Study on Low-Speed Steering Resistance Torque of Vehicles Considering Friction Between Tire and Pavement

Home , Pneumatic trail, Power steering, Self aligning torque, Slip angle, Tire

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Article Study on Low-Speed Steering Resistance Torque of Vehicles Considering Friction between Tire and Pavement

Dong Cao 1, Bin Tang 2,* , Haobin Jiang 1,2, Chenhui Yin 1, Di Zhang 1 and Yingqiu Huang 1 1 School of Automotive and Trafﬁc Engineering, Jiangsu University, Zhenjiang 212013, China; [email protected] (D.C.); [email protected] (H.J.); [email protected] (C.Y.); [email protected] (D.Z.); [email protected] (Y.H.) 2 Automotive Engineering Research Institute, Jiangsu University, Zhenjiang 212013, China * Correspondence: [email protected]; Tel.: +86-0511-8878-2845

Received: 25 December 2018; Accepted: 7 March 2019; Published: 12 March 2019

Abstract: Electric power steering (EPS) systems under existing vehicle power systems cannot provide enough power for heavy-duty commercial vehicles under pivot or low-speed steering conditions. To solve this problem, the paper proposes an EPS system that is based on the hybrid power system constituted by the vehicle power system and the supercapacitor in parallel. In order to provide a theoretical basis for the intervention and withdrawal mechanisms of a super-capacitor in the new EPS, the law of steering resistance torque at a low or extremely low vehicle speed should be explored. Firstly, the finite element model of tire/pavement was established to conduct the simulation and calculation of the low-speed steering friction force between the tire and pavement, and to obtain the fitting expression of the equivalent steering friction coefficient with the running speed of the tire. Secondly, the expression of the steering friction torque was deduced based on the calculus theory and mathematical model of the low-speed steering resistance torque, including the steering friction torque and aligning torques, established to conduct the simulation of the equivalent resistance torque applied on a steering column under low-speed condition. Subsequently, the real vehicle experiments were carried out and comparisons of the experimental results and simulation results was performed. The consistency indicated that the model of low-speed steering resistance torque had a high accuracy. Finally, the law of low-speed steering resistance torque with a vehicle speed and steering wheel angle were analyzed according to the 3D surface plot drawn from the simulation results.

Keywords: heavy-duty commercial vehicle; electric power steering system; hybrid power system; steering resistance torque; steering friction force; aligning torque

1. Introduction With the rapid development of China’s national economy and advanced highway networks, the driving speed of heavy-duty commercial vehicles, such as coaches and lorries, has greatly increased, while high-speed safety has becoming increasingly prominent. The steering system as a key part in vehicle chassis is crucial for the driving safety of vehicles. Nowadays, heavy-duty commercial vehicles are widely equipped with a hydraulic power steering (HPS) system, with an assist characteristic that is relatively single and an assist effort cannot vary with vehicle speed, which would easily cause poor road feel and affect handling stability during high-speed steering. On the other hand, whether the vehicle is going straight or steering, the steering pump in HPS is always running along with the engine, which causes a large amount of energy loss [1,2]. Recently, the electric power steering (EPS) system has been widely used in passenger cars and light commercial vehicles for its advantages of safety, energy saving and environmental protection. The assist effort

