Method of Improving Lateral Stability by Using Additional Yaw Moment of Semi-Trailer

energies

Article Method of Improving Lateral Stability by Using Additional Yaw Moment of Semi-Trailer

Zhenyuan Bai , Yufeng Lu * and Yunxia Li

School of Mechanical and Automotive Engineering, Qilu University of Technology (Shandong Academy of Sciences), Jinan 205353, China; [email protected] (Z.B.); ldyﬀ[email protected] (Y.L.) * Correspondence: [email protected]

Received: 20 October 2020; Accepted: 26 November 2020; Published: 30 November 2020

Abstract: The lateral stability control of tractor semi-trailer plays a vital role for enhancing its driving safety, and the distributed electric drive structure of a hub motor creates opportunities and challenges for realising the lateral stability accurately. Based on the dynamics simulation software TruckSim, a nonlinear dynamic tractor semi-trailer model is established, and a MATLAB/Simulink linear three-degree-of-freedom monorail reference model is established. The upper controller adopts fuzzy proportional–integral–derivative control to export active yaw torque values of the tractor and semi-trailer. The lower controller outputs the driving/braking torque of each wheel according to the target wheel driving/braking rules and torque distribution rules. The tractor produce an active yaw torque through conventional differential braking the hub motor is installed on both sides of the semi-trailer, and the active yaw torque is produced by the coordinated control of the driving/braking torque of the hub motor and the differential braking of the mechanical braking system. To prevent wheel locking, the slip rate of each wheel is controlled. Finally, based on the TruckSim–MATLAB/Simulink cosimulation platform, cosimulation is performed under typical working conditions. The simulation results show that the control strategy proposed in this report is superior to the conventional differential braking control (ESP). It can not only improve the lateral stability of the vehicle more effectively, but also improve the roll stability.

Keywords: tractor semi-trailer; lateral stability; hub motor; diﬀerential braking; coordination

1. Introduction In recent years, tractor semi-trailers have become a major means in the field of transportation due to their economical and efficient advantages [1]. However, because of its characteristics of a long body, complex structure, high centroid at full load, and rear amplification, the tractor semi-trailer is prone to lose lateral stability, for example sideslip; drift; and jackknife, when driving on low adhesion coefficient roads, such as roads affected by rain, snow, and wetness. Since a fully loaded trailer is much heavier than a tractor, the trailer plays a vital role in the lateral stability control of the entire vehicle [2–4]. According to the 2016 statistical report of the Federal Automobile Transportation Administration of the United States Department of Transportation, heavy truck traffic accident deaths accounted for 10% of all traffic accident deaths, and tractor semi-trailers accounted for 61% [5]. The lateral instability of tractor semi-trailers is the main factor in traffic accidents [6]. Increasing environmental pollution and energy depletion have accelerated the research progress of distributed electric vehicles [7–10]. The rear wheel hub drive vehicle is driven by distributed power. Through the accurate control of the wheel hub motor of the rear wheel hub drive vehicle, the handling and safety of the vehicle can be significantly improved [11–13]. Many scholars have researched the lateral stability control of tractor semi-trailers and hub-driven vehicles.

Energies 2020, 13, 6317; doi:10.3390/en13236317 www.mdpi.com/journal/energies Energies 2020, 13, 6317 2 of 23

Seyed Hossein et al. [14] proposed an adaptive control algorithm that combined yaw moment with differential braking based on the Lyapunov direct method. The algorithm was used to estimate unknown parameters of the vehicle, and the final results showed that the adaptive control algorithm can improved the vehicle driving stability effectively. Takenaga et al. [15] proposed a tractor semi-trailer path tracking controller based on the fuzzy control law and built a joint controller of travel path error feedback and driver preview feedforward, effectively ensuring the tracking of the ideal path for the tractor semi-trailer. Kharrazi [16] proposed a robust control policy to improve the yaw stability performance of a heavy multi-trailer considering the variation of parameters, and the control results showed that the control strategy greatly reduced the magnification of the rear end, and the braking system did not interfere with the steering system of the tractor. Yang et al. [17] built a vehicle lateral stability controller based on MATLAB/Simulink. The upper controller and the lower controller are, respectively, responsible for the production of active torque and the distribution of braking torque; the strategy improved the yaw stability and roll stability of the entire vehicle. Eunhyek et al. [18] designed an integrated chassis control strategy for four-wheel braking and front/rear axle drive torque with the goal of improving the tire slip ratio and implemented the optimal allocation of braking and driving moments using a monitor-based hierarchical structure. Chen et al. [19] proposed two different control methods for the stability control of wheel drive electric vehicles, namely, unsteady state control and continuous control, and adopted sliding mode controller to control the stability of the vehicle in unstable state. Chae et al. [20] used the multi-modal sliding mode controller to improve the driving stability of hub-driven electric vehicles and solved the problem of controller robustness under the condition of variable parameters. Huang et al. [21] developed a transverse stability control strategy for vehicles driven by hub motors based on regional pole assignment. The combined action of the driving force of hub motors and the braking force that generates the additional yawing torque is utilized; the simulation results showed that this strategy can improve the vehicle handling stability and achieve torque distribution more reliably. Wang Zhenbo et al. [22] developed the stability control strategy of four-wheel independent drive electric vehicle (FWIA-EV) based on optimal control to improve the vehicle handling stability. This control strategy performs better than the rule-based control strategy and has the ability of online implementation. Changfu Zong et al. [23] used the method of independently defining the understeer gradient of the tractor and the semitrailer to study the handling stability of the semitrailer train and analyzed the impact of vehicle structural parameters on the understeer grads. Xiujian Yang et al. [24] investigated a strategy to independently control the direct yawing torque of the tractor and trailer in order to study the stability of the lower semi-trailer under extreme operation, which reduced the complexity of the brake wheel selection decision and obtain a better control effect. At present, most research on the lateral stability of tractor semi-trailers is based on improving the performance of a single aspect of the vehicle, such as braking, driving, steering, or suspension, which have some deficiencies. Differential braking control is a mature and overall tactics to enhance vehicle driving stability [25]. The wheel hub motor is independent and controllable, with high transmission efficiency and accurate and rapid torque output. In comparison with traditional vehicles, it has unique advantages in stability control. When the regenerative braking condition is met, some of the vehicle’s kinetic energy will be transformed into electric energy for recycling. This can not only improve the energy utilization rate, but also reduce the wear of the mechanical braking system [26–30]. However, at present, the research on improving the lateral stability of vehicles through a hub motor is mainly for passenger cars, and there is little research on tractor semi-trailers. Giving the shortcomings of the existing research, we proposed a coordinated control strategy in the area of hub motor and differential braking. By installing hub motors on both sides of the semi-trailer, taking advantage of its distributed drive structure, the lateral stability of the tractor semi-trailer is improved. Energies 2020, 13, 6317 3 of 23 Energies 2020, 13, x FOR PEER REVIEW 3 of 24

2. Dynamic Model

2.1. Nonlinear Tractor Semi-TrailerSemi-Trailer ModelModel A nonlinear tractor semi semi-trailer-trailer model was established by using the vehicle dynamics simulation software TruckSim, in wh whichich 2A Tractor Tractor w/1A w/1A Van Trailer was selected as the real vehicle model, including a tractor steeringsteering shaft,shaft, aa drivingdriving shaft,shaft, andand semi-trailersemi-trailer axle.axle. Modelling for the overall appearance was carried out, as shown in Figure 11.. TheThe mainmain parametersparameters ofof thethe TruckSimTruckSim modelmodel areare revealed in Table1 1[31 [31].].

Figure 1. EstablishmentEstablishment of TruckSim nonlinear vehicle model.model.

Table 1. Main parameters of TruckSim model [[31]31]..

ValueValue CategoryCategory Unit Unit TractorTractor Semi-TrailerSemi-Trailer No-loadNo-load mass mass4457 4457 60006000 kg kg Yaw inertiaYaw inertia 34 34,823,823 54,00054,000 Kg m2 Kg m2 2 Roll inertiaRoll inertia 2287 2287 10,14010,140 Kg m Kg m2 WheelbaseWheelbase 3500 3500 67006700 (to hinge(to hinge point) point) m m Wheel distance 2030/1863 1863 mm Wheel distanceWheel radius 2030/1863 510/510 5101863 mm mm WheelCentroid radius height510/510 1173 1935510 mm mm CentroidMaximum height brake pressure1173 0.7 0.71935 mpa mm Maximum brakeEngine pressure power 0.7 225 -0.7 kw mpa EngineTransmission power 7-speed225 MT -- - kw Transmission 7-speed MT - - 2.2. Reference Model 2.2. ReferenceIn this paper, Model the reference model adopts a simpliﬁed four degrees of freedom monorail yaw model, whichIn canthis produce paper, the the reference require yaw model rate adopts of tractor a simplified semi-trailer, four as degrees shown of in Figurefreedom2. monorail The standard yaw model,expression which form can of produce the model the is shownrequire inyaw Equations rate of (1)–(4).tractor semi-trailer, as shown in Figure 2. The standard expression form of the model is shown in Equations (1)–(4). m a = F + Fy F (1) 1 y1 y1 2 − hy . I γ = F a Fy b + F lp (2) z1 1 y1 1 − 2 1 hy

m2ay2 = Fy3 + Fhy (3) . Iz γ = F a Fy b (4) 2 2 hy 2 − 3 2

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Y δf

Fy1 · β1

1 x V 1 1 a y V 1 m

Fhy p l 1 l

1 a2 b l2 2 Vx

2 · β 2 Vy 2 m θ 2 b2 Fy

3 X Fy O

FigureFigure 2. 2.Linear Linear four four degrees degrees of of freedom freedom monorail monorail reference reference model. model.

