A parametric formulation of tracking methods: application to chained systems

A PARAMEIRIC FORMULATION OF ‘I’KACKINC ME‘IHODS: APPLICATION TO CHAlNED SYSTEMS F. DiAZ DEL Rio, G. JIMkNE7- J, I,, SEVILLANO, A. CIVI‘T BAI,CELI,S, Escucla Tbcnica Supcrior dc lngcnicria Infonnltica. Univcrsidad dc Scvilla. AV. Kcin, Mcrccdcs sin. 4 ID1 2 Scvilla. SPAIN, Lfdiaz. pj!, seiri. [email protected]/.~.c.r ABS’IKAC’I Whcn following a path, thcrc arc scvcral possibilitics attcnding to thc way in which thc actual robot statc can bc rclatcd with thc wliolc path. In this work wc formulatc diffcrcnt tracking mcthods on thc basc that a mcmorizcd path can bc dcscribcd by a singlc dcscriptor paramctcr (thc objcctivc point is rcadily givcn by this paramctcr). Wc classify path tracking according to thc way in which wc impose or design thc progrcss ofdcscriptor paramctcr. Bcncfits and disadvantagcs of cach incthod arc idcntificd. Wc also sumniarizc how to construct cach mcthod. Dcspitc that this classification is gcncric, application to mobilc robots and chaincd systcms is vcty valuablc duc to thc importancc oftracking problctn in tlicm and bccausc thcir dcsircd patti is usuntly mctnorizcd. KEY WORDS: paramctflc cqualions, diffcrcntjal gcomctry, mobilc robots, nonholonomic constraints, path following, trajcctory tracking, chaincd systems. 1. INTROUUCTlON it is wcll know that thc convcrgcncc of a non-omnidircctional mobitc robot to a fixcd posturc (thc stabilization problcm), can not bc achicvcd through a smooth fccdback stabilization control law duc to Brockctt’s thcorem [I]. On thc othcr hand, in mobilc robots it is usual tha1 thc path or trajcctory is mcmorizcd, and thc point stabilization problcm is vcry diflicrcnt to thc path’s convcrgencc problcm. For most niobilc robots timc dctcrminism is not a rcquircmcnt of thc tracking, and thc control objcctivc can bc thought of as following a path with a ratc that can bc variablc. Thcrc is no doubt that thc nunibcr of mobilc robots applications will bc largc in thc ncxt fcw ycars, cspccially in ficlds such as intclligcnl transportation systcms (ITS), cxplorcr vchiclcs, and pcrsonal or assistant robots. Canscqucntly, in thc last dccadc thcrc has bccn a grcat intercst in finding controllers and tracking mcthodologics Tor thcsc robots. As a rcsult of thc work wc havc donc during thc last ycars in this ficld, wc prcscnt hcrc a tracking classification bascd on a paramctric formulation. Wc must undcrlinc that wc do not considcr ncw control tcchniqucs in this work, ncvcrthclcss wc conccntratc on a prior subjcct: which mcthods to usc whcn following a path. In fact, control law application can bc considcrcd as a sccond stcp in thc tracking problcm dcsign. Morcovcr, although this formulation can be applicd Eor cvcry systcm, wc cniphasizc same aspccts very valuablc for chaincd systcms. In thc ncxt scctions wc will introducc and analyzc thc ncw tracking classification. In scction 2 and 3 wc formulatc thc differcnt tracking mcthods. In scction 4 wc cxplain thc tracking classification bascd on an cxamptc and finatly wc cxposc thc conclusions. 