Fuzzy Backing Control of Truck and Two Trailers

Fuzzy Backing Control of Truck and Two TrailersAndri Riid Alar Leibak Ennu RüsternDepartment of Computer Control Department of Mathematics Department of Computer ControlTallinn University of Technology Tallinn University of Technology Tallinn University of TechnologyEhitajate tee 5, Tallinn 19086 Ehitajate tee 5, Tallinn 19086 Ehitajate tee 5, Tallinn 19086Estonia Estonia Estoniaandri@dcc.ttu.ee alar@staff.ttu.ee ennu.rystern@dcc.ttu.eeAbstract – For a number of years, truck backer-upperproblem has served as a benchmark for control among thepractitioners of computational intelligence. In our worksfrom the past we have shown how decomposition of thecontrol problem and subsequent segmentation of the controlsystem leads to substantial control performanceimprovement. In current paper, this control principle isextended to the much more complex two-trailer backingproblem with great success.IINTRODUCTIONThe number of scientific contributions utilizing Nquyenand Widrow truck backer-upper [1] as the test object hasgrown rapidly and continuously in recent years. Attemptsto handle more complex, i.e. multi-trailer systems, on theother hand, are still quite rare. Perhaps the best known ofthem are the applications of linear matrix inequalitiesbased control for two-, three-, five- and ten-trailer systems[2-4] by Tanaka et. al. Kinjo et. al. [5, 6] have used geneticalgorithms for tuning the neural controllers in theexperimental setting very similar to Tanaka’s works. Theseworks aim at the stabilization of the system along the x-axis. Few remaining applications [7-9] have either morelimited goals or the experimental setting and algorithmicplatform is not described in sufficient detail to fullyappreciate the contribution of the authors.From the experiments with the trailer-less truck backeruppersystem [10], it has been our understanding thatdecomposition of the control problem and effective use offuzzy logic provides an elegant and efficient solution tothis challenging task. Current paper that appears as alogical development in the trails of [10-12], presents thefuzzy logic enhanced control system that is able to take fullcontrol responsibility over the truck and two-trailer systemand deliver it from an arbitrary initial position to a freelypositioned loading dock with smooth trajectories andminimal control effort, thus providing the solution to theproblem that no sane driver would ever attempt in real life.II PROBLEM DEFINITIONThe driving system (or the car as we shortly address it inthis paper) consists of the cab part and two attached trailers(Fig. 1) and is described by five state variables – thecoordinates x, y of the reference point placed at the end ofthe second trailer and Φ 0 , Φ 2 , Φ 4 that are the anglesbetween x-axis and the cab part, first trailer and the secondtrailer, respectively. The length (l) and the width (w) of thecab part are both 2m and the dimensions of the trailers are2×4m.The challenge in this paper is to design a control systemto provide appropriate steering angle θ so that the car willultimately be positioned at x = x f , y = y f , Φ 4 = Φ f (prespecified"loading dock").The kinematics of the car are governed by followingequations (taken from [2]):v ⋅ ∆tΦ0( k + 1) = Φ0( k)+ tanθ( k)lΦ ( k)= Φ ( k)−Φ( k)1v ⋅ ∆tΦ2( k + 1) = Φ2( k)+ sin ΦLΦ ( k)= Φ ( k)−Φ( k)30224( k)v ⋅ ∆tΦ4( k + 1) = Φ4( k)+ sin Φ3( k)L⎛ Φ4( k + 1) + Φ4( k)⎞y(k + 1) = y(k)+ v ⋅ ∆tcosΦ3( k) sin⎜⎟⎝ 2 ⎠⎛ Φ4( k + 1) + Φ4( k)⎞x(k + 1) = x(k)+ v ⋅ ∆tcosΦ3( k) cos⎜⎟⎝ 2 ⎠where -90° ≤ Φ 1 ≤ 90° is the difference between the anglesof the cab and the first trailer and -90° ≤ Φ 3 ≤ 90° is thesimilar difference between the angles of the first andsecond trailer; L (5m) is the added length of a trailer and itsjoint, v = -1m/s is backward driving speed and ∆t = 0.1s isthe sampling time. Also note that steering angle θ isrestricted to [-70°, 70°].yxFigure 1. Truck with two trailers and corresponding state variables1Φ 4Φ 2θΦ 0Φ 3Φ 1,