In the formula, m1 and m2 are the mass of the tractor and semi-trailer, respectively; Iz1 and Iz2 are 푚1푎푦1 = 퐹푦1 + 퐹푦2 − 퐹ℎ푦 (1) the yaw moment of inertia of the tractor and semi-trailer, respectively; γ1 and γ2 are the yaw rate of the tractor and semi-trailer, respectively; Fy1, Fy2, and Fy3 correspond to the lateral tire forces of the 퐼푧1훾̇1 = 퐹푦1푎1 − 퐹푦2푏1 + 퐹ℎ푦푙푝 (2) front axle of the tractor, the rear axle of the tractor, and the semi-trailer tires; Fhy is the lateral pulling force between the tractor and the semi-trailer; a1 and b1 are the distances from tractor centroid to its 푚2푎푦2 = 퐹푦3 + 퐹ℎ푦 (3) front and rear axles; a2 and b2 are the distances from trailer centroid to towing point and trailer axle; lp is the distance from tractor centroid to towing point; ay1 and ay2 are the lateral acceleration of the tractor and semi-trailer, respectively,퐼푧 and2훾̇2 = 퐹ℎ푦푎2 − 퐹푦3푏2 (4)

In the formula, 푚1 and 푚2 are the mass of. the tractor and semi-trailer, respectively; 퐼푧1 and ay1 = vy1 + vx1γ1 퐼푧2 are the yaw moment of inertia of the tractor and semi-trailer, respectively; 훾1 and 훾2 are the .. . yaw rate of the tractor and semi-trailer. , respectively. ; 퐹푦1, .퐹푦2, and 퐹푦3 correspond to the lateral tire ay2 = vy1 + vx1γ1 lpγ1 a2 γ1 + θ + vx1θ forces of the front axle of the tractor, the rear axle− of −the tractor, and the semi-trailer tires; 퐹ℎ푦 is the lateralIn pulling the following, force between approximate the tractorassumptions and thevx1 semi= vx2-trailer= vx.; Here,푎1 andvx1 is푏1 the are longitudinal the distances speed from of tractortractor, centroidvx2 is the to longitudinal its front and speed rear axles of trailer.; 푎2 and 푏2 are the distances from trailer centroid to towing pointThe and yaw trailer rate axle conforms; 푙푝 is the to the distance following from formula: tractor centroid to towing point; 푎푦1 and 푎푦2 are the lateral acceleration of the tractor and semi-trailer, respectively,. and γ2 = γ1 + θ (5) 푎푦1 = 푣̇푦1 + 푣푥1훾1 The lateral force of the axletrees of the tractor semi-trailer can be approximated as the following formula: 푎푦2 = 푣̇푦1 + 푣푥1훾1 − 푙푝훾̇1 − 푎2(훾̇1 + 휃̈) + 푣푥1휃̇ Fyi = Kyiαi (6) In the following, approximate assumptions 푣푥1 = 푣푥2 = 푣푥. Here, 푣푥1 is the longitudinal speed of tractor,In the 푣 formula,푥2 is thei longitudinaltakes 1, 2, 3; speedKy1, K yof2, trailer.Ky3, respectively, represent the cornering characteristics of the tractor’sThe yaw front rate andconforms rear axletree to the following wheels and formula: the trailer’s tires; α1, α2, α3, respectively, represent the sideslip angles of the tractor’s front and rear axle wheels and the trailer’s wheels, and 훾2 = 훾1 + 휃̇ (5) (v +α γ ) The lateral force of the axletrees of theα =tractorδ semiy1 -trailer1 1 can be approximated as the following(7) 1 f vx formula: − (b γ v ) 퐹 α ==퐾 훼1 1− y1 (8) 푦𝑖 2 푦𝑖 𝑖 vx (6) . In the formula, 푖 takes 1, 2, 3; 퐾푦1 , v퐾푦2(,lp +퐾l 푦)3γ , l respectivelyθ , represent the cornering y1− 2 1− 2 (9) α3 = θ v characteristics of the tractor's front and rear− axletreex wheels and the trailer’s tires; 훼1 , 훼2 , 훼3 , respectively, represent the sideslip angles of the tractor's front and rear axle wheels and the trailer’s wheels, and

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(푣푦1 + 훼1훾1) 훼1 = 훿푓 − (7) 푣푥 (푏1훾1 − 푣푦1) 훼2 = (8) 푣푥 Energies 2020, 13, 6317 5 of 23 (푣푦1 − (푙푝 + 푙2)훾1 − 푙2휃̇) 훼3 = 휃 − (9) 푣푥 FromFrom Equations Equations (1)–(9), (1)–(9), the the steady-state steady-state response response expression expression of the of tractor’s the tractor’s yaw rate yaw of ratethe tractor of the semi-trailertractor semi can-trailer be obtained can be obtained as: as: 푣 v 훾 ∗ =γ = 푥 x 훿 δ 1 1∗ l (1+K2v2)푓 f (10)(10) 푙1(1 +1 퐾푠푣푥s)x amongamong them,them, 푏 푙 푚 + (b푏l m− +푙 ()b푏 l푚p)b m 푎 푙 a푚l m++(푎a ++lp)푙b )m푏 푚 1 2 1 112 1 푝 1−2 22 2 1 21 21 1 11 푝2 22 2 퐾푠 = Ks = 2 − 2 (11)(11) 2 l l2Ky1 l2l2Ky2 푙1 푙2퐾푦1 1 − 푙11푙2퐾푦2 ItIt cancan bebe obtainedobtained fromfrom EquationEquation (5)(5)that that thethe yawyaw raterate ofof thethe semi-trailersemi-trailer andand thethe tractortractor isis equalequal ∗ ∗ whenwhen turningturning inin a a steady steady state, state, that that is, is,γ 2훾∗2 ==γ훾11∗

2.3.2.3. HubHub MotorMotor ModelModel

TheThe hubhub motormotor producesproduces activeactive yawyaw torquetorque byby applyingapplying requirementrequirementtorque torque to tothe the tiretireto to enhanceenhance the trailer’s lateral stability. We choose the permanent magnet synchronous motor as the hub motor the trailer’s lateral stability. We choose the permanent magnet synchronous motor as the hub motor because of its high power density, high power, and small torque ripple. The idea expression is because of its high power density, high power, and small torque ripple. The idea expression is =1 1 푇 =Tm τs+푇1 Te (1(12)2) 푚 휏푠 + 1 푒 where 푇 is the motor output torque, 푇 is the expected torque of the motor, and 푠 is the Laplace where Tm푚is the motor output torque, Te푒 is the expected torque of the motor, and s is the Laplace transformtransform operator.operator. ItIt isis assumed assumed thatthat thethe motor motor controlcontrol systemsystem cancan achieveachieve thethe desireddesired torquetorque accurately,accurately, but but the the phase phase didifferenceﬀerence intervalinterval is τ;τ; in in addition, addition, the the motor’s motor’s output output torque torque is restricted is restricted by bythe the maximum maximum torque. torque.

−푇푚푎푥 ≤ 푇푚 ≤ 푇푚푎푥 (13) Tmax Tm Tmax (13) − ≤ ≤ 3.3. DesignDesign ofof ControlControl SystemSystem

3.1.3.1. TheThe GeneralGeneral ArchitectureArchitecture ofof thethe ControlControl SystemSystem TheThe referencereference modelmodel takestakes thethe frontfront wheelwheel angleangle outputoutput byby TruckSimTruckSim asas thethe input,input, outputsoutputs thethe ideal value of yaw rate of the tractor and semi-trailer, and the yaw rate deviations e and e are ideal value of yaw rate of the tractor and semi-trailer, and the yaw rate deviations e1 and e22 are obtained,obtained, respectively,respectively,after after determiningdetermining thethe didifferenceﬀerence fromfrom thethe actualactual yawyaw raterate outputoutput of TruckSim. TheThe upperupper controllercontroller takestakes the eeand andec ecas as input, input, and and outputs outputs the the additional additional tractor tractor and and semi-trailer semi-trailer yaw moments ∆M and ∆M , respectively. The lower controller allots the driving torque and braking yaw moments Δ푀11 and Δ푀2, respectively. The lower controller allots the driving torque and braking torquetorque generatedgenerated byby thethe wheelwheel hubhub motormotor andand thethe mechanicalmechanical brakingbraking torquetorque ofof thethe semi-trailer.semi-trailer. Finally,Finally, toto preventprevent wheelwheel locking,locking, thethe slipslip raterate ofof eacheach wheelwheel isis monitoredmonitored byby thethe slipslip raterate controller. TheThe controlcontrol logic logic structure structure is is illustrated illustrated in in Figure Figure3 .3.

