Conflict and Sensitivity Analysis of Articulated Vehicle Lateral Stability Based on Single-Track Model

Hubei Key Laboratory of Advanced Technology for Automotive Components, Wuhan University of Technology, Wuhan 430070, China School of Automotive Engineering, Wuhan University of Technology, Wuhan 430070, China Hubei Center for Quality Inspection of Special Purpose Vehicles, Suizhou 441300, China Automotive and Tractors Engineering Department, Minia University, El-Minia 61519, Egypt Suizhou-WUT Industry Research Institute, Suizhou 441300, China


Introduction
e articulated vehicle is widely used in engineering vehicles due to its high mobility and low steering radius [1]. e disadvantage of the articulated vehicle is the lack of lateral stability.
us, it is necessary to investigate the key parameters that affect lateral stability during the maneuvers. Some researchers use the active brake system and torque distribution system based on control algorithms to improve vehicle stability. e PID (proportional integral derivative), fuzzy control [2], LQR (linear quadratic regulator) [3], MPC (model predictive control) [4], and NMPC (nonlinear model predictive control) [5] are initiated in this area. Since the vehicle is a complicated dynamic system with multiple DOFs (degrees of freedom), these control algorithms are based on the simplified control model and validated by the models with 8, 10, or 14 DOFs. e control model is usually with 2 or 3 DOFs based on the focus of the researchers.
According to the current literature, the accuracy of the simplified 2 or 3 DOFs; model is validated [6]. To analyze the influence of tire and vehicle parameters on the vehicle lateral stability, the 2 or 3 DOFs' model is used in the regular vehicle area. For example, Hassan et al. use a two-state linear bicycle model to accomplish the conflict and sensitivity analysis of vehicular stability of a passenger car [7].
As the assembly that transfers the shock and vibration from the road surface to the vehicle, the tire characteristics play a significant role in vehicle stability. e cornering stiffness is taken as the indicator to describe the correlation between the sideslip angle and lateral force of the tire. erefore, cornering stiffness contributes effectively to the lateral direction stability. A bicycle model is established to illustrate the influence of tire characteristics on the handling stability trends under normal and extreme maneuvering conditions [8].
e simulation results show that the tire cornering stiffness is a dominant factor in vehicle lateral stability [9,10]. e impact of the partial/full tire tread separations on handling performance is conducted with a wide range of speeds and different maneuvers. e findings show that the dynamic response of the vehicle depends on the tread separation and the vehicle speed, as well as the location of the separation event [11,12]. Besides, according to the works of Deng et al., nonpneumatic tires with large cornering stiffness have been proposed, which have great potential in improving the lateral stability of the vehicle [13,14].
To investigate the influence of the vehicle parameters on the vehicle lateral stability, Wade et al. initiate an analysis to investigate the correlation of roll/yaw moment of inertia and CG (Center of gravity) height with vehicle geometry parameters such as weight, length, width, and height [15]. e results show that vehicle performance is strongly affected by vehicle dimension parameters. Li et al. use a scale wheel loader to validate the established 7 DOFs' dynamic model and analyse the vehicle stability [16]. e nonlinear singletrack model with a simplified piece-with linear tire model-is built to analyze the effect of load transfer on the vehicle performance [17]. e model is validated by the simulation result of CarSim. Whitehead et al. evaluate the influence of the CG location in both longitudinal and vertical directions on the vehicle rollover performance based on a simplified 3DOFs' model [18]. For the electrical vehicles, the chassis layout is more flexible than the vehicle with ICE (internal combustion engine), and the battery pack is a heavy assembly. us, researches have been done to investigate the CG position and payload of the battery pack on the vehicle dynamic performance [19]. e effect of the wheelbase variation on the steering stability, yaw rate gain, and steering error is analyzed by Wang et al. via numerical simulation [20]. Based on the result, a six-wheel vehicle with a variable wheelbase is designed to enhance the lateral antidisturbance capability of the vehicle [21].
Most of the current works focus on the passenger vehicle with the Ackman steering system [22]. Researchers have conducted studies to analyze the effect of the vehicle parameters, including the vehicle inertial properties on the lateral stability and the sensitivity to each parameter; others only focus on the influences of the tire characteristics on the lateral dynamics. e gross vehicle mass of the passenger car is much lower than the typical engineering vehicle with an articulated steering system. To investigate the characteristic of the vehicle with an articulated steering system, the parametrical conflict analysis is introduced to evaluate the vehicle's lateral stability in terms of the lateral acceleration, yaw rate, vehicle sideslip angle, lateral force, and sideslip angles of the tires. e parametrical conflict analysis includes the tire cornering stiffness, vehicle payload, wheelbase, and speed. is paper tends to define the conflict and trade-off between the vehicle parameters and lateral stability.

