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    Fuzzy Control for Platoons of Smart CarsJulie Dickerson,HyunMunKim. and Bart Kosko

    Signaland ImageprocessingInstituteUnivmity of Southem CaliforniaLosAngeles, California 90089-2564

    Department of Electlid Engineeringsystem

    Abstract-An addi tive fuzzy system cancontrol the throttle of cars in single laneplatoons. The system uses fuzzy contro llersfor velocity control and gap control. Fuzzycontrol lers create, maintain, and divideplatoon s on the highway. Each carscontroller uses data fro m its car and the car infront of it. Car s dro p back during platoonmaneuvers to avoid the slinky effect oftightly coupled platoons. Some car and enginetypes need their own fuzzy rules and sets. Ahybrid neural fuzzy system can learn the fuzzyrules and sets from input-output data. Wecompute new fuzzy rules and sets for a truckvelocity controller. The learne d systemcontrols the velocity of the truck with noovershoot or slow response. We tested thefuzzy gap controller first with a car model andthen with a real car on highway 1-15 inCalifornia. Each cars control ler used dataonly from sensors on the car. We used thiscontroller to drive the smart car on thehighway in a two-car platoon.

    I. PLATOONSOFSMARTCARSTraffic clogs highways around the world.Platoons of cars can increase the flow and mean speed

    on freeways. Platoons are high-speed groups of smartcars in single lanes on freeways of the future.Electronic links tie the cars together. Computer controlspeeds response times to road hazards so that cars cantravel more safely on their own or in groups.A platoon is a group of carswith a lead carandone or more follower cars that travel in the same lane.The lead car plans the course for the platoon. It picksthevelocity and car spacing and picks which maneuversto perform. Platoons use four maneuvers: merge,split, velocity change, and lane change [5]. A mergecombines two platoons into one. A split splits oneplatoon into two. A lane change moves a single carinto an adjacent lane. combinations help cars movethrough traffic.Standard control systems use an input-outputmath model of the car and its environment. Fuzzysystemsdonot use an input-outputmath model or exactcar parameters. A fuzzy system is a set of fuzzy rulesthat maps inputs to outputs [4]. The fuzzy platoon

    0-7803-1896-X/94 $4.00 01994 EEE

    controller uses rules that act like the skills of a humandriver. The rules have the form If input conditionshold to some degree, then output conditions hold tosome degree or If x is A, then y isB for fuzzy setsA andB. Each fuzzy rule defines a fuzzy patch or aCartesian product AxB in the input-output state spaceXxY. To approximate the function the fuzzy systemcovers its graph with fuzzy patchesand averages patchesthatoverlap [3].The fuzzy platoon controller (FPC) is adistributed control system for future freeways that drivesa car in or out of a platoon. The FPC includes anintegrated maneuver controller (IMC) for courseselection and an individual vehicle controller (IVC) forthrottle, brake, and steering control as in Figure 1.We designed a fuzzy throttle controller (FTC)for velocity and gap control in smart platoons [l]. Thecontroller uses the throttle only and has thre subsystemsfor control of velocity and gap distance as in Figure 2.The FTC gets information from its own sensors, the carahead,and from the platoon goals. We tested the fuzzygap controller on the highway. The controller gotinformation from its own sensor and from the platoongoals. The controlled car followed the lead car as itchanged speed and went over hills. The system

    Platoons travel at high speedsand need preciselongitudinal control for safety. We used a hybridneural-fuzzy system that finds the ellipsoidal rule-performed smmthly in all cases.

    PlatoonGoals

    Integrated MawuveiController

    Course Selection- Merge- Velocity Change- Lane Change- split

    IntegratedVehicle

    Controller- Throttle- Brakes- steering

    Z S e n s o r Data7Data from Car AheadFigure 1. Block diagram of the fuzzy platoon leadersystem.1632

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    - ControllerAcceleration, Caror

    II.ADDITIVEFUZZY SYSTEMSWITH ELLPSOIDM..RULESAdditive fuzzy systems can uniformlyapproximate continuous [4] or measurable [5] functionswith fuzzy rule patches. It gives a model-free estimate

    of a continuous function since it does not us e an input-output math model of the function. A fuzzy systemcontains a set of rules of the form If input conditionshold, then output conditions hold or If X is A, then Yis B for fuzzy sets A and B. Each fuzzy rule defines afuzzy patch or a Cartesian product AxB as in Figure3.Toapproximate the function the fuzzy system covers itsgraph with fuzzy patches and averages patches thatoverlap. Centroidal defuzzification with correlationproduct inference [41 gives the output value yk at timek:

    II

    A1Figure 3. The fuzzy rule patch If X is fuzzy set AI, thenY is fuzzy set B1 is the fuzzy Cartesian product AlxBlin the input-output product space XxY.