FigureFigure 3.3. ControlControl ﬂowchartflowchart ofof laterallateral stabilitystability ofof tractortractor semi-trailer.semi-trailer.

3.2. Upper Controller The control of the tractor semi-trailer needs to be real time and complex; compared with single proportional–integral–derivative (PID) control and fuzzy control, the fuzzy proportional– integral–derivative (PID) control can not only introduce people’s control experience and methods and Energies 2020, 13, x FOR PEER REVIEW 6 of 24

3.2. Upper Controller

EnergiesThe2020 control, 13, 6317 of the tractor semi-trailer needs to be real time and complex; compared with single6 of 23 proportional–integral–derivative (PID) control and fuzzy control, the fuzzy proportional–integral– derivative (PID) control can not only introduce people's control experience and methods and standardize, but also realize the automatic adjustment of parameters K , K , and K to meet the control standardize, but also realize the automatic adjustment of parametersp 퐾i 푝, 퐾𝑖, andd 퐾푑 to meet the ofcontrol nonlinear of nonlinear system [32 system], so the [32], upper so controller the upper adopts controller fuzzy adopts PID control. fuzzy The PID upper control. controller The upper takes thecontroller deviation takese and the variationdeviationrate e andec ofvariation the actual rate yaw ec rateof the and actual reference yaw rate yaw and rate refe of therence tractor yaw andrate trailer as the input, and the additional yaw moment ∆M and ∆M of the tractor and trailer as the of the tractor and trailer as the input, and the additional 1yaw moment2 Δ푀1 and Δ푀2 of the tractor output.and trailer The as fuzzy the output. PID control The fuzzy adaptive PID structurecontrol adaptive is depicted structure in Figure is depicted4[33,34]. in Figure 4 [33,34].

Fuzzy Fuzzification defuzzification reasoning

e ec Kp Ki Kd yd y e PID object controller e de/dt ec

FigureFigure 4.4. Fuzzy proportional–integral–derivativeproportional–integral–derivative (PID)(PID) controlcontrol structurestructure [[3333,,3434]]..

TheThe fuzzy fuzzy universe universe of ofe and푒 andec is 푒푐 [ 3,is3], [−3, which 3], which obeys a obeys Gaussian a Gaussian distribution. distribution. The fuzzy The universe fuzzy − ofuniverse the outputs of theK outputsp, Ki, and 퐾푝K,d 퐾is𝑖, [0,and 1], 퐾 which푑 is [0, obeys 1], which a trigonometric obeys a trigonometric function. The function. fuzzy inputThe fuzzy and outputinput and language output variableslanguage arevariables ml (minus are ml large), (minus mm large), (minus mm middle), (minus middle), ms (minus ms (minus small), zesmall), (zero), ze ps(zero), (plus ps small), (plus pm small), (plus pm middle), (plus and middle), pl (plus and large)—seven pl (plus large) fuzzy—seven language fuzzy sets. language The membership sets. The functionsmembership of e, functions ec, Kp , K ofi, ande, ec,K d퐾are푝 , 퐾 shown𝑖 , and in퐾푑 Figure are shown5. The in surface Figure relationship 5. The surface between relationship input parameters and output parameters of fuzzy PID control is shown in Figure6. The fuzzy rules of Energiesbetween 2020 input, 13, x FORparameters PEER REVIEW and output parameters of fuzzy PID control is shown in Figure 76. of The 24 outputsfuzzy rulesKp, Kofi, outputs and Kd are퐾푝, presented 퐾𝑖, and 퐾 in푑 Tablesare presented2–4[33,35 in]. Tables 2–4 [33,35].

ml mm ms ze ps pm pl ml mm ms ze ps pm pl 1 1

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

0 0

-3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 e ec

ml mm ms ze ps pm pl ml mm ms ze ps pm pl 1 1

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

0 0

-1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 Kp Ki ml mm ms ze ps pm pl 1

0.8

0.6

0.4

0.2

-1 -0.5 0 0.5 1 Kd

Figure 5. Membership functions of e, ec, Kp, Ki, and Kd [33,34]. Figure 5. Membership functions of e, ec, 퐾푝, 퐾𝑖, and 퐾푑 [33,34].

Figure 6. Surface relationship between input and output.

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ml mm ms ze ps pm pl ml mm ms ze ps pm pl 1 1

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

0 0

-3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 e ec

ml mm ms ze ps pm pl ml mm ms ze ps pm pl 1 1

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

0 0

-1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 Kp Ki ml mm ms ze ps pm pl 1

0.8

0.6

0.4

0.2

-1 -0.5 0 0.5 1 Kd

Energies 2020, 13, 6317 7 of 23

Figure 5. Membership functions of e, ec, 퐾푝, 퐾𝑖, and 퐾푑 [33,34].

FigureFigure 6.6. SurfaceSurface relationship between between input input and and output. output.

Table 2. Fuzzy rule table of Kp.

ec Kp ml mm ms ze ps pm pl ml pl pl pl pm ps ze ze mm pl pl pm ps ps ze ms ms pm pl ps ps ze ms ms e ze pm pm ps ze ms mm mm ps ps ps ze ms ms mm mm pm ps ze ms mm mm mm ml pl ps ze mm mm mm ml ml

Table 3. Fuzzy rule table of Ki.

ec Ki ml mm ms ze ps pm pl ml pl pl pl pm ps ze ze mm pl pl pm ps ps ze ms ms pm pl ps ps ze ms ms e ze pm pm ps ze ms mm mm ps ps ps ze ms ms mm mm pm ps ze ms mm mm mm ml pl ps ze mm mm mm ml ml

Table 4. Fuzzy rule table of Kd.

ec Kd ml mm ms ze ps pm pl ml ms ms ml ml ml mm ps mm ps ms ml mm mm ms ze ms ze ms mm mm ms ms ze e ze ze ms ms ms ms ms ze ps ze ze ze ze ze ze ps pm pl ms ps ps ps ps pl pl pl pm pm pm ps ps pl Energies 2020, 13, 6317 8 of 23

3.3. Lower Controller General brake torque distribution rules have some shortcomings: single-wheel braking may cause a larger braking torque to concentrate on one wheel, easily resulting in locking. Single-side braking eﬃciency is low. Full braking makes the semi-trailer unable to produce additional yaw moment, which easily causes sideslip, jackkniﬁng, and other unstable phenomena. The lower-level torque distribution strategy proposed determines the target drive/brake wheels and distributes the drive/brake torque according to the additional yaw moment, steering direction, yaw rate deviation, wheel vertical load, and other signals output by the upper-level controller.

3.3.1. Determination of Braking Torque 1. Moment allocation of tractor The active yaw torque ∆M1 of the tractor is produced by diﬀerential braking. The relation between theEnergies active 2020 yaw, 13, x torque FOR PEER and REVIEW the braking torque of tractor semi-trailer is indicated in Figure7. 9 of 24

Figure 7. Relationship between yaw torque and braking force of tractor semi-trailer.semi-trailer.