Appl. Sci. 2019, 9, 1015; doi:10.3390/app9051015 www.mdpi.com/journal/applsci Appl. Sci. 2019, 9, 1015 2 of 16 of EPS can vary with vehicle speed, which is conducive to improving handling stability. In addition, the motor in EPS provide steering assist effort directly, thus EPS hardly consumes electric energy under no-steering conditions, which is more energy saving [3,4]. Nevertheless, EPS is not suitable for heavy-duty commercial vehicles due to a large steering resistance torque and the power limitation of existing power supply systems in heavy-duty commercial vehicles, especially under the condition of pivot or low-speed steering [5]. It is known that heavy-duty commercial vehicles travel at medium or high speeds most of the time. In this case, the steering resistance torque is lower and the power supply system can meet the steering power demand. So the issue of insufficient power provided by existing vehicle power systems during pivot or low-speed steering needs to be emphatically solved to promote the application of EPS in heavy-duty commercial vehicles. Supercapacitors (SC) have been widely used in electric vehicles and the braking system of trains for its advantages of high power density, short charging time, large current discharge capacity, etc. [6–8] Therefore, a new type of EPS system based on the hybrid power system constituted by the vehicle power system and the supercapacitor, which is defined as SC-EPS here, is proposed in this paper. When vehicle speed is low, the vehicle power system and supercapacitor jointly provide electric power for EPS. When the vehicle speed is high, the vehicle power system solely provides electric power for EPS while charging the supercapacitor. In order to provide a theoretical basis for the intervention and withdrawal mechanisms of a supercapacitor in the new EPS structure, the law of steering resistance torque at a low or extremely low vehicle speed needs to be explored first. In other words, the power supply modes of the SC-EPS system are determined by the change laws of the steering resistance torque. Domestic and foreign scholars have carried out relevant research on the compositions, calculations and influences of the steering resistance torque. The pivot steering resistance torque is related to various factors such as the front axle load, the alignment parameters and steering angle of the front wheel, the tire/pavement friction, etc. [9–12] For example, empirical formulas are usually used to calculate the pivot steering resistance torque at present [13,14], however, empirical formulas do not reflect the relationship between the steering resistance torque and the steered wheel angle. Wang, Y.C. et al. analyzed the effects of the steered wheel angle, friction coefficient, tire pressure and vertical load on the pivot steering resistance torque based on a vehicle road test [15], they also considered the contact between the tire and pavement in their research on the pivot steering resistance torque [16], however, the effect of the kingpin offset on the steering resistance torque is neglected in modeling. Zhuang, Y. et al. introduced the LuGre friction model in their research on the pivot steering resistance torque, and the parameter identification and simulation calculation was carried out [17]. Zhao, Y.X. proposed the hypothesis of gradual distribution of tire load starting from contact and friction between the tire and pavement, and calculated the pivot steering resistance torque [18]. Ma, B. et al. proposed an estimation method of the static steering torque including the tire sliding torque and gravity aligning torque [19]. For the steering resistance torque under a medium and high speed steering condition, it is mainly composed of aligning torques caused by the alignment parameters of the front wheel, which is related to vehicle speed, the side-slip characteristics of the tire, pneumatic trail, the alignment parameters and steering angle of the front wheel, etc. [20–22] For example, Liu, Z. et al. analyzed the effects of vehicle speeds (10 km/h, 20 km/h, 30 km/h, 40 km/h, 50 km/h) on the steering torque based on modeling and simulation [23]. Kim, S.H. et al. took the tire lateral force, the aligning torques due to kingpin inclination and the pneumatic trail into account in their research on hardware-in-the-loop simulations of an electrohydraulic power steering system [24]. Wei, Y.T. et al. analyzed the relationships between the lateral force, the aligning torques of the tire and side-slip angle, the tire’s rolling speed by establishing the simulation model of a rolling tire, and presented the change curve of the steering torque [25]. To sum up, domestic and foreign scholars mainly studied the steering resistance torque under pivot steering and medium or high speed steering conditions, but did not involve their research on low-speed steering resistance torque. According to analysis, the low-speed steering resistance torque mainly includes the tire/pavement friction torque and aligning torques caused by the alignment parameters of the front wheel. When the Appl. Sci. 2019, 9, x FOR PEER REVIEW 3 of 16 steeringAppl. Sci. 2019 resistance, 9, 1015 torque mainly includes the tire/pavement friction torque and aligning torques3 of 16 caused by the alignment parameters of the front wheel. When the vehicle steers at a low speed, the frictionvehicle generated steers at a between low speed, the thetire frictionand pavement generated includes between the the components tire and pavement of slidingincludes and rolling the friction.components The proportion of sliding and of the rolling two friction.friction components The proportion will of transfer the two to friction each other components as the vehicle will transfer speed changes,to each other and asthus the the vehicle dynamic speed process changes, is andquite thus complex. the dynamic There processare no mature is quite models complex. to Theredescribe are suchno mature dynamic models processes to describe so far. suchJudging dynamic from the processes above gap so far.analysis, Judging the fromstudy the on abovelow-speed gap analysis,steering resistancethe study ontorque low-speed not only steering can provides resistance a torquetheoretical not onlybasis can for providesthe intervention a theoretical and basiswithdrawal for the mechanismsintervention andof a supercapacitor withdrawal mechanisms in SC-EPS, of but a supercapacitor can fill in the blank in SC-EPS, in the but research can fill field in the on blank low-speed in the steeringresearch resistance field on low-speed torque, providing steering resistance a useful torque,reference providing for the design a useful of reference assist characteristics for the design and of controlassist characteristics strategies for andothe controlr new EPS strategies systems for as other well. new EPS systems as well. Considering that that the the low-speed low-speed steering steering resistance resistance torque torque of of heavy-duty heavy-duty commercial commercial vehicles vehicles is notis not convenient convenient for forexperimental experimental verification, verification, the themodel model of low-speed of low-speed steering steering resistance resistance torque torque of a compactof a compact car was car established was established and verified and verified by real by ve realhicle vehicle experiments experiments in this paper. in this Then paper. the Then proved the modelingproved modeling method method of low-speed of low-speed steering steering resistance resistance torque torque would would be begeneralized generalized to to heavy-duty heavy-duty commercial vehicles. vehicles. Firstly, Firstly, the finite finite element model modelss of the tire and pavement were established to conduct the the simulation and and calculation calculation of of a a low-sp low-speedeed steering steering friction friction force force between between the the tire and pavement and to obtain the fitting fitting expression of the equivalent steering friction coefficient coefficient with the runningrunning speed of the tire. Subsequently, Subsequently, the math mathematicalematical model of low-sp low-speedeed steering resistance torquetorque including the the steering steering friction friction torque torque and and aligning torques were established to conduct the simulation of of the the equivalent equivalent resistance resistance torque torque a appliedpplied on on a asteering steering column column in in vehicle vehicle speeds speeds of of0 km/h,0 km/h, 1 km/h, 1 km/h, 2 km/h, 2 km/h, 3 km/h, 3 km/h, 4 km/h, 4 km/h, 5 km/h, 5 km/h,10 km/h 10 and km/h 20 km/h, and 20 respectively, km/h, respectively, which are which verified are byverified real vehicle by real experiments. vehicle experiments. Finally, the Finally, law of the low-speed law of low-speed steering steeringresistance resistance torque with torque vehicle with speedvehicle and speed steering and steeringwheel angle wheel was angle analyzed was analyzedaccordingaccording to the 3D surface to the3D plot surface drawnplot from drawn simulation from results.simulation results.

2. Structure Structure and and Working Working Principles of the SC-EPS System The structure of the SC-EPS system is shown in Figure1 1,, includingincluding thethe torque/angletorque/angle sensor,sensor, recirculatingrecirculating ball ball steering steering gear, gear, assist assist motor, motor, controller, controller, DC–DC DC–DC converter converter and supercapacitor. and supercapacitor. The supercapacitorThe supercapacitor was wasconnected connected in parallel in parallel with with the the vehicle vehicle power power system system through through the the DC–DC converter to form the hybrid power system.

Figure 1. StructureStructure of of the SC-EPS system. Appl. Sci. 2019, 9, 1015 4 of 16

The SC-EPS system has three power supply modes: Hybrid power supply mode, vehicle power supply mode and supercapacitor power supply mode. When the vehicle speed is low, the SC-EPS system is in hybrid power supply mode. In this case, the vehicle power system and supercapacitor jointly provided electric power to the assist motor, the controller determines the mode and proportion of power distribution according to the signals, such as the vehicle speed and steering wheel angle. When the vehicle speed is high, the SC-EPS system is in vehicle power supply mode, which means the vehicle power system solely provides electric power to the assist motor. Meanwhile, the supercapacitor is in charging state to ensure that the stored energy can be used during low-speed steering. When the vehicle power system fails, the SC-EPS system is in supercapacitor power supply mode, under which the supercapacitor emergently provides electric power to the assist motor to maintain the steering assist for a short time.

3. Simulation of Low-Speed Steering Friction Force between the Tire and Pavement In this section, the finite element models of tire and pavement based on ABAQUS software are established to carry out the simulation and calculation of steering friction force under a low-speed steering condition. The law of steering friction force with running speed of tire is explored. The focus of this section is to study the friction between two surfaces, therefore, a few simplified treatments are made in tire modeling to reduce the difficulty of software calculation: The lateral tread pattern is ignored and only the longitudinal tread pattern is retained; the internal complex structure of tire is ignored and only the tread, carcass, cord layer and belt layer of the tire are retained; moreover, the rim would constrain the tire’s deformation and should not be ignored, which is simplified into a flat-bottomed steel ring with a regular shape and assembled with the tire. Since the rubber material, which makes up the tire tread, has extremely complex characteristics, it is usually defined as the isotropic, incompressible and hyper-elastic material in the model. The Mooney–Rivlin material model is used in this section to describe the constitution of rubber and the material parameters of each component, as shown in Table1[ 26,27]. When modeling, the steel wires of the cord layer and belt layer are simulated by a rebar unit, which is firstly defined on the surface unit and then embedded into the corresponding entity unit of rubber. The tread and carcass of the tire are constrained by a tie. The rim, which is not deformed, is rigidly coupled with the central point of carcass’ rotation axis, thus it can move with the tire carcass. The finite element models of the tire and pavement are shown in Figure2. To clarify, the kingpin inclination and caster are usually not considered in 3D finite element models to simplify the models and facilitate calculation, and the tire just rolls and deflects around the central point which is coupled with the periphery of the rim.

