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Jun 30, 2018 - Choi, S.B.; Bang, J.H.; Cho, M.S.; Lee, Y.S. Sliding mode control for anti-lock ... Dugoff, H.; Fancher, P.S.; Segel, L. An Analysis of Tire Traction ...
actuators Article

Mathematical Simulations and Analyses of Proportional Electro-Hydraulic Brakes and Anti-Lock Braking Systems in Motorcycles Che-Pin Chen and Mao-Hsiung Chiang * Department of Engineering Science and Ocean Engineering, National Taiwan University, No. 1, Sec. 4, Roosevelt Rd., Taipei City 10617, Taiwan; [email protected] * Correspondence: [email protected]; Tel.: +886-2-3366-3730 Received: 21 May 2018; Accepted: 28 June 2018; Published: 30 June 2018

 

Abstract: In the motorcycle industry, the safety of motorcycles operating at high speeds has received increasing attention. If a motorcycle is equipped with an anti-lock braking system (ABS), it can automatically adjust the size of the brake force to prevent the wheels from locking and achieve an optimal braking effect, ensuring operation stability. In an ABS, the brake force is controlled by an electro-hydraulic brake (EHB). The control valve inside the EHB was replaced with a proportional valve in this study, which differed from the general use of a solenoid valve. The purpose for this change was to precisely control the brake force and prevent hydraulic pressure oscillating in the piping. This study employed MATLAB/Simulink and block diagrams to establish a complete motorcycle ABS simulation model, including a proportional electro-hydraulic brake (PEHB), motorcycle motion, tire, and controller models. In an analysis of ABS simulation results, when traveling on different road surfaces, the PEHB could effectively reduce braking distance and solve the problem of hydraulic pressure oscillation during braking. The research demonstrated that this proportional pressure control valve can substitute the general solenoid valve in commercial braking systems. This can assist the ABS in achieving more precise slip control and improved motorcycle safety. Keywords: slip control; anti-Lock braking system; proportional electro-hydraulic brake; proportional pressure control valve

1. Introduction The brake modules of modern motorcycles are generally equipped with an ABS to enable vehicles to stop within the shortest distance and motorcycle handles to be controlled to avoid obstacles and ensure driver safety. Brake performance is affected by the level of tire adhesion during the braking process. Tire adhesion refers to the acting force of the road surface on the tires and is also known as tire friction, which is divided into longitudinal and lateral directions. The sliding ratio between tires and the ground surface is defined as the slip [1–6]. S=

Vv − Vw Vv

(1)

where S is the brake slip, Vv is the vehicle speed, and Vw is the wheel speed. The curve in Figure 1 shows that optimal maneuverability was yielded when the slip was 0 (that is, when the vehicle speed was equal to the wheel speed) [7]. Conversely, the brake was locked if the slip was 1; thus, the wheel speed was 0 with no maneuverability. The highest point of longitudinal

Actuators 2018, 7, 34; doi:10.3390/act7030034

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longitudinal force Opboundary served asfor thethe boundary forunstable the stable and as unstable as well as braking force braking Op served as the stable and zones, well as zones, the target for the longitudinal forcemaneuverability. Op served as the boundary for the stable and unstable zones, as well as the target forbraking the optimal optimal maneuverability. the target for the optimal maneuverability.

1.0

S  Lateral-force coefficient Lateral-force coefficient  S Braking-force coefficient Braking-force coefficientHF HF

1.0

Stable

Unstable

Stable

Unstable

 HF  HF

Op

0.8

Op

0.8 0.6

0.6 0.4

S S

0.4 0.2

0.2

0 0

0.2

0.4

0.2

0.4

0.6

0.8

Brake 0.6Slip S 0.8 Brake Slip S

1.0 1.0

Figure 1. 1. Relationship Relationship between between tire tire adhesion adhesion coefficient coefficientand andbrake brakeslip slipcurve. curve. Figure Figure 1. Relationship between tire adhesion coefficient and brake slip curve.

When the rider presses the brake handle, the master cylinder generates brake pressure that is When the the rider rider presses presses the the brake handle, handle, the master master cylinder generates generates brake brake pressure pressure that that is is When transmitted to the brake caliperbrake of a wheel. Atthe this time, cylinder the wheel speed sensor is responsible for transmitted to the brake caliper of a wheel. At this time, the wheel speed sensor is responsible for transmitted to the brake caliper of asignals wheel.and At this time, the wheel speed sensor control is responsible for constantly measuring wheel speed providing these to the electronic unit (ECU) constantly measuring measuringwheel wheel speed speedsignals signalsand andproviding providingthese theseto tothe the electronic electronic control control unit unit (ECU) (ECU) constantly of the ABS. After calculations by the microcomputer inside the ECU, control signals are sent to the of the the ABS. After After calculations by by the the microcomputer microcomputer inside inside the the ECU, control control signals are are sent to to the the of EHB. ABS. The EHB iscalculations responsible for converting electrical controlECU, signals fromsignals the ECU sent into actual EHB. The EHB for converting electrical control signals from from the ECU actual control EHB. The EHBisof isresponsible responsible converting control the into ECU intoevent actual control actions the solenoid for valve to controlelectrical brake pressure insignals the wheel cylinder. In the of actions of the solenoid valve to control brake pressure in the wheel cylinder. In the event of wheels control of the the brake solenoid valve to brake in theIfwheel cylinder. In the event of wheelsactions locking, pressure is control controlled to pressure a fixed level. the wheel speed continues to locking,locking, the brake pressure is controlled to a fixed level. If the wheel speed continues to decrease, wheels the brake pressure is controlled to a fixed level. If the wheel speed continues to decrease, the brake pressure will also decrease. When the ECU has determined that the wheels have the brakethe pressure also decrease. When theWhen ECU has determined that the wheels resumed decrease, brake will pressure willincrease also decrease. the ECU determined thataction thehave wheels have resumed rotating, it will again the brake pressure to has perform a braking and repeat rotating, it will again increase the brake pressure to perform a braking action and repeat these actions resumed rotating, willvehicle again increase the brake to perform a braking andbetween repeat these actions untilitthe has stopped [8–10].pressure The control circuit of an EHBaction actuator until the vehicle has stopped [8–10]. The control circuit of an EHB actuator between masterbetween cylinder these actions until the vehicle has stopped [8–10]. The control circuit of an EHB actuator master cylinder and wheel cylinder is shown in Figure 2 [11]. and wheel cylinder shown in Figure 2 [11]. in Figure 2 [11]. master cylinder and is wheel cylinder is shown Master Cylinder Master Cylinder Pump Pump

Inlet Valve

EHB Actuator EHB Actuator Check Valve Check Valve

M Motor M Motor Accumulator

Inlet Valve

Accumulator

Orfice Orfice

Outlet Valve Outlet Valve

Module Module Pressure Pressure Up Up Hold Hold Down Down

Inlet Inlet Valve Valve OFF OFF ON ON ON ON

Outlet Outlet Valve Valve OFF OFF OFF OFF ON ON

Motor Motor Pump Pump ON ON ON ON ON ON

ECU ECU

Wheel Cylinder Wheel Cylinder Speed Sensor Speed Sensor

(a) (a)

(b) (b)

Figure 2. The control circuit of an EHB actuator: (a) structure; and (b) ABS control mode. Figure 2. The control circuit of an EHB actuator: (a) structure; and (b) ABS control mode. Figure 2. The control circuit of an EHB actuator: (a) structure; and (b) ABS control mode.

The general ABS control method drives the solenoid valve to increase, hold, and relief brake TheHowever, general ABS the solenoid valvecomponent to increase,that hold,can andberelief brake force. the control solenoidmethod valve drives is a discrete actuation controlled force. However, solenoid actuation component that of canthebeproportional controlled intermittently to the adjust brakingvalve forceisinaandiscrete approximate manner. The design intermittently to adjust braking force in an approximate manner. The design of the proportional

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The2018, general Actuators 7, 34 ABS control method drives the solenoid valve to increase, hold, and relief brake3 force. of 21 However, the solenoid valve is a discrete actuation component that can be controlled intermittently pressure control force valveininanthis study combined the design inlet and outlet valves into a three-port to adjust braking approximate manner. The of the proportional pressure control two-position solenoid valve, which wasand capable accurately rapidlytwo-position yielding the solenoid optimal valve in this study combined the inlet outletofvalves into aand three-port braking forcewas through theofcontinuous adjustment the valve opening.braking Figure force 3 shows a novel valve, which capable accurately and rapidly of yielding the optimal through the control circuit for a proportional electro-hydraulic brake a(PEHB) actuator of afor motorcycle [12]. continuous adjustment of the valve opening. Figure 3 shows novel control circuit a proportional Normal braking involves the rider pressing brake handle to Normal pressurize the master cylinder, and electro-hydraulic brake (PEHB) actuator of the a motorcycle [12]. braking involves the rider the pressure entershandle through the inlets of reversing valve valve and is pressing the brake to pressurize thethe master cylinder, andand the proportional pressure enters through theoutput inlets from the caliper to drive the wheel valve cylinder implement braking. proportional is of the reversing valve and proportional and to is output from the caliperThe to drive the wheel valve cylinder activated when the pressure is too high,valve and the inlet andwhen outletthe openings adjusted to to implement braking. The proportional is activated pressureare is too high, according and the inlet the level solenoid are force to achieve an appropriate pressure drop force in thetocaliper, yielding and outletofopenings adjusted according to the level of solenoid achievethereby an appropriate the optimal slip pressure drop in control. the caliper, thereby yielding the optimal slip control. Master cylinder B Group Circuit

A Group Circuit (5)

(5)

M (2)

(3)

(2)

(4)

(4)

(1)

(1)

Rear Front PEHB Wheel Wheel Actuator Brake Brake (1)Proportional Valve (2)Accumulator (3)Motor (4)Pump (5)Reversing Valve High Pressure Pipeline

Low Pressure Pipeline

Calipers Pressure Pipeline

Figure 3. The novel control circuit for a proportional electro-hydraulic brake (PEHB) actuator of Figure 3. The novel control circuit for a proportional electro-hydraulic brake (PEHB) actuator of a motorcycle. a motorcycle.