III CONTROL SYSTEM COMPONENTSIt was demonstrated in [10] that the control task of drivinga simple truck backwards to the loading dock can besolved with less effort if distinction between the tasks oftrajectory planning and further determination of thesteering wheel angle that would force the car to follow aspecified trajectory, is made. Trajectory determination canbe carried out quite efficiently in indirect way by defininga desired car orientation angle (that in current applicationwould obviously apply to the second trailer angle Φ 4 ) foreach point in input space (x, y) and it was suggested in [10]that a simple fuzzy logic system called trajectorymanagement unit (TMU) can do the job. TMU actuallycomes in two parts – the upper one that defines carbehaviour for the positive (in respect to y) half plane (Fig.2) and lower one for the negative half plane (Fig. 3). Notethat sole reason for these blocks being implemented asseparate pieces is the need to cancel undesirable co-effectsfrom the interpolation of antagonistic rules. For the simpletruck navigation the task can then be completed bycomparing the desired orientation angle supplied by TMUwith the actual orientation of the car and this difference(i.e. orientation error) is supplied to the simple PDcontroller, which determines appropriate steering angle asa function of the error.However, as the experimentation with the truck andtrailer system in [11] proved, things are not so simple anymore if we add a trailer to the truck because of the jointissue. The angle between the cab and trailer parts needs tobe manipulated if we expect any control performance andcalls for a further segmentation of the control system.Interaction of the cab and trailer parts can be coordinatedby a joint controller, which in this case produces anoptimal angle between the cab and trailer partscorresponding to the (trailer) orientation error. Iforientation error ϕ is positive, the angle between the trailerand the cab (ξ) must also be positive (and vice versa) andthus the dependence must be monotonously increasing.This single-input single-output functional block isimplemented using fuzzy logic in order to obtain anonlinear mapping that is necessary to achieve expectedcontrol performance (Fig. 4). Finally, the differencebetween expected joint angle and its actual value will beused as an error function for the PD controller defining thesteering angle.Needless to say, in case of truck and two-trailer systemthings are even more complicated as there are two joints(and respective angles Φ 3 and Φ 1 that need to bemanipulated) because of what not one but two jointcontrollers have to be inserted into the control system. Firstone governs the relationship between orientation error Φ r4 -Φ 4 and expected Φ r3 , whereas the second one takes theinput from the first and compares it to the actual value ofΦ 3 to produce desired Φ r1 . Error function for the PDcontroller will be computed by Φ r1 - Φ 1 . In summary, thisreasoning leads to the principal scheme of the controllerdepicted in Fig. 5.mf3mf2mf10-10 -5 0 5 10Figure 2. 15 TMU rules responsible for trajectory management wheny > 0. Each small arrow indicates optimal angle of the car correspondingto the given ruleξmf1 mf2 mf3 mf4 mf5If x is mf3 and y is mf2 thenΦ r is 90°mf1 mf2 mf3 mf4 mf5 mf6-15 -10 -5 0 5 10 15x604020Figure 3. Trajectory mapping unit (lower part)mf1 mf2 mf3 mf4 mf50-20-40-60If ϕ is mf1 then ξ is mf1If ϕ is mf2 then ξ is mf2If ϕ is mf3 then ξ is mf3If ϕ is mf4 then ξ is mf4If ϕ is mf5 then ξ is mf5-150 -100 -50 0 50 100 150ϕxFigure 4. Joint controller.252015105mf1mf2mf3mf4mf5y