TheThe twotwo tirestires withwith adjacentadjacent coaxialcoaxial lineslines areare simpliﬁedsimplified toto aa singlesingle tire.tire. Assuming thatthat the braking momentmoment ofof thethe wheel when the wheel is not locked is approximately proportional to its vertical load load;; thethe brakingbraking momentmoment ofof everyevery tiretire ofof thethe tractortractor isis shownshown below.below. (i)(i) WhenWhen ∆ΔM푀1 >> 0,0, it is necessary to brake the left wheel, and the braking moments ofof each wheel onon thethe leftleft sideside ofof thethe tractortractor areare asas follows:follows: 퐹푍퐿1 4∆푀1 푇퐿1 = 퐹푋퐿1푅1 = FZL∙ 1 4∆M1∙ 푅1 (14) TL1 = FXL퐹1푍퐿R1 += 퐹푍퐿2 푡푤1 + 푡푤2 R1 (14) FZL1+FZL2 · tw1+tw2 · 퐹푍퐿2 4∆푀1 푇 = 퐹 푅 = ∙ ∙ 푅 퐿2 푋퐿2 2 F 4∆M 2 (15) T = F 퐹푍퐿R1 += 퐹푍퐿2ZL푡2푤1 + 푡푤21 R (15) L2 XL2 2 FZL1+FZL2 tw1+tw2 2 where 퐹푋퐿1 and 퐹푋퐿2 are the longitudinal braking force of· the left· wheels on the front and rear axle F F whereof the tractorXL1 and, respectivelyXL2 are the; 퐹푍퐿 longitudinal1 and 퐹푍퐿2 brakingare the vertical force of loads the left on wheelsthe left wheel on the of front the front and rearand axlerear F F ofaxis the of tractor, the tractor respectively;, respectivelyZL1;and 푡푤1 andZL2 are푡푤2 the are vertical the track loads widths on the of leftthe wheelfront and of the rear front axles and of rear the t t axistractor of, therespectively tractor, respectively;; and 푅1 and w푅1 2and are thew2 are rolling the trackradius widths of the offront the and front rear and axle rear wheels axles ofof thethe tractor,tractor, respectively;respectively. and R1 and R2 are the rolling radius of the front and rear axle wheels of the tractor,(ii) respectively. When Δ푀1 < 0, it is necessary to brake the right wheel, and the braking moments of each wheel(ii) on When the right∆M 1side< 0, of it the is necessarytractor are to as brake follows: the right wheel, and the braking moments of each wheel on the right side of the tractor are as follows:퐹푍푅1 4∆푀1 푇푅1 = 퐹푋푅1푅1 = ∙ ∙ 푅1 (16) 퐹푍푅1 + 퐹푍푅2 푡푤1 + 푡푤2 FZR1 4∆M1 T = F R 퐹푍푅=2 4∆푀1 R (16) R1 XR1 1 FZR1+FZR2 tw1+tw2 1 푇푅2 = 퐹푋푅2푅2 = ∙ · ∙ 푅· 2 (17) 퐹푍푅1 + 퐹푍푅2 푡푤1 + 푡푤2 FZR2 4∆M1 where 퐹푋푅1 and 퐹푋푅2 are the longitudinalTR2 = FXR2 brakingR2 = force of the rightR2 wheels on the front and rear (17)axis FZR1+FZR2 · tw1+tw2 · of the tractor, respectively, and 퐹푍푅1 and 퐹푍푅2 are the vertical loads on the right wheel of the front and rear axle of the tractor, respectively. 2. Moment Allocation of semi-trailer The tire on both sides of the semi-trailer are equipped with hub motors. First, the torque distribution control method combining differential drive and differential braking used as realize part of additional yaw torque—that is, the differential drive of one wheel hub motor and the differential braking of the other wheel hub motor are adopted. In comparison with the differential braking or differential drive alone, the adhesion requirements of the road surface are significantly reduced, the torque output of the wheel is reduced, and the adhesion between the tire and the ground is improved. The principle display as Figure 8.

Energies 2020, 13, 6317 9 of 23

where FXR1 and FXR2 are the longitudinal braking force of the right wheels on the front and rear axis of the tractor, respectively, and FZR1 and FZR2 are the vertical loads on the right wheel of the front and rear axle of the tractor, respectively. 2. Moment Allocation of semi-trailer The tire on both sides of the semi-trailer are equipped with hub motors. First, the torque distribution control method combining differential drive and differential braking used as realize part of additional yaw torque—that is, the differential drive of one wheel hub motor and the differential braking of the other wheel hub motor are adopted. In comparison with the differential braking or differential drive alone, the adhesion requirements of the road surface are significantly reduced, the torque output of the wheel is reduced, and the adhesion between the tire and the ground is improved. The principle display as Figure8. EnergiesEnergies 20202020,, 1133,, xx FORFOR PEERPEER REVIEWREVIEW 1010 ofof 2424

FigureFigure 8.8.8. PrinciplePrinciple ofof yawyaw momentmoment producedproduced byby brakingbraking forceforce andandand driving drivingdriving force forceforce at atat the thethe same samesame time. time.time.

BecauseBecauseBecause of ofof the thethe small smallsmall braking brakingbraking (driving) (driving)(driving) capacity capacitycapacity of of theof thethe motor motormotor in the inin thethe power powerpower system ssystemystem of the ofof wheel thethe wheelwheel hub motor,hubhub motor,motor, the ability thethe abilityability to generate toto generategenerate yaw momentyawyaw momentmoment is limited, isis limited,limited, so mechanical soso mechanicalmechanical braking brakingbraking is adopted isis adoptedadopted for the forfor part thethe thatpartpart does thatthat doesdoes not meet notnot meetmeet the requirements. thethe requirements.requirements. The collaborativeTheThe collaborativecollaborative control controlcontrol strategy strategystrategy is shown isis shownshown in Figure inin FigureFigure9. At 9.9. the AtAt samethethe samesame time, time,time, the designthethe designdesign capacity capacitycapacity of the ooff the brakethe brakebrake can cancan be reduced bebe reducedreduced because becausebecause of the ofof thethe electric electricelectric braking brakingbraking capacity capacitycapacity of theofof thethe wheel wheelwheel motor. motor.motor.

ssttaarrtt

NN ||ΔΔMM22|| ||ΔΔMMmm--mmaaxx||

ΔΔMMmm==ΔΔMMmm--mmaaxx ΔΔMMdd==（（||ΔΔMM22||——||ΔΔMMmm||))ssggnn((ΔΔMM22)) ΔΔMMmm==ΔΔMM22

FigureFigureFigure 9. 9.9. CooperativeCooperative controlcontrolcontrol strategy strategystrategy chart. chart.chart.

Here,Here,Here, ∆ ΔΔM푀푀m푚푚−−max푚푎푥푚푎푥is isis the thethe maximum maximummaximum yaw yawyaw moment momentmoment provided providedprovided by outputbyby outputoutput torque torquetorque of hub ofof motorshubhub motorsmotors on both onon − sidesbothboth sides ofsides semi-trailer, ofof semisemi--trailer,trailer,∆Mm ΔisΔ푀푀 the푚푚 is activeis thethe activeactive yaw torque yawyaw torquetorque to be produced toto bebe producedproduced by the byby hub thethe motor, hubhub motor,motor, and ∆M andandd is ΔΔ the푀푀푑푑 activeisis thethe active yawactive torque yawyaw torquetorque to be produced toto bebe producedproduced by the byby semi-trailer thethe semisemi--trailer mechanicaltrailer mechanicalmechanical braking brakingbraking system. system.system. TheThe maximummaximum yawyaw torquetorque toto thethe semisemi--trailertrailer centroidcentroid whenwhen drivendriven byby aa singlesingle wheelwheel hubhub motomotorr isis asas follows:follows: Δ푀 푇 푡 ̂ Δ푀푚푚−−푚푎푥푚푎푥 푇푚푎푥푚푎푥푡푤푤33 ΔΔ̂푀푀푚푚 == == (18)(18) 22 22푅푅33 wherewhere 푇푇푚푎푥푚푎푥 isis thethe largestlargest outputoutput torquetorque ofof thethe hubhub motor,motor, 푡푡푤푤33 isis thethe wheelwheel tracktrack ofof thethe semisemi--trailer,trailer, andand 푅푅33 isis thethe wheelwheel rollingrolling radiusradius ofof thethe semisemi--trailtrailer.er. TakingTaking ΔΔ푀푀22 >> 00 asas an an example, example, the the torque torque distribution distribution strategy strategy is is further further explained. explained. It It stipulatesstipulates thatthat thethe brakingbraking momentmoment isis positivepositive andand thethe drivingdriving momentmoment isis negative.negative. AtAt thisthis time,time, L3L3 isis brakingbraking andand R3R3 isis drivingdriving (when(when ΔΔ푀푀22 << 00,, itit isis onlyonly necesnecessarysary toto changechange thethe leftleft andand rightright roundsrounds inin thethe strategystrategy andand considerconsider thethe absoluteabsolute valuevalue ofof ΔΔ푀푀22 forfor thethe calculation).calculation). ̂ (i)(i) WhenWhen ΔΔ̂푀푀푚푚 ≥≥ 00..55ΔΔ푀푀22,, thethe yawyaw momentmoment aa ofof thethe semisemi--trailertrailer ΔΔ푀푀22 isis providedprovided byby thethe wheelwheel hubhub motor,motor, andand thethe torquetorque distridistributedbuted byby thethe wheelwheel ofof thethe semisemi--trailertrailer isis asas follows:follows: ΔΔ푀푀22 ∗∗푅푅33 푇푇퐿퐿33 == −−푇푇푅푅33 == (19)(19) 푡푡푤푤33

Energies 2020, 13, 6317 10 of 23

The maximum yaw torque to the semi-trailer centroid when driven by a single wheel hub motor is as follows: ∆Mm max Tmaxtw3 ∆ˆMm = − = (18) 2 2R3 where Tmax is the largest output torque of the hub motor, tw3 is the wheel track of the semi-trailer, and R3 is the wheel rolling radius of the semi-trailer. Taking ∆M2 > 0 as an example, the torque distribution strategy is further explained. It stipulates that the braking moment is positive and the driving moment is negative. At this time, L3 is braking and R3 is driving (when ∆M2 < 0, it is only necessary to change the left and right rounds in the strategy and consider the absolute value of ∆M2 for the calculation). (i) When ∆ˆMm 0.5∆M , the yaw moment a of the semi-trailer ∆M is provided by the wheel ≥ 2 2 hub motor, and the torque distributed by the wheel of the semi-trailer is as follows:

∆M2 R3 TL3 = TR3 = ∗ (19) − tw3 Energies 2020, 13, x FOR PEER REVIEW 11 of 24 (ii) When ∆ˆMm < 0.5∆M2, the active yaw torque is ﬁrst provided by the hub motor, and the insuﬃ(ii)cient When additional Δ̂푀푚 < yaw0.5Δ moment푀2, the isactive provided yaw bytorque mechanical is first braking.provided The by momentthe hub allocatedmotor, and by the wheelsinsufficient of the additional semi-trailer yaw are moment as follows: is provided by mechanical braking. The moment allocated by the wheels of the semi-trailer are as follows: TR3 = Tmax (20) 푇푅3 = −푇푚푎푥− (20) 2푅 (Δ푀 − 2Δ̂푀 ˆ ) 3 22R3∆M2 2∆푚Mm 푇퐿3 = 푇T푚푎푥L3 =+T푇max푑 =+ Td = t − (21)(21) 푡푤3 w3 Here, T푇d푑 isis the the mechanical mechanical braking braking torque torque of of the the semi-trailer semi-trailer wheel. wheel.

