modeling and control of electromagnetic brakes for enhanced braki

7/27/2019 Modeling and Control of Electromagnetic Brakes for Enhanced Braki

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Faculty Publications (ECE) Electrical & Computer Engineering

1-1-1997

Modeling and control of electromagnetic brakes forenhanced braking capabilities for automated

highway systemsMing QianPushkin KachrooUniversity of Nevada, Las Vegas, [email protected]

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Repository CitationM. Qian, and P. Kachroo, "Modeling and control of electromagnetic brakes for enhanced braking capabilities for automated highwaysystems,"IEEE Conference on Intelligent Transportation Systems, , pp. 391-396, January, 1997.

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MODELING AND CONTROL OF ELECTROMAGNETIC BRAKES FORENHANCED BRAKING CAPABILITIES FOR AUTOMATED HIGHWAYSYSTEMSMing QianVirginia Tech Center for Transportation Research1700 Kraft Drive, Suite 2000Blacksburg, VA 24061Ph.#(540)23 1-7509, [email protected] Pushkin KachrooBradley Department of Electrical and Computer EngineeringVirginia Tech, Blacksburg, VA 24061Ph.#(540)231-8340, [email protected] Keywords auxillary brakes, sliding mode controlABSTRACTIn automatic highway systems, a faster responseand a robust braking system are crucial part ofthe overall automatic control of the vehicle. Thispaper describes electromagnetic brakes as asupplementary system for regular frictionbrakes. This system provides better responsetime for emergency situations, and in generalkeeps the friction brake working longer andsafer.A modified mathematical model forelectromagnetic brakes is proposed to describetheir static characteristics (angular speed versusbrake torque). The performan ce of themodified m athematical model is better than th eother three models available in the literature.To control the brakes, a robust sliding modecontroller is designed to maintain the wheel slipat a given value. Simulations show that thecontroller designed is capable of controlling thevehicle with parameter deviations anddisturbances .INTRODUCTIONThe principle of braking in road vehiclesinvolves the conversion of kinetic energy intoheat. This high energy conversion dem ands alarge rate of heat dissipation so that stableperformanc e can be maintained. By using theelectromagnetic brake as supplementaryretardation equipment, the friction brakes can beused less frequently and therefore never reachhigh temperatures. Th e brake linings can have alonger life span, and the potential brake fadeproblem can be avoided. In this paper, a newmathematical model fo r electromagnetic brakesis proposed to describe their static characteristics(angular speed versus brake torque). Th eperformance of th e new mathematical model is

better than the other three models available inthe literature in a least-square sense. A robustsliding mode controller is designed that achieveswheel-slip control for vehicle motion. T heobjective of this brake control system is to keepthe wheel slip at an ideal value so that the tirecan still generate lateral and steering forces aswell as shorter stopping distances. The systemshows t h e nonlinearities and uncertainties.Hence, a nonlinear control strategy based o nsliding mode, which is a standard approach totackle the parametric and modeling uncertaintiesof a nonlinear system, is chosen for slip control.Simulation will be performed to confirm theeffectiveness of the controller.GENERAL PRINCIPLE OFELECTROMAGNETIC BRAKESThe conventional friction brake can absorb andconvert enormous energy values, but only on t h econdition that t h e temperature of the frictioncontact materials is controlled. Electromagneticbrakes work in a relatively cool condition an dsatisfy all the energy requirements of braking athigh speeds. Electromagnetic brakes can beapplied separately completely without the use offriction brakes. Due to their specific method ofinstallation, electromagn etic brakes can avoidproblems that friction brakes face as wementioned before. Typically, electromagn eticbrakes have been mounted in the transmissionl ine of vehicles [1,5]. Th e prop eller shaft isdivided and fitted with a sliding universal jointand is connected to t h e coupling flange on thebrake. The brake is fitted into the chassis of thevehicle by means of anti-vibration mounting.The working principle of the electric retarder isbased on the creation of eddy currents within ametal disc rotating between two electromagnets,