The The brake brake performance performance of of an an ABS ABS is is determined determined through through control control logic, logic, which which is is applied applied to to overcome numerous uncertain parameters such as the time-varying characteristics of braking overcome numerous uncertain parameters such as the time-varying characteristics ofdynamics braking as well as as environments, roads, and coefficients of friction. Various controlling strategies have dynamics well as environments, roads, and coefficients of friction. Various controlling strategies been proposed and confirmed to control wheel slip effectively. The relevant mechanisms include have been proposed and confirmed to control wheel slip effectively. The relevant mechanisms optimal controller [10,13], [10,13], fuzzy learning/logic controller [14–16], sliding modemode controller [17], [17], and include optimal controller fuzzy learning/logic controller [14–16], sliding controller proportional–integral–derivative (PID)(PID) control [18,19]. and proportional–integral–derivative control [18,19]. Of the relevant mechanisms, the PID controller is the the most most widely widely adopted adopted across across industries industries [20]. [20]. Of the relevant mechanisms, the PID controller is Various adjustment methods for PID have been proposed as described in [21]. Among them, Various adjustment methods for PID have been proposed as described in [21]. Among them, the the adjustment method for internal (IMC) was proposed in 1986 [22]. adjustment method for internal modelmodel controlcontrol (IMC) was proposed by RiverabyinRivera 1986 [22]. The main The main operating principle of IMC is through feedback compensation. In addition to being operating principle of IMC is through feedback compensation. In addition to being able to able to compensate for disturbances and eliminate the uncertainty of the approximate model, the compensate for disturbances and eliminate the uncertainty of the approximate model, the IMC can IMC can provide closed-loop stability and can also adjust to changes in the approximate model provide closed-loop stability and can also adjust to changes in the approximate model parameters. parameters. The modelapproach matchingwas approach was employed [23] to design directlythe design IMC according The model matching employed in [23] to in directly IMCthe according to the to the robust performance of the system. In the present study, a new type of brake actuator with robust performance of the system. In the present study, a new type of brake actuator with aa proportional developed. Compared withwith the general solenoid valve with on/off proportionalsolenoid solenoidvalve valvewas was developed. Compared the general solenoid valveanwith an control, a proportional solenoid valve involves the advantage of continuous pressure servo control. on/off control, a proportional solenoid valve involves the advantage of continuous pressure servo To commercialize this invention, a basic PIDacontroller was adopted,was and adopted, a bang-bang was control. To commercialize this invention, basic PID controller andcontroller a bang-bang installed in front of the integrator so that the system can achieve precise and rapid ABS slip control. controller was installed in front of the integrator so that the system can achieve precise and rapid ABS slip control.

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2. Proportional Pressure Control Valve The proportional pressure control valve consists of a proportional valve body and a 2. Proportional Pressure Control Valve proportional electromagnet. The proportional pressure control valve consists of a proportional valve body and a proportional electromagnet. 2.1. Proportional Valve Body The proportional valve body is composed of a proportional iron shell, a shuttle shaft with an 2.1. Proportional Valve Body iron core, and a valve body. Its design is shown in Figure 4 [24,25]. Outlet A and Inlet B are The proportional body is composed of caliper. a proportional iron shell, a shuttle with anvalves iron interconnected, and valve both are connected to the The openings of the spoolshaft and needle core, and a valve body. Its design is shown in Figure 4 [24,25]. Outlet A and Inlet B are interconnected, are adjusted through the shuttle shaft. The opening sizes of the two valve ports are proportional to and both are connected to the caliper. of the spool and needle valves are adjusted each other. The needle valve port The mustopenings be tightly closed when it is not actuated to avoidthrough internal the shuttle shaft. The opening sizes of the two valve ports are proportional to each other. The needle leakage. Therefore, the cone valve design was adopted, with the check valve in the shuttle shaft valve port must be tightly closed when it is not actuated to avoid internal leakage. Therefore, the cone being used as a safety device. In the event that the shuttle shaft cannot be reset and the needle valve valve design wasthe adopted, the check in released the shuttle shaft being used asvalve. a safety device. In shaft the malfunctions, caliperwith pressure can valve still be through the check The shuttle event that the shuttle shaft cannot be reset and the needle valve malfunctions, the caliper pressure can force (Fm) equation is as follows: still be released through the check valve. The shuttle shaft force (Fm ) equation is as follows: Fm = Fem + Fcal (2) Fm = Fem + Fcal (2) In addition, Fm = Pm  Asp , Fcal = Pcal  ( Asp − Aol )

InSubstituting addition, FmEquation = Pm · A(2) sp , Fcal = Pcal · ( Asp − Aol ) Substituting Equation (2) Pcal = ( Fm − Fem ) / ( Asp − Aol ) Pcal = ( Fm − Fem )/( Asp − Aol )

(3) (3)

where is the proportional solenoid cal is the caliper force; Pm is the master cylinder where FemFem is the proportional solenoid force;force; Fcal is F the caliper force; Pm is the master cylinder pressure; pressure; P cal is the caliper pressure; Asp is the spool valve cross-sectional area; and Aol is the needle Pcal is the caliper pressure; Asp is the spool valve cross-sectional area; and Aol is the needle valve hole valve hole cross-sectional area. cross-sectional area.

Fem

Valve Body Iron Core Shuttle Shaft Fm

Pm

Check Valve A

Spool Valve

Pcal

B Oil Filter

Needle Valve Fcal

Figure4.4.The Thestructure structureofofthe theproportional proportional valve body a PEHB. Figure valve body forfor a PEHB.

2.2. Proportional Electromagnet 2.2. Proportional Electromagnet The actuation principle of the proportional electromagnet is the same as the general solenoid The actuation principle of the proportional electromagnet is the same as the general solenoid valve: the operating current passing through the coil causes the internal soft magnetic components valve: the operating current passing through the coil causes the internal soft magnetic components to to magnetize and generate a magnetic effect, thereby achieving the objective of mechanical magnetize and generate a magnetic effect, thereby achieving the objective of mechanical movement movement due to electric excitation through energy conversion. Figure 5 shows the structure of the due to electric excitation through energy conversion. Figure 5 shows the structure of the proportional proportional electromagnet. The magnetic field direction and magnetic induction force are electromagnet. The magnetic field direction and magnetic induction force are controlled by the current controlled by the current in the input coil, in turn causing the materials, such as base and iron core, in the input coil, in turn causing the materials, such as base and iron core, to generate attraction due to

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generateattraction attractiondue duetotoexcitation. excitation.The Thedisplacement displacementofofthe the valveshaft shaftdetermines determinesthe theflow flow totogenerate excitation. Thedirection, displacement of the valve shaft determines the flowvalve volume and direction, whereas volume and whereas the spring provides the compressive force required for closing the volume and direction, whereas the spring the compressive forceport required for the spring provides thecurrent compressive force provides required forpurpose closing the valve when noclosing currentthe is valve port when no is supplied. The main of the brass ring is to change the valve port when no current is supplied. The main purpose of the brass ring is to change the supplied. The main purpose of the brass ring is tothem change the direction ofcore thethrough magneticthe field lines in directionof ofthe the magnetic field lines inorder order for enter theiron ironcore guide tube, direction magnetic field lines in for them toto enter the through the guide tube, order for them to enter the iron core through the guide tube, after which they are divided into two afterwhich whichthey theyare aredivided dividedinto intotwo twodirections, directions,entering enteringthe theflange flangeand andaxial axialcenter centerofofthe thebase base after directions, entering the flange and axial center of the base through the radial and axial directions, throughthe theradial radialand andaxial axialdirections, directions,respectively. respectively.The Thestopper stopperisisthe thecomponent componentcontrolling controllingthe the through respectively. The stopper isshaft the component controlling the movement of the shuttle shaft in the linear movement of the shuttle in the linear region of operation. movement of the shuttle shaft in the linear region of operation. region of operation.

Base Base

BrassRing Ring Brass

Coil Coil

GuideTube Tube Guide

Stopper Stopper

Shell Shell

Figure5.5.The Thestructure structureofofthe theproportional proportionalelectromagnet electromagnetfor foraaPEHB. PEHB. Figure Figure 5. The structure of the proportional electromagnet for a PEHB.

Six types types ofof simplified simplified magnetic magnetic flux flux paths paths (Figure (Figure 6)6) were were employed employed inin this thisstudy study as as the the Six design basis for the magnetic induction force, to deduce the direction for the magnetic field lines Six types of simplified magnetic flux paths (Figure 6) were employed in this study as the design design basis for the magnetic induction force, to deduce the direction for the magnetic field lines inin theair air gapmagnetic betweenthe themetals, metals, shown Table [26,27].Here Here basis forgap the induction force, to deduce the11direction for themagnetic field the linespermeability in the air gapof 0 represents the between shown ininTable [26,27]. 0 represents the permeability of between the metals, shown in Table 1 [26,27]. Here µ represents the permeability of air, g represents 0 air, gg represents represents the the air air gap, gap, rr represents represents the the iron iron core core radius, radius, hh represents represents the the right side side air, right the air gap, r represents the iron core radius, h represents the right side distance between iron core andz t distance between iron core and brass ring, represents the thickness of brass ring, and distance between iron core and brass ring, t represents the thickness of brass ring, and z brass ring, t represents the thickness brass ring, and z represents the distance from base to iron core. represents thedistance distance from basetotoof iron core. represents the from base iron core.

Figure Six simplified Figure6. Sixtypes typesof simplifiedmagnetic magneticflux fluxpaths. paths. Figure 6.6.Six types ofofsimplified magnetic flux paths.