3 DOFs' Single-Track Model
e vehicle is a complicated system with multiple DOFs. Researchers usually use simplified models to analyze the vehicle characteristics based on their focus. In the vehicular lateral stability area, the single-track vehicle system is widely used to analyze vehicle performances. For a single-track model, the left and right wheels on the same axle are assumed to have the same characteristics. us, a four-wheel vehicle can be simplified as a single-track model with two wheels. e roll motion, pitch motion, and longitudinal tire slip are also neglected in this model. e typical single-track model for a regular vehicle has 2 or 3 DOFs. For the 2 DOFs' model, the lateral motion and yaw motion are the concern of the researchers; for the 3 DOFs' model, the longitudinal motion is added based on the 2 DOFs' model [23]. For the vehicle with an articulated steering system, the 2 DOFs' model has the same definition as the regular vehicle model. In this model, the articulated angle is taken as a constant value during the steering process [24]. For the 3 DOFs' model, it has the degree of lateral motion and yaw motion of the front and rear parts of the articulated vehicle [25]. Figure 1 illustrates the single-track model of the articulated vehicle.
In Figure 1, m 1 and m 2 represent the mass of the front and rear part of the vehicle, L f1 and L r1 are the distance between the front mass center to the front axle and the rear mass center to the rear axle, L f2 and L r2 are the distance from the articulated point to the front and rear mass centers, ω f and ω r are the yaw rate of the front and rear part of the vehicle, and v 1 and v 2 are the velocity of the front and rear part of the vehicle. e lateral motion of the vehicle and yaw motion of the front and rear parts of the vehicle can be expressed by the following equations: In (1), the lateral accelerations of the front part, _ v y1 , and rear part, _ v y2 , are caused by the lateral tire forces of the front, F yf , and rear axle, F yr . e tire forces can be gained by In the above equation, α f and α r are the sideslip angles of the front and rear tire. To obtain them, the relationship of the vehicle velocity, v 1 , front tire velocity, v f , and rear tire velocity, v r , is established according to where L rf � L f2 + L r1 + L r2 and β means the vehicle sideslip angle. When the articulation angle θ is within the range of ±15 degrees, 0.98|tan θ| ≤ |θ| is essential. erefore, 2 Shock and Vibration tan θ ≈ θ is used to simplify the equations. us, based on (5) and (6), the sideslip angle of the tires can be acquired by e torque generated by the liquid steering system is calculated by [26] T � −K l θ, (8) where K l is the equivalent torsional stiffness of the steering system, which can be obtained by [27] where β e is the comprehensive elastic modulus considering the oil and pipe of the steering system, V e is the mean volume of the cavities with and without the rod when the articulation angle is 0 degree, A 1 and A 2 are the areas of cavity cross section with and without rod, respectively, and h o is the arm of the force of the steering rod when the articulation angle is 0 degree. Based on the equations mentioned above, the 3 DOFs' model of the vehicle with an articulated steering system can be established by Matlab/Simulink.

Simulation and Discussion
e single-track model can be used to evaluate parametrical conflict and sensitivity of the vehicle handling stability [27]. e established model is validated before the parametrical conflict and sensitivity analysis.

Simulation Validation.
e vehicle parameter is gained from the research of Gao et al. [3] and Xu et al. [28]. e parameters in this model are illustrated in Table 1.
During the current research, the snaking steering process is usually a typical test scenario, as shown in Figure 2.
During the field test, the fully loaded articulated vehicle runs on a cement road. e vehicle steering angle variation range is from -10 to 10 degrees (as shown in Figure 3). e vehicle velocity also varies in the time domain, as shown in Figure 4. According to the input in Figures 3 and 4, the yaw rate of the vehicle can be calculated. e simulated yaw rate of the vehicle and measured yaw rate in the field test are compared in Figure 5.
e MAPE (maximum absolute percentage error) and RMSE (root mean square error) of the simulated value based on the measured value are calculated to validate the accuracy of the established model. e values of MAPE and RMSE are 1.35% and 0.42°/s, respectively. e simulation results obtained in the built single-track model are in considerable agreement with those obtained in the current research [29][30][31][32]. us, the established model can be used in further analysis.
Based on the validated single-track model, the input velocity during the following simulation is changed into a constant value of 15 km/h to eliminate the effect of velocity variation in the time domain. e lateral acceleration, yaw rate of the front and rear parts of the vehicle, vehicle sideslip angle, and tire sideslip angle are taken as the indicators to evaluate the vehicle performance. To evaluate more signals in the time domain more intuitively, the peak value and RMS (root mean square) of these signals are calculated and compared in the following analysis.