3910-7803-4269-0/97/S10.00 0 1998 IEEEAuthorized licensed use limited to: University of Nevada Las Vegas. Downloaded on April 14,2010 at 20:05:22 UTC from IEEE Xplore. Restrictions apply.
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which activate a force opposing the rotation o fthe disc. If the electromagn et is not energized ,the rotation of the disc is unaffected by theretarder and accelerates under the action of theweight to which its shaft is connec ted. When th eelectromagnet is energized, the rotation of thedisc is retarded and the energy absorbed isconverted into heating of the disc. A typicalretarder consists of a stator and a rotor. Th estator holds induction coils, energized separatelyin groups (e.g. four coils in a group). The statorassembly is supported through anti-vibrationmountings on the chassis frame of the vehicle.The rotor is made up of two discs, whichprovide the braking force when subject to theelectromagnetic influence when the coils areexcited [ 5 ] .It was found that electromagnetic brakes candevelop a negative power which representsnearly twice the maximum power output of atypical engine, and at least three times thebraking power of an exhaust brake [l] . Byusing t h e electromagnetic brake assupplementary retardation equipment, thefriction brakes can be used less frequently, andpractically never reach high temperatures. Th ebrake linings last considerably longer beforerequiring maintenance, and the potential "brakefade" problem can be avoided. In researchconducted by a truck manufacturer, it wasproven that the electromagnetic brake assumed80 percent of t h e duty which would otherwisehave been dema nded of the regular servicebrake [ l] . On the other hand, theelectromagnetic brake prevents the dangers thatcan arise from the overuse of friction brakesbeyond their capability to dissipate heat.MATHEMATICAL MODEL OFELECTROMAGNETIC BRAKESThe electromagnetic brake is a relativelyprimitive mechanism, yet it employs complexelectromagne tic and thermal phenom ena. As aresult, the calculation theory is mainly empirical.There are three models proposed in the literatureon eddy current brakes [2-41. Sm yth e'sapproach [2] is to treat the rotating part as a discof finite radius and obtain a closed-formsolution by means of a reflection procedurespecifically suited to the geometry of theproblem. Th e first step is to calculate themagnetic induction, B, produced by the eddycurrents induced in a rotating disk by a longright circular cylinder. After deriving t h e streamfunction, which is the current flowing throughany cross section of the rotating disk from a

point to its edge, the torque can be calcu latedby integrating the prod uct of the radialcomponent of the current by the magneticinduction and by the lever arm and integratingover the area of the pole piece. Sinc e there is ademagnetizing effect such that permeable polepieces of an electromagnet short-circuit the fluxof th e eddy current, the total flux in motionwould be

where $(, is the flux penetrating the rotatingdisk at rest, and p y o $ !R represents thedemagnetizing flux attained through dividingthe demagnetizing magnetomotive force by thereluctance of the electromagne t. Th e finalintegration result of the brake torque is:

2 2 2

2 22 2 2 2

oYR $0 D (2)T = ay@D=whereT = brake torqueo = angular velocity@(, = flux penetrating the rotating disk at restD = constant coefficient, depending on polearrangementR =p = constant coefficienty=lO-'//p, where p is the volume resistivity of thedisk

( R + P Y o 1

reluctance of the electrom agnet

This model is good at low speed but decreasesto o fast in high speed compared with theexperimental curve (see Figure 1). T h easymptotic behavior shows a fall-off of thetorque more rapid than o -l in the high speedregion, which is in contradiction withexperimental results. Smythe pointed out thatthis behavior could be due to other conditions,such as the degree of saturation of the iron inthe magnet which will upset the assumedrelations between magnetomotive force and f l u x( 4 ) and may modify equations (1 ) an d (2).Schieber adapted a general method of solutionto a rotating system which is different fromSmythe's approach [ 3 ] . The result is for low-speed only:

A

1 ( 3 )1 2 2 2 ( R / a ) L2 11- m / a ) 2 } 2T = -~ o 6 o 7 c R m B [ l -where(3 = electrical conductivity

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6 = sheet thicknesso = angular velocity.n = coefficientR = radius of electromagnetm = distance of disc axis from pole-face centera = disk radiusBz = z component of magnetic flux densityBased on the works of Schieber andSmythe, Wouterse tried to find the globalsolution for the high-speed region as well as thelow-speed region [4]. Wouterse proposed thefollowing expression for low speed:

1 7 t 2 24 PFe=--D dBocv

where Fe is the bciking fo k e and Y is the speed.The other variables are parameters that can beevaluated based on different types of eddycurrent brakes. The formula completely agreeswith Sm ythe s result in the low-speed regio n.Wouterses study on the air gap magnetic fieldat different speeds produced three remark ablep h e n o m e n a :- At very low speeds, the field differs onlyslightly from the field at zero speed.- At the speed at which the maximum draggingforce is exerted, the mean induction under thepole is already significantly less than B ,,.- At higher speeds, the magnetic induction tendsto further decrease.Based on this observation, Wouterse prop osedthe following solution at the high speed region:F e ( v ) = Fe with2v k-+-

wherep = specific resistance of disc materiald = disc thicknessD = diameter of soft iron pole, for noncircularpole shape D denotes the diameter of the circlewith the same area as pole face6 = ratio of zone width, in asymptotic currentdistribution around poles, to air gapc = ratio of total contour resistance to resistanceof contour part under polev = tangential speed, measured at center of pole