Table 1. Material properties of the tire and pavement.

Mooney Coefﬁcients Elasticity Poisson’s Density Part (MPa) Modulus (MPa) Ratio (g/cm3) C10 C01 Tread —— —— 1.14 2.0477 1.1859 Carcass 794 0.45 1.39 —— —— Belt 55,000 0.3 7.64 —— —— Pavement 1400 0.35 2.4 —— —— Appl. Sci. 2019, 9, x FOR PEER REVIEW 4 of 16

3. Simulation of Low-Speed Steering Friction Force between the Tire and Pavement In this section, the finite element models of tire and pavement based on ABAQUS software are established to carry out the simulation and calculation of steering friction force under a low-speed steering condition. The law of steering friction force with running speed of tire is explored. The focus of this section is to study the friction between two surfaces, therefore, a few simplified treatments are made in tire modeling to reduce the difficulty of software calculation: The lateral tread pattern is ignored and only the longitudinal tread pattern is retained; the internal complex structure of tire is ignored and only the tread, carcass, cord layer and belt layer of the tire are retained; moreover, the rim would constrain the tire’s deformation and should not be ignored, which is simplified into a flat-bottomed steel ring with a regular shape and assembled with the tire. Since the rubber material, which makes up the tire tread, has extremely complex characteristics, it is usually defined as the isotropic, incompressible and hyper-elastic material in the model. The Mooney–Rivlin material model is used in this section to describe the constitution of rubber and the material parameters of each component, as shown in Table 1 [26,27]. When modeling, the steel wires of the cord layer and belt layer are simulated by a rebar unit, which is firstly defined on the surface unit and then embedded into the corresponding entity unit of rubber. The tread and carcass of the tire are constrained by a tie. The rim, which is not deformed, is rigidly coupled with the central point of carcass’ rotation axis, thus it can move with the tire carcass. The finite element models of the tire and pavement are shown in Figure 2. To clarify, the kingpin inclination and caster are usually not consideredAppl. Sci. 2019 in, 9 ,3D 1015 finite element models to simplify the models and facilitate calculation, and the5 tire of 16 just rolls and deflects around the central point which is coupled with the periphery of the rim.

Appl. Sci. 2019, 9, x FOR PEER REVIEW 5 of 16

Table 1. Material properties of the tire and pavement.

Elasticity Modulus Poisson’s Density Mooney Coefficients (MPa) Part 3 (MPa) Ratio (g/cm ) C10 C01 Tread —— —— 1.14 2.0477 1.1859 Carcass 794 0.45 1.39 —— —— Belt 55,000 0.3 7.64 —— —— Pavement 1400 0.35 2.4 —— —— FigureFigure 2. 2. FiniteFinite element element models models of of the the tire tire and and pavement. pavement.

In the simulation of low-speed steering friction, the tire pressure was set as 0.25 MPa, the vertical load on the tire was set as 3000 N and thethe friction coefﬁcientcoefficient ofof surface-to-surfacesurface-to-surface contact was taken as 0.7, which is the initial value set in thethe software,software, indicating the contact property between two surfaces. TheThe steered steered process process of theof tirethe wastire simulatedwas simulated by the tire’sby the forward tire’s rollingforward and rolling the pavement’s and the pavement’shorizontal tuning, horizontal respectively. tuning, respectively. The simulations werewere respectivelyrespectively performedperformed inin tiretire running running speeds speeds of of 1 1 km/h, km/h, 22km/h, km/h, 44 km/h,km/h, 6 km/h, 8 8 km/h km/h and and 10 10 km/h. km/h. The The first ﬁrst 3 3simulation simulation re resultssults of of the the steering steering friction friction force force are are shown in Figure 33a.a. ItIt cancan bebe seenseen thatthat thethe frictionfriction forceforce atat eacheach speedspeed roserose graduallygradually andand thenthen keptkept aa steadysteady state. The steering friction force in the steady state was averaged in all simulation results, and then the average value was treated as the steering frictionfriction force at each corresponding speed as shown in Table2 2.. ItIt cancan bebe seenseen thatthat thethe steeringsteering frictionfriction forceforce obviouslyobviously decreaseddecreased withwith thethe runningrunning speedspeed ofof the tire.

(a) (b)

Figure 3. Simulation results of low-speed steering friction force: ( (aa)) Curves Curves of of steering steering friction friction force; force; and (b) ﬁttingfitting curve of the equivalentequivalent frictionfriction coefﬁcientcoefficient withwith thethe runningrunning speedspeed ofof thethe tire.tire.

Table 2. Simulation results of low-speed steering friction force. Table 2. Simulation results of low-speed steering friction force. Running Speed u (km/h) 1 2 4 6 8 10 Running Speed u (km/h) 1 2 4 6 8 10 SteeringSteering friction friction force (N) force (N) 1504.52 1504.52 1230.89 1230.89 997.21 997.21 847.82 847.82 741.94741.94 660. 660.3939 Equivalent friction Equivalent friction coefficient0.5015 μ 0.4103 0.3324 0.2826 0.247373 0.22011 coefﬁcient µ 0.5015 0.4103 0.3324 0.2826 0.24 0.220

Table 2 also shows the equivalent friction coefficient at different corresponding speeds, which was obtained through dividing the steering friction force by the vertical load. Subsequently, the data bx⋅ in Table 2 are fit in the form of y =a⋅ e +c (a , b, c are the fitting coefficients) [28], according to the variation trend and the fitting curve, which is shown in Figure 3b. Finally, the function of the equivalent friction coefficient with running speed is:

-0.4603⋅u μ=f()u = 0.4511⋅ e + 0.2376 (1) where μ is the equivalent friction coefficient between the tire and pavement under a low-speed steering condition; u is the running speed of the tire, km/h. Appl. Sci. 2019, 9, 1015 6 of 16