    Figure 4. A positive definite matrix A defines an ellipsoidabout the center m of the ellipsoid. The eigenvalues ofA define the length of the axes. The projections of theellipsoid onto the axes define the input and output fuzzysets.

    Area (I$)Centroid( E ; )i = l

    Area(Ej)j= 1

    9 is the area of the jth output set. cyj is the centroid ofthejth output set. mBj(xk) scales the output set Bj . r isthe number of output fuzzy sets Ej.

    A fuzzy patch can take the form of an ellipsoid[2]. A positive-definitematrix A defines an ellipsoid inthe q-dimensional input-output state space, whereq = n + p (Figure 4), n is the number of inputs to thefuzzy system, and p is the number of outputs. Theellipsoid is the locus of all z that satisfy [7]

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    a2 =(z-m)TA(z-m) (2)where ais a positive realnumber and m is the center ofthe ellipsoid in Rq. The eigenvalues of A are Ai, . .,Aq. The eigenvectors define the ellipsoid axes. TheEuclidean half-lengths of the axes equala/,&, ...,a/&. The projections of the rulesonto the input axes form the fuzzy sets.

    III. m Y T H R N T R 0 ~In a platoon each car tries to travel at thedesired platoon velocity and maintain a fixed gapbetween cars. The IMC system in the leader carchooses the desired platoon velocity and gaps between

    cars. When the platoon travels at a constant velocityeach car uses its own velocity controller to maintainthe desired platoon velocity. These systems use thevelocity and acceleration data of the controlled car.Each carmeasures thevelocity and acceleration dataofthe car in front of it. The system output is thechange in throttle angle.The fuzzy throttle controller (FTC) performslongitudinal maneuvers for the platoon. Separate fuzzysubsystems for velocity and gap control change thethrottle angle for cars in the platoon as in Figure 2.The velocity controller controls the speed of the platoonleader. The gap controller controls splits, merges, andchanges in spacing for follower cars. The subsystemswork together to control the platoon.The gap controller corrects the distance errorwhen it is too large. The gap controller for platoonfollowers uses the differences in acceleration andvelocity between cars and the distanceefco~toachieve ormaintain a constant spacing. The distance errorM&t)is the difference between the desired gap between thecarsand the actual gap. A range-finding system on eachcar in the platoon measures the distance between thecars. Equations 3, 4, and 5 give the inputs for the ithcar [l]:

    (3)

    In this analysis the follower cars only get datafrom their own sensors. The sensor measures thedistanceand the velocity difference between a carand thecar in front of it. We estimate the acceleration input in(5)with the d iffmce of the velocity measurements:

    ThrottleAngle

    DrivetrainrsyiTransmissionI + TransmissionTorqueBrakeTorque

    Aerodynamics+ DrivetrainCarMassAcceleration speed

    Figure 5. Longitudinal car model block diagram.[6]

    AUi(tk)= sign(AVi(tk)- AVj(tk-1)) c a(6)The accelerationmeasurements are noisy so we use onlythe scaled sign of the acceleration. c, equals the inverse

    of the system update rate zS- l . We used zS= 0.05.wcmds.The output of the fuzzy controller in the ith caris AQi. We use a low pass filter to smooth the throttleinput to the car [9]. The input to the low pass filter Oi(tk) is

    eiLP (tk-1) is the previous output of the filter. Thesmoothed throttle input is

    We stored the fuzzy controller as a look-up table basedon the fuzzy setsand rules. We scale the outputofthelook-up table by In. Scalingdecreasesmunddf errorssince the microprocessor in the car only uses integeroperations.

