Continuous Damping Control for Rollover Prevention with Optimal Distribution Strategy of Damping Force

applied sciences

Article Continuous Damping Control for Rollover Prevention with Optimal Distribution Strategy of Damping Force

Xu Zhang 1, Chuanxue Song 1, Shixin Song 2, Jingwei Cao 1 , Da Wang 1, Chunyang Qi 1, Zhinan Ma 3 and Feng Xiao 1,4,* 1 State Key Laboratory of Automotive Simulation and Control, Jilin University, Changchun 130022, China; [email protected] (X.Z.); [email protected] (C.S.); [email protected] (J.C.); [email protected] (D.W.); [email protected] (C.Q.) 2 College of Mechanical and Aerospace Engineering, Jilin University, Changchun 130022, China; [email protected] 3 BAIC Group Oﬀ-Road Vehicle Co., Ltd., Beijing 101300, China; [email protected] 4 Taizhou Automobile Power Transmission Research Institute, Jilin University, Taizhou 225322, China * Correspondence: [email protected]

Received: 8 September 2020; Accepted: 12 October 2020; Published: 16 October 2020

Abstract: Vehicle rollover has always been a highly dangerous condition that can cause severe traffic casualties. In this work, a 14-degree-of-freedom vehicle model in MATLAB/Simulink is constructed with the vehicle suspension system dynamics. The validity of the model is verified by comparing with the CarSim model. Then an optimal distribution of damping force strategy with continuous damping control is proposed by combining the traditional lateral load transfer ratio control with optimized equations of suspension damping force. The damping force compensation of the left and right sides is the core of the optimal distribution of damping force strategy. The effectiveness and optimization effect of the optimal distribution of damping force strategy is proved by the simulation results under the fishhook and crosswind tests. The result shows that continuous damping control has evident control effects on vehicle rollover compared with passive suspension. The optimal distribution of the damping force strategy with continuous damping control has a great better performance than traditional continuous damping control, and it provides a certain assistance to vehicle handling stability.

Keywords: rollover prevention; continuous damping control; lateral load transfer ratio; optimal distribution of damping force

1. Introduction Vehicle rollover is a very dangerous type of road accident. With the increasing complexity of the traffic environment and the continuous growth of passenger car ownership, traffic accidents have become commonplace. In 2014, the National Highway Traffic Safety Administration (NHTSA) recorded that 18.9% percent of traffic fatalities were caused by rollovers (7592 of 40,164) [1]. In 2018, these rollovers caused 22.4% of the total traffic fatalities according to the NHTSA statistics (8111 of 32,675) [2]. Although automobile research and design are constantly improving, the fatality rate remains at a very high level. One major reason is that drivers have difficulty predicting the risk of rollover directly. In the event of a rollover, cars are difficult to control due to the great decline of vehicle handling and the fluctuation of the drivers’ psychological state. To protect people’s lives and to ensure the safety of property, rollovers must be detected as soon as they occur. Moreover, an active anti-rollover control method must be adopted. One research hotspot in the field of the active safety control of vehicles is predicting the vehicle driving state and determining the real-time state parameters transmitted to prevent vehicle rollover

Appl. Sci. 2020, 10, 7230; doi:10.3390/app10207230 www.mdpi.com/journal/applsci Appl. Sci. 2020, 10, 7230 2 of 21 accidents [3,4]. In terms of time sequence, two steps can be taken for vehicle rollover prevention: vehicle rollover warning and vehicle rollover prevention action [5]. Many indexes are used to predict a rollover, and they are also used as vehicle rollover warning parameters. The static stability factor (SSF) is the first basic index that shows the inherent tendency of a vehicle to rollover [6]. It was used by Winkler et al. to explain the relationship between vehicle geometry and rollover stability and to describe the anti-rollover performance of vehicles [7]. As one of the indexes for New Car Assessment Program in evaluating vehicle rollover performance, SSF is defined as half of the ratio of the wheel distance to the mass gravity center height. It ignores the dynamic performance of suspension, and its control strategy has limitations in practical application. The dynamic stability factor (DFC) was indicated by Huston and Kelly to solve the problem in which vehicle dynamic information cannot be reflected in SSF [8]. Energy utilization is another index, which uses vehicle kinetic energy compared with the maximum potential energy [9]. Lateral load transfer ratio (LTR), according to the force between the tire and the road surface, is defined as the ratio of the difference between the vertical loads of the left and right wheels and the sum of these loads [10]. It has been widely used because of its simple expression and good portability [11–14]. Phanomchoeng et al. and Ataei et al. indicated that the tire vertical acceleration and other information are required for an index based on LTR, which can simultaneously identify tripped rollover and un-tripped rollover [15]. To improve the prediction ability of traditional LTR, Larish et al. put forward predictive LTR (PLTR) through a theoretical derivation of the linear vehicle dynamics model [16]. The rollover index (RI) is an experience index proposed by Yoon et al., which uses dynamic simulation and has high precision and certain prediction ability [17]. For RI, a large number of experimental data need to be compared and accumulated. An improved algorithm of RI was formulated by Chou and Chu by using the gray model [18], which adds the prediction function of RI [19]. Moreover, Mashadi et al. proposed a method for judging the rollover of wheels after leaving the ground by studying the dynamic characteristics of vehicles with wheels off the ground [20]. To improve the prediction effect, Chen and Peng proposed a strong prediction ability index called time to rollover (TTR) [21]. TTR is defined according to the lateral pressure and roll angle, depending on the state estimation of the linear vehicle model; it can be realized by different algorithms [22]. In addition, Stankiewicz et al. used the concept of zero moment point (ZMP) to predict the rollover risk of vehicles [23]. Ataei et al. proposed a general rollover index for the detection of rollover in both tripped and un-tripped cases in 2019 [24]. When a rollover index is detected as the risk of rollover, the active anti-rollover control is triggered immediately so that the dynamic behavior of the vehicle is restrained to ensure the rollover stability of the vehicle. Anti-rollover has four basic control methods: active steering, active/semi-active suspension, differential braking, and active stabilizer bar [25]. Among these methods, suspension control is the most direct and effective way to control vehicle roll motion [26,27]. Meanwhile, an active stabilizer bar or active/semi-active suspension is often used in suspension systems. The active stabilizer bar can provide a large roll stiffness and can achieve a very good anti-rollover effect [28,29]. It also needs enough installation space. An active and semi-active suspension system is a kind of automobile suspension mechanism that ensures that people in the car have maximum comfort with suitable control strategy [30]. A slow active suspension is used by Westhuizen et al. to change the roll stiffness of the vehicle and avoid un-tripped rollover by reducing the lateral stability of the vehicle [31]. Chu Duanfeng et al. designed the anti-roll control of semi-active suspension and active suspension using a sliding mode algorithm [5]. However, the ideal control effect cannot be obtained by a single method. To guarantee the lateral and yaw stability of the vehicle, chassis integrated control has been increasingly used. Yoon et al. integrated continuous damping control (CDC) and electronic stability control (ESC) to improve vehicle rollover stability [32]. Yoon et al. also established the CDC, ESC, and active front steering (AFS) integrated control method; the three-mode switching control is proposed to make use of the advantages of AFS and ESC [33]. Tchamna et al. integrated ESC and active suspension, which can prevent vehicle rollover by directly controlling suspension travel [34]. Appl. Sci. 2020, 10, 7230 3 of 21

In the field of automatic driving, many new integrated anti-rollover control methods have also been proposed in recent years [35–37]. This paper includes the following three parts: the vehicle dynamics model and model verification (Part 1), the design of the optimal distribution strategy of the damping force controller (Part 2), and the simulation and result analysis (Part 3). In Part 1, a 14-degree-of-freedom model, including the longitudinal, lateral, vertical, roll, pitch and yaw direction of vehicle, and rotation and jump of each vehicle wheel, is constructed in MATLAB/Simulink with numerous equations with vehicle suspension system dynamics. The basic parameters of the vehicle are set, and the same parameters are selected for simulation in both MATLAB/Simulink and CarSim. The validity of the model is verified by comparing it with the CarSim model. In Part 2, an optimal distribution of a damping force strategy with continuous damping control is proposed by combining the traditional lateral load transfer ratio control with the optimized equations of suspension damping force. The damping force compensation of the left and right sides is the core of the optimal distribution of the damping force strategy. In Part 3, the effectiveness and optimization effect of the optimal distribution of the damping force strategy is proven by the simulation results under the fishhook and crosswind tests. The result shows that continuous damping control has an evident control effect on vehicle rollover compared with passive suspension. The optimal distribution of the damping force strategy with continuous damping control has better performance than the traditional continuous damping control. Moreover, it provides a certain assistance to vehicle handling stability.