Therefore, when a magnetic force is present and the magnetic flux is Φ, the magnetomotive force Therefore, when magneticforce forceisispresent presentand andthe themagnetic magneticflux fluxisis  themagnetomotive magnetomotive , ,the in theTherefore, air gap (Fair ) is asaafollows: when magnetic forceininthe theair airgap gap( (FFair) )isisas asfollows: follows: Fair = ΦPair −1 (4) force air

−1 be acquired by multiplying the magnetic In soft magnetic materials, the magnetomotive force −can 1 (4) air (4) FFairair==PP air field intensity (H) and the current path length (l). The magnetic field intensity can be determined In soft soft magnetic materials, the magnetomotive magnetomotive forceflux candensity be acquired acquired by multiplying multiplying the according to its corresponding relationship with the magnetic (B) through the B–H curve In magnetic materials, the force can be by the magneticfield fieldintensity intensity( (HH) )and andthe thecurrent currentpath pathlength length( (l l).).The Themagnetic magneticfield fieldintensity intensitycan canbe be magnetic B determined according to its corresponding relationship with the magnetic flux density ( ) through determined according to its corresponding relationship with the magnetic flux density ( B ) through

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of each material. When a magnetic flux is present, the magnetomotive force of a soft magnetic material (Fsteel ) is as follows: i

Fsteel =

∑ Hj l j

(5)

j =1

Table 1. The air gaps magnetic flux path model calculations. Flux Path Model

Mean Path Length

Permeance (Pair )

I II III IV

g 1.22g 1.22g p g(g + t) p g(g + r) z

2πµ0 (r + 2 )h/g g 3.3µ0 (r + 2 ) g 3.3µ0 (r + 2 ) p g+t 4µ0 (r + g(g + t)) ln( g ) p g+r 4µ0 (r + g − g(g + r)) ln( g )

V VI

g

µ0 π 2 z (r + g − 2z/π )

According Equations (4) and (5), magnetic flux and total magnetomotive force (Fsum ) are generated when the number of coil turns (N) and the operating current (I) are input. In addition, Equation (6) shows that total magnetomotive force can be obtained from electrical energy through energy conversion. The relationship between the iron core displacement and the solenoid force (Fem ) can be obtained through the principle of virtual work (W), as shown in Equations (7) and (8). N I = Fsum = Fair + Fsteel 1 2 Φ Pair −1 2

(7)

∂W Φ 2 d −1 Φ2 −2 d = · ( Pair ) = − · [ Pair P ] ∂z 2 dz 2 dz air

(8)

W= Fem =

(6)

According to the above derivation, when the proportional electromagnet is completed, the magnetic flux can be calculated by inputting the operating current, thereby acquiring the magnetic flux density and magnetic field intensity of soft magnetic materials. The solenoid force can be obtained from the linear region of operation and used as a basis for the subsequent proportional valve control. 3. Mathematical Model and Controller 3.1. Mathematical Model of Motorcycle Motion The motorcycle motion model provides calculations of vehicle dynamics that can be used to calculate relevant data during braking such as speed, slip, and braking distance. Figure 7a presents a simplified motorcycle motion model [28].

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Mv

Tb

I

Vv

Tt

FD

(a)

N (b)

Figure 7. A simplified motorcycle structure: (a) motion model; and (b) single wheel braking model. Figure 7. A simplified motorcycle structure: (a) motion model; and (b) single wheel braking model.

Its positive force N acting on the wheel is expressed as follows: Its positive force N acting on the wheel is expressed as follows:   N f = 1 − e−t / t f ( M v  g  Lb + Fint  H G − Fw  H w ) / L −t/t f  N f ( Mv · g · Lb + Fint · HG − Fw · Hw )/L  = 1 − e − t / tr   N = 1 − e M  g  L − Fint  H G + Fw  H w ) / L ( − t/t  Nr =r 1 − e r ( Mv · g · a L a − Fint · HG + Fw · Hw )/L v

)

(

(

)

(9) (9)

In Equation (9), the subscript f represents the action on the front wheel, whereas the In Equation (9), the subscript f represents the action on the front wheel, whereas the subscript r M vtotal subscript represents the rider total represents rtherepresents action on the wheel. Among these actions, mass of the the rear action on the rear wheel. Among M these actions, the v represents and the motorcycle, indicates the wheelbase between the front and rear wheels ofand the rear motorcycle, L indicates mass of the rider andLthe motorcycle, the wheelbase between the front wheels and L and L , respectively, denote the distances between the front and rear wheels and the center of a b of the motorcycle, and La and Lb , respectively, denote the distances between the front and rear mass of the vehicle. HG represents the height of the center of the mass of the vehicle, Hw indicates the wheels and the center of mass of the vehicle. H G represents the height of the center of the mass of average height of the wind force acting on the motorcycle, t represents time, t f denotes the time-delay the vehicle, indicates the average of the the wind force acting on the motorcycle, H wfront t constant of the shock absorber, andheight tr represents time-delay constant of the rear shock absorber. F represents the inertial force of the vehicle and F represents the longitudinal force of the w represents int time, t f denotes the time-delay constant of the front shock absorber, and t r represents wind. Their mathematical formulas are expressed as follows: the time-delay constant of the rear shock absorber. Fint represents the inertial force of the vehicle and

Mv · aTheir mathematical formulas are expressed (10) Fw represents the longitudinal force ofFintthe=wind.

as follows:

Fw =

1 · ρw · Cw · Aw · (Vv + u0 )2 2 F = M a

(11) (10) Among these, ρw is air density, Cw is the coefficient of air resistance, Aw is the frontal area of the vehicle, Vv is the vehicle speed, andFuw0=is1the   wwind  Cw  speed. Aw  (Vv +Inu0this (11)is )2 paper, the effect of wind speed 2 ignored, that is, it is assumed that u0 = 0 and wind force is Fw = 0.155 · Vv2 . Among these,  w is air density, Cw is the coefficient of air resistance, Aw is the frontal area 3.2. Wheel Braking Model of the vehicle, Vv is the vehicle speed, and u0 is the wind speed. In this paper, the effect of wind Figure 7b presents an analysis of the torque of a vehicle during braking, where Tt represents the speed is ignored, that is, it is assumed that u0 = 0 and wind force is Fw = 0.155 Vv2 . torque generated by the adhesive force of the road surface on the wheel and Tb denotes the torque generated by the brake caliper on the wheel. These are expressed as follows: 3.2. Wheel Braking Model int

v

Tt = Fof (12) w D ·aRvehicle Figure 7b presents an analysis of the torque during braking, where Tt represents the torque generated by the adhesive force of the road surface on the wheel and Tb denotes the Tb = Kb · Pcal (13) torque generated by the brake caliper on the wheel. These are expressed as follows: Tt = FD  Rw

(12)

Tb = Kb  Pcal

(13)

where FD represents the longitudinal adhesion value of the ground on the tire, Rw is the tire radius, and K b is a torque constant and represents the constant of proportionality between the

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where FD represents the longitudinal adhesion value of the ground on the tire, Rw is the tire radius, and Kb is a torque constant and represents the constant of proportionality between the brake pressure and brake torque. The torque balance equation of the wheel is as follows: .

Iw · ω = Tt − Tb

(14)

where Iw indicates the moment of inertia of the wheel and ω indicates its angular velocity. After obtaining the longitudinal force balance ( Fint = Fw + FD ), the acceleration value of the vehicle can be expressed as follows:   a = Fw + FD f + FDr /Mv

(15)

The integral of the aforementioned formula can be used to obtain the speed value of the vehicle, whereas integrating the formula twice can obtain braking distance. The motorcycle motion of symbols, parameters and values which are used in our simulation are described in Table 2. Table 2. The motorcycle motion data. Symbol

Parameter

Value

Mv HG Hw L La tf tr ρw Cw Aw Rw

total mass of the rider and the motorcycle height of the center of the mass of the vehicle average height of the wind force acting on the motorcycle wheelbase between the front and rear wheels of the motorcycle distances between the front wheels and the center of mass of the vehicle time-delay constant of the front shock absorber time-delay constant of the rear shock absorber air density coefficient of air resistance frontal area of the vehicle tire radius

220 Kgf 0.6 m 0.7 m 1.2 m 0.7 m 0.2 0.1 1.18 kg/m3 0.48 0.55 m2 0.21 m

3.3. Tire and Ground Model Several mathematical models can be applied to measure the acting forces of the ground and tires (hereafter referred to as tire force). This study employed the tire model presented by Dugoff to describe the relationship between longitudinal tire force FD , slip s, and vehicle speed Vv . The equations for this Dugoff’s model are as follows [1,29]:   Cn ·s i FD = 1−hs  N µ − µ 2 N (1− s ) 4Cn ·s

when when

Cn ·s 1− s Cn ·s 1− s

< >

µ· N 2 µ· N 2

(16)

where Cn is the longitudinal stiffness of the tire, µ is the friction coefficient, and N is the normal force of the tire. The friction coefficient µ in the following equation, according to the experimental results of Dugoff’s, assumes that the friction coefficient and slip speed Vv · s change linearly under normal conditions, that is, µ = µno (1 − An · Vv · s) (17) where µno is the friction coefficient during no slip, and An is the adhesion reduction coefficient. On wet road surfaces, the friction coefficient decreases exponentially with an increase in slip speed, that is,   Vv · s µ = µno · exp − (18) Vc where Vc is the characteristic speed and is a constant; its size is related to the root mean square texture height.

 Vv  s    Vc 

 = no  exp  − where

(18)

V is the characteristic speed and is a constant; its size is related to the root mean square 9 of 21

c 7, 34 Actuators 2018,

texture height.

3.4. 3.4. PEHB PEHB Mathematic Mathematic Model Model Analysis Analysis The (Figure 3) is composed of a proportional controlcontrol valve, The PEHB PEHBdesign designconfiguration configuration (Figure 3)primarily is primarily composed of a proportional motor pump, accumulator, caliper set, and hydraulic pipes. First, this study established a relationship valve, motor pump, accumulator, caliper set, and hydraulic pipes. First, this study established a database of proportional force,solenoid current, force, and stroke (Figure The system obtain solenoid relationship database of solenoid proportional current, and 8). stroke (Figurecan 8). The system can force through thedata look-up tablethe after input table current andinput stroke. obtaindata solenoid force through look-up after current and stroke.