Table2 also shows the equivalent friction coefficient at different corresponding speeds, which was obtainedAppl. Sci. 2019 through, 9, x FOR dividing PEER REVIEW the steering friction force by the vertical load. Subsequently, the6 dataof 16 in Table2 are fit in the form of y = a · eb·x + c (a, b, c are the fitting coefficients) [28], according to the variationThe value trend of μ and in theEquation fitting (1) curve, can be which treated is shown as the in friction Figure coefficient3b. Finally, between the function the tire of theand equivalentpavement under friction pivot coefficient steering with condition running when speed u is:=0, which is consistent with the value of 𝑓 in the commonly used empirical formula of the pivot steering resistance torque [13]. µ = f (u) = 0.4511 · e−0.4603·u + 0.2376 (1) 4. Modeling of Low-Speed Steering Resistance Torque Considering Tire/Pavement Friction where µ is the equivalent friction coefficient between the tire and pavement under a low-speed steering condition;On theu premiseis the running that the speed friction of theinside tire, the km/h. steering system itself is ignored, the steering resistance torqueThe mainly value ofconsistsµ in Equationof 4 parts (1) under can be a treatedlow-speed as thesteering friction condition: coefficient The between low-speed the tiresteering and pavementfriction torque under and pivot aligning steering torques condition caused when by theu =kingpin0, which inclination, is consistent kingpin with caster the value and pneumatic of f in the commonlytrail. used empirical formula of the pivot steering resistance torque [13].

4.4.1. Modeling Low-Speed of Steering Low-Speed Friction Steering Torque Resistancebetween the TorqueTire and Considering Pavement Tire/Pavement Friction On the premise that the friction inside the steering system itself is ignored, the steering resistance There is surface contact between the tire and pavement, as shown in Figure 4. Where δt is the torque mainly consists of 4 parts under a low-speed steering condition: The low-speed steering friction deformation or subsidence of the tire, l and b are respectively the length and width of the torque and aligning torques caused by thet kingpin inclination,t kingpin caster and pneumatic trail.

tire/pavement contact region. The values of δt , lt and bt can be calculated by the following 4.1.empirical Low-Speed formulas Steering [29]: Friction Torque between the Tire and Pavement

There is surface contact between the tire and pavement,0.85 as shown in Figure4. Where δt is Fz the deformation or subsidence of the tire,δtttl=ckand b are respectively the length and width of the(2) t BD0.7t 0.43 P 0.6 tire/pavement contact region. The values of δt, lt and bt can be calculated by the following empirical formulas [29]: s 0.85δ lD= 2 Fzt (3) δt = ctktt (2) B0.7D0.43D P0.6 s δt −tδ lt = =−2D()t (3) bBt 1D e (4) = − −tδ=t = where ct is the parameter related to thebt tireB type,1 ect 1.15 for bias tires and ct 1.5 for radial(4) =+ wheretires, kBctt is0.015 the parameter 0.42 , F relatedz is the to vertical the tire load type, onc tthe= 1.15tire withfor bias the tires unit andof 10ct N,= 1.5B for is the radial width tires, of ktiret = with0.015 theB + unit0.42 of, Fcm,z is theD vertical is the diameter load on of the the tire tire with with the the unit unit of of 10 cm, N, PB is is the thewidth tire pressure of tire withwith thethe unitunit of of cm, 100D iskPa the and diameter s and of thet tireare withthe empirical the unit of coefficients, cm, P is the tirewhich pressure are 0.557 with theand unit 122.7 of 100respectively. kPa and s and t are the empirical coefﬁcients, which are 0.557 and 122.7 respectively.

FigureFigure 4.4. Diagram ofof contactcontact betweenbetween thethe tiretire andand pavement.pavement.

It is known that the steered wheel does not turn around the center of the tire/pavement contact It is known that the steered wheel does not turn around the center of the tire/pavement contact region rather than around the kingpin axis when the vehicle is steering. There is a certain distance region rather than around the kingpin axis when the vehicle is steering. There is a certain distance between the intersection of the kingpin axis to the pavement and the center of the contact region, between the intersection of the kingpin axis to the pavement and the center of the contact region, which is called the kingpin offset, as shown in Figure 5a. On the other hand, when the pressure is constant, the tire with a low load has a circular shape near the contact region center. As the load increases, the tire will come into contact with the pavement over the entire width, and the contact region shape will become approximately elliptical and rectangular. Therefore, the shape of the Appl. Sci. 2019, 9, x FOR PEER REVIEW 7 of 16

Appl.contact Sci. 2019 region, 9, 1015 is taken as a rectangle for investigation in this paper. Considering the influence 7of of the 16 kingpin offset on the steering resistance torque, the geometric model of the low-speed steering friction torque between the tire and pavement is established in the coordinate system with the whichintersection is called of the the kingpin kingpin axis offset, to the as shownpavement in Figure as origin,5a. Onas shown the other in Figure hand, 5b. when the pressure is constant,Figure the 5c tire shows with athe low corresponding load has a circular longitudinal shapenear pressure the contact distribution region of center. the tire, As theand load the increases,expression the is shown tire will as come follows: into contact with the pavement over the entire width, and the contact region shape will become approximately elliptical and rectangular. Therefore, the shape of the contact n region is taken as a rectangle for investigation in+ this2n paper.Fl Considering the inﬂuence of the kingpin ()=−n 1 zt n pxy, +  y (5) offset on the steering resistance torque, the geometricn lbn model1 2 of the low-speed steering friction torque tt between the tire and pavement is established in the coordinate system with the intersection of the kingpinwhere n axis is toa constant, the pavement n = 2~10 as origin, in general, as shown which in Figurecan be 5approximatelyb. n = 4 for the radial tire.

(a) (b) (c)

FigureFigure 5.5. SteeringSteering modelmodel ofof thethe steeredsteered wheel:wheel: ((aa)) SchematicSchematic diagramdiagram ofof thethe kingpinkingpin axisaxis andand kingpinkingpin offset;offset; ((bb)) geometricgeometric modelmodel ofof thethe low-speedlow-speed steeringsteering frictionfriction torque;torque; andand ((cc)) longitudinallongitudinal pressurepressure distributiondistribution ofof thethe tire.tire.