2. Vehicle Dynamics Model and Model Veriﬁcation

2.1. Fourteen-Degree-of-Freedom Vehicle Model A 14-degree-of-freedom model was constructed to indicate a nonlinear dynamic vehicle [38]. Numerous sub-models and equations were involved in the whole vehicle, such as the steering model, driving and braking model, wheel rotating model, and tire model by magic formula. Only the equations related to vehicle roll dynamics and suspension dynamics are listed below.

2.1.1. Vehicle Dynamics Equations As shown in Figure1, the vehicle dynamics equations of an automobile in a horizontal, longitudinal symmetry plane and a transverse symmetry plane passing through a mass center can be written as the following equations:

. mt(u + qw rv) . − (1) = msh (q + pr) + (F + F + F + Fxr ) F Fw F 0 xl1 xr1 xl2 2 − i − − f . . mt(v + ru pw) = msh0(p + qr) + (F + F + F Fyr ) (2) − − yl1 yr1 yl2 2 . B1 Izzr = (Ixx Iyy)pq + L1(Fyl1 + Fyr1) L2(Fyl2 + Fyr2) + 2 (Fxl1 Fxr1) − B2 − − (3) + (F Fxr ) 2 xl2 − 2 where mt and ms are the vehicle total and sprung mass, respectively. u, v, and w are the longitudinal, lateral, and vertical speed of the vehicle, respectively. r, p, and q are the yaw angle rate, roll angle rate, and pitch angle rate of the vehicle body, respectively. Ixx, Iyy, and Izz are the moment inertia of the vehicle around the X-axis, Y-axis, and Z-axis, respectively. In addition, Fxij and Fyij are the longitudinal and lateral force of each wheel, respectively. Lj is the longitudinal distance from each axle to the mass center, while Bj is the wheel distance of each axle. h0 is the vertical distance from the mass center to the roll axis. Fi,Fw, and Ff are the slope, aerodynamics, and rolling resistance force, respectively. Subscript Appl.Appl. Sci.Sci.2020 2020, ,10 10,, 7230 x FOR PEER REVIEW 44 of of 21 22

the roll axis. Fi, Fw, and Ff are the slope, aerodynamics, and rolling resistance force, respectively. i (l or r) indicates the left and right sides of the wheel, while subscript j (1 or 2) indicates the front and Subscript i (l or r) indicates the left and right sides of the wheel, while subscript j (1 or 2) indicates the rear axles. front and rear axles.

FigureFigure 1. 1.A A 14-degree-of-freedom 14-degree-of-freedom vehiclevehicle modelmodel(including (including longitudinal,longitudinal, lateral,lateral, vertical,vertical, roll,roll,pitch pitch andand yaw yaw direction direction of of vehicle, vehicle, and and rotation rotation and and jump jump of of each each wheel). wheel). 2.1.2. Body Motion Equations 2.1.2. Body Motion Equations When the vehicle is moving, the motion of the vehicle body can be regarded as a rigid body. When the vehicle is moving, the motion of the vehicle body can be regarded as a rigid body. According to rigid body motion theory and angular momentum theory, the mass of a car body in According to rigid body motion theory and angular momentum theory, the mass of a car body in motion satisﬁes the following diﬀerential equations: motion satisfies the following differential equations: Pitch motion equation:

Pitch .motion equation: Iyysq + (Ixxs Izzs)pr − Fvl1 Fvr1 Fvl2 Fvr2 (4) 퐼푦푦푠푞̇ + (퐼푥푥푠 − 퐼푧푧푠)푝푟= L1( + ) + L2( + ) + h(Fxl1 + Fxr1 + Fxr1 + Fxr2) − Dl1 Dr1 Dl2 Dr2 퐹푣푙1 퐹푣푟1 퐹푣푙2 퐹푣푟2 (4) Roll motion equation:= −퐿1 ( + ) + 퐿2 ( + ) + ℎ(퐹푥푙1 + 퐹푥푟1 + 퐹푥푟1 + 퐹푥푟2) 퐷푙1 퐷푟1 퐷푙2 퐷푟2 . Ixxsp + (Izzs Iyys)qr Roll motion equation:− = (Al1Fvl1 + Ar1Fvr1 + Al2Fvl2 + Ar2Fvr2) + ms gh0 sin ϕ (5) . 퐼 푝̇ + (퐼 − 퐼 )푞푟 msh0(v + ru pw) 푥푥푠 푧푧푠 푦푦푠 − − ′ = (퐴푙1퐹푣푙1 + 퐴푟1퐹푣푟1 + 퐴푙2퐹푣푙2 + 퐴푟2퐹푣푟2) + 푚푠푔ℎ 푠𝑖푛 휑 (5) aij + bij ′ = − 푚푠ℎ (푣̇ +D푟푢ij − 푝푤) (6) bij

 alj  푎푖푗 + 푏푖푗  nlj + (nlj blj) a +b , on the le f t wheel =  − −퐷푖푗 = lj lj (6) Aij  arj 푏푖푗 (7)  nrj (nrj brj) , on the right wheel  − − arj+brj where h is the height of the mass center. F푎vij푙푗is the vertical force of each wheel. Ixxs, Iyys, and Izzs are the −푛푙푗 + (푛푙푗 − 푏푙푗) , 표푛 푡ℎ푒 푙푒푓푡 푤ℎ푒푒푙 moment inertia of the vehicle sprung mass푎푙푗 + around푏푙푗 the X-axis, Y-axis, and Z-axis, respectively. g is the 퐴푖푗 = 푎 (7) acceleration of gravity. Aij, Dij, aij, bij, and푟푗nij are the geometry parameters related to the suspension 푛푟푗 − (푛푟푗 − 푏푟푗) , 표푛 푡ℎ푒 푟𝑖푔ℎ푡 푤ℎ푒푒푙 connection structure{ of each wheel. ϕ푎is푟푗 the+ 푏 roll푟푗 angle of the vehicle.

where h is the height of the mass center. Fvij is the vertical force of each wheel. Ixxs, Iyys, and Izzs are the moment inertia of the vehicle sprung mass around the X-axis, Y-axis, and Z-axis, respectively. g is Appl. Sci. 2020, 10, x FOR PEER REVIEW 5 of 22

the acceleration of gravity. Aij, Dij, aij, bij, and nij are the geometry parameters related to the suspension connection structure of each wheel. φ is the roll angle of the vehicle.