Figure 8. 8. The The relationship relationship database database of of proportional proportional solenoid solenoid force, force, current, current, and and stroke. stroke. Figure

When the direction of the solenoid force of the proportional solenoid is defined as positive, the When the direction of the solenoid force of the proportional solenoid is defined as positive, equation of motion on the shuttle shaft is as follows: the equation of motion on the shuttle shaft is as follows: (19) Fem = Fm + M s  ..X + B  X. + K ( X c + X ) Fem = Fm + Ms · X + B · X + K ( Xc + X ) (19) where M s is the shuttle shaft and iron core mass; B is the damping coefficient; K is the spring

where Ms isXthe shuttle shaft and iron core mass; B is the damping coefficient; K is the spring constant; constant; c is the spring initial compression value; and X is the shuttle shaft stroke. Xc is The the spring initialiscompression and X isproportional the shuttle shaft stroke. valve body a variation ofvalue; a three-port solenoid valve. Its mathematic model The valve body is a variation of a three-port proportional solenoid valve. mathematic comprises two functions, namely those of the inlet valve and outlet valve. TheIts inlet valve is amodel spool comprises namely those that of the inlet valve outlet The inlet valve is a spool valve. The two flowfunctions, rate–pressure equation describes the and brake fluidvalve. flowing through the orifice is valve. The flow rate–pressure equation that describes the brake fluid flowing through the orifice is  d 2 Q = Cd  (20) ( d − X )  s ( P1 − P2 ) 4  π·d 2 Q = Cd · (20) (d − X ) · ( P1 − P2 ) 4 ρ where Q is the flow rate; d is the spool valve hole diameter;  is the fluid mass density; Cd is the discharge and Pvalve the pressure before after the fluid flows P1 spool where Q is thecoefficient; flow rate; dand is the hole diameter; ρ is the fluidand mass density; Cd is the 2 represent discharge coefficient; and P and P represent the pressure before and after the fluid flows through the 2 1 through the orifice, respectively. The brake fluid may flow from either end to the other depending orifice, respectively. The brake fluid may flow from either end to the other depending on which side on which side exhibits the higher pressure. exhibits the higher pressure. The outlet valve parameters are identical to those of the inlet valve. The only difference between The valvethe parameters are is identical those valve of the is inlet valve. valve The only the inlet outlet valve and outlet valve that thetooutlet a needle thatdifference normally between remains the inlet valve and the outlet valve is that the outlet valve is a needle valve that normally remains closed. After excitation, the degree to which the outlet valve opens is adjusted by the shuttle shaft closed. Afterwith excitation, degree to which the valvecomponent opens is adjusted bythe theoutlet shuttle shaft interlocked the ironthe core. The framework of outlet the interior blocks of valve is interlocked with the iron core. The framework of the interior component blocks of the outlet valve is identical to that of the inlet valve. The coning angle of the needle valve is θ. Accordingly, the area of the port is as follows: A0 = π · d · X · sin(θ/2) (21)

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The major component of the caliper is a piston. The brake fluid from the EHB pushes the piston, causing the caliper to clench the brake disc and stop the car. The spring serves to restore the caliper to its original position. Therefore, the equation of the caliper is as follows: Fcal = Kc · XP

(22)

Because Fcal = Pcal · Ac and X p = Vc /Ac , the previous equation can be rewritten as follows: dPcal =

dVc · Kc A2c

(23)

where Kc is the coefficient of caliper spring; XP is the stroke of caliper piston; Vc is the volume of caliper piston; and Ac is the caliper piston cross-sectional area. The strokes of the caliper involve two stages. In the first stage, the caliper does not make contact with the brake disc. In the second stage, the caliper is regarded as a rigid hydraulic chamber, described through the following equation: dVc dPcal = β · (24) Vc where β is the bulk modulus of the liquid. Once the ABS decompression cycle begins, the two ports adjust their degrees of opening. The brake fluid flows through the outlet valve and out the caliper, entering the oil return pipe. Subsequently, the brake fluid is stored in the pressure accumulator. The return pump is activated to draw the fluid back into the brake pipe to avoid a pressure increase in the oil return area, which may result in brake-release failure. The structure of the pressure accumulator is similar to that of the caliper in that the pressure accumulator is also composed of a piston and a restoring spring; however, its spring constant and cross-sectional area are smaller. A reciprocating fixed-displacement pump is positioned between the pressure accumulator and the oil return pipe. The return pump is activated by the DC motor. In the mathematic model, a fixed-frequency square wave represents the motor. In the mathematical equation representing the return pump function, the difference between the inlet pressure and the atmospheric pressure is derived and then multiplied by the displacement gain, and the maximum discharge is then added. The PEHB model of symbols, parameters and values which are used in our simulation are described in Table 3. Table 3. The PEHB model data. Symbol

Parameter

Value

Ms B K Xc d θ β Pcal /Vc

shuttle shaft and iron core mass damping coefficient spring constant spring initial compression value spool valve hole diameter coning angle of the needle valve bulk modulus of the liquid Caliper pressure per unit volume

0.1 Kgf 0.20 Kgf-s/mm 0.18 Kgf/mm 0.3 mm 1.2 mm 25 190 Kgf/mm2 0.06 Kgf/mm5

3.5. Traditional Discrete Switch Control The switch control of a traditional solenoid valve, which must be open or closed, can be represented by Figure 9. This type of discrete switch control has a control signal of 1 when the index is greater than 0. Applied to this paper, this represented full supercharge; the solenoid valve remained open during the entire sampling time. Conversely, if the index is less than 0, the control

3.5. Traditional Discrete Switch Control The switch control of a traditional solenoid valve, which must be open or closed, can be represented index Actuators 2018, 7,by 34Figure 9. This type of discrete switch control has a control signal of 1 when the11 of 21 is greater than 0. Applied to this paper, this represented full supercharge; the solenoid valve remained open during the entire sampling time. Conversely, if the index is less than 0, the control signal 1, signifying signifying full full depressurization; depressurization; itit remains remains in in aa depressurized state during signal is is − −1, depressurized state during the the entire entire sampling time. In such a situation, the control signal is not fine enough to achieve the desired sampling time. In such a situation, the control signal is not fine enough to achieve the desired control control effect. this performance, the sampling periodperiod can becan shortened (i.e., increase sampling effect. To Toimprove improve this performance, the sampling be shortened (i.e., the increase the frequency) to achieve immediate control. sampling frequency) to achieve immediate control.

u 1

Se 0

 1 if se  0 u= −1 if se  0

-1

Figure 9. Method for driving the traditional solenoid valve switch. Figure 9. Method for driving the traditional solenoid valve switch.

3.6. Proportional–Integral–Derivative Controller 3.6. Proportional–Integral–Derivative Controller The proportional–integral–derivative (PID) controller, which has a long history of use, is a The proportional–integral–derivative (PID) controller, which has a long history of use, is a feedback control method commonly found in industrial control and is widely used in various fields. feedback control method commonly found in industrial control and is widely used in various fields. The PID has become a common and reliable technical tool in industrial control because of its simple The PID has become a common and reliable technical tool in industrial control because of its simple structure, favorable stability, and ease of operation and adjustment. It is composed of a proportional structure, favorable stability, and ease of operation and adjustment. It is composed of a proportional unit, integral unit, and derivative unit. By adjusting the parameters of controller K , Ki , and Kd , unit, integral unit, and derivative unit. By adjusting the parameters of controllerpK p , K i , and Kd , the control control system system can can be be adjusted adjusted to to satisfy satisfy design design requirements. requirements. The responding result result can can be be the The responding expressedininterms termsof of how rapidly the controller responds to anthe error, thetodegree the expressed how rapidly the controller responds to an error, degree which to thewhich controller controller overshoots, and the degree of system oscillation. The input to the PID controller is overshoots, and the degree of system oscillation. The input to the PID controller is the error valuethe or error valuederived or the signal derived from value. The output (controlled variable) thesum result the the signal from this value. Thethis output (controlled variable) is the result ofisthe of of three sum of threeDefining algorithms. as the control the PID algorithm canasbeEquation expressed as u ( t ) output, algorithms. u(t)Defining as the control the PIDoutput, algorithm can be expressed (25); its block diagram is presented in is Figure 10. in Figure 10. Equation (25); its block diagram presented

+K Kii u(ut()t )==KKppee(t()t )+

d

d  ++KK d d ee((tt)) 00 ee((τ))ddτ dtdt

Z tt

(25) (25)

The transfer transfer function function of of PID PID controller controller can can be be described described as: as: The U (s) U s Gc ((ss)) = ( ) = K =+ EK(si) +=KK ps + Ksi + Kd s G c p d  E s s  K = K( p ) 1 + T1i s + Td s Where Ti = Kpi , Td = KKdp Kp   K 1 = K p  1and + T +is Tdthe s  derivative Where Ti time = constant. , Td = d where Ti is the integral time constant d Ki Kp  Ti s 

where Ti is the integral time constant and Td is the derivative time constant.

(26) (26)

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K pe (t ) K pe (t )

P P r (t ) r (t )

+ +

e (t ) e (t )

 

t

- ym ( t ) ym ( t )

D D

u (t ) u (t )

 

I Ki 0t e ( ) d I Ki 0 e ( ) d

y (t ) y (t )

Plant Plant

d Kd d e (t ) dt Kd e (t ) dt

Sensor Sensor Figure Figure 10. 10. Block Block diagram diagram of of the the PID PID controller. controller. Figure 10. Block diagram of the PID controller.