FigureIn Figure5c shows5b, an infinitesimal the corresponding element longitudinal with dx as length pressure and distribution dy as width of at the an tire,arbitrary and point the expression is shown as follows: ()xy, is taken in the tire/pavement contact region, as a consequence, the load on the infinitesimal element is: n + 1 2nF l n ( ) = z t − n p x, y n+1 y (5) n =⋅⋅lt ()bt 2 d,ddFpxyxyz (6) where n is a constant, n = 2~10 in general, which can be approximately n = 4 for the radial tire. When the infinitesimal element rotates around the original point O on the pavement, the friction In Figure5b, an infinitesimal element with dx as length and dy as width at an arbitrary point force between the two is: (x, y) is taken in the tire/pavement contact region, as a consequence, the load on the infinitesimal element is: =⋅ =⋅() ⋅ ⋅ ddFfzμ F μ pxy ,dd x y (7) dFz = p(x, y) · dx · dy (6) where μ is the friction coefficient between the tire and pavement. When the infinitesimal element rotates around the original point O on the pavement, the friction 22 forceSince between the thedistance two is: from this arbitrary point ()xy, to point O is rxy=+, the friction torque of the infinitesimal element to pointdFf O= is:µ · dFz = µ · p(x, y) · dx · dy (7) where µ is the friction coefficient between=⋅ the = tire22 and + pavement. ⋅⋅() ⋅ ⋅ ddMrFff xyμ pxy ,dd x yp (8) Since the distance from this arbitrary point (x, y) to point O is r = x2 + y2, the friction torque of the infinitesimal element to point O is: The steering friction torque of a single tire can be calculated by integrating dM f on the whole

contact region, which is denoted as M − andq can be expressed as: fs 2 2 dM f = r · dFf = x + y · µ · p(x, y) · dx · dy (8)  bt − l c t ==2 2 ()22 + MMfs− dd,d f  xl μpxy x y y (9)  −+bt − t c S 2 2 Appl. Sci. 2019, 9, 1015 8 of 16

The steering friction torque of a single tire can be calculated by integrating dM f on the whole contact region, which is denoted as M f −s and can be expressed as:

( bt − ) lt R 2 c R 2 p 2 2 M f −s = dM f = dx µp(x, y) x + y dy (9) −( bt + ) − lt sS 2 c 2 where c is the kingpin offset. Under a low-speed steering condition, the friction coefﬁcient µ between the tire and pavement can be represented approximately by Equation (1). Considering the direction of the steering friction torque, the expression of the low-speed steering friction torque can be ﬁnally written as:

 ( bt − ) lt dδ dδ R 2 c R 2 p 2 2  M f = −sign · 2 · M f −s = −sign · 2 · b dx l µp(x, y) x + y dy  dt dt −( t +c) − t  2 2 µ = f (u) = 0.4511 · e−0.4603·u + 0.2376 (10)  4 4  5 2 Fz lt 4  p(x, y) = 5 − y  4 lt bt 2

dδ where M f is the low-speed steering friction torque, δ is the steered wheel angle, “−sign dt ”, which indicates that the direction of M f is always opposite to the turning direction of the steered wheel and u is the vehicle speed. When u = 0, the value of M f in Equation (10) can be regarded as the friction torque under the pivot steering condition.

4.2. The Aligning Torque Caused by the Kingpin Inclination The existence of the kingpin inclination makes the steering system work against the front axle load when steering and makes the steered wheel return to the center automatically as well [30]. The corresponding aligning torque can be expressed as:

D δ M = −G · c + − δ tan θ · sin(2θ) · sin (11) θ 1 2 t 2 where θ is the kingpin inclination angle, Mθ is the aligning torque caused by the kingpin inclination, “−” indicates that the direction of Mθ is always opposite to the direction of steered wheel angle, G1 is the front axle load and c is the kingpin offset.

4.3. Aligning Torques Caused by the Kingpin Caster and Pneumatic Trail

Due to the centrifugal effect, the lateral force Fy exists at the center of steered wheel when the vehicle steers. Accordingly, the lateral reaction force FY, which is also known as the cornering force, is generated by the pavement on the steered wheel. The cornering force produces the aligning torques which is related to the kingpin caster trail and pneumatic trail. The detailed solution procedure of the aligning torques caused by the kingpin caster and pneumatic trail is shown in Figure6. The 2-degree-of-freedom (2-dof) vehicle dynamic model is employed here to calculate the ideal side-slip angle and yaw rate. It can also be used to identify and estimate the actual side-slip angle. Under the 2-dof vehicle model, the lateral force and the aligning torque are considered acting on the front axle as a whole, which is consistent with the processing ideas in the calculation of M f and Mθ in Sections 4.1 and 4.2. Appl. Sci. 2019, 9, 1015 9 of 16 Appl. Sci. 2019, 9, x FOR PEER REVIEW 9 of 16

𝒖—Longitudinal 𝜷—Side-slip Angle Speed of Vehicle of Mass Center 2-dof

𝜹—Steering Angle Vehicle Model 𝝎 —Yaw Rate of Front-wheel 𝒓

𝒂𝝎 𝜶 𝜹𝝃 𝜷 𝒓 𝜹 𝟏 𝒖

𝑴 —Aligning Torque 𝜶 —Side-slip Angle 𝜸 𝑭 —Lateral Force 𝟏 Caused by Kingpin Caster 𝒚 of Front-wheel ’Magic Formula’ Tire Model 𝑴𝒁—Aligning Torque 𝑭𝒛—Vertical Load Caused by Pneumatic Trail on Tire