Appl.2.1.3. Sci. Suspension2020, 10, 7230 Motion Equations 5 of 21 Suspension model is shown in Figure 2. As shown in Figures 1 and 2, the body and wheel vertical 2.1.3.motion Suspension equations Motion are as follows: Equations

Suspension model is shown in Figure2. As shown퐹푣푙1 in퐹푣푟 Figures1 퐹푣푙12 and퐹푣푟2,2 the body and wheel vertical 푚푠(푤̇ + 푝푣 − 푞푢) = + + + (8) motion equations are as follows: 퐷푙1 퐷푟1 퐷푙2 퐷푟2 퐹 . Fvl1 Fvr1 Fvl2 Fvr2 푣푙1 푚푢푙m1푤(̇ 푢푙w1+=pv퐾푢푙1qu(푍)푟푙1 =− 푍푢푙1+) + 퐶푢푙1+(푤푟푙1 −+푤푢푙1) − s 퐷 (8) − Dl1 Dr1 Dl2 Dr2 푙1 퐹푣푙2  푚푢푙2.푤̇ 푢푙2 = 퐾푢푙2(푍푟푙2 − 푍푢푙2) + 퐶푢푙2(푤푟푙2 − 푤푢푙2) −Fvl1  mul1wul1 = Kul1(Zrl1 Zul1) + Cul1(wrl1 wul1) 퐷푙2  Dl1  . − − − F (9)  = ( ) + ( ) vl2퐹푣푟1  푚mul2푤ẇ ul2 = 퐾Kul2 (Z푍rl2 −Z푍ul2 ) +C퐶ul2 w(푤rl2 −wul푤2 )D− 푢푟1 . 푢푟1 푢푟1 푟푟1− 푢푟1 푢푟1 푟푟−1 푢푟−1 l2 (9)  Fvr1퐷푟1  mur1wur1 = Kur1(Zrr1 Zur1) + Cur1(wrr1 wur1) D  . − − − r퐹1 푣푟2  푚 푤̇ = 퐾 (푍 − 푍 ) + 퐶 (푤 − 푤 ) −Fvr2 mur푢푟22wur푢푟22= Kur푢푟22(Zrr푟푟22 Zur푢푟2)2 + Cur푢푟2(2wrr푟푟2 2 wur2푢푟)2 D { − − − r2퐷푟2

Figure 2. Suspension model. The unsprung mass of the suspension components and the wheels is Figure 2. Suspension model. The unsprung mass of the suspension components and the wheels is replaced by an equivalent unsprung mass in this model. replaced by an equivalent unsprung mass in this model. The force on the suspension by each wheel can be expressed by the following equations: The force on the suspension by each wheel can be expressed by the following equations: . Fvl1 = Ksl1(Zul1 Zs + L1 sin θ + nl1 sin ϕ) + Csl1(Zul1 w + L1q cos θ + nl1p cos ϕ) (10) 퐹푣푙1 = 퐾푠푙1(푍푢푙1−− 푍푠 + 퐿1 sin 휃 + 푛푙1 sin 휑) + 퐶푠푙1(푍푢푙̇ 1−− 푤 + 퐿1푞 cos 휃 + 푛푙1푝 cos 휑) (10) . F = K (Z Zs L sin θ + n sin ϕ) + C (Z w L q cos θ + n p cos ϕ) (11) 퐹vl푣푙22 = 퐾sl푠푙22(푍ul푢푙22 − 푍푠 − 퐿22 sin 휃 + 푛l2푙2 sin 휑) + 퐶sl푠푙22(푍ul푢푙̇22 − 푤 − 퐿2 2푞 cos 휃 + l푛2푙2푝 cos 휑) (11) − − . − − F = K (Z Z + L sin θ n sin ϕ) + C (Z w + L q cos θ n p cos ϕ) (12) vr1 퐹푣푟sr1 1= 퐾ur푠푟11(푍푢푟s 1 − 푍1 푠 + 퐿1 sinr1휃 − 푛푟1 sin 휑sr)1 + 퐶ur푠푟1 1(푍푢푟̇ 1 −1푤 + 퐿1푞 cosr1 휃 − − − . − − (12) 푛푟1푝 cos 휑) Fvr = Ksr (Zur Zs L sin θ nr sin ϕ) + Csr (Zur w L q cos θ nr p cos ϕ) (13) 2 2 2 − − 2 − 2 2 2 − − 2 − 2 퐹푣푟2 = 퐾푠푟2(푍푢푟2 − 푍푠 − 퐿2 sin 휃 − 푛푟2 sin 휑) + 퐶푠푟2(푍푢푟̇ 2 − 푤 − 퐿2푞 cos 휃 − where mu is the unsprung mass, and θ is the pitch angle of the vehicle. Ksij and Kuij are the stiffness(13) 푛푟2푝 cos 휑) coefficients of the spring and the wheel, respectively. Csij and Cuij are the damping coefficients of thewhere shock mu is absorber the unsprung and the m wheel,ass, and respectively. θ is the pitchZsij angleand ofZuij theare vehicle. the sprung Ksij and and Kuij unsprung are the stiffness mass displacementcoefficients of of the each spring wheel. andZ therij is wheel, the road respectively. roughness C ofsij eachand wheel.Cuij are the damping coefficients of the shock absorber and the wheel, respectively. Zsij and Zuij are the sprung and unsprung mass 2.2.displacement Model Verfication of each wheel. Zrij is the road roughness of each wheel. The vehicle model is constructed in MATLAB/Simulink (abbreviated as Simulink below). The graphical simulation model generated by Simulink is mainly based on the dynamic equation of the system built by users. Compared with the multi-body dynamic model, the biggest advantage of the Simulink model is that the combination of the model and the control system is more convenient. Moreover, the performance of the vehicle before and after the control can be compared and studied intuitively. Appl. Sci. 2020, 10, 7230 6 of 21

Given the high simulation accuracy of the software CarSim, the CarSim model is regarded as the actual vehicle in the model veriﬁcation. The model in Simulink is veriﬁed by inputting the same vehicle parameters under the same sine input of the steering wheel both in Simulink and CarSim as shown in Table1. The comparison results are shown in Figure3 and Table2 below.

Table 1. Main parameters of vehicle model.

Vehicle Parameters Value Unit

Appl. Sci. 2020, 10, x FOR PEER REVIEW mt 900 kg 6 of 22 ms 700 kg 2 2.2. Model Verfication Ixx 268 kg m · 2 The vehicle model is constructedI yyin MATLAB/Simulink708 (abbreviatedkg m as Simulink below). The · 2 Izz 708 kg m graphical simulation model generated by Simulink is mainly based· on the dynamic equation of the h 700 mm system built by users. Compared with the multi-body dynamic model, the biggest advantage of the B 1469 mm Simulink model is that the combination1 of the model and the control system is more convenient. B2 1430 mm Moreover, the performance of the vehicle before and after the control can be compared and studied L1 970 mm intuitively. L2 903 mm Given the high simulation accuracyg of the software 9.8 CarSim, them /Cars2 Sim model is regarded as the actual vehicle in the model verification.Ksl1/K sr1The model in14 Simulink N is/mm verified by inputting the same vehicle parameters under the sameK sinesl2/K inputsr2 of the steering18 wheel N/mm both in Simulink and CarSim as shown in Table 1. The comparison results are shown in Figure 3 and Table 2 below.

(a) (b)

Figure 3. Model comparison results (red solid line with little square is the result of the Simulink Figure 3. Model comparison results (red solid line with little square is the result of the Simulink simulation; blue solid line is the result of the CarSim simulation except in (d)). (a) is the input steering simulation; blue solid line is the result of the CarSim simulation except in (d)). (a) is the input steering angle; (b) is the comparison of the lateral acceleration; (c) is the comparison of the roll angle; (d) is the angle; (b) is the comparison of the lateral acceleration; (c) is the comparison of the roll angle; (d) is the comparison of the vertical force of each wheel. comparison of the vertical force of each wheel. Table 1. Main parameters of vehicle model. Table 2. Main results of comparison. Vehicle Parameters Value Unit 1 Test Results (Unit)mt CarSim900 Simulinkkg Percentage Point maximumms 1.882669700 1.9110172kg 1.50% Roll angle (deg) minimumIxx 1.140343268 kg·1.149754m2 0.83% − − 2 averageIyy 0.3543169708 kg 0.3560246·m 0.48% maximumIzz 0.2252319708 kg 0.2512166·m2 11.5% Lateral acceleration (g’s) minimumh 0.1377398700 mm0.1543274 12.0% − − averageB1 0.044965581469 0.04642835mm 3.25% B2 1430 mm 1 Percentage point = ((Simulink) (CarSim))/(CarSim) 100%. L1 − 970 mm × L2 903 mm g 9.8 m/s2 Ksl1/Ksr1 14 N/mm Ksl2/Ksr2 18 N/mm

Appl. Sci. 2020, 10, x FOR PEER REVIEW 7 of 22

Table 2. Main results of comparison.