3.3.2. Formulation of Target Wheel BrakingBraking (Driving)(Driving) RulesRules The curve of the relationship between the additional yaw torque and the portrait ground braking force ofof everyevery tiretire isisindicated indicated in in Figure Figure 10 10.. The The Figure Figure 10 10 indicates indicates that that applying applying the the braking braking torque torque to theto the outer outer front front wheel wheel and and outer outer rear rear wheel wheel leads leads to understeering; to understeering; the outer the frontouter wheelfront haswheel a higher has a ehigherﬃciency. efficiency. In contrast, In contrast, applying applying the braking the braking torque torque to the innerto the frontinner wheel front andwheel inner and rearinner wheel rear leadswheel to leads excessive to excessive steering; steering; the inner the rearinner wheel rear wheel has a higherhas a higher eﬃciency efficiency [36]. It [36]. is ruled It is ruled that the that front the wheelfront wheel turns turns left in left a positivein a positive direction, direction, and and the yawthe yaw velocity velocity and and active active yaw yaw torque torque are are positive positive in thein the counterclockwise counterclockwise direction. direction. The The decision decision rules rules for for the the target target wheel wheel of theof the tractor tractor are are presented presented in Tablein Table5, and 5, and those those for thefor the semi-trailer semi-trailer are presentedare presented in Table in Table6. 6.

To the inside 5000 Inner rear wheel

Outer rear wheel

M(N·m) Δ

5000 Inner Longitudinal braking force

front wheel Fx(N) Yaw moment Yaw

Outer front wheel

-5000 To the outside

Figure 10. VariationVariation curve of vehicle yaw moment and wheel longitudinal braking force.

Table 5. Decision rules for target wheel of tractor.

Reference Yaw Actual Yaw Yaw Rate Deviation Steering Target Brake ∗ ∗ Rate (후ퟏ) Rate (후ퟏ) (풆ퟏ = 후ퟏ − 후ퟏ) Characteristics Wheel + + + understeer L2 + + — oversteer R1 + + 0 neutral steer \ + 0 + understeer L2 + — + understeer L1 0 — + oversteer L1 0 + — oversteer R1 0 0 0 neutral steer \ — — — understeer R2 — — + oversteer L1 — — 0 neutral steer \ — + — understeer R1 — 0 — understeer R2

Energies 2020, 13, 6317 11 of 23

Table 5. Decision rules for target wheel of tractor.

Reference Yaw Rate Actual Yaw Rate Yaw Rate Deviation Steering Target Brake (γ ) (γ ) (e = γ γ ) Characteristics Wheel 1∗ 1 1 1∗ − 1 + + + understeer L2 + + oversteer R1 − + + 0 neutral steer \ + 0 + understeer L2 + + understeer L1 − 0 + oversteer L1 − 0 + oversteer R1 − 0 0 0 neutral steer \ understeer R2 − − − + oversteer L1 − − 0 neutral steer − − \ + understeer R1 − − 0 understeer R2 − −

Table 6. Decision rules for target wheel of semi-trailer.

Reference Yaw Rate Actual Yaw Rate Yaw Rate Deviation Steering Target Wheel (γ∗ ) (γ ) (e2 = γ∗ γ2) Characteristics 2 2 2− Drive Brake + + + understeer R3 L3 + + oversteer L3 R3 − + + 0 neutral steer \\ + 0 + understeer R3 L3 + - + understeer R3 L3 0 - + oversteer R3 L3 0 + oversteer L3 R3 − 0 0 0 neutral steer \\ understeer L3 R3 − − − + oversteer R3 L3 − − 0 neutral steer − − \\ + understeer L3 R3 − − 0 understeer L3 R3 − −

4. Slip Rate Controller After determining the driving (braking) torque distribution of the target wheel, to ensure that the wheel does not lock and further improve the lateral stability, slip rate control is applied to each wheel. The real-time slip rate can be obtained by the speed u and wheel speed n output by TruckSim. The slip ratio calculation formula is as follows: u rω λ = −u (22) where u is the vehicle linear velocity, r is the tire radius, and ω is the wheel angular speed. Guo et al. [37] pointed out that the variation range of the tire slip rate λ corresponding to the maximum longitudinal adhesion coeﬃcient ϕLmax is generally between 0.08 and 0.3. The curves of the tire longitudinal and lateral adhesion coeﬃcient versus slip rate are depicted in Figure 11. Considering the safety of the vehicle in extreme conditions, the wheel slip ratio of the tire is controlled between 0.15 and 0.20. The values of the increase rate Ki and decline rate Kd are

( 5000Nm Ki = s 5000Nm . (23) Kd = s Energies 2020, 13, x FOR PEER REVIEW 12 of 24

Table 6. Decision rules for target wheel of semi-trailer.

Reference Yaw Actual Yaw Yaw Rate Deviation Steering Target Wheel ∗ ∗ Rate (후ퟐ) Rate (휸ퟐ) (풆ퟐ = 휸ퟐ − 휸ퟐ) Characteristics Drive Brake + + + understeer R3 L3 + + — oversteer L3 R3 + + 0 neutral steer \ \ + 0 + understeer R3 L3 + - + understeer R3 L3 0 - + oversteer R3 L3 0 + — oversteer L3 R3 0 0 0 neutral steer \ \ — — — understeer L3 R3 — — + oversteer R3 L3 — — 0 neutral steer \ \ — + — understeer L3 R3 — 0 — understeer L3 R3

4. Slip Rate Controller Energies 2020, 13, 6317 12 of 23 After determining the driving (braking) torque distribution of the target wheel, to ensure that the wheel does not lock and further improve the lateral stability, slip rate control is applied to each The output value T of the adjusted braking torque is wheel. The real-time slipD rate can be obtained by the speed u and wheel speed n output by TruckSim.  The slip ratio calculation formula is as follows:= +  T(t+1) 푢T(−t) 푟휔KiTs, S < 0.15  휆 = (22)  T(t+1) = T(t), 0.15 < S < 0.20 (24)  푢 where 푢 is the vehicle linear velocity, T 푟 is =theT tire radius,K Ts, S and> 0.20 휔 is the wheel angular speed. (t+1) (t) − d Guo et al. [37] pointed out that the variation range of the tire slip rate 휆 corresponding to the where T is the output value of the braking torque at moment t, T is the output value of the maximum(t) longitudinal adhesion coefficient 휑퐿푚푎푥 is generally between(t+1 )0.08 and 0.3. The curves of thebraking tire longitudinal torque at moment and lateralt + 1, adhesion and Ts is coefficient the calculation versus step slip size. rate are depicted in Figure 11.

1.0

t 0.8 n e i c i f f

e 0.6 o c

n o i

s 0.4 e h d A

0.2

0 20 40 60 80 100 Slip ratio S（％） FigureFigure 11. 11. RelationRelation of of tire tire longitudinal longitudinal and and lateral lateral adhesion adhesion coefficient coeﬃcient versus slip rate. 5. Simulation Analysis Considering the safety of the vehicle in extreme conditions, the wheel slip ratio of the tire is controlledA nonlinear between vehicle 0.15 and model 0.20. was The builtvalues in of TruckSim, the increase and rate its S-function퐾𝑖 and decline was outputrate 퐾푑 to are MATLAB / 5000푁푚 Simulink. Double lane change and steering퐾𝑖 = wheel angle⁄푠 step were selected as the input conditions for { . (23) the joint simulation of TruckSim and Simulink5000푁푚 퐾푑 = ⁄푠 The output value 푇 of the adjusted braking torque is 5.1. Double-Lane-Change퐷 Condition