Figure 6. Solution procedure of the aligning torques caused by the kingpin caster and pneumatic trail. Figure 6. Solution procedure of the aligning torques caused by the kingpin caster and pneumatic trail. The corresponding vehicle motion differential equations of the 2-dof vehicle model in Figure6 The corresponding vehicle motion differential equations of the 2-dof vehicle model in Figure 6 are expressed as Equation (12) [31]: are expressed as Equation (12) [31]: ( 1 . (k1+ k2)β + (ak1 1 − bk2)ωr − k1δ = m v + uωr ()++u () − −=+ () . (12)  kk12β 1 ak2 1 bk 22ωrr k 1δ mv uω (ak1 − bk2)β +uu a k1 + b k2 ωr − ak1δ = IZωr  (12) 1  ()ak−+ bk β ()ak22 + bk ω −=ak δ I ω where a and b are respectively the12 distance fromu the 1 front 2 andrZr rear 1 axle to the center of mass, k1 and k2 are respectively the cornering stiffness of the front and rear wheel, u and v are respectively the wherelongitudinal a and velocity b are respectively along the x-axis the anddistance the lateral from the velocity front alongand rear the axley-axis to ofthe the center vehicle, of mass,IZ is thek1 rotational inertia around the z-axis, β is the side-slip angle of the mass center, ωr is the yaw rate and δ and k2 are respectively the cornering stiffness of the front and rear wheel, u and v are is the steered wheel angle. respectively the longitudinal velocity along the x-axis and the lateral velocity along the y-axis of the According to the 2-dof vehicle motion differential equations in Equation (12), the simulation vehicle,model is establishedIZ is the rotational in the MATLAB/Simulink inertia around the z environment,-axis, β is the which side-slip takes angle the forwardof the mass velocity center,u andωr issteered the yaw wheel rate angleand δδ as is the input, steered and wheel takes theangle. side-slip angle of mass center β and yaw rate ωr as output.According The desired to the side-slip 2-dof anglevehicle of motion the front differential wheel α1 can equations be obtained in Equation by one more (12), stepthe calculation,simulation whichmodel is:is established in the MATLAB/Simulink environment, which takes the forward velocity u aω and steered wheel angle δ as input,α =and− (takesδ − ξ )the= side-slipβ + r −angleδ of mass center β and yaw rate(13) 1 u ω α r asWhen output. the The vehicle desired is steering, side-slip theangle tire of easily the front gets wheel into a nonlinear1 can be obtained state. Combined by one more with step the calculation,side-slip angle which of theis: tire and vertical load, the ‘Magic Formula’ tire model is used to calculate the lateral force F and aligning torque caused by the pneumatic trail M , so as to accurately describe the y aω Z mechanical characteristics of the tire. Theαδξβδ=− expressions() − = of + ther lateral − force F and aligning torque (13)M 1 u y Z are as follows [32]: When the vehicle( is steering, the tire easily gets into a nonlinear state. Combined with the side- F = D sin{C arctan[B (1 − E )α + E arctan(B α)]} slip angle of the tire and verticaly 1 load, the1 ‘Magic1 Formula’1 tire1 model is1 used to calculate the lateral(14) MZ = D2 sin{C2arctan[B2(1 − E2)α + E2arctan(B2α)]} force Fy and aligning torque caused by the pneumatic trail MZ , so as to accurately describe the wheremechanicalD1, B 1characteristics, C1 and E1 are of respectively the tire. The the expressions peak factor, of stiffness the lateral factor, force shape Fy factorand aligning and curvature torque factor for the solution of lateral force. D2, B2, C2 and E2 are respectively the peak factor, stiffness factor, MZ are as follows [32]: shape factor and curvature factor for the solution of the aligning torque and α is the side-slip angle of the tire.  =−+{ () ()}  FDy 11sin C arctan B 111 1 Eα EB arctan 1α Under a single steering condition, the coefﬁcients of the ‘Magic Formula’ can be referred to(14) in MD=−+sin{} C arctan B() 1 Eα EB arctan ()α the literature [33]. The curves Z of the22 lateral forceFy 222and the aligning torque 2 caused by the pneumatic trail MZ with the side-slip angle of the tire α are drawn according to Equation (14) in the MATLAB, whichwhere areD1 shown, B1 , inC1 Figure and 7Ea,b.1 are respectively the peak factor, stiffness factor, shape factor and curvature factor for the solution of lateral force. D2 , B2 , C2 and E2 are respectively the peak Appl. Sci. 2019, 9, x FOR PEER REVIEW 10 of 16

factor, stiffness factor, shape factor and curvature factor for the solution of the aligning torque and α is the side-slip angle of the tire. Under a single steering condition, the coefficients of the ‘Magic Formula’ can be referred to in

the literature [33]. The curves of the lateral force Fy and the aligning torque caused by the pneumatic

Appl.trail Sci.M2019Z with, 9, 1015 the side-slip angle of the tire α are drawn according to Equation (14) 10in ofthe 16 MATLAB, which are shown in Figure 7a,b.

(a) (b)

Figure 7. The relation curves obtained by ‘Magic Formula’: (a) The lateral force with the side-slip angle of tire; and (b) the aligning torque with the side-slip angle of tire. Figure 7. The relation curves obtained by ‘Magic Formula’: (a) The lateral force with the side-slip Furthermore,angle of tire; and the (b) aligning the aligning torque torq causedue with bythe theside-slip kingpin angle caster of tire. can be expressed as:

D Furthermore, the aligningM torque= − Fcaused· l = byF ·thel = kingpinF · sin casterγ cos canδ be expressed as: (15) γ Y y y 2 =− ⋅ = ⋅ = ⋅D where γ is the kingpin caster angle,MFlFlF “γ−” indicatesYyy that the directionsinγδ cos of M is opposite to the direction(15) 2 γ of FY and l is the distance from the contact point of the steered wheel and pavement to the kingpin axiswhere in theγ vehicle’sis the kingpin longitudinal caster plane.angle, “−” indicates that the direction of Mγ is opposite to the F 4.4.direction Model of of theY Low-Speedand l is the Steering distance Resistance from the Torque contact point of the steered wheel and pavement to the kingpin axis in the vehicle’s longitudinal plane. Since the components of the steering resistance torque have taken into account directions in their equations,4.4. Model of they the justLow-Speed need to Steering be added Resistance up. The Torque low-speed steering resistance torque is equal to the sum of the low-speed steering friction torque between the tire/pavement and the aligning torques caused by theSince kingpin the components inclination, kingpinof the steering caster resistance and pneumatic torque trail, have as taken shown into in account Equation directions (16): in their equations, they just need to be added up. The low-speed steering resistance torque is equal to the

sum of the low-speed steering frictionMr =torqueM f + beMtweenθ + M theγ + tire/pavementMZ and the aligning torques(16) caused by the kingpin inclination, kingpin caster and pneumatic trail, as shown in Equation (16): In the case of pivot steering, only the aligning torque caused by the kingpin inclination needs to MMMMM=+++ be considered in the calculation of the aligningrf torque.θγZ (16)

5. ModelIn the Simulation case of pivot and steering, Validation only by the Experiments aligning torque caused by the kingpin inclination needs to be considered in the calculation of the aligning torque. 5.1. Model Simulation 5. Model Simulation and Validation by Experiments The simulation models of the pivot and low-speed steering resistance torque are respectively established5.1. Model Simulation in the MATLAB/Simulink environment according to Equation (16). In the simulation models, the input signal is the steering wheel angle which is simulated by a periodic signal of triangularThe simulation wave, and models the output of the signal pivot is and the low-speed equivalent steering resistance resistance torque appliedtorque are on therespectively steering column.established Moreover, in the MATLAB/Simulink the equivalent resistance environment torque onaccording the steering to Equation column and(16). steering In the simulation resistance torquemodels, have the theinput following signal relations:is the steering wheel angle which is simulated by a periodic signal of triangular wave, and the output signal is the equiMvalentr resistance torque applied on the steering Tr = (17) iω0 · η+ where Tr is the equivalent resistance torque on the steering column, Mr is the steering resistance torque, iω0 is the angle ratio of the steering system and η+ is the forward efﬁciency of the steering gear. The simulation model of the low-speed steering resistance torque is shown in Figure8. Appl. Sci. 2019, 9, x FOR PEER REVIEW 11 of 16

column. Moreover, the equivalent resistance torque on the steering column and steering resistance torque have the following relations: M T = r r ⋅ (17) iω0 η+

where Tr is the equivalent resistance torque on the steering column, Mr is the steering resistance

torque, iω0 is the angle ratio of the steering system and η+ is the forward efficiency of the steering gear. Appl. Sci. 2019, 9, 1015 11 of 16 The simulation model of the low-speed steering resistance torque is shown in Figure 8.