Appl. Sci. 2020, 10, 7230 Test Results (unit) CarSim Simulink Percentage Point 1 7 of 21 maximum 1.882669 1.9110172 1.50% Roll angle (deg) minimum −1.140343 −1.149754 0.83% Despite the relatively small diffaverageerences (related0.3543169 to the0.3560246 number of degrees0.48% of freedom and other physical parameters, such asm tireaximum stiffness 0.2252319 and damping), 0.2512166 the change11.5% trend of the model is correct, andLateral the simulation acceleration parameters (g’s) minimum related to−0.1377398 vehicle roll−0.1543274 are very accurate12.0% (such as roll angle). The 14-degree-of-freedom model in Simulinkaverage has0.04496558 good simulation 0.04642835 accuracy 3.25% and can thus be used for simulation. 1 Percentage point = ((Simulink) - (CarSim))/(CarSim) × 100%. Despite the relatively small differences (related to the number of degrees of freedom and other 3. Design of Optimal Distribution Strategy of Damping Force Controller physical parameters, such as tire stiffness and damping), the change trend of the model is correct, 3.1.and Lateralthe simulation Load Transfer parameters Ratio related to vehicle roll are very accurate (such as roll angle). The 14- degree-of-freedom model in Simulink has good simulation accuracy and can thus be used for The lateral load transfer rate (LTR) is the mainstream rollover evaluation index, and it is commonly simulation. used in the rollover control field [39]. When the vehicle is rolling, its LTR can be expressed by vertical force3. Design as the of following Optimal equation,Distribution as shown Strategy in Figureof Damping4[12]: Force Controller

Fzl Fzr 3.1. Lateral Load Transfer Ratio LTR = − (14) Fzl + Fzr The lateral load transfer rate (LTR) is the mainstream rollover evaluation index, and it is where F , F are the left and right vertical forces of sprung mass, which can be expressed as commonlyzl usedzr in the rollover control field [39]. When the vehicle is rolling, its LTR can be expressed  by vertical force as the following equation, as shown in FFigurevl1 F vl42 [12]:  Fzl = Fzl1 + Fzl2 = +  Dl1 Dl2 (15)  퐹푧푙 − 퐹F푧푟vr1 Fvr2  Fzr = F + Fzr = + 퐿푇푅zr1 = 2 Dr1 Dr2 (14) 퐹푧푙 + 퐹푧푟

Figure 4. VeVehiclehicle rollover diagramdiagram (front(front view).view).

whereIt canFzl, F bezr are seen the from left Equationand right (14)vertical that forces when of the sprung vehicle mass, runs which in a straight can be lineexpressed without as steering wheel input, the vehicle has no load transfer and the LTR is 0. When the input of the steering wheel 퐹푣푙1 퐹푣푙2 increases gradually, the vehicle body will퐹푧푙 = roll퐹푧푙 due1 + 퐹 to푧푙2 the= eﬀect+ of suspension and the vertical forces of 퐷푙1 퐷푙2 outside wheels will gradually increase. Assuming that the lateral acceleration is constant, the vertical(15) 퐹푣푟1 퐹푣푟2 forces of the inside wheel will decrease퐹푧푟 correspondingly,= 퐹푧푟1 + 퐹푧푟2 = and+ the LTR value will gradually increase. { 퐷푟1 퐷푟2 When the rollover critical moment is reached, the vertical forces of the inside wheels inside will be 0. The wheelsIt can be on seen another from side Equation carry all (14) the that mass when of the the vehicle, vehicle and runs the in LTR a straight reaches line the without maximum steering value ofwheel+1 orinput, the minimumthe vehicle value has no of load1. transfer At this time,and the if theLTR input is 0. ofWhen steering the input wheel of anglethe steering continues wheel to − increase,increases the gradually, rollover the accident vehicle will body occur. will Therefore, roll due to the the value effect range of suspension of LTR is [and1, + the1]. vertical forces of outside wheels will gradually increase. Assuming that the lateral acceleration− is constant, the vertical forces of the inside wheel will decrease correspondingly, and the LTR value will gradually Appl. Sci. 2020, 10, 7230 8 of 21

3.2. Optimal Distribution Strategy of Damping Force According to the relevant theories of LTR, the ideal anti-rollover damping moment can be expressed as B1 B2 MxDn roll = (FDl1 FDr1) + (FDl2 FDr2) = Fyh = Mx (16) − 2 − 2 − − − where MxDn-roll is the anti-rollover moment provided by the damping force. FDij is the damping force of each wheel. Mx is the roll moment around the X-axis, which can be solved by the vehicle dynamic model constructed by Equations (1)–(8). Considering the theory of semi-active suspension, the power of the shock absorber must be positive. CDC can only convert mechanical energy into thermal energy without the injection of external energy, that is,  . . . F (w Z ) = C w(w Z ) w(w Z )  Dij uij nij roll uij , uij > 0 PDij roll =  − − − − . (17) −  0,w(w Z ) 0 − uij ≤ . F Sign(Z ) 0 (18) Dij Dij ≥ where the function Sign(x) is described as:   1, x > 0  Sign(x) =  0, x = 0 (19)    1, x < 0 − . . Z = w Z (20) Dij − uij where ZDij is the displacement of the shock absorber of each wheel; x is a variable; PD-roll is the power of anti-rollover shock absorber. Cn-roll is the damping coefficient of the anti-rollover shock absorber. In this paper, the damping force is a free variable to be distributed. Only when the vertical forces are optimized, the distribution result of vertical forces is better, but the damping force are greatly different. Large suspension dynamic displacements caused by damping force control will affect vehicle stability. Thus, the vertical load difference of each wheel between the dynamic and static condition should not be too large to avoid the large wheel dynamic load caused by the semi-active suspension control, which worsens the vehicle handling stability [40,41]. The optimized performance index of the vertical dynamic load is defined as:

2 2 2 2 Fvl1o Fvl2o Fvr1o Fvr2o Jvd = tl1( ) + tl2( ) + tr1( ) + tr2( ) (21) Fvl1 Fvl2 Fvr1 Fvr2 where t is the weighting factor; Fvijo is the static vertical force of each wheel. The constrained optimization problem can be deﬁned as:

. minJvd(MxDn roll, ZDij, FDij) (22) FDij −

Equation (19) can be simpliﬁed by using the deﬁned variables, αij and βij, as

α α α α J = l1 + l2 + r1 + r2 (23) vd 2 2 2 2 (FDl1 + βl1) (FDl2 + βl2) (FDr1 + βr1) (FDr2 + βr2)

2 αij = tFvijo (24) β = F F (25) ij vij − Dij Appl. Sci. 2020, 10, 7230 9 of 21

The Lagrange multiplier method is used to solve the optimization problem in Equation (22). The optimization objective function considering the constraints of Equations (16) and (18) is constructed as

B1 B2 Lmultiplier = Jvd + γ(MxDn roll 2 (FDl1 FDr1) 2 (FDl2 FDr2)) P − − . − − − (26) λ (F Sign(Z ) ε 2) − ij Dij Dij − ij where γ and εij are arbitrary real numbers; λij represent nonnegative real numbers. When the objective variable, Jvd, takes the extreme value, the partial derivative of each parameter is equal to 0. The partial diﬀerential equation can be written as

.  ∂Lmultiplier 2α  = − l1 ( ) =  ∂F 3 γ FDl1Sign ZDl1 0  Dl1 (FDl1+βl1) − −  .  ∂Lmultiplier 2α  = − l2 ( ) =  ∂F 3 γ FDl2Sign ZDl2 0  Dl2 (FDl2+βl2) − −  ∂L .  multiplier = 2αr1 + ( ) =  ∂F − 3 γ FDr1Sign ZDr1 0  Dr1 (FDr1+βr1) −  ∂L . (27)  multiplier = 2αr2 + ( ) =  ∂F − 3 γ FDr2Sign ZDr2 0  Dr2 (FDr2+βr2) −  ∂L .  multiplier = ( ) =  ∂F FDijSign ZDij 0  λij −  ∂Lmultiplier  = 2λijεij = 0 ∂Fεij

In one solution, all εij values that are equal to 0 are excluded because the variable leads to Fdij = 0, thus conﬂicting with semi-active control theory. Another solution in which all λij are 0 is as follows:

2αlj γ = − (28) 3 (Fdlj + βlj)

2αrj γ = (29) 3 (Fdrj + βrj) Equations (29) and (30) may not be able to solve for γ. Instead, the solution should be ( λ = 0, ε = 0 lj rj (30) λrj = 0, εlj = 0

For the un-tripped rollover investigated in this study, the movement of the left and right sides of the vehicle suspension is always on the contrary, and the damping force on the left and right sides of the suspension is always reversed. The front and rear wheel distance is usually the same or similar, supposing B = B1 because the wheel distance should be determined by the outermost wheel in the anti-rollover control [42]. The range of feasible solutions is shown in Figure5. The theoretical extreme value of the performance index Jvd exists only when the damping force of the shock absorber is in the range of the feasible solution shown in Figure5. In addition, the damping force of the shock absorber may be distributed in quadrants I and III. However, for the un-tripped rollover investigated in this research, the damping force is distributed in quadrants II and IV. The constraint equation of Equation (19) can be written as ( . FDljSign(ϕ) 0 . ≤ (31) F Sign(ϕ) 0 Drj ≥ Appl. Sci. 2020, 10, 7230 10 of 21 Appl. Sci. 2020, 10, x FOR PEER REVIEW 10 of 22

FigureFigure 5. 5. FeaFeasiblesible region region of of optimal optimal distribution distribution of of damping damping force. force.

When λrj = 0, Equations (29) and (31) are substituted into Equation (27): When λrj = 0, Equations (29) and (31) are substituted into Equation (27):

휕퐿∂L푚푢푙푡푖푝푙푖푒푟multiplier −22α훼lj푙푗 −2α훼rj푟푗 . == − 3 ++ − 3 + 퐹F퐷푙푗DljSignSign((휑ϕ̇ ))= =00 (32) (32) 휕F퐹퐷푙푗 (퐹퐷푙푗 + 훽푙푗)3 (퐹퐷푟푗 + 훽푟푗)3 ∂ Dlj (FDlj + βlj) (FDrj + βrj)

When λrj and α rj are not negative, then. 휑̇ > 0. By the same token, when λlj = 0, then. φ̇ < 0When. Substitutingλrj and Equationsαrj are not (28) negative, and (31) then into ϕEquation> 0. By (27): the same token, when λlj = 0, then ϕ < 0. Substituting Equations (28) and (31) into Equation (27): 휕퐿푚푢푙푡푖푝푙푖푒푟 −2훼푟푗 −2훼푙푗 = 3 + 3 + 퐹퐷푟푗Sign(휑̇ ) = 0 (33) ∂Lmultiplier휕퐹퐷푟푗 (퐹퐷푟푗2+αrj훽푟푗) (퐹퐷푙푗 2+αlj훽푙푗) . = − + − + F Sign(ϕ) = 0 (33) ∂F 3 3 Drj The theoretical solutionDrj of the( FoptimizationDrj + βrj) problem(FDlj + βinlj )Equation (22) can be expressed as

3 2√훼푙푗 The theoretical solution of the optimization3 problem in Equation3 (22) can be expressed as − 푀푥퐷푛−푟표푙푙 + √훼푙푗 ∑ 훽푙푗 − 훽푙푗 ∑ √훼푙푗 1 − 푆𝑖푔푛(푀 ) 퐹 = 퐵 [ 푥퐷푛−푟표푙푙 ] 퐷푙푗−푡 2 3 α 3 √ lj ∑ 훼푙푗 P P 2  MxDn roll√+ √3 αlj βlj βlj √3 αlj 1 Sign(M )  − B − xDn roll  FDlj3 t = − P − − (34)  2 훼 √3 αlj 2 √− 푟푗 3 3 (34)  − 푀2 √3 αrj + 훼 ∑P훽 − 훽P ∑ 훼  푥퐷푛−푟표푙푙M √+ 푟푗3 푟푗 푟푗 3 √ 푟푗 1 + 푆𝑖푔푛(푀푥퐷푛−푟표푙푙)  퐵 B xDn roll √αrj βrj βrj √αrj 1+Sign(MxDn roll) 퐹퐷푟푗−푡 = = − − P − [ − ]  FDrj t 3 3 α 2 { − ∑ √훼푟푗√ rj 2 ConsideringConsidering the the limitation limitation of of the the CDC CDC shock shock absorber, absorber, the the damper damper on on the the other other side side needs needs to to outputoutput a certain certain compensation compensation damping damping force force when when the theoretical the theoretical solution solution exceeds exceeds the shock the absorber shock absorberlimitation. limitation. Such force Such needs force to needs meet orto approachmeet or approach the constraint the constraint equation equation of Equation of Equation (16) as much (16) as as muchpossible as possible to ensure to the ensure rollover the stability rollover of stability the vehicle. of the The vehicle. limitation The characteristicslimitation characteristics of CDC is indicated of CDC isas indicated follows: as follows:  FDmin, FDij < FDmin  퐹퐷푚푖푛, |퐹퐷푖푗| <| |퐹퐷푚푖푛| |  sat(FDij) =  FDij, FDmin < FDij < FDmin < FDmax (35) sat(퐹퐷푖푗) = {  퐹퐷푖푗,||퐹퐷푚푖푛| |< |퐹퐷푖푗| <| |퐹퐷푚푖푛| |<| |퐹퐷푚푎푥| | (35)  F퐹Dmax,,F|퐹Dmax <| < F|Dij퐹 | 퐷푚푎푥 | 퐷푚푎푥| 퐷푖푗 F F wwherehere FDminDmin andand FDmaxDmax areare the the lower lower and and upper upper limits limits of the of the CDC CDC shock shock absorber, absorber, they they can can be beshown shown inin Figure Figure 66.. Appl. Sci. 2020, 10, 7230 11 of 21 Appl. Sci. 2020, 10, x FOR PEER REVIEW 11 of 22