2. PARAMETRIC FORMULATlON A memorized, refcrcncc or dcsircd path (or mcrcly pnrh) can bc dcscribcd by a singlc dcscriptor paramctcr [13], namcly P, and it can bc cxprcsscd as a vcctor of statc coordinatcs qacl(r). As a rcsult, thc tracking progcss can bc idcntificd with thc progrcss of r. Although thc paramctcr may bc timc, in cascs whcrc timc dcpcndcncc is nob rclcvant, many othcrs arc possiblc. For cxamplc, in diffcrcntial gcomchy thc natural arc paramctcr 1131, which makcs the lincar spced cqual to I, is gcncrally prcfcrrcd. Furthcrmorc, whcn studying convcrgcncc to a path, it is usual to supposc that dcsircd trajcctory has no cnd. 37 I Whcn studying thc tracking problcm, wc can think of r as a ncw statc coordinatc, which must bc addcd to thc other n coordinatcs that dctinc thc state q(fl of thc rcal systcm. Thcrcforc wc incrcasc thc nunibcr of coordinalcs in 1 if wc want lo cxprcss thc siatc of the tracking: {r. 9). In addition, it is usual lo cxprcss systcm stalc through crror coordinatcs e&), which arc somc kind of relation bctwccn rcal nnd dcsircd statcs. Errurs will easily cxprcss how far thc system is from its convcrgcticc objcctivc. Thcrcforc, wc can classify thc tracking of thc path according to Ihe way in which wc imposc or design (whcn possiblc) paramctcr r. From this point of vicw, wc can considcr a ncw systcm input: ihc onc that govcms paramctcr’s evolution, which wc will call a Wc will SCC that for some tracking mcthods, U is obvious, but for somc others, osclcction can introducc intcrcsting fcaturcs on thc tracking. To sum up, wc can cxprcss tracking for whectcd mobilc robots as: wlicrc U is the vcctur of input variables, which has a liiicar dation for chained systcms. Oncc thc tracking mcthod has bccn sclcctcd, a sccond stcp is finding a convcrgcncc. Dcpcnding on thc sclcctcd tracking mcthod, control law extraction may bc different. 2. TRACKING METHODS’ CLASSIFICATION Many rcscarchcrs havc studicd various tracking methods whcn the dcsircd path is mcmorizcd or prcviously gcncratcd. In tlic ficld of mobilc robots two main tracking mcthods havc bccn studicd. In a first group we find those that considcr hc cxplicitly in thc tracking [2,3,4j (usually callcd “/rajecioq, trading” (TT)), and try to approach the robot to a moving objcctivc point. A usual case of TT is tracking in scrvosystcms, whcrc wc track a mobilc systcm or target at thc timc it IIIOYC~. The dcsircd cuordinatcs arc simply qba(l), bccause r=t, and crror coordinates may bc dcfincd as eq(t)q(i)-qdedt). In scrvosystcms, time is critical and this is thc only possibility wc havc. Noncthclcss, when wc Iry to track a mcmorizcd path, thc tracking mcthodology can bc dcsigncd opcnly, as wc know a priori the wholc trajectory. Of coursc classical scrvosystcm tracking can bc applicd just by idcntifying thc paramctcr rassociatcd to rhc path with timc, that is r(t)=t. Furthcrmorc, IT can bc cxtcndcd in a morc gcncral assumption than scrvosystcms: Ict the paramctcr I’ bc rrjhcfion uj’timc r=r(l). An asyniptolically stablc control law may guarantcc that thc systcm will convcrgc to a point qdes(r) in thc dcsircd path in a dctcrministic timc, cxccpt for thc inherent pcrturbations that il may suffer. According to Eq. (I), U will bc simply I for servosysrcms, and a function of limc Tor a morc gcncral TT: a=u(t). In a sccond group, wc find lhosc methods that do not considcr timing rcquircmcnts and try to convcrgc to a path [4,5,6,7,8], which arc usually callcd “pflthfolk~wing”(PF) for mobilc robots. WE can also find scvcral cxccllcnt compcndia of both mcthods in [2,9]. PF Is based on some dation bctwccn actual systcm’s stalc q(l) and thc wholc mcmorizcd path. This rclation or projeclicrn will givc us thc dcsircd point qdJr), i.c. thc dcscriptor paramctcr Y as a function of thc actual