xyΦ 4Φ 2Φ 0-16TMUΦ r4upperTMU0.40.3++yxΦ r4+-lowerTMU0.35JC1Φ r3+-JC2-+xyFigure 6. TMUΦ 4coordinatetransformationx ’ y ’x fy fΦ fΦ r1--+TMUΦ ’ r4PDangletransformationθΦ r4Figure 7. TMU interfaceFigure 5. Control system (principal scheme)IV CONFIGURING THE CONTROL SYSTEM.In order to achieve good control performance add-onsand further adjustments to the control system depicted inFig. 5 are required that will be explained in detail in thissection. First of all, we need to coordinate the cooperationof upper and lower TMUs with a switching block (Fig. 6)that will see to it that for positives values of y, Φ r4 will besupplied by upper TMU and for negative values by itsrespective counterpart. The effect from using hard limitersat the inputs of fuzzy systems is twofold. First, they areabsolutely necessary to guarantee that input values arewithin the operating range of TMUs. Secondly and veryconveniently, these limiters also ensure that car navigationwill be governed by appropriate rules even when x and yappear to be outside of the scope of original TMUs.Scaling factors applied to the same inputs, on the otherhand, allow us to adjust the TMUs to the substantiallyincreased dimensions of the driving system (compared tothe single truck to which the TMUs have been originallyoptimized for) and with each other. Exactly the samereason calls for the corrective measure of -16 that will bededucted from the value of y fed to the upper TMU – itvirtually shifts the rules of the TMU higher so that there isenough space for the u-turn the car is required to makewhen returning from the negative half-plane.Secondly, fuzzy systems in Figs 2 and 3 are constructedunder the assumption that x f = 0, y f = 0, Φ f = 90°. In orderto save ourselves from redesigning these blocks each timethe position and/or the orientation of the loading dock ischanged, an interface is built between the actual values ofx, y, Φ r4 and those that will be used for the computations inthe subsystem in Fig. 6 (which are denoted by x’, y’ andΦ’ r4 in Fig. 7).The values of x’ and y’ are obtained using the followingformulas:'ox = r cos(90 −Φf+ Φ4), (1)'oy = r sin(90 −Φf+ Φ4)wherer22= ( x − xf) + ( y − yf) . (2)Setpoint value corresponding to the given destination (x f ,y f , Φ f ) is computed by'oΦr 4= Φr4+ Φf− 90 . (3)In result, through all these adjustments, the TMU blockin Fig. 5 is updated to the current driving goal and physicalcharacteristics of the controlled object and, in a manner ofspeaking, generates a virtual force field in the driving areaas depicted in Fig. 8.

140JC2120Φ r3 - Φ 32.0JC -1.2Φ r110050y80Φ r10-5060-50 0 50Φ r3 - Φ 3Figure 8. "Force field", generated by TMUScaling factors come also very handy for adjusting thejoint controllers. It can be deducted that for minimizing thedifference between Φ 4 and Φ r4 we need a negative angle ofΦ 3 if Φ r4 - Φ 4 is positive (and vice versa), thus the functionperformed by the joint controller 1 should bemonotonously decreasing. This can be accomplished by anegative scaling factor on the respective controller output(Fig. 9). The same applies for the second joint controller -for minimizing the difference between Φ 3 and Φ r3 we needa negative angle of Φ 1 if Φ r3 - Φ 3 is positive (and viceversa). The absolute values of scaling factors, on the otherhand, enable us to impose limitations on Φ 1 and Φ 3 andspecify sensitivity of joint controllers. It turns out that thefirst joint should be less receptive to control actions thatpropagate from the wheels of the cab and thus requireslower values for both scaling factors. In fact, appropriateselection of the output scaling factor value of first jointcontroller is critical to the overall control performance –even a small decrease e.g. a value of -0.8 leads toproportionally higher maximum values of Φ 3 that make thesystem unstable and jeopardize the control goal.