Figure 8.8. Simulation model of the low-speed steeringsteering resistanceresistance torque.torque. 5.2. Real Vehicle Experiments 5.2. Real Vehicle Experiments Taking a compact car as the test object, the experiments of the pivot and low-speed steering Taking a compact car as the test object, the experiments of the pivot and low-speed steering resistance torque were carried out. resistance torque were carried out. The experimental apparatus mainly included the MSW DTI sensor-Universal measurement The experimental apparatus mainly included the MSW DTI sensor-Universal measurement steering wheel and the Correvit S-Motion DTI Non-contact optical sensor. Detailed information on the steering wheel and the Correvit S-Motion DTI Non-contact optical sensor. Detailed information on main experimental apparatus is shown in Table3. When the driver turned the measurement steering the main experimental apparatus is shown in Table 3. When the driver turned the measurement wheel, which is tightly mounted on the vehicle steering wheel through brackets, the real-time steering steering wheel, which is tightly mounted on the vehicle steering wheel through brackets, the real- wheel torque/angle could be measured. The Correvit S-Motion DTI Non-contact optical sensor was time steering wheel torque/angle could be measured. The Correvit S-Motion DTI Non-contact optical ﬁxed on the vehicle door by suckers. After relevant parameters were conﬁgured, the real-time vehicle sensor was fixed on the vehicle door by suckers. After relevant parameters were configured, the real- speed, lateral acceleration and yaw rate could be measured in the process of vehicle movement. time vehicle speed, lateral acceleration and yaw rate could be measured in the process of vehicle The experimental apparatus was connected to the power supply and the notebook computer, and all movement. The experimental apparatus was connected to the power supply and the notebook the measured data were recorded by DEWESoft software in the computer. computer, and all the measured data were recorded by DEWESoft software in the computer.

Table 3. Detailed information of main experimental apparatus. Table 3. Detailed information of main experimental apparatus. Apparatus Application Type Manufacturer Apparatus Application Type Manufacturer Universal measurement steering wheelUniversal for measurement measurement of the steering wheel MSW DTI sensor steering torque, steering angle and UniversalMSW DTI measurement sensor for measurement of the steering torque,5612A KISTLER steering speed; for vehicle driving Universalsteering measurement wheel steering angle and steering speed; for 5612A KISTLER dynamics tests like ISO 4138, steering wheel steady-statevehicle driving circular dynamics course drive. tests like ISO 4138, steady-state circular course drive. High-precision, slip-free measurementHigh-precision, of: slip-free measurement of: Correvit S-Motion DTI • Distance; • Distance; Correvit S-Motion DTI 2055A KISTLER Non-contact optical sensor • • Speed (x,x y y); Non-contact optical Speed ( , ); 2055A KISTLER • Acceleration and angular rates; sensor • Acceleration and angular rates; • GPS position data and time; • Pitch and roll angle.

• GPS position data and time; • Pitch and roll angle.

Before the experiments, the assist power of vehicle steering system needed to be cut off. The experimental procedures were as follows: Make the vehicle respectively maintain a static state and Appl. Sci. 2019, 9, 1015 12 of 16 uniform linear motion state at different speeds (1 km/h, 2 km/h, 3 km/h, 4 km/h, 5 km/h, 10 km/h, 15 km/h and 20 km/h), turn the measurement steering wheel to the left and right with a constant speed andspeed then and collect then collect the data the dataof the of thesteering steering wheel wheel torque/angle, torque/angle, vehicle vehicle speed speed as as well well as as the the lateral acceleration and yaw rate of the vehicle, and then conduct several experimentsexperiments forfor eacheach condition.condition. The experimental scene is shown in Figure9 9..

Figure 9. The experimental scene. 5.3. Comparison of Simulation and Experiment Results 5.3. Comparison of Simulation and Experiment Results On the premise of no steering assist power, it can be known from Newton’s Third Law that the On the premise of no steering assist power, it can be known from Newton’s Third Law that the steering wheel torque and equivalent resistance torque on the steering column are equal and opposite steering wheel torque and equivalent resistance torque on the steering column are equal and opposite in direction when turning the steering wheel at a constant speed, as shown in Equation (18): in direction when turning the steering wheel at a constant speed, as shown in Equation (18): Mr Ts =TT−=−Tr = =−− r (18) sriω0 ·⋅ η+ (18) iω0 η+ where T is the steering wheel torque and T is the equivalent resistance torque on the steering column. where Ts is the steering wheel torque andr T is the equivalent resistance torque on the steering Thus,s the measured steering wheel torquer and the simulated equivalent resistance torque on the column.steering column can be contrasted to verify the accuracy of the low-speed steering resistance torque model.Thus, The the comparative measured steering results are wheel shown torque in Figure and the 10 .simulated For the pivot equivalent steering resistance experiment, torque the on driver the steeringturned the column measurement can be contrasted steering wheelto verify with the a ac constantcuracy of speed the low-speed when the vehiclesteering was resistance under atorque static model.state, causing The comparative the steering results wheel are to shown reach the in Figure limit positions 10. For the on pivot two sides. steering As experiment, a result, there the are driver two turnedmutations the onmeasurement the curve in steering Figure 10 wheela. For with the low-speeda constant steeringspeed when experiments, the vehicle the was driver under turned a static the state,measurement causing steeringthe steering wheel wheel when to the reach vehicle the limit maintained positions a state on two of uniform sides. As linear a result, motion there at differentare two mutationsspeeds (1 km/h,on the 2curve km/h, in 3Figure km/h, 10a. 4 km/h, For the 5 km/h,low-speed 10 km/h, steering 15 experiments, km/h and 20 the km/h). driver The turned steering the wheelmeasurement did not reachsteering the limitwheel position when onthe either vehicle side main for safety.tained Therefore, a state of it isuniform roundat linear the corner motion of theat differentcurves from speeds Figure (1 km/h, 10b–h. 2 km/h, In addition, 3 km/h, the 4 km/h, decreasing 5 km/h, height 10 km/h, of these 15 km/h curve and rings 20 km/h). from Figure The steering 10a–h, whichwheel did means not thereach steering the limit resistance position torque on either decreases side for with safety. the vehicleTherefore, speed. it is Itround can be at seen the corner from the of theﬁgures curves that from the simulationFigure 10b–h. results In addition, are consistent the decreasing with the experimentalheight of these results, curve indicatingrings from that Figure the 10a–h,established which model means of the steeringsteering resistanceresistance torquetorque hasdecreases a high accuracy.with the vehicle speed. It can be seen from the figures that the simulation results are consistent with the experimental results, indicating that the established model of the steering resistance torque has a high accuracy. Appl. Sci. 2019, 9, 1015 13 of 16 Appl. Sci. 2019, 9, x FOR PEER REVIEW 13 of 16