Figure 6. Range of variation of continuous damping control (CDC) damping force. Figure 6. Range of variation of continuous damping control (CDC) damping force. When the theoretical solution of Equation (34) does not meet the condition of Equation (16), the When the theoretical solution of Equation (34) does not meet the condition of Equation (16), the compensation damping force of the other shock absorber is optimally distributed according to the compensation damping force of the other shock absorber is optimally distributed according to the performance index of Equation (23). The expected damping force considering the shock absorber limit performance index of Equation (23). The expected damping force considering the shock absorber and rollover stability constraints is obtained as follows: limit and rollover stability constraints is obtained as follows:    3 2 √3 αlj P P 2 훼  M + 3 α β β 3 α   √ 푙푗  B xDn3 roll √ lj lj lj 3√ lj 1 Sign(MxDn roll)   − = 푀 − + − 훼P ∑ 훽 − 훽− ∑ 훼 − −  FDlj d sat푥퐷푛−푟표푙푙 √ 푙푗 3 α 푙푗 푙푗 √ 푙푗 1 − 푆𝑖푔푛2 (푀푥퐷푛−푟표푙푙) 퐹 = 푠푎푡 { − 퐵  √ lj [  ]} 퐷푙푗−푑   3  (36)  2 √3 αrj ∑ √훼푙푗 2   3 P P 3    B MxDn roll+ √αrj βrj βrj √αrj 1+Sign(MxDn roll)    − − −   FDrj d = sat P 3 − (36)  3  √αrj 2  −2√훼푟푗   − 푀 + 3 훼 ∑ 훽 − 훽 ∑ 3 훼 푥퐷푛−푟표푙푙 √ 푟푗 푟푗 푟푗 √ 푟푗 1 + 푆𝑖푔푛(푀푥퐷푛−푟표푙푙) 퐹 = 푠푎푡 { 퐵 [ ]} 퐷푟푗−푑 3 When the expected resistance force is∑ not√훼푟푗 within the regulation range, the2 saturation constraint of the shock{ absorber should be considered and compensated by the shock absorber on the other side. This conditionWhen the meets expected the expected resistance moment force is constraint not within of the anti-rollover regulation preferentially. range, the saturation The distribution constraint of theof the CDC shock damping absorber force should is optimized, be considered and it eandffectively compensated avoids the by problemthe shock regarding absorber theon the optimization other side. ofThis online condition solving meets constraints. the expected The real-time moment performanceconstraint of ofanti the-rollover control preferentially. algorithm is improved. The distribution of the CDC damping force is optimized, and it effectively avoids the problem regarding the 4. Simulation and Result Analysis optimization of online solving constraints. The real-time performance of the control algorithm is impTheroved. simulation analysis was carried out in MATLAB/Simulink. The passive suspension and traditional CDC LTR damping on/off control (abbreviated as CDC-LTR) were compared with the CDC optimal4. Simulation distribution and Result damping Analysis force strategy (abbreviated as CDC-ODDF) to illustrate the optimization effect. The CDC-LTR to prevent rollover simply sets the absolute threshold value of LTR parameters. The simulation analysis was carried out in MATLAB/Simulink. The passive suspension and When the threshold is reached, the anti-roll control is adopted for CDC suspension. Given that this traditional CDC LTR damping on/off control (abbreviated as CDC-LTR) were compared with the researchCDC optimal mainly distribution focuses on tripped damping rollover force while strategy driving (abbreviated a vehicle, ESC as CDC differential-ODDF) braking to illustrate and other the methodsoptimization of stability effect. controlThe CDC are- removedLTR to prevent to highlight rollover the simply control sets effect the of aabsolute single CDC threshold on rollover. value of LTRFor parameters. tripped rollover, When the the threshold lateral force is reached, is the main the anti source-roll ofcontrol the roll is adopted force. The for lateralCDC suspension. force may beGive causedn that by this the research turning ofmainly the vehicle focuses itself on tripped in the driving rollover process, while driving or it may a vehicle, be caused ESC by differential the larger lateralbraking wind and duringother methods the driving of stability process. control Therefore, are removed the following to highlight is divided the intocontrol two effect conditions of a single for stabilityCDC on testing rollover. simulation analysis: fishhook and crosswind. TheFor realtripped vehicle rollover, selected the for lateral the fundingforce is the project main is source an A0 of class the high-centroid roll force. The vehicle. lateral force Although may be it hascaused a high by centroid,the turning its massof the is vehicle not large. itself Moreover, in the driving it is more process, prone or toit sideslipmay be rathercaused than by the rollover larger whenlateral that wind vehicle during has the no driving ESC system process. while Therefore, driving. the Therefore, following the is selection divided ofinto its two parameters conditions is the for criticalstability condition testing simulation obtained by analysis repeated: fishhook simulation, and which crosswind. can show the effect of vehicle control under a limitedThe state. real vehicle selected for the funding project is an A0 class high-centroid vehicle. Although it has a high centroid, its mass is not large. Moreover, it is more prone to sideslip rather than rollover when that vehicle has no ESC system while driving. Therefore, the selection of its parameters is the critical condition obtained by repeated simulation, which can show the effect of vehicle control under a limited state. Appl. Sci. 2020, 10, 7230 12 of 21

Appl. Sci. 2020, 10, x FOR PEER REVIEW 12 of 22 4.1. Fishhook Test 4.1. Fishhook Test The basic parameters of the ﬁshhook test simulation analysis are shown in Table3. The basic parameters of the fishhook test simulation analysis are shown in Table 3. Table 3. Main parameters of ﬁshhook test. Table 3. Main parameters of fishhook test.

TestTest Parameters Parameters Value Unit Unit LongitudeLongitude target target speed speed 80 km/h km /h FrictionFriction coe coefficientﬃcient 0.85 - - SteerSteer input input angle angle peak peak value value 2 29494 deg deg SimulationSimulation time time 20 second second RoadRoad condition condition straight - - |Threshold|Threshold of of CDC-LTR CDC-LTR control control|| 00.9.9 - - CDCCDC-LTR,-LTR, continuous continuous damping control-lateralcontrol-lateral load load transfer transfer ratio. ratio.

TheThe results results of of the the ﬁshhook fishhook test test simulation simulation areare shownshown in Figure 77..

(a)

(b)

Figure 7. Cont. Appl. Sci. 2020, 10, 7230 13 of 21

Appl. Sci. 2020, 10, x FOR PEER REVIEW 13 of 22

(c)

(d)

FigureFigure 7. 7.Roll Roll angle, angle, roll roll angle angle rate, rate, LTR,LTR, andand lateral acceleration of of fishhook ﬁshhook test. test. The The result result of ofpassive passive suspensionsuspension is is expressed expressed by by the the black black dotteddotted line.line. The The result result of of CDC CDC-LTR-LTR is is expressed expressed by by the the blue blue chainedchained line. line. The The result result of of CDC- CDC- optimal optimal distributiondistribution damping force force (ODDF (ODDF)) is is expressed expressed by by the the red red solidsolid line. line. (a )(a is) is the the comparison comparison result result ofof thethe rollroll angle.angle. ( (bb)) is is the the comparison comparison result result of of the the roll roll angle angle rate.rate (c. )(c is) is the the comparison comparison result result of of LTR.LTR. ((dd)) isis the comparison result result of of lateral lateral acceleration. acceleration. To To observe observe thethe comparison comparison of of the the two two CDC CDC control control strategiesstrategies more conveniently, conveniently, a a local local view view in in which which passive passive suspensionsuspension is is removed removed is is added. added.

TheThe results results of of the the roll roll angle, angle, rollroll velocity,velocity, andand LTR of these three three modes modes show show that that both both CDC CDC controlcontrol methods methods can can eff effectivelyectively prevent prevent rollover. rollover. TheThe result result of of the the roll roll angle angle demonstrates demonstrates that the roll angle angle rate rate of of the the vehicle vehicle with with passive passive suspensionsuspension reaches reaches 6−6 rad rad/s/s in in approximately approximately 3 s s,, and the the roll roll angle angle exceeds exceeds the the limit, limit, thus thus causing causing − seriousserious rollover. rollover. From From the the result result ofof LTR,LTR, thethe vehiclevehicle with passive passive suspension suspension reaches reaches the the rollover rollover thresholdthreshold at at the the first first turningturning left left in in 1 1s. s.Given Given the theinfluence influence of some of some factors, factors, such as such sideslip as sideslip and small and smallinertia, inertia, the the vehicle vehicle is in is anin an unstable unstable state state without without rollover. rollover. However, However, after after the the 1.57th 1.57th second second of of continuouscontinuous right right turning, turning, the the vehicle vehicle cancan nono longerlonger maintain driving. driving. Finally, Finally, at at approximately approximately the the 2.2th second, LTR jumps directly from −1 to +1, indicating that rollover occurs. At approximately the 2.2th second, LTR jumps directly from 1 to +1, indicating that rollover occurs. At approximately the 3rd second, the parameter has no significance.− However, some LTR values with a discontinuous 3rd second, the parameter has no significance. However, some LTR values with a discontinuous value value of +1 still remain. of +1 still remain. Comparing the CDC-ODDF control with the CDC-LTR control, the overall parameter Comparing the CDC-ODDF control with the CDC-LTR control, the overall parameter performance performance of the vehicle seems to have little difference. However, it is determined by the rollover of thecritical vehicle condition. seems toLTR have also little reaches difference. +1 from However, the 2.1th it issecond determined to the by2.2th the second rollover under critical these condition. two LTRcontrol also reachesstrategies.+ 1The from rollover the 2.1th cannot second happen to because the 2.2th the second vehicle under is in a thesecritical two unstable control state. strategies. The rolloverThis cannot result happen can bebecause compared the through vehicle the isin verti a criticalcal and unstable damping state. forces in Figure 8. This result can be compared through the vertical and damping forces in Figure8. Appl. Sci. 2020, 10, 7230 14 of 21

Appl. Sci. 2020, 10, x FOR PEER REVIEW 14 of 22

(a)

(b)

(c)

(d)

Figure 8. Vertical and damping forces of each wheel in ﬁshhook test with CDC-LTR and CDC-ODDF. The details can be seen in the legend. (a) is the comparison result of the vertical forces of wheels on the left. (b) is the comparison result of the vertical forces of wheels on the right. (c) is the comparison result of the damping forces of wheels on the left. (d) is the comparison result of the damping forces of wheels on the right. L is short for left, and R is short for right. Here, 1 represents the front axle, and 2 denotes the rear axle. Appl. Sci. 2020, 10, 7230 15 of 21