    IV GAPTROLTEWWe tested the gap controller on highway 1-15in Escondido, California. First we tested smallplatoons with a realisticcarmodel [6].Then we put thecontroller in a real car from V O W Incorporated. In

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    ActualDiatana(ft)0 10 20 30 40

    Time(s)(a)

    10 1

    i a a i n g ~ a l ~(IVS)-100 10 20 30 40Time (s)@)80

    ThrMlk (")-

    20I00 10 20 30 40Time (s)

    (C)

    - 1

    -61 . . I0 6 12 18 24 30Time (I)

    @)-" I -(9 I

    0 10 20 30Time (s)(C)

    Figure 6. Follower car data as a two-car platoon Figure 7. Follower car data as a two-car platoon climbsaccelerates on the highway . (a) The distance between a hill. (a) The distance between the lead and followe rthe lead and follower cars. (b) Closing rate of the cars. (b) Closing rate of the velocity differencesvelocity differences between the cars. (c) Throttle between the cars. (c) Throttle values for the followervalues for the follower cars. cars.this test the follower cars got data from only theirsensors. Figure 5shows the basic subsystems of thevalidated car model that we used to test our controller.The inputs to this model are the throttle angle and thebrake torque. The outputs are the speed and accelerationof the car. This study looks only at throttle control.We set the brake torque to zero. The system has amechanical delay of 0.25 seconds.We used a pulse-doppler radar system. Theradar measures the distance and the velocity differencebetween the computer controlled car and the car ahead.The radar has a measurement delay of 0.05 seconds.The radar locks on a targeton a car ahead.We tested the gap controller in a two-carplatoon on highway 1-15. The desired gap between thecars was 125 feet. Figure 6 shows the platoon as itaccelerates onto the highway. The cars started with aseparation of 10 feet. The fuzzy controller started whenthe follower car reached 25 miles per hour. The cruisecontrol does not work below this speed. The platoon

    accelerated to 55 miles per hour in 20 seconds. Figure6 a shows the follower car gap as the platoonaccelerates. Figure 6 b shows the closing rate betweenthe cars. Figure 6 c shows the throttle value as the caraccelerates. The follower car slowly falls back to thedesired gap as the platoon reaches the desired velocity.The follower car overshot the desired gap and thenconverged to the proper spot.In the second test the platoon went up anddown hills. The desired gap is 125 feet. The followercar drops back as the platoon starts up the hill. Figure7 a shows the gap distance as the platoon goes up ahill. The follower drops back and then moves to theproper gap. Figure 7 b shows the closing rate betweenthe lead and follower cars. The "spike" at 15 secondsshows the radar sensor losing the lead car in theplatoon. When this happens the follower car maintainsa constant throttle until the sensor detects a new target.Figure 7 c shows the throttle values as the car ascendsthe hill.

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    We tested a hybrid system that combinedunsupervised and supervised learning. The hybridsystem used unsupervised learning to quickly pick thef i t set of ellipsoidal fuzzy rules-it picks them inshape, orientation, and number. Then supervisedlearning tuned the rules. The fmt choice of ellipsoidparameters dictates how well the supervised systemperfonns and how fast it converges.A . Unsupervised Rule Estimation

    First and second order statistics of the data canestimate the fuzzy patches as ellipsoids. Vectorquantizerscan use competitive learning to estimate thelocal conditional covariancematrix for each pattern class[4]. The covariancematrix defmes an ellipsoid in the q-dimensional input-output space centered at thequantization vector or centroid whereq is the combinednumber of inputs n and outputs p of the fuzzy system.Regions of sparse data lead to large ellipsoids and thusless certain rules. Thequantizing vector hopsmore asitmatchesor quantizes more diversedataAdaptive vector quantization (AVQ) systemsadaptively cluster pattem data in a state space. Anautoassociative AVQ system combines the input x andthe output y of the data to form zT=[xT I yT]. The jthneuron wins if z is closest in Euclidean distance.Competitive learning estimates the first andsecond order statistics of the data with the stochasticdifferenceequationsfor thewinning neuron 141

    The coefficientsck and dk must satisfy the convergenceconditions in [4]. In practice they decrease linearly intime. The initial choices of m/(O) should equal sampledata z.B. Supervised E llipsoidal Learning

    Supervised learning can also find and tune theellipsoidal fuzzy patches. The backpropagationalgorithm[4] learns the ellipsoidal patches by locallyminimizing the mean-squared error function. Theerror is the desired output minus the output of theadditive fuzzy system. It performs stochastic gradientdescent on the instantaneousmean-squared errorSEk [41:

    dk is the desired output of the system. yk(xk) is theoutput of the additive fuzzy system at time k for theinput xk in (1). Gradient descent estimates theeigenvalues, rotation angles, and centroids of theellipsoidal patches [2]. The partial derivatives andupdateequationsare found in [21.