position and path: Nq, r)=O, whcrc IT is somc kind of projcction of thc actual position to lhc path. Thcn thc real systcm must try to follow this point qdn(r) instcad of thc one given by the othcr approach. For cxamplc, the dcsircd point is usually sctcctcd to bc thc ”closest point on thc path” to the actual robot’s position [13]. Thc tcrm “closest” mcans thc path’s point that makcs certain distancc crilcrion minimum. Tlic crror coordinates may rcsult also as eq(pq(+qdm(r). Of coursc using this approach, it is not guarantccd that thc systcm will reach a point of the dcsircd trajcctoty in a dctcrministic timc. But thc main problcm with PF is that projection uniqucncss has not bccn guarantied yct, hcncc no PF havc bccn dcsigncd that can be applicable for all possiblc paths (to thc authors’ knowlcdgc). Wc must notc that PF projcction can bc considcrcd as an imposed holonomic constraint on thc whole state {q, r}, and this mcans lhat thc systcm will immcdiatcly toosc onc of its state coordinatcs. Input D can bc obtaincd by difl’crcntiating thc projcction with rcspcct to timc, substituting statc cquation (I), and then solving for IT= i. : 372 Thcrcforc PF introduces a dcpendencc likc U yu(q, U) = /(q) U. Note that if denominator is null in Eq. (2), variation of r is undcfincd. We havc shown in 141 that this caw is cqirivalcnt to thc non-uniqucncss of thc choscn projcction. In othcr words, PF is not appticablc for a11 kind of paths. Thc key advantage of PF is that it is morc suitablc for many situations in which timc is not a critical paramctcr [5,6,10]. This can bc understood iY wc considcr the following cxamplc: if pcrturbations forcc thc mobile robot to bc at rcst, dcsircd point for TT will movc unavoidably, This mcans that crrors will grow up to somc valuc that may introducc instability. On thc othcr hand, if PF wcrc uscd, dcsircd point will be the samc in spitc of thcsc pcrturbations, bccausc path’s shapc and rcal robot shtc rcmain thc samc (cris lincar to inputs U Tor chaincd systems). A diffcrcnt tracking mcthod was proposcd in a prcvious papcr of ours [I I]. Wc namcd it error adoprive trocking (EAT), bccausc tracking adapts to systcm crrors. Wc showcd that it tctains thc cxposcd advantagcs of PF and it can bc applicd to all scts of paths. Its dcsign is similar to that of TT but it is intcndcd for path rccovcring without strict timing rcquircmcnts. Note that for most mobitc robots (including advanced whcclchairs [ 121) timc dctcnninism is not a rcquircmcnt ofthe tracking. Whcn timc is not critical, wc can “dcsign” the variation of I’, and wc can rcgard robot statc, i.c. taking crrors into account. Thus, contrasting thc IT‘S rigid variation of r, that is o=o(/,), wc proposcd o=g(e,J, whcrc g(eJ is a “convcnicnt” function of thc crrors. Hcrc “convcnicnt” is rcfcrrcd to thc dcsigncr objcctivcs, but also it should mcan that tracking is done correctly, and the prcviously cxplaincd advantagcs of PF are prcscrvcd. That is, function g(eJ should fulfill: If crrors arc small, g{eJ should tcnd to I (or niorc cxactly to lul/1udea(r)l if wc want to movc along thc path at anothcr pacc), so the tracking rescmbtcs IT. ff crrors arc largc, thc rcfcrcncc robot should “wait for” thc actual robot. That is, g(e.J should bc small until a good canvcrgcncc IS rcachcd (D should tcnd to zero). Of coursc in this situation, no dctcrministic following is cxpccrcd. Of coursc many passible functions g(eJ can bc dcsigncd