4. Simulation and Results Analysis 4. 4. Simulation Simulation and and Results Results Analysis Analysis For this analysis, MATLAB/Simulink was used to construct a PEHB system. The simulation For analysis, MATLAB/Simulink was used used to construct PEHB system. simulation For this thiswere analysis, MATLAB/Simulink was construct aaStep PEHB system. The The simulation commands primarily step and triangular wavetocommands. commands confirmed the commands were primarily step and triangular wave commands. Step commands confirmed the commands were primarily step and triangular wave commands. Step commands confirmed the response time and triangular wave commands confirmed the following status and linearity, and this response time and triangular wave commands confirmed the following status and linearity, and this response time and triangular wave commands confirmed the following status and linearity, and this depressurization result was analyzed to determine whether it could satisfy ABS control depressurization result analyzed to determine whetherwhether it could satisfy ABS satisfy control requirements. depressurization resultwas was analyzed to determine could ABS control requirements. Subsequently, a complete motorcycle ABS simulationitmodel was established, which Subsequently, a complete motorcycle ABS simulation model was established, which comprised a requirements. Subsequently, a complete motorcycle ABSasimulation established, comprised a motorcycle motion model, tire model, and controller model model.was A comparison ofwhich EHB motorcycle motion model,motion tire model, andtire a controller model. A comparison of EHB and PEHB of models comprised motorcycle model, model, a controller model. A comparison EHB and PEHB amodels was also conducted. The EHB and model used a bang-bang controller for slip was also conducted. The EHB model used a bang-bang controller for slip feedback control, whereas and PEHB models was also conducted. The EHB model used a bang-bang controller for slip feedback control, whereas the PEHB model employed a PID controller for slip feedback control. the PEHB control, model employed athe PIDPEHB controller foremployed slip feedback control. Figurefor 11 shows a block diagram feedback whereas model a PID controller slip feedback control. Figure 11 shows a block diagram of the anti-lock braking control system. Additionally, different of the anti-lock braking control system. Additionally, different road surface condition control results Figure 11 shows a block diagram of and the anti-lockdistance braking control system. different road surface condition control results braking comparisons wereAdditionally, analyzed. and braking distance comparisons were analyzed. road surface condition control results and braking distance comparisons were analyzed. S ref + S ref + − −

Controller Controller

u u

PEHB PEHB

Pcal Pcal

Brake Brake Torque Torque

Tb − Tb −

V

Tt Tt

Rw Rw

FD FD

w− Wheel Vw − Wheel Dynamics + Dynamics

+ +

+

Vehicle Vehicle Dynamics Dynamics

Car body Car body model model

 

1/Vv 1/Vv

Slip Slip

Vv Vv Tire & Road Tire & Road Model Model

Figure 11. Block diagram of the anti-lock braking control system. Figure 11. Block diagram of the anti-lock braking control system.

4.1. PEHB System Simulations and Analyses 4.1. PEHB System Simulations and Analyses In this study, a mathematical model was first constructed and analyzed for a single-wheel thispresented study,a mathematical a in mathematical model first constructed and analyzed for a single-wheel In this study, model first constructed and analyzed single-wheel PEHB, as Figures 12 andwas 13.was MATLAB/Simulink was usedfor toa connect each PEHB, block PEHB, as presented in Figures 12 and 13. MATLAB/Simulink was used to connect each block as presented in Figures 12 and 13. MATLAB/Simulink was used to connect each block excluding excluding the coil and iron core, which were voltage signals; the primary parameters considered excluding the coil andwhich ironpressure, core, which signals;aparameters the primary parameters considered the coilairand iron core, were voltage signals; thesimulate primary considered were air valve gaps, were gaps, hydraulic andwere flow.voltage To braking situation, the inlet were air gaps, hydraulic pressure, and flow. To simulate a braking situation, the inlet valve hydraulic flow. of To 100 simulate a braking situation, thethe inlet valve produced a constant produced apressure, constantand pressure bars. After the coil received excitation command, the iron produced a constant pressure of 100 bars. After the coil received the excitation command, the iron pressure of 100 bars. After the coil received the excitation command, the iron core began to move and core began to move and the inlet and outlet valves were adjusted to a certain opening degree. When core began to move and the inlet and outlet valves were adjusted to a certain opening degree. When the inlet and outlet valves were adjusted to a certain opening degree. When the hydraulic pressure the hydraulic pressure used for the shaft was balanced with the solenoid force, the opening degree the hydraulic pressure used forwith the shaft was balanced with the solenoid force, thethat opening degree used for the shaft wasthe balanced thedecreased solenoid force, the opening degree was maintained, and the was maintained, and brake caliper the caliper pressure moderately; is, skidding was maintained, and the brake caliper decreased the caliper pressure moderately; that is, skidding brake caliper decreased the caliper pressure moderately; that is, skidding due to locking from the due to locking from the application of excessive braking force could be prevented, and the fluid due to locking from thevalve application of excessive force be prevented, and the valve fluid application of the excessive braking force could be prevented, and thecould fluid flowing from the outlet flowing from outlet was carried away by braking the motor pump. flowing from the outlet valve was carried away by the motor pump. was carried away by the motor pump.

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  Figure 12. Block diagram of a single‐wheel PEHB system input and output.  Figure Block diagramofofaasingle-wheel single-wheel PEHB input andand output. Figure 12. 12. Block diagram PEHBsystem system input output. Figure 12. Block diagram of a single-wheel PEHB system input and output.

 

Figure 13. Block diagram of single-wheel PEHB system. Figure 13. Block diagram of single-wheel PEHB system.

Figure 13. Block diagram of single-wheel PEHB system. Figure 13. Block diagram of single‐wheel PEHB system.  Figure 14 shows step wave command relief simulation results. It shows that when the voltage is step wave results. shows that when the voltage is lowerFigure than 114 V,shows the solenoid forcecommand is short torelief opensimulation the valve port. TheItcorresponding caliper pressure Figure 14 shows step wave command relief simulation results. It shows that when the voltage is  Figure 14 shows step wave command relief simulation results. It shows that when the voltage is lower than 1 V, the solenoid force is short to open the valve port. The corresponding caliper pressure drops and iron core stroke for 2, 3, 4, 5, 6, 7, 8, 9, and 10 V are, respectively, 5.5, 19.8, 36.6, 55.4, 76.9, lower than 1 V, the solenoid force is short to open the valve port. The corresponding caliper pressure  drops and iron core stroke for 2, 3, 4, 5, 6, 7, 8, 9, and 10 V are, respectively, 5.5, 19.8, 36.6, 55.4, 76.9, lower than 1 V, the solenoid force is short to open the valve port. The corresponding caliper pressure 98.0, 99.3, 99.4 and 99.5 bars and 0.17, 0.29, 0.38, 0.45, 0.55, 0.73, 0.78, 0.79 and 0.8 mm. The simulation 98.0, 99.4 and 99.5caliper bars2,and 0.45, 0.73, 0.78,saturated 0.79 and 0.819.8, mm. The simulation drops and iron core stroke for 2, 3, 4, 5, 6, 7, 8, 9, and 10 V are, respectively, 5.5, 19.8, 36.6, 55.4, 76.9,  drops and99.3, iron core stroke for 3,pressure 4,0.17, 5, 6,0.29, 7,and 8,0.38, 9,iron and 100.55, Vstroke are, respectively, 5.5, 36.6, 55.4, 76.9, results show that the core are when the command is 98.0, results show that the caliper pressure and iron core stroke are saturated when the command is 98.0, 99.3, 99.4 and 99.5 bars and 0.17, 0.29, 0.38, 0.45, 0.55, 0.73, 0.78, 0.79 and 0.8 mm. The simulation  99.3, 99.4 andthan 99.58 bars andproportional 0.17, 0.29, 0.38, 0.55, 0.73, 0.78, stable 0.79 and 0.8pressure mm. The simulation results greater V. The valve0.45, simulation remains after relief and meets greater than 8 V. The proportional valve simulation remains stable after pressure relief and meets results  show  the  caliper  and pressure  and be iron  stroke  are when saturated  when  the  is  with application requirements, and can usedcore  as asaturated reference for thethe internal controller of the than show that the that  caliper pressure ironit core stroke are command is command  greater with application requirements, and for it can be used as acalculation. reference for the internal controller of the subsequent ABS module as the basis the slip control greater than 8 V. The proportional valve simulation remains stable after pressure relief and meets  8 V. The proportional valve simulation remains stable after pressure relief and meets with application subsequent ABS module as the basis for the slip control calculation.

Command(V) Command(V)

Command(V)

with  application  and  it  can  be  used  as  a  reference  for ofthe  controller  of  the  requirements, andrequirements,  it can be used as a reference for the internal controller theinternal  subsequent ABS module (a) subsequent ABS module as the basis for the slip control calculation.  as the basis for the slip control calculation. (a)

10

5

0 0

10 10

(a)

5 5 0 0 0 0

1 1

1

2 2

2

3 3

3

4 4

5 Time(s) 5 Time(s)

4

5 Time(s)

6 6

Figure 14. Cont.

6

7 7

8 8

7

9 9

8

10 10

9

10

 

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Stroke(mm)

1

0.5 0.5

0 0 0 0

1 1

2 2

3 3

4 4

5 Time(s) 5 Time(s) (c)

6 6

7 7

8 8

9 9

10 10

 

 

(c) Brake Pressure(bar)

Brake Pressure(bar)

Stroke(mm)

1

100 100 50 50 0 00 0

1

1

2 2

3 3

4 4

5 5 Time(s) Time(s)

6

6

7

7

8

8

9

9

10

10

   

Figure 14. The proportional valve relief simulation in 100 bars: (a) step wave command; (b) iron core Figure 14. The proportional valve relief simulation in 100 bars: (a) step wave command; (b) iron core  Figure 14. The proportional valve relief simulation in 100 bars: (a) step wave command; (b) iron core  stroke; and (c) caliper pressure. stroke; and (c) caliper pressure.  stroke; and (c) caliper pressure. 