v, = critical speedB(,:= air gap induction at zero speedx = air gap between pole faces including discthickness or coordinate perpendicular to air gapR = distance from center of disc to center ofpoleWouterse also made use of another knownphenomenon of the high-speed region in hisproposal: the drag force becomes proportionalto V-. The model turns out to be much closerto the experimental result in the high-speedregion.While Wouterses model gives a global solutionwhich is good at high speed as well as at lowspeed, it must use two different expressions forthe low-speed and high-speed regions. From asimulation or control perspective, there aredifficulties involved in determining the criticalspeed or transitional region at which to split thelow- and high- speed region. As Woutersepointed out, the proportionality factor 6 inequation ( 5 ) is not exactly know n; It isestimated to have a value of about u nity. T he10-20% estimate error of 5would cause about a10% error in equation ( 5 ) . A uniform model isneeded to represent the function at both regionsin one expression and reduce the estimationerror further.Our approach is to modify Smy thes mo delaccording to Wo uterses observation. AsSmythe pointed out himself [2], his model givestoo rapid a falling off at high speeds because thedegree of saturation of the iron in the magnetupset many of his assumptions. To overcom ethis problem, we treat reluctance (R in formula(2)) as a function of speed instead of a constantfor representing the aggregate result of all thoseside effects that upset Smythes assumptions todeduct his formula. This aggregate effect canbe called reluctance effect. The expressionof reluctance should also reflect Woutersesobservations on the high-speed region:(a) The drag force becomes proportional tov-, and (b) The original magne tic induc tionunder the pole tends to be canceled by thecurrent induced around it in the disc. We fou ndthat to represent reluctance as

2=that satisfies th e above requirement.

c 1+c2w+c30 3is a good approximation1+c40

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Substituting this reluctance function in Smythesformula, we are proposing the followinguniform model which conforms to theexperimental values for the electromagneticbrake operation at low as well as high speed:

1+ k p k p 3 This model agrees with all the proposed modelsexcept for the high velocity range of Smythesformula, which is inaccurate anyway. Thismodel represents the correct behavior at highspeeds. Parameters kl -k 5 can be evaluated fo rspecific types of electromagnetic brakes.The models have been evaluated based on a datachart given by Omega T echnologies. The CC250 type of electromagnetic brake is chosen todo the evaluation. Since the model is nonlinearand five unknown parameters need to be solved,the least squares method is used to obtain thevalue of k l - k 5 .By applying the least squares method on the newproposed model, Smyth es model, and J.H.Wouterses model (at high speed), we obtain theresults shown in Figure 1. It can be seen that thenew mod el has better performance inapproximating the original curve in the least-squares sense. For simulation and controlpurposes of this paper, the new model can be

(least-square results)curve -- original data; dark curve --proposed model; thin lines -- Smythes modeland Wouterses model (Smythes model has ahigher peak than Wouterses model).

*

VEHICLE SYSTEM DYNAMICS [7]

The dynamic equation for the angular motion ofthe wheel isW w = [T, - Tb - RwFt - R w F w ] / Jw(7)where the variables are defined in [7]. The t iretractive (braking) force is given byTh e adhesion coefficient p[ ) is a function ofwheel slip. Wheel slip is defined as

(9)where oV V /I? is the vehicle angularvelocity which is defined as equal to the linearvehicle velocity, V, divided by the radius of thewheel. The variable is defined as

Ft = p ( h ) N v (8 )

h = ( o w -m v ) / 0 , o # 0W

In our simulation, the functionp[h)= is used for a nomin al curve,2pPhPh .h p + hwhere p an d h are the peak values. Thepeak value for the adhension coefficient mayhave values between 0.1 (icy road) and 0.9 (d ryasphalt and concrete).The dynamic equation fo r the vehicle motion isRefer to [7] for variable definitions. Usingequations (7) an d ( l l ) , and defining the statevariables as

P P

V = [NwFt - F v ] / M v . (1 1)

x = V / R (12)1 Wx = o w (13)

and denoting x = m a x ( x x ), we obtain2 17 2Xl = -f 1 1X ) + b l N p(h ) (14)X 2 = -f2 (x,) - 2 N p ( h ) + b3T (15 )Wheel slip is chosen as the controlled variablefor braking control algorithms because of itsstrong influence on the braking force betweenthe tire and the road. By controlling the wheelslip, we control the braking force to obtain thedesired output from the system. In order tocontrol the wheel slip, we can have systemdynamic equations in terms of wheel slip.During deceleration, condition x < x2 - 1(x # 0) is satisfied, and therefore wheel slip isdefined as: h = ( x - x ) / x and1 2 1 1



