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FORWARD PATH TRACKING FOR MOBILEROBOTS WITH SEVERAL TRAILERS1J.L. Mart nez, J. Morales and A. MandowUniversity of M alaga. Dept. of System Engineering andAutomation. E.T.S. Ingenieros IndustrialesPlaza El Ejido s/n, 29013-M alaga, SpainPhone:+34 95 2131408. Fax:+34 95 2131413E-mail: jlmartinez, jmorales, amandowctima.uma.esAbstract: In this paper, kinematics of train-like vehicles consisting of an au-tonomous tractor and several passive trailers with on/off-axle hitching is employedto avoid collisions between the different bodies during forward path tracking. Thisis achieved by limiting the tractors curvature according to the propagation ofthe relative constraints between consecutive bodies of the chain, which allows theuse of conventional path tracking algorithms. This technique has been successfullytested in the tracked mobile robot Auriga- with two attached trailers. Copyrightc ? 2004 IFACKeywords: path tracking, mobile robots, articulated vehicles, under-actuatedsystems, kinematics.1. INTRODUCTIONArticulated train-like vehicles consisting of a trac-tor car and one or several trailing units increasetransportation efficiency, which is why they arefound in a variety of applications, such as airportluggage carriers, passenger and sightseeing vehi-cles, load-haul-dump mining, or multiple-trailertrucks. However, it is well known that this kind ofnon-holonomic system poses a complex non-linearand under-actuated control problem, which makesthem difficult to operate even for trained humandrivers.Motion control of non-holonomic vehicles usuallyrequires that a path tracking algorithm issuesreferences to a low-level controller in order toachieve the desired path. Thus, the aim of pathtracking is that the vehicle follows a specified1This work has been partially supported by the spanishproject DPI 2002-04401-C03-01. Second author is a doc-toral researcher with a spanish MCYT grant.obstacle-free path in spite of perturbations bymanipulating the tractors speed and steering.Path tracking methods have to use fast poseestimations based on the vehicles sensor system.However, motion control strategies for single ve-hicles (e.g., mobile robots) cannot directly beadapted to articulated vehicles because of wellknown problems such as jackknife (i.e., unstabil-ity in backward motion) and off-tracking of thetrailers (i.e., the deviation from the towing unitspath). Therefore, path tracking for articulated ve-hicles implies dealing with specific constraints re-garding stability and collision avoidance betweenunits.In general, this problem has been approached bymeans of linearization techniques. Thus, Lamound(1993) has proven the controllability for the on-axle n-trailer system. This case, in which it isassumed that trailer hitches lie on the axle ofthe preceding vehicle, complies with the flatness0LifLibLi+1fii+1xi-1x0y0xGyGxixi+1vivi-1vi+1i-1ii+1L0bv00Fig. 1. Parameters of the kinematic chain.and chained form properties (Kolmanovsky andMcClamroch, 1995). The general off-axle caseposes a more complex control problem. This iswhy Bolzern et al. (1998) proposed path trackingbased on linearization of a virtual on-axle vehiclewhich shares some properties with the actual one.Altafini (2003) offered a path tracking solutionaimed at minimizing off-tracking which is basedon special local frames for each unit. Bushnell etal. (1994) developed theorems to establish upperbounds for off-tracking, so that conventional pathplanning among obstacles could be applied to anenlarged tractor.The work presented in this paper first analyzesgeneral on/off-axle n-trailer kinematics and itstransitory and stationary responses. The noveltyof this approach arises from applying limitationson the path tracking steering commands for thetractor. This has been proved to be useful inorder to avoid jackknife and collisions with asingle trailer for both forward and rear motion(Mart nez et al., 2002). This means that thetractor can be controlled with the same pathtracking algorithms as when it does not tow anytrailer. The method has been validated with atwo trailer setup attached to the Auriga- mobilerobot.The paper is organized as follows. Next sectionstates a kinematic model for general n-trailer ve-hicles. Then, stability of the entire system is ad-dressed in section 3. The trailers control problemis presented in section 4. Section 5 deals withimplementation details and experimental resultswith Auriga-. Finally, conclusions and futureworks are discussed in section 6.2. KINEMATIC MODEL OF A VEHICLEWITH SEVERAL TRAILERSThe parameters of the kinematic chain for a vehi-cle with several passive trailers can be observed inFig. 1. Distances Lifand Libare either positiveor null constants (but not at the same time). IfLi1b= 0 then for ithtrailer is on-axle, otherwiseit is off-axle.iis the variable that represents the relativeangle between i 1thunit and ithtrailer (positiveif counterclockwise). 