Figure 15 shows the triangular wave The proportional Figure  15 15  shows  the  triangular  wave  command  relief simulation simulation results. results.  The  proportional  Figure  shows  the  triangular  wave command command relief relief  simulation  results.  The  proportional  electromagnet is not actuated at command 2.5 V or less, and the pressure drop is 100 bars above 8 V. electromagnet is not actuated at command 2.5 V or less, and the pressure drop is 100 bars above 8 V.  electromagnet is not actuated at command 2.5 V or less, and the pressure drop is 100 bars above 8 V.  The pressure drop between 2.5 2.5  V2.5  and 8and  V is8  and theand  relationship between voltagevoltage  and pressure The  pressure  drop  between  V and  8 linear, V  is  linear,  the  between  and  The  pressure  drop  between  V  V  linear,  and  the relationship  relationship  between  voltage  and  drop can be derived as 18.2 bars/V. The caliper pressure follows the command well and linearly, and pressure  drop  can  be  derived  as 18.2  bars/V.  The  caliper  pressure  follows  the  command  well  and  pressure  drop  can  be  derived  as 18.2  bars/V.  caliper  pressure  follows  the  command  well  and  linearly, and the pressure does not have vibration when the valve is fully open.      the pressure does not have vibration when the valve is fully open. linearly, and the pressure does not have vibration when the valve is fully open.  (a) (a)

Command(V) Command(V)

10 10

5

5

0 0 0 0

0.5 0.5

1 1

1.5 1.5

2 2

2.5 2.5 Time(s)

3

3

3.5

4

3.5

4.5

4

5

4.5

5

Time(s)

 

(b)

Stroke(mm) Stroke(mm)

1

 

(b)

1

0.5 0.5 0 0

0 0

0.5

0.5

1

1

1.5

1.5

2

2

2.5 Time(s)

2.5 Time(s)

3

3

3.5

4

3.5

4.5

4

5

4.5

5

 

 

Brake Pressure(bar) Brake Pressure(bar)

(c)

(c) 100

100 50 50 0 0

0 0

0.5

0.5

1

1

1.5

1.5

2

2.5 Time(s)

3

2

2.5 Time(s)

3

3.5

4

4.5

5

  3.5

4

4.5

5

  Figure 15. 15.  TheThe  proportional valve relief simulation in 100 bars: (a) (a)  triangular wave command; (b)(b)  iron Figure  proportional  valve  relief simulation  in  100  bars:  triangular  wave  command;  coreiron core stroke; and (c) caliper pressure.  stroke; and (c) caliper pressure. Figure  15.  The  proportional  valve  relief simulation  in  100  bars:  (a)  triangular  wave  command;  (b)  iron core stroke; and (c) caliper pressure. 

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4.2. Motorcycle ABS Simulation Model with an EHB 4.2.Motorcycle MotorcycleABS ABSSimulation SimulationModel Modelwith withan anEHB EHB 4.2.

Figure 16 depicts the brake mathematical model of the bang-bang controller using Figure 16 16 depicts depicts the brake mathematical mathematical model ofof the the bang-bang bang-bang controller usingof Figure brake model using MATLAB/Simulink, whichthe represents the action characteristic (i.e., it must be controller open or closed) MATLAB/Simulink, which represents the action characteristic (i.e., it must be open or closed) of the which actiontransfer characteristic (i.e.,was it must be open closed) of the theMATLAB/Simulink, solenoid valve. After the represents hydraulic the pressure function applied andor the hydraulic lag solenoidvalve. valve. Afterthe thehydraulic hydraulicpressure pressuretransfer transferfunction functionwas wasapplied appliedand andthe thehydraulic hydrauliclag lag solenoid time constant TB isAfter 0.5 s, it was integrated to obtain pressure change. Finally, it was applied to the brake timeconstant constantTB TBisis0.5 0.5 s,ititwas wasintegrated integratedtotoobtain obtainpressure pressurechange. change.Finally, Finally,ititwas wasapplied appliedtotothe the time caliper and the brake forces, was adjusted. Figure 17 depicts the establishment of a complete motorcycle brake caliper and the brake force was adjusted. Figure 17 depicts the establishment of a complete brake caliper and the which brake force was adjusted. depicts themotion establishment complete ABS simulation model, comprised an EHBFigure model,17motorcycle model, of tirea model, and motorcycleABS ABSsimulation simulationmodel, model,which whichcomprised comprisedan anEHB EHBmodel, model,motorcycle motorcyclemotion motionmodel, model,tire tire motorcycle integrated bang-bang controller model. model,and andintegrated integratedbang-bang bang-bangcontroller controllermodel. model. model, TheThe slipslip command was fixed at 0.2 and the vehicle speed and front frontand andrear rearwheel wheelspeeds speeds commandwas wasfixed fixedatat0.2 0.2and andthe theinitial initialvehicle vehiclespeed speedand and The slip command initial front and rear wheel speeds were controlled at 60 km/h. the wheel and vehicle speeds. The Thespeed speed were controlled 60 km/h.The Thebrake brakeforce forcechanged changed the the wheel wheel and and vehicle vehicle speeds. speeds. were controlled atat60 km/h. The brake force changed The speed difference between the two was converted into slip, that is, the feedback factor of the ABS control differencebetween betweenthe thetwo twowas wasconverted convertedinto intoslip, slip,that thatis, is,the thefeedback feedbackfactor factorofofthe theABS ABScontrol control difference loop. TheThe simulation results are presented ininFigure 18; the entire process was wasapproximately approximately2.42.4 s, loop. simulation results are presented Figure 18; the entire process loop. The simulation results are presented in Figure 18; the entire process was approximately 2.4 s,s, braking distance was approximately 22 m, and during the process the wheel speed clearly oscillated braking distance was approximately 22 m, and during the process the wheel speed clearly oscillated braking distance was approximately 22 m, and during the process the wheel speed clearly oscillated because ofof the on-off control. After 0.5 atatapproximately approximately 0.2±±± 0.1 and the front because ofthe the on-off control. After 0.5s,s,s,the theslip sliposcillated oscillatedat approximately0.2 0.2 0.1 and the front because on-off control. After 0.5 the slip oscillated 0.1 and the front wheel exhibited greater inertial force and therefore hadaaahigher higher frequency. After 2 s, wheel exhibited greater inertial forcedue dueto brakingand andtherefore thereforehad had higher frequency. After wheel exhibited greater inertial force due totobraking braking frequency. After 22s,s, the vehicle speed was decreased to less than 10 km/h; the increased due to loop gain and was thethe vehicle speed was decreased to less than 10 km/h; the slip increased due to loop gain and was vehicle speed was decreased to less than 10 km/h; the slip increased due to loop gain and was unstable and divergent. This wasan aninherent inherentphenomenon phenomenonof ofthe ABS; thus, general ABS controls, unstable and divergent. This was of theABS; ABS;thus, thus,inin in general ABS controls, unstable and divergent. This was an inherent phenomenon general ABS controls, the ABS function must be turned off when the vehicle speed is less than approximately 10 km/h. thethe ABS function speed isis less lessthan approximately approximately km/h. ABS functionmust mustbebeturned turnedoff offwhen when the the vehicle vehicle speed 1010 km/h. First-order damping was applied the bang-bang bang-bang controller alleviate the pressure pressure output First-order damping waswas applied to the controller to alleviate the pressure output changing First-order damping applied totobang-bang the controller toto alleviate the output changing intoaaramp ramp change. Generally, thebraking braking process affected byinertia, inertia, resulting an changing into change. Generally, the process affected by resulting ininin an into a ramp change. Generally, the braking process is affected byisisinertia, resulting in an increase the increase in the normal force of the front wheel and a decrease in the rear wheel. By controlling the increase in of thethe normal the afront wheelinand decrease thecontrolling rear wheel.the Bybrake controlling the normal force front force wheelofand decrease thearear wheel.inBy component brakecomponent component measurements thesame sameway, way, thesimulations simulations demonstrated that thefront front wheel brake measurements the the that the measurements the same way, the simulations demonstrated thatdemonstrated the front wheel would skidwheel earlier, would skid earlier, and the slip would continue to oscillate and be unable to stabilize. In other would skid earlier, and the slip would continue to oscillate and be unable to stabilize. In other and the slip would continue to oscillate and be unable to stabilize. In other words, under normal words,under under normalABS ABScontrol, control,the theaverage averagedischarge dischargepressure pressureofofthe the frontwheel wheel ishigher higherthan than words, ABS control, thenormal average discharge pressure of the front wheel is higherfront than that ofisthe rear wheel. that of the rear wheel. The average pressure of the simulation results was approximately 50 bars for that of the rear wheel. pressure of was the simulation results50was approximately 50 barswhich for The average pressure ofThe the average simulation results approximately bars for the rear wheel, the rear wheel, which was greater than that of the front wheel (40 bars). thegreater rear wheel, which greater than (40 thatbars). of the front wheel (40 bars). was than that ofwas the front wheel

Figure 16.Simplified Simplifiedsolenoid solenoidvalve valvebrake brakeactuator. Figure 16. valve brake actuator. Figure 16. Simplified solenoid

Figure17. 17.Motorcycle MotorcycleABS ABSsimulation simulationmodel model withan an EHB. Figure Figure 17. Motorcycle ABS simulation modelwith with anEHB. EHB.

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Speed (Km/h)

(a) Vehicle Front Wheel Rear Wheel

60 40 20 0 0

0.5

1

1.5 Time (s)

2

2.5

3

 

(b) 1

Slip

Front Wheel Rear Wheel 0.5

0 0

0.5

1

1.5 Time (s)

2

2.5

3

 

Brake Pressure (bar)

(c) Front Wheel Rear Wheel

100

50

0 0

0.5

1

1.5

2 Time (s)

2.5

3

3.5

 

Figure 18. 18. EHB EHB  performing  anti‐lock  braking  a  dry  surface:  (a)  vehicle  and speed; wheel  Figure performing anti-lock braking on aon  dry roadroad  surface: (a) vehicle speed speed  and wheel speed; (b) slip; and (c) brake pressure.  (b) slip; and (c) brake pressure.