Φ r4 - Φ 4Φ r340200-20-60 -40 -20 0 20 40JC10.8JC -0.5-50 0 50Φ r4 - Φ 4Figure 9. Joint controller No. 1xΦ r3Figure 10. Joint controller No. 2Finally, the block that translates the difference betweenΦ r1 and Φ 1 into appropriate steering angle θ is in thisapplication yet another scaling block (or a P-controllerwith K p = -3). The latter has a negative sign because weneed negative steering angle if Φ r1 - Φ 1 > 0 (and viceversa). Note that direct linear transformation is valid in therange [-23.3°, 23.3°] of Φ r1 - Φ 1 because of the physicalrestriction on θ.V RESULTSThis section presents the backing results from 10 randomlyselected initial positions (Table I) to the loading docksituated at (x f = 0, y f = 70) with Φ f = 120°. In backing upthe truck and trailer systems there is one particularlyunfortunate position well known to drivers calledjackknife, which means that either Φ 3 or Φ 1 equals -90° or90°. In jackknife position car starts moving along a circle,never leaving it, never arriving to the loading dockand.thus becoming uncontrollable car. Since the jointcontrollers keep the values of Φ 1 and Φ 3 well under-90°/90°, it follows that whether the car goes to jackknifeor not, depends on the initial values of Φ 1 and Φ 3 only.With the help of selected simulations it was found outexperimentally that car goes jackknife if the absolutevalues of Φ 1 and Φ 3 both exceed 60° (if they have the samesign) or 50° (if they have different signs) and as aprecaution, in initial conditions the absolute values of Φ 1and Φ 3 are therefore kept below 45 degrees. Backingtrajectories (for the sake of the clarity of the illustration,only the trajectory of the second trailer is drawn at each20 th sampling step) from these initial positions are depictedin Fig. 11. Backing quality is evaluated in terms of backingerrors (a weighted sum of distance and orientation errors atthe loading dock), computed byε = ε d+ 0. 0267ε Φ, (4)where⎪⎧2εd= ( xf− x(Tf)) + ( y⎨⎪⎩ εΦ= abs( Φf−Φ4( Tf))f− y(T ))f2, (5)where T f is the duration of the backing. These backingerrors for all 10 test drives are visualized in Fig. 12.

One must note that at current backing speed (1m/s) caradvances 10cm within one sampling interval, and thisuncertainty alone may be the reason why backing error isas high as 0.1 even if the car arrives at the loading dock atperfect angle. As it turns out, however, actual error valuesremain the within the range [0.1, 0.7]. When we recall thattotal length of the driving system is 12m, such inaccuracyis really marginal, besides, the backing trajectories appearto be very smooth. In order to demonstrate that suchcontrol quality is accomplished with minimal controlenergy, control action θ along with the dynamics of Φ 1 andΦ 3 for the one of the curviest backing trajectories (driveNo. 4) are depicted in Fig. 13.TABLE ITEST DRIVE INITIAL CONDITIONSNo. x(0) y(0) Φ 0 (0) Φ 2 (0) Φ 4 (0)1 28.3 124.9 115.2 145.7 103.22 18.1 21.8 345.6 357.2 340.43 29.4 138.8 268.0 277.7 294.24 -44.1 114.7 96.5 108.2 71.55 10.3 72.7 158.4 146.7 104.96 -45.0 79.8 336.0 342.8 352.97 -8.5 45.7 246.0 241.7 251.48 46.7 61.5 162.2 153.3 184.39 37.0 38.7 324.6 335.8 346.710 -49.0 43.0 2.0 23.0 43.8Φ 1,Φ 3θε0.060.040.02Figure 12. The backing results in terms of final positioning errors6040200-20-40Φ 1Φ 3-600 10 20 30 40 50t60 70 80 90806040200-20-4001 2 3 4 5 6 7 8 9 10drive No.-600 10 20 30 40 50 60 70 80 90tFigure 13. Angles Φ 1 and Φ 3 (above) and steering angle (below) duringthe backing (drive No. 4)14041 31201005y 8068604010dock97220-80 -60 -40 -20 0 20 40 60xFigure 11. The backing trajectoriesThough the control system is optimized for and theexperiments carried out at the fixed backing speed of 1m/s, it is capable of controlling the car at considerablyhigher speeds. It can be observed though that controlbecomes gradually less stable as speed increases. First sucheffects become apparent if speed reaches 5m/s and thoughthe car is still able to follow the trajectory to the loadingdock, the action at the steering wheel becomes more andmore frantic from this point. The system becomescompletely