0is the tractors heading.Accordingly, subindex zero refers to the tractorvehicle, subindex one to the first trailer, and soon.Under the assumption of small velocities andaccelerations as well as light loads on board, thekinematic model of a mobile robot with n-trailerscan be obtained by applying the rolling withoutslipping constraint and the method of relativevelocities. So, the ithtrailers relative angularspeed is defined by:didt= vi1sini+ (Lif+ Li1bcosi)Pi1j=0jLif(1)that can be expressed in function of vehiclescurvature by using i=?Pij=0j?/vi:didt= vi1?i1+siniLif+i1Li1bcosiLif?(2)where the longitudinal speed of the ithunit isdefined by:vi=vi1cos(i) Li1bsin(i)i1Xj=0j=vi1(cos(i) i1Li1bsin(i)(3)Equation (2) can be transformed into a functionof the distance travelled by ithunit siinstead ofthe elapsed time t:didsi= ?i1+siniLif+i1Li1bcosiLif?(4)Each units curvature can be obtained as thequotient between its angular and linear speeds,given by Eq. (1) and (3) respectively:i=i1Li1bcos(i) + sin(i)i1Li1bLifsin(i) Lifcos(i), Li1b6= 0tan(i)/Lif,Li1b= 0(5)When length Lifis null, the model defined byEq. (1) becomes static, and a direct relationbetween angle iand curvature i1appears:i= arctan(i1Li1b)(6)In this case, the curvature is given by:i= i1/q1 + L2i1b2i1(7)3. STATIONARY AND TRANSIENTRESPONSESWith constant speed v0sand curvature 0sset-points for the tractor, all units can tend to followconcentric circumferences, as shown in Fig. 2. Thisequilibrium only occurs when the steady statevalue is, defined geometrically by:is=i1sr1 + 2i1s?L2i1b L2if?(8)is a real number. This happens either when Li1bis greater or equal than Lif, or when the followingequation is satisfied:|i1s| 1qL2if L2i1b(9)2s3s1s1/3s1/2s1/1sFig. 2. Stationary response with constant actua-tions.Two equilibrium points for the ithtrailers angleisare possible:is= atan(i1sLi1b) atan(isLif)(10)is= atan(i1sLi1b) + atan(isLif) (11)The state plane for the ithtrailer, assuming con-stant curvature for the i 1thunit, is shown inFig. 3. During forward motion, the equilibriumpoint near given by (11) (i.e., jackknife po-sition) is unstable whereas the equilibrium pointnear zero given by (10) is asymptotically stable.In addition, the space-constant in the vicinityof the equilibrium point, i.e. the slope of thetrajectory represented in Fig. 3 near (10), isproportional to Lif. In case that Lif= Li1bthen the space-constant is equal to Lif.00id/dsEq. (11)Eq. (10)Eq. (3)iiFig. 3. State plane for ithtrailer with constantcurvature of i 1thunit.Assuming that the response of the tractors low-level control loop to curvature set-point changes0sis overdamped, the transients of 1and 1for the first trailer, given by Eq. (1) and (5), arealways overdamped. For instance, if the responseof tractors curvature is modelled by a first-orderlinear system, the equilibrium point given by Eq.(10) is a stable node, whereas equilibrium ofEq. (11) becomes a saddle point (see Fig. 4).Consequently, the transients of consecutive unitsiand iare also overdamped.0123450.40.420.440.460.480.50.520.540.560.581 (rad)0 (m-1)Eq.(10)Eq.(11)0sFig. 4. State plane for first trailer with constantcurvature set-point for the tractor.4. PATH TRACKING CONSTRAINTS FORSEVERAL TRAILERSThe objective of this section is to ensure thatthe trailers show a safe behavior while effectivelycontrolling the tractor to follow a path. Then,the steering control output provided by the path-tracker can be modified according to these condi-tions.4.1 Steering set-point limitationsFirstly, to ensure the stability of the entire sys-tem, it is necessary that every trailer tends to anequilibrium point (i.e., Eq. (8) has real solutionfor every i). In order to solve this problem dur-ing forward path tracking, the propagated cur-vature from the previous i 1thunit (as statedby Eq. (5) i1should always be maintained inabsolute value below the limit value of Eq. (9):i1m1=1qL2if L2i1bLi1b LifLi1b Lif(12)Secondly, in order to avoid reaching a physicallimit im, related to the collision between i 1thand ithtrailer or the breaking of their link (seeFig. 5), a further limit on trailer curvature canbe imposed by solving i1in Eq. (10) with themaximum values im:i1m2=?sin(im)Lif+ Li1bcos(im)?(13)imi-1iFig. 5. Physical limit of turning angle i.Thirdly, a new curvature constraint i1m3isinherited from the propagation of more restrictivelimitation from the following unit im. i1m3canbe solved numerically by means of Eq. (5) when iis chosen as the worst case, by changing the signto the value of Eq. (10).To sum up, the ithtrailer imposes three curvatureconstraints to the previous i 1thunit: the existence of equilibrium points in the ithunits motion: i1m1, the collision avoidance between the ithandthe i 1thunits: i1m2, and the constraint inherited from i + 1thunit:i1m3.Among these, the most restrictive one is chosen:i1m= min(i1m1,i1m2,i1m3)(14)This procedure eventually results on the safestcurvature limitation for the tractor 0m.4.2 Relaxing steering limitationsThe safest limitation 0mcan be too severe be-cause it is calculated for the worst cases. Conse-quently, this curvature limit can deteriorate thepath tracking performance. This section profitsfrom the transition time of trailer kinematics inEq. (1) in order allow for set-points values 0sgreater than 0m. This results in a more relaxedlimit 0r.Usually, the inherited limitation 0m3is the mostrestrictive. Therefore,0m 0r min(0m1, 0m2).