attcnding to thc characteristics and purposes of cach systcm, but in [I I] wc showcd that a vcry suitablc function is U = g(eJ = ap(- K#i) ; whcrc K,>O is a scalc factor. With thcsc conditions thc tracking cannot bc dctcrministic and rcminds that of PF. Morcovcr, a sccond kind of EAT can bc dcsigncd. If wc nccd somc aspccts of timc dctcrminism, variation of r can bc cxtcndcd to include the “inaccuracy in thc dctcrministic tracking”, that is, thc diffcrcncc bctwccn thc dcscriptor paramctcr r (that indicatcs thc targct of our control dcsign qdlr(r)) and timc 1 (that indicatcs thc targct or thc timing rcquircmcnts qde,(i)). Hcncc awoutd bc g(r.e,J: a combined function or crrors eg and thc difrcrcncc bctwccn paramctcr rand timc t. Oncc again there arc many possiblc functions g(eJ, but an intcrcsting strategy [I I] is intcndcd to gct a dctcrministic tracking at long last (that is, a “rclaxcd” dctcrministic tracking). Paramctcr r will rcmain “at rest” whcn the robot is far from thc path (in spitc of thc incrcasc of diffcrcncc r-t). Whcn robot “rccupcratcs” and approachcs thc path, wc conccntratc on rcducing thc diffcrcncc r-/. Thcrcrorc g(t,e,J can not bc boundcd by I, bccausc in thc sccond casc I- must approach /. Morcovcr at thc origin (/-r=O. e,+), g must bc I, Thus wc showcd that a convcnicnt choice for this more complcx EAT can bc: g(t,eJ- exp(-K,eqz) (I+K,prc/on(/-r;)), ; whcrc KpO, K,,W arc scalc factors that indicatcs how fast thc convcrgcncc of r to t is. Hcrc g(r,eJ is uppcr boundcd by I + K,~ .and lowcr boundcd by I - K,~ 2 2 or 0. Thc formulation advantage of EAT is that it can obviously bc applicd to all scts of paths, and implies a rclnlion Tor thc input cr Iikc U =cr(q) or cr =u(t, 9). Wc can scc this rclation as a (in gcncral) non-intcgrablc dation bctwccn tlic whole statc givcn by {q, r}. To sum up, wc can find in thc currcnt litcraturc the following proposals for r, which supposc a dircct rclation constraint bctwccn {q, r}: 373 1. 7T:r=r(r). * u=a(t). 2. PF: n(q, r)=O u=o(q,u). In IT, wc “loose” uoordinatc r, and in PF WE usually loosc onc of thc error coordinatcs e,, [4], Morcovcr thc two proposals for EAT arc non-intcgrablc relations bctwccn coordinatcs {q, r): 3. EAT: a--i=o(q) 4. EAT: ff=r:=a(r,q). 5. Q= a(q, U). 7. u=cT(r, U). s. u=a(u). If wc cambinc all possiblc dcpcndcnccs of q wc noticc that thcrc are four cases IcA: 6. a=a(f, u,q). Wc must point out that incthod 5 is not thc samc as PF, bccausc wc suppose that rclation 5 is not intcgrablc (In PF it is clcarly intcgrablc, bccausc dq, r)=Ocxists). Morcovcr we will not consider hcrc thc last two tracking mcthods (7 and 8). Thcy do not sccm to bc intcrcsting for tracking purposcs, bccausc variation of paramctcr r docs not dcpcnd on thc statc q. On thc contrary, cases 5 and 6 arc a morc intuitivc and cffcctivc cxtcnsion of thcm, and wc will prcscnt thcir charactcrislics and applications in this paper for the first time. Ncvcrthclcss thcsc mcthods 7 and 8 will be studicd in fulurc work in ordcr to dctcrminc if thcy can bc worthwhile for somc particular applications. In ordcr to fix idcas and cxplain the two new tracking mcthods, wc will analyzc a simplc systcm in thc ncxt section. Dcspite that this examplc is vcry simplc, our classification is gcncric for cvcry systcm; so extraction of control laws for chained systcins can takc advanlagc of using somc of thcsc methods. Somc cxamplcs of thcsc bcncfits hsvc bccn cxainincd in somc ot-our prcvious works [4,1 I]. 