The plannng algorthm as descrbed n ths paper does not account for obstacle avodance; however, t does plan \nce" paths whch may can be used n conuncton wth an obstacle-avodance algorthm to acheve a complete soluton to the pathplannng problem. To steer the system from an ntal pont of (x; y) = (0; 0) to a nal pont of (x; y) = (0; 0) and all of the body angles (ncludng the vrtual angle) are algned wth the horzontal axs n both the ntal and nal conguratons, polynomal nputs are one possble choce for plannng the traectory n the chaned form coordnates. As noted n Secton., polynomal nputs are not mmedately suted to ths type of traectory snce the tme needed to steer the system, computed from equaton (), s zero and thus the algorthm fals. The traectory can be planned n two steps, usng as an ntermedate pont (x; y) = (30; 0), as shown n Fgure SUMMARY In ths paper, a systematc method for convertng the knematc model of a mult-traler system wth n passve tralers and m steerable cars nto a mult-nput chaned form was presented. Many algorthms for steerng systems n chaned form exst (three were descrbed here), and methods for stablzng systems n mult-nput chaned form have also been presented (Walsh and Bushnell, 993). The method for convertng the mult-traler system nto chaned form added vrtual axles to the system n a form of dynamc state feedback. Although the vrtual extenson was not always necessary, t provded a guaranteed method to convert all mult-steerng traler systems nto chaned form, and thus to nd feasble paths for these systems. Acknowledgements Ths research was supported n part by the NSF under grant IRI-9090 and by the ARO under grant FD-DAAL03. D. Tlbury would lke to acknowledge an AT&T Ph.D. Fellowshp for partal support of ths work. The research of O. Srdalen was supported n part by the Japan Socety of the Promoton of Scence and the Center of Martme Control Systems at NTH/Sntef, Norway. The authors would also lke to thank Greg Walsh for hs help n anmatng the smulatons for the vdeotape was presented at the conference. 7. REFERENCES Bushnell, L., D. Tlbury and S. S. Sastry (993). Steerng three-nput chaned form nonholonomc systems usng snusods: The retruck example. In: Proc. of the European Control Conf. pp. 3{37. Canudas de Wt, C. (993). Personal communcaton. Fless, M., J. Levne, P. Martn and P. Rouchon (99). Flatness and defect of nonlnear systems: Introductory theory and examples. Intl. J. of Control. To Appear. Gralt, G. (993). Personal communcaton. Isdor, A. (989). Nonlnear Control Systems. nd ed.. Sprnger-Verlag. Martn, P. and P. Rouchon (993). Systems wthout drft and atness. In: Math. Theory of Networks and Systems. To appear. Monaco, S. and D. Normand-Cyrot (99). An ntroducton to moton plannng under multrate dgtal control. In: Proc. of the IEEE Conf. on Decson and Control. pp. 780{785. Murray, R. M. (99). Nlpotent bases for a class of non-ntegrable dstrbutons wth applcatons to traectory generaton for nonholonomc systems. Math. of Control, Sgnals, and Systems : MCSS. In press. Murray, R. M. and S. S. Sastry (993). Nonholonomc moton plannng: Steerng usng snusods. IEEE Trans. on Auto. Control 38(5), 700{76. Rouchon, P., M. Fless, J. Levne and P. Martn (993). Flatness and moton plannng: the car wth n tralers. In: Proc. of the European Control Conf. pp. 58{5. Srdalen, O. J. (993). Converson of the knematcs of a car wth N tralers nto a chaned form. In: Proc. of the IEEE Intl. Conf. on Robotcs and Auto.. pp. 38{387. Tlbury, D. and A. Chelouah (993). Steerng a three-nput nonholonomc system usng multrate controls. In: Proc. of the European Control Conf. pp. 8{3. Tlbury, D. and S. Sastry (99). On goursat normal forms, prolongatons, and control systems. Tech. Report UCB/ERL M9/6. Unv. of Calforna, Berkeley. Tlbury, D., O. Srdalen, L. Bushnell and S. Sastry (993a). A mult-steerng traler system: Converson nto chaned form usng dynamc feedback. Tech. Report UCB/ERL M93/55. Unv. of Calforna, Berkeley. Tlbury, D., R. Murray and S. Sastry (993b). Traectory generaton for the N-traler problem usng Goursat normal form. In: Proc. of the IEEE Conf. on Decson and Control. pp. 97{977. To appear n IEEE Trans. on Auto. Control. Walsh, G. C. and L. G. Bushnell (993). Stablzaton of multple nput chaned form control systems. In: Proc. of the IEEE Conf. on Decson and Control. pp. 959{96.