The initial velocity was defined as vx1 = 100 km/h, and the road surface adhesion coefficient was set to 0.3. From the comparison of animation simulation in Figure 12, it can be seen that without control, the semi-trailer has obvious sideslip, has deviated from the track, the tractor has a large roll, and the whole vehicle loses the ability to track the path; after the cooperative and differential controls are added, the tractor semi-trailer completed the double-lane-change test, and the roll amplitude was also significantly improved. The steering wheel angle input curve is shown in Figure 13. The yaw rate of the semi-trailer and the maximum sideslip angle of the semi-trailer are important indices for evaluating the yaw stability of the tractor semi-trailer. Figures 14 and 15 show that, without control, the yaw rate of the tractor increases continuously after 8 s and reaches the peak value of 45◦/s at 12 s; the yaw rate of the semi-trailer reaches a peak value of 73.68◦/s at 8 s, and the tractor semi-trailer has serious sideslip. After the cooperative and differential controls are added, the absolute value of the peak yaw rate of the tractor decreases to 28.68 and 30.19◦/s, respectively, and the absolute value of the peak yaw rate of the semi-trailer decreases to 48.54 and 40.43◦/s, respectively. At 10.5 s, the yaw rate of the tractor and semi-trailer converges to 0 and is finally stable. Figures 16 and 17 show that, after adding the cooperative and differential controls, the absolute value of the centroid sideslip angle of the tractor decreases from 92.83◦ before the control to 8.67 and 10.92◦, respectively, and the absolute value of the semi-trailer centroid sideslip angle decreases from 61.02◦ before the control to 25.66 and 39.79◦, respectively. In comparison with the differential braking control, the coordinated control Energies 2020, 13, 6317 13 of 23 makes the centroid sideslip angle of the tractor and semi-trailer converge to 0 faster. The hinge angle is the main index to characterize the jackknife stability of the tractor semi-trailer. Figure 18 shows that, after coordinated control and differential braking control are added, the absolute value of the articulation angle decreases from 93.32◦ (folding instability occurred) before control to 27.11 and 44.76◦, respectively. In comparison with differential braking control, collaborative control can make the hinge angle converge to 0 faster. Figures 19 and 20 show that, without control, the roll angle of the tractor reaches the peak value of 4.17◦ at 7 s, the roll angle of the semi-trailer fluctuates greatly, and reaches the peak value of 6.81◦ at 13 s; after the cooperative and differential controls are added, the roll angle of the peak roll angle of the tractor decreases to 2.01 and 2.29◦, respectively, and the peak roll angle of the semi-trailer decreases to 2.96 and 2.51◦, respectively. At 11s, the roll angle of the tractor semi-trailer converges to 0 and is finally stable. Figures 21 and 22 show that, after adding the cooperative and differential controls, the absolute value of the lateral acceleration of the tractor decreases from 8.14◦ before the control to 7.27 and 7.35◦, respectively, and the absolute value of the semi-trailer centroid Energies 2020, 13, x FOR PEER REVIEW 14 of 24 sideslipEnergies 20 angle20, 13, decreasesx FOR PEER from REVIEW 7.06 ◦ before the control to 4.60 and 5.21◦, respectively. 14 of 24

Figure 12. Simulation animation comparison. FigureFigure 12. 12.Simulation Simulation animationanimation comparison.comparison.

Figure 13. Steering wheel. FigureFigure 13. 13. Steering Steering wheel. wheel.

FigureFigure 14. 14. Yaw Yaw rate rate of of tractor. tractor.

Energies 2020, 13, x FOR PEER REVIEW 14 of 24

Figure 12. Simulation animation comparison.

Energies 2020, 13, 6317 14 of 23 Figure 13. Steering wheel.

Energies 2020, 13, x FOR PEER REVIEW 15 of 24 Energies 2020, 13, x FOR PEER REVIEW 15 of 24 Figure 14. YawYaw rate of tractor.

Figure 15. Yaw rate of semi-trailer. FigureFigure 15. YawYaw rate of semi semi-trailer.-trailer.

Figure 16. Sideslip angle of tractor. Figure 16. Sideslip angle of tractor. Figure 16. Sideslip angle of tractor.

Figure 17. Sideslip angle of semi-trailer. Figure 17. Sideslip angle of semi-trailer.

Energies 2020, 13, x FOR PEER REVIEW 15 of 24

Figure 15. Yaw rate of semi-trailer.

Energies 2020, 13, 6317 15 of 23 Figure 16. Sideslip angle of tractor.

Energies 2020,, 13,, xx FORFOR PEERPEER REVIEWREVIEWFigure 17. Sideslip angle of semi semi-trailer.-trailer. 16 of 24

FigureFigure 18.18. Hinge angle. angle.

FigureFigure 19.19. Roll angle angle of of tractor. tractor.

Figure 20. Roll angle of semi-trailer.

Energies 2020, 13, x FOR PEER REVIEW 16 of 24

Figure 18. Hinge angle.

Energies 2020, 13, 6317 16 of 23 Figure 19. Roll angle of tractor.

Energies 2020, 13, x FOR PEER REVIEW 17 of 24 Energies 2020, 13, x FOR PEER REVIEW FigureFigure 20. 20. RollRoll angle angle of semi-trailer.semi-trailer. 17 of 24

Figure 21. LateralLateral acceleration acceleration of of tractor.

Figure 22. Figure 22. LateralLateral acceleration acceleration of of semi semi-trailer.--trailer.trailer.

In summary, when the control is not added, the tractor semi-trailer has serious sideslip and drift and could even jackknife. The control strategy proposed in the article can effectively restrain the sideslip and jackknife instability, improve the yaw stability and jackknife stability of the tractor semi- trailer, and complete the double-lanelane-change test. Compared with differential braking control, coordinated control has a faster response speed and faster convergence speed, and it improves the laterallateral stabilitystability ofof thethe semitrailersemitrailer moremore obviously.obviously.

5.2. Step Steering Condition 푣 The initial velocity was defined as 푣푥1 = 70 km/h,, and the road surface adhesion coefficient was set to 0.3. From the comparison of animation simulation in Figure 23, it can be seen that without control, the tractor semi-trailer experienced sideslip and even jackknife instability and failed to complete the step steering test; after the cooperative and differential controls are added, the tractor semi-trailer has good path tracking ability, which effectively suppresses the occurrence of sideslip and jackknife. The input curve of the steering wheel angle is shown in Figure 24.. FiguresFigures 25 and 26 show that, without control, the yaw rate of the tractor increases continuously from 7 s to a peak of 46°/s at 14 s. The yaw rate of the semi-trailer increases sharply at 17 s and reaches a peak value of 31.81°/s at 18 s. The tractor semi-trailer has sideslip. After coordinated control and differential braking

Energies 2020, 13, 6317 17 of 23

In summary, when the control is not added, the tractor semi-trailer has serious sideslip and drift and could even jackknife. The control strategy proposed in the article can eﬀectively restrain the sideslip and jackknife instability, improve the yaw stability and jackknife stability of the tractor semi-trailer, and complete the double-lane-change test. Compared with diﬀerential braking control, coordinated control has a faster response speed and faster convergence speed, and it improves the lateral stability of the semitrailer more obviously.

5.2. Step Steering Condition

The initial velocity was defined as vx1 = 70 km/h, and the road surface adhesion coefficient was set to 0.3. From the comparison of animation simulation in Figure 23, it can be seen that without control, the tractor semi-trailer experienced sideslip and even jackknife instability and failed to complete the step steering test; after the cooperative and differential controls are added, the tractor semi-trailer has good path tracking ability, which effectively suppresses the occurrence of sideslip and jackknife. The input curve of the steering wheel angle is shown in Figure 24. Figures 25 and 26 show that, without control, the yaw rate of the tractor increases continuously from 7 s to a peak of 46◦/s at 14 s. The yaw rate of the semi-trailer increases sharply at 17 s and reaches a peak value of 31.81◦/s at 18 s. The tractor semi-trailer has sideslip. After coordinated control and differential braking control are added, the maximum Energies 2020, 13, x FOR PEER REVIEW 18 of 24 value of the yaw rate of the tractor decreases to 19.34 and 19.93◦/s, respectively, and finally tends to 10 /s. The peak value of the yaw rate of the semi-trailer decreases to 17.93 and 17.55 /s, respectively, respectively,◦ and finally tends to 10°/s. The peak value of the yaw rate of the semi-tra◦iler decreases andto finally 17.93 and tends 17.55°/s, to 10◦ / respectively,s. The response and speed finally of tends coordinated to 10°/s. control The response is faster speed than that of coordinated of differential brakingcontrol control. is faster than Figures that 27 of anddifferential 28 show braking that, withoutcontrol. Figures control, 2 the7 and maximum 28 show that, absolute without values control, of the sideslipthe maximum angles of absolute the tractor values and of semi-trailer the sideslip reach angles 85.12 of the and tractor 22.80 ◦and, respectively, semi-trailer resulting reach 85.12 in seriousand sideslip.22.80°, respectively, After coordination resulting control in serious and sideslip. differential After braking coordination control control are added, and differential the absolute braking values ofcontrol the sideslip are added, angles the of absolute the tractor values are of reduced the sideslip to 13.84 angles and of 17.38 the tractor◦, respectively, are reduce andd to converge 13.84 and to 0 before17.38°, 15 respectively, s. The absolute and valuesconverge of theto 0 sideslip before 15 angles s. The of absolute the semi-trailer values of are the reduced sideslip to angles 5.95 and of the 8.12 ◦, respectively,semi-trailer and are convergereduced to to 5.95 0 before and 8.12°, 15 s. Therespectively, response and speed converge and convergence to 0 before speed15 s. The of coordinatedresponse controlspeed are and higher convergence than those speed of diofff erentialcoordinated braking. control Figure are higher29 shows than that, those without of differential control, thebraking. absolute valueFigure of the29 shows hinge anglethat, without reaches control, 92.53◦, the and absolute the jackknife value phenomenonof the hinge angle occurs. reaches After 92.53°, control and is added,the thejackknife absolute phenomenon value of the occurs. maximum After value control of is the added, hinge the angle absolute is reduced value of to the 17◦ maximumand finally value stabilizes of atthe 8◦/ s.hinge The angle convergence is reduced speed to 17° ofand coordinated finally stabilizes control at 8°/s. is higher The convergence than that speed of diff oferential coordinated braking. Figurescontrol 30 is andhigher 31 thanshow that that, of withoutdifferential control, braking. the Figures roll angle 30 and of the 31 tractorshow that, reaches without the control, peak value the of roll angle of the tractor reaches the peak value of 4.52° at 18 s, the roll angle of the semi-trailer 4.52◦ at 18 s, the roll angle of the semi-trailer fluctuates greatly, and reaches the peak value of 7.14◦ at 15.5fluctuates s; after thegreatly, cooperative and reaches and the diff peakerential value controls of 7.14° are at added,15.5 s; after the rollthe anglecooperative of the and peak differential roll angle of controls are added, the roll angle of the peak roll angle of the tractor decreases to 2.45 and 2.40°, the tractor decreases to 2.45 and 2.40◦, respectively, and the peak roll angle of the semi-trailer decreases respectively, and the peak roll angle of the semi-trailer decreases to 3° and 2.95°, respectively. At 15 to 3◦ and 2.95◦, respectively. At 15 s, the roll angle of the tractor semi-trailer converges to 2.5 and s, the roll angle of the tractor semi-trailer converges to 2.5 and is finally stable. Figures 32 and 33 show is finally stable. Figures 32 and 33 show that, without control, the lateral acceleration of the tractor that, without control, the lateral acceleration of the tractor semi-trailer fluctuates greatly and loses semi-trailer fluctuates greatly and loses control; after adding the cooperative and differential controls, control; after adding the cooperative and differential controls, the lateral acceleration of the tractor the lateral acceleration of the tractor semi-trailer converges to 3 m/s2 and is finally stable. semi-trailer converges to 3 m/s2 and is finally stable.