The comparison of the vertical forces demonstrates that the time during the wheel load on the same side at the same time is zero if the vehicle with CDC-ODDF is shorter. The vertical force of the right wheel of the rear axle is evidently reduced from the 1.6th second to the 1.8th second with CDC-ODDF. The vertical force of the left wheel of the front axle also decreases to a certain extent from the 2.3th second to the 2.4th second with CDC-ODDF. It can decrease the load transfer during roll, which is also the reason why the LTR value is less than CDC-LTR control. Moreover, the advantages of CDC-ODDF can be determined by comparing the damping forces, which is the core part of the CDC-ODDF. The comparison of Figure8c,d shows that when the left side needs a very large damping force, the right side of the vehicle with CDC-ODDF also increases correspondingly. The critical state of rollover from the 1st second to the 3.5th second is taken for analysis, and the corresponding local enlarged contrast view is shown in Figure8. In the interval of the 1.4th second to the 1.6th second, the maximum forward steering wheel condition is reached. The roll angle rate fluctuates at approximately the 1.4th second. To ensure the stable driving of the vehicle, the damping forces on the right side of the vehicle with CDC-ODDF achieve the first peak to compensate for the small damping force on the left side. At the same time, the damping forces on the right side with CDC-LTR are only approximately half of the forces with CDC-ODDF because LTR is very small, and CDC-LTR control is in the off state. When the steering system starts to turn right at approximately the 1.57th second, the damping forces with CDC-ODDF react immediately, while the CDC-LTR cannot respond in time and the output of damping force is still very small. At the peak value of LTR at approximately the 2.3th second, the compensation effect with CDC-ODDF is more evident. During this period, the damping forces with CDC-ODDF on the right side of the front axle compensate for the left side once at approximately the 2nd second, while the left side has several large compensations for the right side from the 2.4th second to the 2.6th second. The compensation of the left side of the rear axle and the right side of the front axle is the most evident from the data of the view with CDC-ODDF. Because the damping forces of CDC are controlled by electrical signal, the sudden changes of damping forces often lead to the sudden change of control current. The damping force current signal can be collected and identified by appropriate methods, and the driver can be given a corresponding real-time rollover warning, which improves the rollover prediction effect. The main comparison results of CDC-LTR and CDC-ODDF are shown in Table4.

Table 4. Main comparison results of ﬁshhook test (CDC-LTR, CDC-ODDF).

Test Results (Unit) CDC-LTR CDC-ODDF Percentage Point 1 maximum 0.008001 0.07378 7.79% − minimum 0.09906 0.09281 6.30% Roll angle (rad) − − − average 0.04896 0.04605 5.94% − − − standard deviation 0.03012 0.02856 5.18% − maximum 0.5657 0.4664 5.34% − minimum 0.8282 0.7776 6.11% Roll angle rate (rad/s) − − − average 0.02054 0.01906 7.21% − − − standard deviation 0.09313 0.08186 12.1% − 1 Percentage point = ((CDC-ODDF) (CDC-LTR))/(CDC-LTR) 100%. − ×

4.2. Crosswind Test The crosswind test is important because crosswind is the most common lateral force, except for obstacles on the road, in practical application scenarios [43]. The general wind scale is the wind speed at 10 m above the ground, which prevents a higher wind speed on the ground. However, many overpasses are applied to urban expressways in the construction of modern cities. Sometimes, the height of these overpasses can even reach the height of 20 m from the ground (equivalent to approximately seven ﬂoors). The crosswind at this location is very dangerous to fast vehicles with a large mass and a high mass center on rainy and snowy days. The research condition of crosswind rollover is not only the condition of single lateral external force but also has a certain reference value Appl. Sci. 2020, 10, 7230 16 of 21

in practicalAppl. Sci. application.2020, 10, x FOR PEER The REVIEW basic parameters of the crosswind test simulation analysis are16 shown of 22 in Table5. at 10 m above the ground, which prevents a higher wind speed on the ground. However, many overpasses are applied to urbanTable expressways 5. Main parameters in the construction of crosswind of test. modern cities. Sometimes, the

height of these overpasses canTest even Parameters reach the height ofValue 20 m from Unit the ground (equivalent to approximately seven floors). The crosswind at this location is very dangerous to fast vehicles with a Longitude target speed 80 km/h large mass and a high massFriction center coeﬃ cienton rainy (no crosswind) and snowy days. 0.85 The research - condition of crosswind rollover is not only the conditionFriction coe ofﬃ singlecient (crosswind) lateral external force 0.2 but also -has a certain reference value in practical application. The basicWind parameters speed of the crosswind 8 test simulation m/s analysis are shown in Simulation time 8 second Table 5. Road condition straight - |Threshold of CDC-LTR control| 0.9 - Table 5. Main parameters of crosswind test.

Crosswind conditions are moreTest severe Parameters because the frictionValue coe ffiUnitcient of the crosswind section is Longitude target speed 80 km/h small. Thus, serious instabilityFriction sideslip coefficient occurs (no before crosswind) a rollover. 0.85 Therefore,- these factors are combined with the test results for a briefFriction analysis. coefficient (crosswind) 0.2 - The results of crosswind test simulationWind speed are shown in Figure8 9. m/s According to the results of the rollSimulation angle rate timeand LTR, the8 vehicle second suff ers from the first low-friction Road condition straight - crosswind section from the 0.8th second to the 2nd second, enters the second friction crosswind section |Threshold of CDC-LTR control| 0.9 - from the 3.5th second to the 4.2th second, and the vehicles start to lose stability from the 5.2th second to the 5.5thCrosswind second (theconditions condition are more is mainly severe sideslip; because somethe friction wheels coefficient are off theof the ground, crosswind but section no rollover is small. Thus, serious instability sideslip occurs before a rollover. Therefore, these factors are occurs). The serious sideslip occurs after the 7.2th second, and the vehicles can no longer maintain combined with the test results for a brief analysis. normal driving. The results of crosswind test simulation are shown in Figure 9.

(a)

(b)

Figure 9. Cont. Appl. Sci. 2020, 10, 7230 17 of 21

Appl. Sci. 2020, 10, x FOR PEER REVIEW 17 of 22

(c)

FigureFigure 9. Roll 9. Roll angle angle rate, rate, LTR, LTR and, and lateral lateral acceleration acceleration of of crosswind crosswind test. test. The The result result of passive of passive suspensionsuspension is expressed is expressed by by the the black black dotted dotted line.line. The The result result of of CDC CDC-LTR-LTR is expressed is expressed by the by blue the blue chainedchained line. line. The The result result of of CDC-ODDFCDC-ODDF is isexpressed expressed with with the red the solid red li solidne. (a) line. is the (comparisona) is the comparison result resultof of the the roll roll angle angle rate rate.. (b) ( b is) is the the comparison comparison result result of LTR of LTR.. (c) is(c ) the is the comparison comparison result result of lateral of lateral acceleration.acceleration To. observeTo observe the the comparison comparison of of thethe two CDC control control strategies strategies more more conveniently, conveniently, a local a local view,view, in which in which the passivethe passive suspension suspension is is removed, removed, isis added.