4.3. Motorcycle ABS Simulation Model for New PEHB  4.3. Motorcycle ABS Simulation Model for New PEHB The solenoid valve brake actuator in Figure 19 was replaced with the mathematical models of  The solenoid valve brake actuator in Figure 19 was replaced with the mathematical models of the the  PEHB  and controller PID  controller  for  ABS  control  simulations.  The  block  diagram  of  the  controller  is  PEHB and PID for ABS control simulations. The block diagram of the controller is provided provided  in In Figure  20. controller, In  the  PID  controller,  to  increase  integral acompensation,  a  conditional  in Figure 20. the PID to increase integral compensation, conditional function similar function similar to that of the bang‐bang system was added in front of the integrator. Thus, for the  to that of the bang-bang system was added in front of the integrator. Thus, for the integrator, this integrator, this was equivalent to receiving a gain of a previous order that is changeable. Figure 21  was equivalent to receiving a gain of a previous order that is changeable. Figure 21 illustrates the illustrates  differences  in  simulation  results  for and the  without system  with  and  without  the  bang‐bang  differences the  in simulation results for the system with the bang-bang controller in front controller  in  front  of  the  integrator  in  the  PID  controller.  After  the  bang‐bang  controller  was  of the integrator in the PID controller. After the bang-bang controller was installed, front–rear wheel installed, front–rear wheel slip control with a target value of 0.2 was conducted; the results indicated  slip control with a target value of 0.2 was conducted; the results indicated that during the initial that during the initial stage of the brake, the overshoot of the slip control was small. Subsequently,  stage of the brake, the overshoot of the slip control was small. Subsequently, the steady-state error the  steady‐state  was  eliminated  to  achieve  target the value,  thereby  reducing  the after braking  was eliminated toerror  achieve the target value, thereby the  reducing braking distance. Finally, the distance. Finally, after the controller parameters were adjusted by Ziegler and Nichols method, the  controller parameters were adjusted by Ziegler and Nichols method, the controller parameters can controller parameters can achieved K i = −5, and Kd = −0.5.  achieved K = −40, K = −5, and K = p− = −40, K 0.5. p

i

d

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  Figure 19. Motorcycle ABS simulation model with PEHB.  Figure 19. Motorcycle ABS simulation model with PEHB. Figure ABSsimulation simulationmodel modelwith withPEHB. PEHB. Figure 19. 19. Motorcycle Motorcycle ABS

  Figure 20. PID controller simulation model. Figure 20. PID controller simulation model.  Figure 20. PID Figure PID controller controllersimulation simulationmodel. model. (a) (a) (a)

PID (With Bang-Bang) PID (With Bang-Bang) PID (With Bang-Bang) PID (Without Bang-Bang) PIDPID (Without Bang-Bang) (Without Bang-Bang)

Slip Slip Slip

0.4 0.4 0.4 0.3 0.3 0.3 0.2 0.2 0.2 0.1 0.1 0.1 0 0 000 0

0.5 0.5 0.5

1 11

1.5 Time (s) 1.5 1.5 Time (s)(s) (b) Time (b)

2 2 2

(b)

0

3 3

3

PID (With Bang-Bang) PID Bang-Bang) PID(With (Without Bang-Bang) PID (With Bang-Bang) PID (Without Bang-Bang)

PID (Without Bang-Bang)

Slip Slip Slip

0.4 0.4 0.40.3 0.3 0.30.2 0.2 0.20.1 0.1 0.1 00 0 00

2.5 2.52.5

0.5 0.5

0.5

1 1

1

1.5 Time 1.5 (s) Time1.5 (s)

2 2

2

2.5 2.5

2.5

3 3

3

Time (s)

Figure 21. The difference results with and without the bang-bang controller in the PID controller:  (a) Figure 21. The(b) difference results with and without the bang-bang controller in the PID controller: (a) front wheel; rear wheel. Figure 21. The Figure 21. The difference results with and without the bang‐bang controller in the PID controller: (a)  front wheel; (b)difference rear wheel.results with and without the bang-bang controller in the PID controller: (a) front wheel; (b) rear wheel. front wheel; (b) rear wheel.  

As presented in Figure 22a, under initial conditions of a dry road surface and identical vehicle As presented in Figure 22a, under initial of a dry started road surface vehicle speed and front and rear wheel speeds, the conditions braking sequence at 0.1 and s, atidentical which time slip As presented in Figure 22a, under initial conditions of a dry road surface and identical vehicle  speed and front and rear wheel speeds, the braking sequence started at 0.1 s, and at which time slip As presented in Figure 22a,vehicle under initial conditions a dry road surface identical control was also initiated. The speed and front andofrear wheel speeds decreased steadilyvehicle to a speed  and  front  and  rear  wheel  speeds,  braking  sequence  started  0.1  s,  at  time which  time  control was also initiated. The vehicle speed and front and wheel decreased steadily to aslip  speed front and rear wheel speeds, the the  braking sequence started atspeeds 0.1 at  s, at which slip control halt,and and the braking distance was further reduced to 21 m.rear control was also initiated. The vehicle speed and front and rear wheel speeds decreased steadily to a  halt, and the braking was further reduced to 21wheel m. was also initiated. The distance vehicle speed and front and rear speeds decreased steadily to a halt, and

halt, and the braking distance was further reduced to 21 m.  the braking distance was further reduced to 21 m.

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The slip trends of the front and rear wheels are provided in Figure 22b. The slip was controlled The slip trends of the front and rear wheels are provided in Figure 22b. The slip was controlled  within a stable region, and the slip of the front and rear wheels were precisely controlled at a target within a stable region, and the slip of the front and rear wheels were precisely controlled at a target  value of 0.2 at 0.5 s and 0.8 s, respectively. The pressures of the front and rear wheel calipers are value  of 0.2 at  0.5  s  and 0.8  s,  respectively.  The  pressures  of  the  front  and rear  wheel  calipers are  depicted in Figure 22c; the stable pressure indicates that the front wheel is higher. At the beginning, depicted in Figure 22c; the stable pressure indicates that the front wheel is higher. At the beginning,  because the slip of the front wheel had exceeded the target value at 0.2 s, the brake force had to be because the slip of the front wheel had exceeded the target value at 0.2 s, the brake force had to be  removed as a response. After entering a precise slip control loop, it was revealed that brake pressure of removed as a response. After entering a precise slip control loop, it was revealed that brake pressure  the front wheel was greater than that of that  the rear wheel, whichwhich  was consistent with with  the normal braking of  the  front wheel  was  greater  than  of  the  rear  wheel,  was  consistent  the  normal  modelbraking model of a motorcycle.  of a motorcycle.

Speed (Km/h)

(a) Vehicle Front Wheel Rear Wheel

60 40 20 0 0

0.5

1

1.5 Time (s)

2

2.5

3

  

(b) Front Wheel Rear Wheel

Slip

0.4 0.3 0.2 0.1 0 0

0.5

1

1.5 Time (s)

2

2.5

3

  

Brake Pressure (bar)

(c) Front Wheel Rear Wheel

100

50

0 0

0.5

1

1.5 Time (s)

2

2.5

3

  

22. PEHB  performing  anti‐lock  braking on on aa dry dry  road road  surface:  and and wheel  FigureFigure  22. PEHB performing anti-lock braking surface:(a)  (a)vehicle speed  vehicle speed wheel speed; (b) slip; and (c) brake pressure.  speed; (b) slip; and (c) brake pressure.

As  depicted  in  Figure  23a,  under  the  initial  conditions  of  a  wet  road  surface  and  identical 

As depicted in Figure 23a, under the initial conditions of a wet road surface and identical vehicle vehicle speed and front and rear wheel speeds, the braking sequence started at 0.1 s, at which time  speedthe slip control was initiated. The vehicle speed and front and rear wheel speeds decreased steadily  and front and rear wheel speeds, the braking sequence started at 0.1 s, at which time the to  a  halt;  however,  the  braking  time  increased  by front 0.7  s  compared  that  obtained  for  dry  road  slip control was initiated. The vehicle speed and and rear with  wheel speeds decreased steadily surface conditions.  to a halt; however, the braking time increased by 0.7 s compared with that obtained for dry road The  slip  trends  of  the  front  and  rear wheels  are  presented in  Figure 23b. Skidding  was more  surface conditions. evident in the front wheel, and was at its maximum at approximately 0.32. However, the slips of the  The slip trends of the front and rear wheels are presented in Figure 23b. Skidding was more front and rear wheels were both precisely controlled at a target value of 0.2 at 0.78 s. The pressures of  evident in the front wheel, and was at its maximum at approximately 0.32. However, the slips of the the  front  and  rear  wheel  calipers  are  depicted  in  Figure  23c.  Because  the  slip  of  the  front  wheel  front and rear wheels were both precisely controlled at a target value of 0.2 at 0.78 s. The pressures exceeded the target value at 0.2 s, the brake force had to be removed as a response; at the same time,  of thea precise slip control loop was applied. However, because the road surface was wet and the friction  front and rear wheel calipers are depicted in Figure 23c. Because the slip of the front wheel exceeded the target at to  0.2avoid  s, theskidding,  brake force toforce  be removed a response; at the strong.  same time, coefficient  was  value smaller,  the had brake  applied  as should  not  be  overly  Because the steady‐state pressure was less than that of the dry road surface (approximately 60 bars),  a precise slip control loop was applied. However, because the road surface was wet and the friction the braking distance was longer than that of the dry road surface.  coefficient was smaller, to avoid skidding, the brake force applied should not be overly strong. Because the steady-state pressure was less than that of the dry road surface (approximately 60 bars), the braking distance was longer than that of the dry road surface.

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19 of 21 19 of 21  19 of 21  (a) (a) Vehicle Vehicle FrontWheel Wheel Front RearWheel Wheel Rear

Speed Speed (Km/h) (Km/h)

60 60 40 40 20 20 000 0

0.5 0.5

11

1.5 1.5 Time(s) (s) Time

22

2.5 2.5

33

    

(b) (b) FrontWheel Wheel Front RearWheel Wheel Rear

Slip Slip

0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 000 0

0.5 0.5

11

1.5 1.5 Time(s) (s) Time

22

2.5 2.5

33

    

Brake Brake Pressure Pressure (bar) (bar)

(c) (c) FrontWheel Wheel Front RearWheel Wheel Rear

100 100

50 50

000 0

0.5 0.5

11

1.5 1.5 Time(s) (s) Time

22

2.5 2.5

33

    

Figure  23. PEHB performing anti‐lock braking on a wet road surface: (a) vehicle speed and wheel  Figure  Figure 23.23. PEHB performing anti‐lock braking on a wet road surface: (a) vehicle speed and wheel  PEHB performing anti-lock braking on a wet road surface: (a) vehicle speed and wheel speed; (b) slip; and (c) brake pressure.  speed; (b) slip; and (c) brake pressure.  speed; (b) slip; and (c) brake pressure.