(a) With control (b) Differential braking control (c) coordinated control

FigureFigure 23.23. Simulation animation animation comparison. comparison.

Figure 24. Steering wheel.

Energies 2020, 13, x FOR PEER REVIEW 18 of 24

respectively, and finally tends to 10°/s. The peak value of the yaw rate of the semi-trailer decreases to 17.93 and 17.55°/s, respectively, and finally tends to 10°/s. The response speed of coordinated control is faster than that of differential braking control. Figures 27 and 28 show that, without control, the maximum absolute values of the sideslip angles of the tractor and semi-trailer reach 85.12 and 22.80°, respectively, resulting in serious sideslip. After coordination control and differential braking control are added, the absolute values of the sideslip angles of the tractor are reduced to 13.84 and 17.38°, respectively, and converge to 0 before 15 s. The absolute values of the sideslip angles of the semi-trailer are reduced to 5.95 and 8.12°, respectively, and converge to 0 before 15 s. The response speed and convergence speed of coordinated control are higher than those of differential braking. Figure 29 shows that, without control, the absolute value of the hinge angle reaches 92.53°, and the jackknife phenomenon occurs. After control is added, the absolute value of the maximum value of the hinge angle is reduced to 17° and finally stabilizes at 8°/s. The convergence speed of coordinated control is higher than that of differential braking. Figures 30 and 31 show that, without control, the roll angle of the tractor reaches the peak value of 4.52° at 18 s, the roll angle of the semi-trailer fluctuates greatly, and reaches the peak value of 7.14° at 15.5 s; after the cooperative and differential controls are added, the roll angle of the peak roll angle of the tractor decreases to 2.45 and 2.40°, respectively, and the peak roll angle of the semi-trailer decreases to 3° and 2.95°, respectively. At 15 s, the roll angle of the tractor semi-trailer converges to 2.5 and is finally stable. Figures 32 and 33 show that, without control, the lateral acceleration of the tractor semi-trailer fluctuates greatly and loses control; after adding the cooperative and differential controls, the lateral acceleration of the tractor semi-trailer converges to 3 m/s2 and is finally stable.

(a) With control (b) Differential braking control (c) coordinated control Energies 2020, 13, 6317 18 of 23 Figure 23. Simulation animation comparison.

Energies 2020,, 13,, xx FORFOR PEERPEER REVIEWREVIEW 1919 ofof 24 Figure 24. Steering wheel.

FigureFigure 25. 25. YawYaw rate ofof tractor. tractor.

FigureFigure 26. 26. YawYaw rate rate of tractor of of semi-trailer. semi-trailer.

Figure 27. Sideslip angle of tractor.

Energies 2020, 13, x FOR PEER REVIEW 19 of 24

Figure 25. Yaw rate of tractor.

Energies 2020, 13, 6317 19 of 23 Figure 26. Yaw rate of tractor of semi-trailer.

Energies 2020, 13, x FOR PEER REVIEW 20 of 24 Energies 2020, 13, x FOR PEER REVIEW 20 of 24 FigureFigure 27. 27. SideslipSideslip angle of of tractor. tractor.

Figure 28. Sideslip angle of semi-trailer. FigureFigure 28. 28. SideslipSideslip angle angle ofof semi-trailer. semi-trailer.

FigureFigureFigure 29.29. 29. HingeHinge angle. angle.

Figure 30. Roll angle of tractor. Figure 30. Roll angle of tractor.

Energies 2020, 13, x FOR PEER REVIEW 20 of 24

Figure 28. Sideslip angle of semi-trailer.

Energies 2020, 13, 6317 20 of 23 Figure 29. Hinge angle.

EnergiesEnergies 20202020,, 1133,, xx FORFOR PEERPEER REVIEWREVIEW 2121 ofof 2424 FigureFigure 30. 30. RollRoll angle ofof tractor.tractor.

FigureFigureFigure 31.31. 31. RollRollRoll angleangle angle of of semi-trailer.semisemi--trailer.trailer.

Figure 32. Lateral acceleration of tractor. FigureFigure 32.32. LateralLateral accelerationacceleration ofof tractor.tractor.

FigureFigure 33.33. LateralLateral accelerationacceleration ofof semisemi--trailer.trailer.

InIn summary, summary, after after adding adding the the control, control, the the yaw yaw stability stability and and jackknife jackknife stability stability of of the the entire entire vehiclevehicle are are enhanced, enhanced, and and angle angle step step test test is is completed. completed. In In comparison comparison with with differential differential braking braking control,control, the the coordinated coordinated control control strategy strategy has has a a f fasteraster response response and and faster faster convergence convergence and and can can effectivelyeffectively restrainrestrain sideslipsideslip andand jackknifejackknife instabilityinstability andand significantlysignificantly enhanceenhance thethe laterallateral stabilitystability ofof thethe tractortractor semisemi--trailer.trailer.

Energies 2020, 13, x FOR PEER REVIEW 21 of 24

Figure 31. Roll angle of semi-trailer.

Energies 2020, 13, 6317 21 of 23 Figure 32. Lateral acceleration of tractor.

FigureFigure 33. 33. LateralLateral acceleration acceleration of of semi-trailer. semi-trailer.

In summary, after adding the control, the yaw stability and jackknife stability of the entire In summary, after adding the control, the yaw stability and jackknife stability of the entire vehicle are enhanced, and angle step test is completed. In comparison with differential braking control, vehicle are enhanced, and angle step test is completed. In comparison with differential braking the coordinated control strategy has a faster response and faster convergence and can effectively restrain control,sideslip the and coordinated jackknife instability control and strategy significantly has a enhance faster the response lateral stability and faster of the convergence tractor semi-trailer. and can effectively restrain sideslip and jackknife instability and significantly enhance the lateral stability of the tractor6. Conclusions semi-trailer. Based on TruckSim and MATLAB/Simulink, a cosimulation model was built, and the yaw rate of the tractor and semi-trailer was considered as the control target. The coordinated control strategy of lateral stability based on coordinated control of a wheel motor electronic differential and mechanical brake was proposed. The simulation verification analysis was carried out under limiting operation condition. The final result shows that the torque distribution rule can realize the desired yaw torque accurately, and the coordinated control strategy of lateral stability exhibits a fast transient response and strong dynamic control ability because of the existence of the hub motor and the coordination with differential braking. Under the condition of a low-adhesion-coefficient road surface, the yaw rate and sideslip angle can track the expected value well and significantly reduce the value of the hinge angle. In comparison with traditional differential braking control (ESP), the coordinated control strategy designed in this article can enhance the lateral stability of a tractor semi-trailer more effectively; the coordinated control strategy has a faster response and faster convergence and can effectively restrain sideslip and jackknife instability and significantly improve the roll stability of the tractor semi-trailer. However, the maximum torque and endurance of the hub motor are a challenge to improve the stability of the tractor semi-trailer. The results show that the combined simulation of TruckSim and Simulink is an effective means for the design and development of the stability control system of the tractor semi-trailer. The cooperative control strategy can better improve the yaw stability; at the same time, it can also improve the roll stability of the vehicle and effectively enhance the running safety of the vehicle.