    VI. LEARNINGOFFUZZYRULESWe learned the fuzzy rules for the leadervelocity controller using the car model in [8]. Theleader velocity controller in [l]generated 7500 trainingsamples in 200 mjectories for a sport utility vehicle.The training vectors (U , AV, A@ defined points in the

    three dimensional input-output space. Unsupervisedellipsoid covariance learning clustered the data andcomputed its local statistics. The AVQ system had450synaptic vectors or local pattem classes. The sum ofthe ellipsoid projections onto each axis of the statespace gave a histogram of the density of the pattemclasses. We chose seven sets in each of the inputdimensions. The center of each fuzzy rule setmatchedapeak in the histogram.We then partitioned the state space into a gridof possible rule patches. We found the rules bycounting the number of synaptic vectors in each cell.Clusters of synaptic vectors in fuzzy rule cell definedthe rules [4]. The sets for the truck are larger andcoarserthan those for the smaller and lighter cars.Next we used supervised learning to tune therules. We optimized the rules to minimize mean-squared error for the trainingdata. There were 49 rules.The supervised system took 30,000iterations to refmethedata. Figure 8 shows the results of changing theplatoon velocity for the truck using the unsupervised

    -HybridLeamingY I Y L . Unsupervised

    140 20lo Time (s) 30Figure8. Comparison of the velocity controllerperformance after unsupervised and hybrid learning.The hybrid controller gives a quick response with noovers hoot.

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    and hybrid controllers. Both controllers had noovershoot when the car sped up. The hybrid controllerperformedbetter when thecar slowed down. The hybridcontrollerhad no overshootat the desired speed.

    VII. CONCLUSIONWe designed and tested a fuzzy throttlecontroller for a car platoon. The next phase of the FPLwill add the brake and steering controllersso the platoon

    canmaneuver on the highway.This controller worked well for coupledsystems where a series of objects must track and predictthe object in front of it. Networks of these controllerscould control the rate of message or car traffic flowthrough electronic and physical intersections. Thecoupled systems can differ. The distributed structure ofthe fuzzy controller could apply to factory assemblylines or to robotic limb control.Unsupervised ellipsoidal learning tuned thefuzzy rules and sets forcafs of different sizesandenginetypes. This gives a new way to find a fuzzy systemusing only data from a human driver or other controller.Ellipsoidal learning works for any control system wheninput-output data is available as in the control ofbiological or economic processes. Future research willcompare how well supervised and unsupervised learningfind the optimal ellipsoids and rules for fuzzy systems.On-line adaptive fuzzy control systems with ellipsoidallearning can adapt the system over time as engineparameterschange.ACKNOWLEDGMENT

    The Caltrans PATH Program supported thisresearch (Agreement #20695 MB). VORADIncorDoratedmovided thecarand radar sensor for the gapcon&ller test. The authors thank Jim RowlandJohn Olds of VORAD for their help in testingcontroller.

    andthis

    REFERENCES[l ] Dickerson, J.A., Kosko, B., Ellipsoidal Learningand Fuzzy Throttle Control for Platoons of SmartCars, in Fuzzy Sets, Neural Networks, and SoftComputing, edited by R.R. Yager, L. Zadeh, VanNostrand Reinhold, 1994.[2 ] Dickerson, J.A., Kosko, B., Fuzzy FunctionApproximation with Supervised Ellipsoidal Learning,World Congress on Neural Networks (WCNN 93AVolume 11,9-17, 1993.[3] Kosko, B., FUZZYSystems as UniversalApproximators, IEEE Transactions on Computers,1993;an early version appears in Proceedings of the 1stIEEE International Conference on FuzzySystems(FUZZ-IEEE FUZZ92), 1153-1162, March1992.[4] Kosko, B., Neural Networks and Fuzzy Systems,Prentice Hall, Englewood Cliffs, 1992.[5] Hsu, A., Eskafi, F., Sachs, S., Varaiya, P., TheDesign of Platoon Maneuver Protocols for IVHS,PATH Research Report, UCB-ITS-PRR-91-6, April20, 1991.[6 ]Ioannou, P., Xu, T., Throttle and Brake Control forVehicle Following, 32nd Control Design Conference(32nd CDC ), 1993.[71 Strang, G. , Linear Algebra and Its Applications,Second Edition, Academic Press, 1980.[8] Sheikholesam, S., Desoer, C.A., LongitudinalControl of a Platoon of Vehicles with NoCommunication of Lead Vehicle Information: ASystem Level Study, PATH Technical Memorandum,91-2,1991.[91 Kuo, B.C., Automatic Control Systems, FourthEdition, Prentice Hall, Englewood Cliffs, 1982.

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