(a) Pivot steering (b) 1 km/h

(e) 4 km/h (f) 5 km/h

(g) 10 km/h (h) 20 km/h

FigureFigure 10. 10.Simulation Simulation and and experimental experimental results of the steering steering wheel wheel torque torque at at different different speeds. speeds. Appl. Sci. 2019, 9, x FOR PEER REVIEW 14 of 16

Furthermore, the surface plot of the low-speed steering resistance torque with the vehicle speed and steering wheel angle can be finally drawn based on the simulation results, as shown in Figure 11. It can be clearly seen from the figure that the change laws of the low-speed steering resistance torque are as follows: 1. The low-speed steering resistance torque increased with the steering wheel angle at a certain Appl.speed. Sci. When2019, 9, 1015the vehicle speed was below 10 km/h, the steering resistance torque almost changed14 of 16 linearly with the steering wheel angle. When the vehicle speed was between 10 km/h and 20 km/h, the steering resistance torque varied nonlinearly with the steering wheel angle; Furthermore,2. The low-speed the surfacesteering plot resistance of the low-speedtorque decrea steeringsed with resistance the vehicle torque speed with at the a certain vehicle steering speed andwheel steering angle. wheel When angle the vehicle can be speed ﬁnally was drawn in the based range on theof 0 simulation km/h to 5 results, km/h, the as shown steering in Figureresistance 11. Ittorque can be decreased clearly seen rapidly from thewith ﬁgure the thatvehicle the changespeed. When laws of the the vehicle low-speed speed steering exceeded resistance 5 km/h, torque the arechange as follows: of the steering resistance torque was not obvious.

FigureFigure 11.11. 3D3D surfacesurface plotplot ofof thethe low-speedlow-speed steeringsteering resistanceresistance torque.torque.

6. Conclusions1. The low-speed steering resistance torque increased with the steering wheel angle at a certain speed. When the vehicle speed was below 10 km/h, the steering resistance torque almost changed The steering wheel angle and the vehicle speed had great effects on the steering resistance linearly with the steering wheel angle. When the vehicle speed was between 10 km/h and 20 km/h, torque. The study on low-speed steering resistance torque can provide a theoretical basis for the the steering resistance torque varied nonlinearly with the steering wheel angle; intervention and withdrawal mechanisms of a supercapacitor in the SC-EPS system on the one hand, 2. The low-speed steering resistance torque decreased with the vehicle speed at a certain steering and the fill the gap in the research field on low-speed steering resistance torque on the other hand, wheel angle. When the vehicle speed was in the range of 0 km/h to 5 km/h, the steering resistance providing a useful reference for the design of assist characteristics and control strategies for other torque decreased rapidly with the vehicle speed. When the vehicle speed exceeded 5 km/h, the change new EPS systems. of the steering resistance torque was not obvious. The work done in this paper and the results found from the above analyses have all come to the 6.following Conclusions conclusions: 1. The finite element (FE) model of steering friction force between the tire and pavement was The steering wheel angle and the vehicle speed had great effects on the steering resistance torque. established and the exponential expression of the equivalent steering friction coefficient with the The study on low-speed steering resistance torque can provide a theoretical basis for the intervention vehicle speed was obtained by means of FE simulation and numerical fitting; and withdrawal mechanisms of a supercapacitor in the SC-EPS system on the one hand, and the fill the 2. The expression of the steering friction torque between the tire and pavement was derived gap in the research field on low-speed steering resistance torque on the other hand, providing a useful based on calculus theory. The mathematical model of the low-speed steering resistance torque reference for the design of assist characteristics and control strategies for other new EPS systems. including the steering friction torque and aligning torques was constructed, and the accuracy of the The work done in this paper and the results found from the above analyses have all come to the model was verified by real vehicle experiments; following conclusions: 3. The low-speed steering resistance torque increased with the steering wheel angle at a certain 1. The finite element (FE) model of steering friction force between the tire and pavement was vehicle speed. As the vehicle speed increased, the low-speed steering resistance torque gradually established and the exponential expression of the equivalent steering friction coefficient with the presented nonlinearity with the steering wheel angle; vehicle speed was obtained by means of FE simulation and numerical fitting; 2. The expression of the steering friction torque between the tire and pavement was derived based on calculus theory. The mathematical model of the low-speed steering resistance torque including the steering friction torque and aligning torques was constructed, and the accuracy of the model was verified by real vehicle experiments; Appl. Sci. 2019, 9, 1015 15 of 16

3. The low-speed steering resistance torque increased with the steering wheel angle at a certain vehicle speed. As the vehicle speed increased, the low-speed steering resistance torque gradually presented nonlinearity with the steering wheel angle; 4. The low-speed steering resistance torque decreased with the vehicle speed at a certain steering wheel angle. When vehicle speed was below 5 km/h, the steering resistance torque signiﬁcantly dropped off. When vehicle speed was above 5 km/h, the rate of decrease slowed down gradually.

Author Contributions: D.C. and B.T. conceived of and designed the method. D.C., C.Y., D.Z. and Y.H. performed the experiments and analyzed the experimental data. Finally, D.C. wrote the paper under the guidance of H.J. and B.T. Funding: The work in this paper is supported by National Natural Science Foundation of China (Grant No. 51605199, No. 51675235), Natural Science Foundation of Jiangsu Province (Grant No. BK20160527), China Postdoctoral Science Foundation (Grant No. 2016M590417), Natural Science Foundation for Colleges and Universities of Jiangsu Province (Grant No. 16KJB580001) and Jiangsu Province Postdoctoral Science Foundation (Grant No. 1601222C). Acknowledgments: The authors would like to thank the teachers in the laboratory for their guidance on the use of the experimental apparatus, and Jiangsu Gang Yang Steering Gear Company for providing the experimental site. Meanwhile, many thanks to Chen Zhu, Ziyan Lin and Yue Yin for participating in the experiments, as well as other teachers and students in the research group for participating in the discussion. Conﬂicts of Interest: The authors declare no conﬂict of interest.

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