3. A SIMPLE EXAMPLE Lct us consider a simplc systcm with two statc coordinatcs x={xl, xr}, whose goal is to follow a rcfcrcncc path xd,,=/.rl,jJri, .k-ldw(r)} made by a virtual robot. Statc cquations arc simply: .r, =U* (3) .r, = 11, ; x:.,& = lt,,&(i-) Xi,,&.” = u2,,A, tr) whcrc (‘) holds for dcrivativc with rcspcct to r, Applying chain’s law: u,,k3(Q= i i+,*>(r), As wc arc intcrcslcd in thc tracking wc dclinc thc syslcm’s crrors as thc dij’crence: .j=1,2. And thcn wc havc .ii,,&, =tlj,8h(t) ,/=1,2. edtj =.VI(!) -.rlcle$ir(i)l ; L:, (I)= 1rdt) -?141des(r1 ; (4) .?(I) = a(r) - .r~~~(r(t)) : L;: It) = u?(t) -tuj&Jr): WC havc cxprcsscd explicitly thc depcndcncc of thcsc variables bccausc thc dcsign In a scwosystciii J.={, a=l and a vcry siniplc convergcnt control law can be: If TT is choscn, dcpcndencc r=r(f) will only add a scalc on thc clcctioii oft-, supposing that i>O H. NOIC that in both cascs, thc condition iW Vi implics that thc rcfcrcncc point {xI&(f)). x2dc,(r(g)/ will unavoidably advance in spitc of real robot motion. But if wc could apply PF mcthod, thc situation would bc vcry diffcrcnt. First we must look for the most suitablc projcctjon @x, r). in ordcr to sclcct thc descriptor paramctcr r and hcncc, thc rcfcrcncc point /.vlk,(r), x?,kJr/] at any instant. The most obvious choicc is thc projcction that sclccts thc closcst point on tlic path to thc robot’s position, i.c. r that makes C:_,.,! minimum. For cxamplc, if thc dcsircd path is thc [in, {xJkS(r)=sr, xjb,(r)=O, s>O/ thcn thc closcst point wit1 bc xd6=j.xI. O}, and tbc projcction: F x,/s. DiffcrcntiatIon of projcction givc us thc ratc for r(tJ D =uI/s. Thcn thc tracking will progress only if r incrcascs, ix. if U,># (or uj<O if thc tracking is to mclhodology must now choose bctwccn difkrcnt forms of r(t) or a. r~#) = i?,,,Jtk&e, i t~,~Jr)-K,c,, j=1.2, whcrc K/ >O. 374 bc donc in rcvcrsc ordcr). Thcn wc must impose some condition or “motion cxigcncy”[4] to cnsurc that (rcal and hcncc virtual) robots advancc, and hcncc to guarantcc that thc tracking is being donc. The simplest motion cxigcncy is of coursc u,=comfonf>O, but othcr morc sophisticatcd can bc proposcd. If thc trivial motion cxigcncy is prcfcrrcd (and cnongh for our control purposes) thcn thc prcvious control law still succccds (of coursc only for input 2): it2 (t) = u2 d&) - Kj e? = - K2 el. This is obviously a particular casc. If a gcncric trajectory had to bc trackcd, thcn a morc gcncral motion cxigcncy is prcfcrablc, for cxamplc U/’+ u?’=crin.~ioni>O [4]. This has thc additional advantagc of maintaining inputs within ccrtain valucs, avoiding an cxccssivc incrcasc of inputs (which may introduce instability). Howcvcr, problcms arise whcn wc cannot guarantcc projcction uniqucncss. For cxamplc, if thc path wcrc a circlc and thc prcvious projection wcrc to bc applicd, thcn projection uniqucncss is brokcn whcn thc robot approachcs thc circlc ccntcr. So it is not rully applicablc for all kind of paths, and projcction uniqucncss should bc carcfully analyzed (SCC [4]). In EAT wc can design the most appropriatc tracking ralc, i.e. an cquation for uas a function of thc crrors. As cxplaincd bcforc a vcry simptc and intcrcsting possibility would bc: CT =g(e,J=exp (-Kr (et2+ e:)) ; K,>O is thc scalc factor. This proposal trics that crrors do not incrcasc grcatly. Thcrcforc, it conscrvcs thc PF advautagcs, whilc avoiding thc projcction diflicultics. Bcsidcs, for