Author Contributions: Conceptualization, Y.L. (Yufeng Lu) and Z.B.; methodology, Y.L. (Yufeng Lu); software, Z.B.; validation, Y.L. (Yunxia Li); formal analysis, Z.B.; investigation, Y.L. (Yunxia Li); resources, Y.L. (Yufeng Lu); data curation, Z.B.; writing—original draft preparation, Z.B.; writing—review and editing, Y.L. (Yufeng Lu); visualization, Y.L. (Yunxia Li); supervision, Y.L. (Yufeng Lu). All authors have read and agreed to the published version of the manuscript. Funding: Research on the co construction of enterprises and schools to help local universities develop MOOC (Qilu University of Technology Project 201828). Conﬂicts of Interest: The authors declare no conﬂict of interest. Energies 2020, 13, 6317 22 of 23

References

1. Aripin, M.K.; Sam, Y.M.; Danapalasingam, K.A.; Peng, K.; Hamzah, N.; Ismail, M.F. A review of active yaw control system for vehicle handling and stability enhancement. Int. J. Veh. Technol. 2014, 6, 1–15. [CrossRef] 2. Nie, Z.; Zong, C. Study on parameter identification of simplified model of heavy truck. Automot. Eng. 2015, 6, 622–630. [CrossRef] 3. Yang, W.; Wei, L.; Liu, J. Joint simulation analysis of lateral stability optimization control of commercial vehicles. J. Mech. Eng. 2017, 2, 115–123. [CrossRef] 4. Xu, X.; Zhang, L.; Jiang, Y. Active control on path following and lateral stability for truck–trailer combinations. Arab. J. Sci. Eng. 2018, 44, 1365–1377. [CrossRef] 5. National Highway Traffic Safety Administration. Traffic Safety Facts 2015; National Highway Traffic Safety Administration: Washington, DC, USA, 2016. 6. Liu, C.; Guan, Z.; Du, F. Fuzzy PID control of lateral stability of four-wheel steering semitrailer. Mod. Manuf. Eng. 2016, 5, 47–50. 7. Murata, S. Innovation by in-wheel-motor drive unit. Veh. Syst. Dyn. 2012, 50, 807–830. [CrossRef] 8. Li, C. An optimal coordinated method for EVs participating in frequency regulation under different power system operation states. IEEE Access 2018, 6, 62756–62765. [CrossRef] 9. Wang, Q.; Jiang, B.; Li, B.; Yan, Y. A critical review of thermal management models and solutions of lithium-ion batteries for the development of pure electric vehicles. Renew. Sustain. Energy Rev. 2016, 64, 106–128. [CrossRef] 10. Li, Y.; Li, B.; Xu, X.; Sun, X. A nonlinear decoupling control approachusing RBFNNI-based robust pole placement for a permanent magnet in-wheel motor. IEEE Access 2017, 6, 1844–1854. [CrossRef] 11. Najjari, B.; Mirzaei, M.; Tahouni, A. Constrained stability control with optimal power management strategy for in-wheel electric vehicles. Proc. Inst. Mech. Eng. Part K J. Multi-Body Dyn. 2019.[CrossRef] 12. Asiabar, A.; Kazemi, R. A direct yaw moment controller for a four in-wheel motor drive electric vehicle using adaptive sliding mode control. Proc. Inst. Mech. Eng. Part K J. Multi-Body Dyn. 2019.[CrossRef] 13. Zhang, Z.; Zhao, W.; Wang, C.; Li, L. Stability control of in-wheel motor electric vehicles under extreme conditions. Trans. Inst. Meas. Control 2019, 41, 2838–2850. [CrossRef] 14. Tamaddoni, S.; Taheri, S. Yaw Stability Control of Tractor Semi-Trailers; SAE Technical Paper 2008-01-2595; SAE: Warrendale, PA, USA, 2008. [CrossRef] 15. Takenaga, H.; Konishi, M.; Imai, J. Route Tracking Control of Tractor-Trailer Vehicles based on Fuzzy Controller. In Proceedings of the Fifth International Workshop on Computational Intelligence & Applications, Hiroshima, Japan, 10–12 November 2009; pp. 105–110. 16. Sogol, K.; Mathias, L.; Jonas, F. Robustness analysis of a steering-based control strategy for improved lateral performance of a truck-dolly-semitrailer. Int. J. Heavy Veh. Syst. 2015, 22, 1–20. 17. Yang, X.; Kang, N.; Li, X. Stability control of semitrailer train based on TruckSim Simulink joint simulation. Highw. Transp. Technol. 2013, 30, 141–148. 18. Eunhyek, J.; Kwanwoo, P.; Youngil, K.; Kyongsu, Y.; Kilsoo, K. A tyre slip-based integrated chassis control of front/rear traction distribution and four-wheel independent brake from moderate driving to limit handling. Veh. Syst. Dyn. 2018, 56, 579–603. 19. Chen, Y.; Chen, S.; Zhao, Y.; Gao, Z.; Li, C. Optimized handling stability control strategy for a four in-wheel motor independent-drive electric vehicle. IEEE Access 2019, 7, 17017–17032. [CrossRef] 20. Chae, M.; Hyun, Y.; Yi, K.; Nam, K. Dynamic Handling Characteristics Control of an in-Wheel-Motor Driven Electric Vehicle Based on Multiple Sliding Mode Control Approach. IEEE Access 2019, 7, 132448–132458. [CrossRef] 21. Huang, C.; Lei, F.; Hu, L. Lateral Stability Control Based on Regional Pole Placement of In-wheel-motored Electric Vehicle. Automot. Eng. 2019, 41, 905–914. 22. Wang, Z.; Wang, Y.; Zhang, L.; Liu, M. Vehicle stability enhancement through hierarchical control for a four-wheel-independently-actuated electric vehicle. Energies 2017, 10, 947. [CrossRef] 23. Zong, C.; Lin, F.; Hu, L. Control algorithm for braking force distribution of electric control braking system of motor train. In Proceedings of the 2011 Annual Meeting of Society of Automotive Engineering of China, Beijing, China, 26–28 October 2011. Energies 2020, 13, 6317 23 of 23

24. Yang, X.; Yang, C.; Zhang, X. Yaw stability control of tractor semi-trailer based on active braking. Automot. Eng. 2011, 33, 955–961. 25. Gao, H. Research on Braking Direction Stability and Control Strategy of Semitrailer Train. Ph.D. Thesis, Jilin University, Changchun, China, 2014. 26. Zhang, D.; Liu, G.; Zhou, H.; Zhao, W. Adaptive sliding mode fault-tolerant coordination control for four-wheel independently driven electric vehicles. IEEE Trans. Ind. Electron. 2018, 65, 9090–9100. [CrossRef] 27. Ivanov, V.; Savitski, D. Systematization of integrated motion control of ground vehicles. IEEE Access 2015, 3, 2080–2099. [CrossRef] 28. Mutoh, N.; Nakano, Y. Dynamics of front-and-rear-wheel-independent-drive-type electric vehicles at the time of failure. IEEE Trans. Ind. Electron. 2012, 20, 1488–1499. [CrossRef] 29. Chen, Y.; Wang, J. Fast and global optimal energy-eﬃcient control allocation with applications to over-actuated electric ground vehicles. IEEE Trans. Control. Syst. Technol. 2012, 20, 1202–1211. [CrossRef] 30. Wang, R.; Wang, J. Fault-tolerant control with active fault diagnosis for four-wheel independently driven electric ground vehicles. IEEE Trans. Veh. Technol. 2011, 60, 4276–4287. [CrossRef] 31. Cheng, Z. Research on the PID control of the ESP system of tractor based on improved AFSA and improved SA. Comput. Electron. Agric. 2018, 148, 142–147. [CrossRef] 32. Wang, S.; Shi, Y.; Feng, Z. Research on control method based on Fuzzy PID controller. Mech. Sci. Technol. 2011, 30, 166–172. 33. Jin, X.; Chen, K.; Zhao, Y.; Ji, J.; Jing, P. Simulation of hydraulic transplanting robot control system based on fuzzy PID controller. Measurement 2020, 164, 108023. [CrossRef] 34. Shi, Q.; Lam, H.; Xuan, C.; Chen, M. Adaptive Neuro-Fuzzy PID Controller based on Twin Delayed Deep Deterministic Policy Gradient Algorithm. Neurocomputing 2020, 402, 183–194. [CrossRef] 35. Feng, Y.; Wu, M.; Chen, X.; Chen, L.; Du, S. A Fuzzy PID Controller with Nonlinear Compensation Term for Mold Level of Continuous Casting Process. Inf. Sci. 2020, 539, 487–503. [CrossRef] 36. Liu, Y. Study on Stability of Hydraulic Hybrid Semitrailer Train. Ph.D. Thesis, Qilu University of Technology, Jinan, China, 2019. 37. Guo, K.; Wang, D. Preliminary study on anti slip control theory of automobile drive. J. Jilin Univ. Technol. 1997, 27, 1–5.

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