uncontrollable if speed exceeds 10m/s. It is,however, highly doubtful if anyone has ever witnessed atwo-trailer truck storming backwards toward the loadingdock at such speeds in real life.VI CONCLUSIONSIn this paper we described a control system for the truckand two-trailer system, fully capable of steering the carfrom a virtually arbitrary initial position to the prespecifiedloading dock. Problem decomposition leads to ahierarchical control system that focuses on trajectorymanagement and subsequent trajectory-drivenmanipulation of trailer angles. These principal tasks arecarried out by very simple functional blocks implementedby the means of fuzzy logic, whereas additionalcomponents such as coordinate transformation blocks andscaling factors make the system easily reconfigurable and

extendable if the need arises. The main benefit fromproblem decomposition, however, is that it allows us todeal with resulting smaller problems individually thatultimately leads to the transparent control system withgood control performance and no loss in functionality.Fuzzy systems similarly benefit from having a smallnumber of input variables that keeps the number of fuzzyrules at a reasonably low level that is especially usefulwhen tuning the subsystems for best performance.The results strongly suggest that the control approachadopted in this paper could be further developed tofacilitate the control of trucks with even more trailers.[10] A. Riid and E. Rüstern, "Fuzzy logic in control: truckbacker-upper problem revisited," Proc. 10th IEEEInternational Conference on Fuzzy Systems,Melbourne, Vol. 1, 2001, pp. 513-516.[11] A. Riid and E. Rüstern, "Fuzzy Hierarchical Controlof Truck and Trailer," Proc. 8th Biennal BalticElectronic Conf., Tallinn, Estonia, 2002, pp. 141-144.[12] A. Riid and E. Rüstern, "Car Navigation andCollision Avoidance System with Fuzzy Logic",Proc IEEE International Conference on FuzzySystems, Budapest, Vol. 3, 2004, pp. 1443-1448.VII ACKNOWLEDGEMENTThis work was partially supported by the Estonian ScienceFoundation grant No. 6837.VIII REFERENCES[1] D. Nguyen and B. Widrow, "The truck backer-upper:An example of self-learning in neural network,"IEEE Contr. Syst. Mag., Vol. 10, No. 2, 1990, pp. 18-23[2] K. Tanaka, T. Taniguchi and H.O. Wang, "ModelbasedFuzzy Control for Two-Trailers Problem:Stability Analysis and Design via Linear MatrixInequalities," Proc. 6th IEEE Int. Conf. Fuzzy Syst.,Barcelona, Spain, 1997, pp. 343-348[3] K. Tanaka, S. Hori and H.O.Wang, "MultiobjectiveControl of a Vehicle with Triple Trailers,"IEEE/ASME Trans. Mechatronics, Vol. 7, No. 3,2002, pp. 357-368[4] K. Tanaka, T. Kosaki, H.O. Wang, "Backing ControlProblem of a Mobile Robot with Multiple Trailers:Fuzzy Modeling and LMI-Based Design," IEEETrans. on Systems, Man and Cybernetics, Part C,Vol. 28, No. 3, 1998, pp. 329-337[5] B. Wang, H. Kinjo, K. Nakazono and T. Yamamoto,"Design of Backward Movement Control for a TruckSystem with Two Trailers Using NeurocontrollersEvolved by Genetic Algorithms," Transactions of theInstitute of Electrical Engineers of Japan, Vol. 123,No. 5, 2003, pp.983-990[6] E. Muhando, H. Kinjo, E. Uezato and T. Yamamoto"Hybrid Controller of Neural Network and LinearRegulator for Multi-trailer Systems Optimized byGenetic Algorithms," Proc. Int. Conf. Control,Automation and Systems, Gyeong Gi, Korea, 2005, 6pp.[7] D. F. Hougen, M. Gini, and J. Slagle, "RapidUnsupervised Connectionist Learning for Backing aRobot with Two Trailers", Proc. IEEE InternationalConference on Robotics and Automation, 1997, pp.2950-2955[8] B. Widrow and M.M. Lamego, "Neurointerfaces,"IEEE Transactions on Control Systems Technology,Vol. 10, No. 2, 2002, pp. 221-228[9] K. Alton and M. van de Pamme, "Learning to Steeron Winding Tracks Using Semi-Parametric ControlPolicies," Proc. International Conference on Roboticsand Automation, Barcelona, Spain, 2005, 6 pp.