this trivial systcm thc prcvious TT control law stilt works: ui(t)=up(-KJe,’+ et)) u,,,jJr)-K,e, : j=1.2, K, >O. Thc main diffcrcncc is that whcn crrors arc Iargc cnough, thc convcrgcncc rcscmblcs that of thc stabilization problcm (virtual robot is at rcst). Going furthcr, we proposc herc thc two ncw tracking incthods 5 and 6. Mcthod 5 has thc samc kind of rclation cr = u(q, U) than PF, which tcll us that it will bchavc in a similar fashion to PF. But now we arc frce to “design” this rclation, avoiding thc problcms of thc projcction diflicultics. For cxamplc a form of cr that rcminds us of PF can bc: u.u, n=- 2 (1 + K.WF arctan(r(r,x)) ” dcs whcrc KnvF>O is a scale factor that indicates haw fast the convcrgcncc of r to thc projccting point x,&pb) will bc, and Nx, r)=O is a suitablc projcction. Thc projccting point xde,(rp+j is hcrc thc point of thc path that fulfils nfx, ,-)=O. In gcncral for points xaeS(r) that arc ncar of xdes(rpp), thc sign and magnitudc of Ns, r) givc us an idca of how far is xdes(r) from it (in 121 some cxamplcs can bc consulted). Thcn thc sccond factor of cr will bc I when PF objcctivc has bccn achicvcd (r cquals rpF), and will bc lcss or biggcr othcrwisc, trying to approximatc I’ to rpF~ Function ardarr has bccn choscn for similar rcasons to that of EAT mcthod. Thc first factor has bccn choscn similar to thc dcpcndcncc that can bc found in thc c7 of PF. Of coursc thc control law has to dccidc thc cxact form of U, but we must rcalizc that if robot approachcs sd,(r}, U will bc parallcl to uaS. In this casc parameter r will movc at thc pace that control law dccidcs. Finally notc also that whcn robot is stoppcd (inputs U arc null), Y docs not progrcsa. Likc in PF with this mcthod it is not prcdictablc whcn thc robot will rcach a path’s point in thc gcncral casc, and it may bc morc stablc than TT whcn crrors arc big. On tfic othcr hand, projcction uniqucncss is not a problcm: if it wcrc satisficd for a wholc path, thc only conscqocncc is that thc sccond factor of U is 1. In addition, anotlwr form of crthat rcminds us of EAT may bc: whcrc KA+O is a scalc Factor that indicatcs how fast thc convcrgcncc ofr to objcctivc point xdrs(r) is. The prcvious analysis is valid again (considcring EAT fcaturcs). As this mcthod can bc considcrcd as a PF with adaptation to crrors, wc can namc it “Aduprii~ Path Fdollowing” (APF). 375 For thc sainc dcsircd path dcscribcd for PF (a straight linc) wc can usc Lc samc projection: n(x, r)= r - x,Ls -U. Thcn wc can usc tbc samc control law of prcvious PF: imposc somc "motion CXigC~lCy"[4] to cnsiirc that real and virtual robots advance, and imposc thc controt law for input 2. Hcrc again a inorc sophistiratcd motion cxigcncy likc U,'+ u2'=conSstant>0 would havc thc addilional advantagc of maintaining inputs within certain values. Morcovcr thcrc may bc othcr possibilitics for dcsigning control laws that can bc considcrcd in futurc works Finally wc can dcsign a tracking with thc rclation: a =U (1, U, q). Following thc samc mcthodology prcscntcd for thc EAT with U= g(t,eJ, and for thc two APF arclations, a pair of intcrcsting cascs arc givcn by: Thc prcvious considcrslions shown for EAT and APF will apply now combining both factors. CONCLUSIONS Wc havc forinulatcd dirt'crcnt tracking mcthods of mcmorizcd paths according to thc way in which we hpme 01- design thc progress of dcscriptor paramctcr. 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