(15)This new limit can be found via simulation. First,a prospective value for 0ris proposed. Startingfrom the stationary for 0s= 0r, a 100% (i.e.,worst case) change in the set point 0s= 0risapplied. If the transient of the trailers for i 2 isacceptable (i.e., stable and without inter-collision)a greater value can be tested. Otherwise, a smallervalue would be considered.It must be noted that if the initial conditions areoutside the ranges of the stationary angles for 0r,then it is necessary to impose 0muntil reachingback into this range. Then, 0rcan always beapplied.5. IMPLEMENTATION5.1 Description of Auriga-The method presented in the previous section hasbeen tested on the Auriga- mobile robot (seeFig. 6). Its dimensions are 1.24m (l), 0.75m (w)and 0.84m (h), and weights approximately 258Kg.This vehicle is powered by an on-board petrol-fedAC generator, although a power cable socket isalso available for indoor tests.Locomotion is based on the independent per-formance of two DC motors, each one with agear-reduction and incremental shaft encoders fordead-reckoning. The top speed of each track is 1m/s.Two control systems can be distinguished in themobile robot Auriga-:Fig. 6. The mobile robot Auriga- and its two trailers. The motor interface is based on a DSP, whichaccepts track speed setpoints and providesactual track speeds. Due to differential steer-ing, the speed and curvature time-constantshave the same value. The navigation system, which includes thelow-level controller and the path tracker. Anindustrial PC based on a Pentium-IV mi-croprocessor at 2.2GHz governs navigationunder a real-time operating system.The dimensions of the two wheeled trailers aresimilar to the vehicle. The first trailer is employedfor carrying loads, while the second one is forspraying plants (see Fig.6).The employed car-like joint allows an easy attach-ment and detachment of the trailers. Each angleiis measured by a displacement sensor with arotary head, whose wire is also simple to detach.Lengths and maximum angles values of this par-ticular setup are the following: L0b= 0.71m,L1f= 0.99m, L1b= 0.61m, L2f= 0.81m,1m= 70and 2m= 55.5.2 Path tracking resultsIn the following experiments, the pure-pursuittracking strategy has been employed due to itssimplicity. This is a well-proven technique forsingle mobile robots where the vehicle changes itscurvature by repeatedly fitting circular arcs to thegoal point (Amidi, 1990).Since non-holonomic vehicles cannot correct theerrors with respect to the nearest point in the pathbeing followed, the goal point is chosen a distanceahead from the nearest point (Ollero et al., 1994).Using Eq. (9) 1m1= 1.876m1, and by meansof Eq. (13) 1m2= 0.706m1. As it is shownin Fig. 7 the propagated value is 0m3=0.255m1. For trailer-1, 0m1=1.449m1,0m2= 0.762m1and 0m3= 0.255m1, so |0|must be limited with 0.255m1.-0.5-0.4-0.3-0.2-0.100.10.20.30.40.5-2.5-2-1.5-1-0.500.511.522.50 (m-1)1 (m-1)0.706-0.706-0.2550.255Eq. (5)Fig. 7. Curvature constraint propagation.Fig. 8 shows the tracking of a straight line at themaximum speed starting far from the path with0= 1= 2= 0. This situation, in combinationwith a short look-ahead distance of 0.3 m repre-sents a worst case path tracking scenario. Whenstrictly applying the safest limitation 0m, a long35 m settling-space can be observed.In this case, a maximum relaxation in the cur-vature limit can be set to 0r= 0.495m1. Thestationary angles associated to this limits are 47ofor the first trailer and 42ofor the second.In Fig. 9 it is shown the application of thissmoother limit, resulting in a better tracking,while avoiding collision between units.However, in case that the tractors curvature islimited to 0m= 0m2, i.e. not considering thesecond trailer, a stroke between the two trailersappears as shown in Fig. 10.6. CONCLUSIONSA general on/off-axle kinematic model for articu-lated vehicles has been considered in this work. It-3-2-1012345-5051015202530354045x (m)y (m)tractortrailer1trailer2pathstartFig. 8. Forward path tracking with curvaturelimitation 0m= 0.255m1.3210123455051015202530354045x (m)y (m)tractortrailer1trailer2startpathFig. 9. Forward path tracking with curvaturelimitation 0r= 0.495m1.-4.5-4-3.5-3-2.5-2-1.5-1-0.500.51-60-40-200204060801 (m-1)2 (degrees)2 m2 m -1m1mstrokeFig. 10. State plane 1 2.has been shown that special constraints based onkinematic parameters can be imposed on steeringcommands of the tractor in order to avoid inter-unit collisions and to guarantee stability of thetrailers. Thus, efficient path tracking methods forsingle non-holonomic vehicles can be adapted toseveral trailer systems by introducing operationallimitations in their curvature control set-points.Experimental results have shown the successfulapplication of curvature restrictions for pure pur-suit path tracking with two off-axle trailers. Thepath tracker issues admissible control commandsso that the vehicle converges to paths startingfrom arbitrary tracking error poses.Future work includes optimization of path plan-ning and p
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