The LTRAccording of a passive to the results suspension of the vehicleroll angle experiences rate and LTR, a jump the vehicle from + suffers1 to 1from at approximately the first low- the friction crosswind section from the 0.8th second to the 2nd second, enters− the second friction 5.2th second, which means that the vehicle has a “jump” on all wheels. Although no rollover occurs, crosswind section from the 3.5th second to the 4.2th second, and the vehicles start to lose stability the vehicle completely loses control. The vehicle with CDC is also unstable. Nevertheless, not all the from the 5.2th second to the 5.5th second (the condition is mainly sideslip; some wheels are off the wheelsground, come but off theno rollover ground. occurs). Notably, The a vehicleserious sideslip with CDC-ODDF occurs after never the 7.2th reaches second, the and rollover the vehicles limit ( |LTR| < 1 fromcan no start longer to the maintain end). normal driving. Moreover,The LTR the of a comparison passive suspension of CDC-LTR vehicle experiences and CDC-ODDF a jump from in two + 1 to crosswind − 1 at approximately regions with the low friction5.2th and second, the firstwhich instability means that region the vehicle is analyzed has a “jump” on theon all basis wheels. of vertical Although and no rollover damping occurs, forces in Figurethe 10 vehicle. completely loses control. The vehicle with CDC is also unstable. Nevertheless, not all the Inwheels the firstcome crosswind off the ground. section Notably, (from thea vehicle 0.8th with second CDC to-ODDF the 2nd never second), reaches the the change rollover of limit damping forces(|LTR| on the < left1 from side start with to the the end). two CDC control strategies is basically the same, while the damping Moreover, the comparison of CDC-LTR and CDC-ODDF in two crosswind regions with low forces on the right side is evidently compensated for the left side with CDC-ODDF. Similarly, the friction and the first instability region is analyzed on the basis of vertical and damping forces in adjustment of the vehicle with CDC-ODDF is faster and more sensitive than that with CDC-LTR. In the Figure 10. second crosswind section (from the 3.5th second to the 4.2th second), the compensation of the left damping force with CDC-ODDF is also evident (especially on the left side of the rear axle). This effect is more prominent at the first instability period. At approximately the 5.28th second, the right damping forces of the vehicle with CDC-ODDF has a compensation effect on the left side. This compensation effect eventually leads to the axle load transfer without exceeding the rollover threshold. Although the vehicle with CDC-ODDF still loses stability in the end, the smaller axle load transfer brings a better contact effect between the tires and the road surface. This result can also be seen from the comparison of vertical forces. Under the extreme instability condition at approximately the 3.35th second, the CDC-ODDF control can ensure that the two tires of the vehicle touch the ground, while CDC-LTR, which only has the wheel on the right side of the rear axle, has vertical force. Although the effect of lateral force on the roll control is not evident (caused by sideslip), the CDC suspension still has great advantages compared with passive suspension. The strategy of CDC-ODDF is better than CDC-LTR. The optimal control strategy(a) is adopted is not only to control the rollover, but also to change the distribution of damping forces and even the vertical forces of the vehicle. In addition, the redistribution of vertical forces can be combined with vehicle ESC control to achieve the goal of assisting ESC lateral stability control. Meanwhile, CDC plays an important role in driving stability with auxiliary effects. Appl. Sci. 2020, 10, x FOR PEER REVIEW 17 of 22

(c)

Figure 9. Roll angle rate, LTR, and lateral acceleration of crosswind test. The result of passive suspension is expressed by the black dotted line. The result of CDC-LTR is expressed by the blue chained line. The result of CDC-ODDF is expressed with the red solid line. (a) is the comparison result of the roll angle rate. (b) is the comparison result of LTR. (c) is the comparison result of lateral acceleration. To observe the comparison of the two CDC control strategies more conveniently, a local view, in which the passive suspension is removed, is added.

According to the results of the roll angle rate and LTR, the vehicle suffers from the first low- friction crosswind section from the 0.8th second to the 2nd second, enters the second friction crosswind section from the 3.5th second to the 4.2th second, and the vehicles start to lose stability from the 5.2th second to the 5.5th second (the condition is mainly sideslip; some wheels are off the ground, but no rollover occurs). The serious sideslip occurs after the 7.2th second, and the vehicles can no longer maintain normal driving. The LTR of a passive suspension vehicle experiences a jump from + 1 to − 1 at approximately the 5.2th second, which means that the vehicle has a “jump” on all wheels. Although no rollover occurs, the vehicle completely loses control. The vehicle with CDC is also unstable. Nevertheless, not all the wheels come off the ground. Notably, a vehicle with CDC-ODDF never reaches the rollover limit (|LTR| < 1 from start to the end). Moreover, the comparison of CDC-LTR and CDC-ODDF in two crosswind regions with low Appl. Sci. 2020, 10, 7230 18 of 21 friction and the first instability region is analyzed on the basis of vertical and damping forces in Figure 10.

Appl. Sci. 2020, 10, x FOR PEER REVIEW (a) 18 of 22

(b)

(c)

(d)

Figure 10. FigureVertical 10.and Vertical damping and damping forces forces of each of each wheel wheel in crosswindin crosswind test test withwith CDC CDC-LTR-LTR and and CDC CDC-ODDF.- ODDF. The details can be seen in the legend. (a) is the comparison result of the vertical forces of The detailswheels can be on seenthe left in. ( theb) is legend.the comparison (a) is result the comparison of the vertical resultforces of of wheels the vertical on the right forces. (c) is of the wheels on the left. (b)comparison is the comparison result of the result damping of theforces vertical of wheels forces on the of left wheels. (d) is on the the comparison right. ( cresult) is the of the comparison result of thedamping damping forces forces of wheels of wheels on the right. on theL is left.short (ford) left, is the and comparisonR is short for right. result 1 represents of the damping the front forces of wheels on theaxle, right. and 2 denotes L is short the rear for axle. left, and R is short for right. 1 represents the front axle, and 2 denotes

the rear axle.In the first crosswind section (from the 0.8th second to the 2nd second), the change of damping forces on the left side with the two CDC control strategies is basically the same, while the damping forces on the right side is evidently compensated for the left side with CDC-ODDF. Similarly, the adjustment of the vehicle with CDC-ODDF is faster and more sensitive than that with CDC-LTR. In Appl. Sci. 2020, 10, 7230 19 of 21

5. Conclusions Vehicle rollover is an instantaneous process. Once the rollover threshold is reached, the actual rollover event is very close. CDC-ODDF can adjust the damping force of the suspension by calculating the best damping force in real time. Through the simulation analysis, CDC-ODDF achieves the same control effect as CDC-LTR, which is to effectively prevent rollover. We can find that the vehicle with CDC-ODDF has advantages in statistics, such as maximum value, minimum value, average value, and standard deviation, even under extreme conditions compared with CDC-LTR. In a certain degree, the key parameters are optimized, which shows good performance with CDC-ODDF. The optimal control strategy is adopted is not only to control the rollover but also to change the distribution of damping forces and even the vertical forces of the vehicle. In addition, the redistribution of vertical forces can be combined with vehicle ESC control to achieve the goal of assisting ESC lateral stability control. The damping forces on the left and right sides with CDC-ODDF can automatically compensate each other to hinder or prevent the occurrence of a rollover trend. Thus, this kind of compensation function embodies the biggest design advantage of CDC-ODDF. The damping forces with CDC-ODDF can be adjusted adaptively to prevent roll motion because such an event can be calculated with the lateral force and roll moment. Because the damping forces of CDC are controlled by an electrical signal, the sudden changes of damping forces often lead to the sudden change of control current. The damping force current signal can be collected and identified by appropriate methods, and the driver can be given a corresponding real-time rollover warning, which improves the rollover prediction effect. In practical application, the damping force of suspension can be adjusted by CDC-ODDF to prevent rollover. In fact, by further designing the optimization target, the damping force distribution of the front and rear axles can be adjusted in real time, and the pitch during braking or other motions can be taken into account, so that the passengers can experience a better feeling.

Author Contributions: X.Z. conceived the control method, completed the modeling, performed simulation experiments, analyzed the results, and finished the manuscript. C.S. provided overall guidance for the study and managed the project. S.S. and F.X. collected the data and reviewed and revised the paper. J.C. gave crucial suggestions about the 14-degree-of-freedom model and participated in model verification. D.W. gave crucial suggestions about the LTR and optimal damping force control strategy. C.Q. and Z.M. put forward the idea and debugged the model in MATLAB/Simulink. All authors have read and agreed to the published version of the manuscript. Funding: This research is supported by the Science and Technology Development Plan Program of Jilin Province (grant no. 20200401112GX), Industry Independent Innovation Ability Special Fund Project of Jilin Province (grant no. 2020C021-3). Conflicts of Interest: The authors declare no conflict of interest.

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