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gives the wheel slip dynamic equation fordeceleration . This equation is nonlinear a ndinvolves uncertainties in its parameters. Th enonlinear characteristics of the equation are dueto the following:-the relationship of wheel slip with velocitywhere velocity is nonlinear-the p-h relationship is nonlinear-there are multiplicative terms in the equation-functions f i ( x i ) an d f (X2) are nonlinear.2SLIDING MODE CONTROLLER DESIGNThe electromagnetic brake dynamics can bedescribed as a first-order system:T = ( K b P b - T ) / I; (1 7 )where T denotes th e braking torque generated,Kb denotes the brake gain or brake

effectiveness (e.g. Kb =O . 1 psi/m/s/s denotesthat for a pressure of loop si, one o btains adeceleration of 0.1 m/s/s). Electromagneticbrakes are different from regular brakes in thatbrake gain Kb is related to the speed due to itsstatic characteristic. pb is the brake pedalpressure and I; is the rise time characteristic ofthe brake dynamics. For regular friction brakes,I; usually has th e value of 150-200ms [6] wheretime delays are caused mainly by the timesneeded to fill the calipers, booster spring pre-loads, and the reaction washer hysteresis. Due todifferent working mechanism, the time delaysfor electromagnetic brakes come from solenoidresponse time and the response time of excitingcurrent. Generally, it can be assumed thatelectromagnetic brake have similar dynamictiming characteristics of regular friction brakes.By taking the derivative of (1 7) and substituting(18) as T, he dynamic equation can be writtenas :X = f + b . u where

-[(l+k)fi (xi )-fz(X2 )-[b2~ +(l+h)*biNl*~(h)+b3Tl*ilf =

4

For a second-order system in the form ofx( t ) = f (x , t ) + b(x , t ) u ( t )where b(x,t) is bounded as (20)O I b ( X 7 t ) I b ( x 7 t ) I b m a x ( x , t ) ,minit can be proven that the control lawu(t) = 6(x7t)-[6(t) - k(x , t ) sgn(~( t ) ) ]21)withsatisfies the sliding condition [7] whereb(x , t )=Jb (x , t )bmax(x , t ) i s the bestestimate of control gain,;(t) = --;(x, t) + X (t) -@(t) is the bestestimate of the equivalent control. For chatteringsgn(s) in (21) is replaced by s / $ inside theboundary where s is the sliding variable (errorfrom the desired wheel slip), and bo un da rythickness, Q , s taken to be 0.005.

kIx, t) 2 PIx , tV Ix , t ) + f l + IPIx,t) -1)FIt) I (22)..

mind

SIMULATION RESULTFigures 9-12 show the results of simulations forwhich the initial and desired values of the wheelslip are 0.02 and -0.12. Different roadconditions have been simulated by usi ngdifferent pp an d h p in functionp[h)= . Simulations are pe rform ed2P p hphp + hon dry concrete road ( p -0.8,h =0.2) andslippery road ( p -0.2, h -0.15) [7] to showP - P -extreme road conditions. Figure 9 shows thesimulation performed on a slippery road, andfigure 10 shows the simulation performed ondry concrete. We chose nominal road conditionfor the control design purposes such that the

values would be the average of thevalues for extreme conditions. Figure 11 showsthe simulation performed on a nominal road( pp=O.S,h -0.175). All figures show goodwheel slip tracking in spite of modeling errors inthe parameters (we assumed 10% estimationerror on b and f in the simulation ). Figure 12shows the simulation result on the vehicledecelerating along the dry concrete road at thefirst second, along th e slippery road at t h e nextsecond, and along the nominal road at th e third

P - P

an d hFP P

P -

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second. Every time the road condition changes,the controller output is quickly changed tocompensate. Based on the simulation result, thecontroller performance is satisfactory.

CONCLUSIONThe sliding mode controller gives satisfactoryresults for the application of electromagneticbrake control. By using electromagnetic brakes,the braking system has an increased capability toproduce the required torque.REFERENCES[ 1] Retarder Applications in CommercialVehicles Conference, London , 1974.[2] Smythe, W.R., O n Eddy Currents in aRotating Disk, AIEE Transaction, Vol. 61,[3] Schieber, D., Braking Torque on Rotating

Sheet in Stationary Magnetic Field, IEEProceedings, V01.121, No.2, Feb 19%[4] Wouterse, J.H., Critical Torque of E d d yCurrent Brake with Widely IEEEProceedings, Vol. 13 8, July 1991, pp.153 -1 5 8 .[6] Technical literature of electromagne ticbrakes, Omega T echnologies[7] Kachroo P. and Tomizuka, M., VehicleTraction Control and its Applications,Technical Report UIPRR-94-08, Universityof California at Berkeley, Institute ofTransportation, 1994.

1942, pp.681-684.

p p . 1 1 7 -1 2 2 .

[ 5 ] Separated Soft Iron Poles, -

modeling and control of electromagnetic brakes for enhanced braki

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