4.4. Analyses of Simulation Results  4.4.4.4. Analyses of Simulation Results  Analyses of Simulation Results Figure 24 presents a comparison of the braking distances for four braking models, including the  Figure 24 presents a comparison of the braking distances for four braking models, including the  Figure 24 presents a comparison of the braking distances for four braking models, including the braking distances of a traditional EHB and the PEHB of this study on dry and wet road surfaces, as  braking distances of a traditional EHB and the PEHB of this study on dry and wet road surfaces, as  braking distances of a traditional EHB and the PEHB of this study on dry and wet road surfaces, as well well as the braking distance without the addition of the ABS. The simulation results revealed that the  well as the braking distance without the addition of the ABS. The simulation results revealed that the  as the braking distance without the addition of the ABS. The simulation results revealed that the PEHB PEHB could effectively reduce braking distance, and that vehicles without an ABS had the longest  PEHB could effectively reduce braking distance, and that vehicles without an ABS had the longest  could effectively reduce distance, that vehicles withoutin anskidding.  ABS hadA  the longest braking braking  distances  due braking to vehicle  vehicle  brake and locking,  which resulted  resulted  summary  of the  the  braking  distances  due  to  brake  locking,  which  in  skidding.  A  summary  of  distances due to vehicle brake locking, which resulted in skidding. A summary of the parameters of parameters of each simulation is shown in Table 4.  parameters of each simulation is shown in Table 4.  each simulation is shown in Table 4. 35 35 30 30

PEHB+PID(Dry (DryRoad) Road) PEHB+PID PEHB+PID(Wet (WetRoad) Road) PEHB+PID EHB+Bang-Bang(Dry (DryRoad) Road) EHB+Bang-Bang WithoutABS ABS Without

Stopping StoppingDistance Distance (m) (m)

25 25 20 20 15 15 10 10 55 000 0

0.5 0.5

11

1.5 1.5 Time(s) (s) Time

22

2.5 2.5

Figure 24. Comparison of various braking distances.  Figure 24. Comparison of various braking distances.  Figure 24. Comparison of various braking distances.

33

  

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Table 4. A summary of the parameters of each simulation. Control Module and Mode

Road State

Braking Time (s)

Stopping Distance (m)

Slip (Steady State)

PEHB + PID PEHB + PID EHB + Bang-Bang Without ABS

Dry Wet Dry Dry

2.29 2.97 2.40 >3

20.88 26.42 21.73 >34.35

0.2 ± 0.06 0.2 ± 0.03 0.2 ± 0.1 1

5. Conclusions This study primarily integrated the inlet and outlet valves of a market EHB into a set of proportional pressure control valves to achieve precise brake force control. MATLAB/Simulink was used to establish the mathematical model of the PEHB actuator. The simulation results demonstrated that this actuator had achieved a stable adjustment of depressurization control as well as favorable linear precision and repeatability; therefore, it can be applied to the ABS for slip control. Additionally, a complete motorcycle ABS simulation model was established, and simulations and analyses were performed for the wheel speed, slip, and brake force of the EHB and PEHB on different road surfaces during braking. Through the simulation results, it was demonstrated that the PEHB could achieve more favorable stability and speed during breaking, in addition to precisely achieving the target values during slip control and effectively reducing braking distance. Author Contributions: All authors have participated to the design of simulations, analysis of data and results, and writing of the paper. Funding: This research received no external funding. Conflicts of Interest: The authors declare no conflict of interest.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

12.

Wu, M.C.; Shih, M.C. Simulated and experimental study of hydraulic anti-lock braking system using sliding-mode PWM control. Mechatronics 2003, 13, 331–351. [CrossRef] Qiuetal, Y.; Liang, X.; Dai, Z. Backstepping dynamic surface control for an anti-skid braking system. Control Eng. Pract. 2015, 42, 140–152. Tanelli, M.; Sartori, R.; Savaresi, S.M. Combining Slip and Deceleration Control for Brake-by-wire Control Systems: A Sliding-mode Approach. Eur. J. Control 2007, 6, 593–611. [CrossRef] Lv, C.; Zhang, J.; Li, Y.; Yuan, Y. Novel control algorithm of braking energy regeneration system for an electric vehicle during safety–critical driving maneuvers. Energy Convers. Manag. 2015, 106, 520–529. [CrossRef] Patra, N.; Datta, K. Observer based road-tire friction estimation for slip control of braking system. Procedia Eng. 2012, 38, 1566–1574. [CrossRef] Bhandari, R.; Patil, S.; Singh, R.K. Surface prediction and control algorithms for anti-lock brake system. Transp. Res. Part C 2012, 21, 181–195. [CrossRef] Choa, J.R.; Choia, J.H.; Yoo, W.S.; Kim, G.J.; Woo, J.S. Estimation of dry road braking distance considering frictional energy of patterned tires. Finite Elem. Anal. Des. 2006, 42, 1248–1257. [CrossRef] Ivanov, V.; Savitski, D.; Augsburg, K.; Barber, P.; Knauder, B.; Zehetner, J. Wheel slip control for all-wheel drive electric vehicle with compensation of road disturbances. J. Terramech. 2015, 61, 1–10. [CrossRef] Aksjonov, A.; Augsburg, K.; Vodovozov, V. Design and Simulation of the Robust ABS and ESP Fuzzy Logic Controller on the Complex Braking Maneuvers. Appl. Sci. 2016, 6, 382. [CrossRef] Mirzaei, M.; Mirzaeinejad, H. Optimal design of a non-linear controller for anti-lock braking system. Transp. Res. Part C 2012, 24, 19–35. [CrossRef] Wang, B.; Huang, X.; Wang, J.; Guo, X.; Zhu, X. A robust wheel slip ratio control design combining hydraulic and regenerative braking systems for in-wheel-motors-driven electric Vehicles. J. Frankl. Inst. 2015, 352, 577–602. [CrossRef] Chen, C.P.; Chiang, M.H. Development of Proportional Pressure Control Valve for Hydraulic Braking Actuator of Automobile ABS. Appl. Sci. 2018, 8, 639. [CrossRef]

Actuators 2018, 7, 34

13. 14. 15. 16. 17.

18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29.

21 of 21

Drakunov, S.; Özgüner, Ü.; Dix, P.; Ashrafi, B. ABS control using optimum search via sliding modes. IEEE Trans. Control Syst. Technol. 1995, 3, 79–85. [CrossRef] Keshmiri, R.; Shahri, A.M. Intelligent ABS Fuzzy Controller for Diverse Road Surfaces. Int. J. Mech. Syst. Sci. Eng. 2007, 1, 257–262. Precup, R.-E.; Preitl, S.; Faur, G. PI predictive fuzzy controllers for electrical drive speed control: Methods and software for stable development. Comput. Ind. 2003, 52, 253–270. [CrossRef] Chaoui, H.; Sicard, P. Adaptive fuzzy logic control of permanent magnet synchronous machines with nonlinear friction. IEEE Trans. Ind. Electron. 2012, 59, 1123–1133. [CrossRef] Choi, S.B.; Bang, J.H.; Cho, M.S.; Lee, Y.S. Sliding mode control for anti-lock brake system of passenger vehicles featuring electrorheological valves. Proc. Inst. Mech. Eng. Part D J. Automob. Eng. 2000, 216, 897–908. [CrossRef] Jiang, F.; Gao, Z. An application of nonlinear PID control to a class of truck ABS problems. In Proceedings of the 40th IEEE Conference on Decision and Control, Orlando, FL, USA, 4–7 December 2001; pp. 516–521. Aparow, V.R.; Ahmad, F.; Hudha, K.; Jamaluddin, H. Modelling and PID control of antilock braking system with wheel slip reduction to improve braking performance. Int. J. Veh. Saf. 2013, 6, 265–296. [CrossRef] Astrom, K.; Hagglund, T. PID Controller: Theory, Design and Tuning; Instrument Society of America: Research Triangle Park, NC, USA, 1995. Astrom, K.; Hagglund, T. Advanced PID Design; ISA-The Instrumentation Systems, and Automation Society: Research Triangle Park, NC, USA, 2005. Rivera, D.E.; Morari, M.; Skogestad, S. Internal model control. 4. PID controller design. Ind. Eng. Chem. Process Des. Dev. 1986, 25, 252–265. [CrossRef] Jin, Q.B.; Liu, Q. IMC-PID design based on model matching approach and closed-loop shaping. ISA Trans. 2014, 53, 462–473. [CrossRef] [PubMed] Chen, C.P.; Tung, C.; Chen, C.A. A Proportional Electro-Hydraulic Braking Control Valve. Taiwan, 2010. Patent No. I320374, 11 February 2010. Chen, C.A.; Tung, C.; Chen, C.P. The Proportional Solenoid Module for a Hydraulic System. Patent No. I474350, 21 February 2015. Lee, C.O.; Song, C.S. Analysis of the solenoid of a hydraulic proportional compound valve. In Proceedings of the 35th National Conference on Fluid, Chicago, IL, USA, 13–15 November 1979; pp. 21–30. Chen, Y.N.; Kuo, W.H. Analysis and Design of Proportional Pressure Control Valve. Master’s Thesis, National Taiwan University, Taipei, Taiwan, 1987. Lu, C.Y.; Shih, M.C. Design and Control of the Hydraulic Anti-Lock Brake System for a Light Motorcycle. Ph.D. Thesis, National Cheng Kung University, Tainan, Taiwan, 2005. Dugoff, H.; Fancher, P.S.; Segel, L. An Analysis of Tire Traction Properties and Their Influence on Vehicle Dynamic Performance; SAE Paper No. 700377; SAE International: Warrendale, PA, USA, 1970. © 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).