Impact of the Cornering Stiffness Coefficient.
According to the work of Long and Chen [33] and Qi et al. [34], tire characteristics are crucial in terms of vehicle handling performance. e cornering stiffness of the tire can be adjusted to satisfy the requirement of the vehicle [35]. In the single-track model, the cornering stiffness coefficient is used to describe the behavior of the tire. To evaluate the effect of the cornering stiffness on both stability and handling performance, three cases are proposed for simulation.
Case A: the cornering stiffness of the front tire is varied by ±30% based on the current value, while the rear tire cornering stiffness stays the same. Case B: the cornering stiffness of the rear tire is varied by ±30% based on the current value, while the front tire cornering stiffness stays the same. Case C: the cornering stiffness of the rear tire varies from -30% to 30% based on the current value, while the rear tire cornering stiffness decreases from 30% to -30%. e lateral stability of the three cases is shown in Figures 6-8. Figure 6 shows the vehicle performance as the changing of the front tire cornering stiffness. e RMS value of the indicators is shown on the left axis while the peak value is shown on the right axis. e RMS and peak value of tire sideslip angle, vehicle sideslip angle, the yaw rate of the front and rear part, and rear tire lateral force decrease with the increase of front tire stiffness. Only the front tire lateral force increases with the front tire stiffness.
In Figure 7, the RMS and peak value of the front tire sideslip angle increase with the rear tire cornering stiffness while the value of the rear tire decreases with it. e variation range of the front tire sideslip angle is lower than the rear tire. e RMS and peak value of the vehicle sideslip angle drop with the increasing of rear tire stiffness, from 4.8 to 4.1 Shock and Vibration degrees and 10.7 to 9.4 degrees, respectively. e RMS and peak value of the yaw rate and rear tire lateral force climb with the increase of the rear tire stiffness. e front tire lateral force varies slightly as the rear tire stiffness changes.
In Case C, the rear tire cornering stiffness drops with the increase of the front tire stiffness. erefore, in Figure 8, the ratio of front and rear tire cornering stiffness is taken as the x-axis to compare the vehicle response. e front tire sideslip angle decreases with the increase of the cornering stiffness ratio dramatically while the rear tire sideslip angle changes slightly. e vehicle sideslip angle and yaw rate of the vehicle drop with the increasing ratio. e lateral force on the front and rear has the opposite trend; the front tire lateral force increases with the ratio while the rear tire lateral force drops with it. Compared to the tire and vehicle sideslip angle, the yaw rate and lateral force vary with the cornering stiffness ratio in a nonlinear trend.

Influence of the Payload Variation.
e axle load of the vehicle is generally seen as a key factor related to the vehicle's dynamic performance [36]. For the articulated vehicles, most of them operate in the engineering area. e axle load varies in a bigger range than the passenger car. ere are two methods to vary the payload, changing the load or the distance between the load gravity center and articulation hinge. In this paper, the payload is set in three cases to investigate the influence of the payload on the vehicle's lateral stability.
Case A: the load of the front axle varies from −30% to 30%, while the load on the rear axle stays the same. Case B: the load of the rear axle varies from −30% to 30%, while the load on the front axle stays the same.            Shock and Vibration Case C: the load of the front axle varies from −30% to 30%, while the load on the rear axle varies from 30% to −30%.
e corresponding vehicle performance of the three cases is shown in Figures 9-11.
In Figure 9, the tire sideslip angle, vehicle sideslip angle, and yaw rate increase with the front load at a steady rate. e lateral force on the front and rear tire increases slightly.
As shown in Figure 10, the tire sideslip angle and vehicle sideslip angle increase with the rear axle load. For the front tire sideslip angle, the increasing rate decreases with the improvement of the rear load, and the peak value of the front tire decreases slightly when the rear part load variation is in the 10% to 30% range. e yaw rate drops with the increase of the load. e lateral force on the front tire almost stays the same when the load changes, while the rear tire's lateral force increases with it.
In Figure 11, the RMS and peak value of the front tire sideslip angle have different trends: the RMS value decreases with the improving of the load ratio while the peak value drops with it. e RMS of the vehicle sideslip angle decreases from 4.46 to 4.28 degrees as the load ratio changes. e vehicle sideslip angle increases at first and then drops when the load ratio is in the range of 1.1 to 1.7. For the yaw rate of the vehicle, all of them increase with the load ratio at a different rate. e lateral force on the front tire holds the same while the rear tire drops with the improving of the load ratio.

Influence of the Wheelbase.
e wheelbase is also a vital factor of the vehicle's lateral stability [37]. ree cases are set to analyze the influence of this parameter.
Case A: the distance between the front axle and the articulation point varies from −30% to 30%, while the rear part stays the same. Case B: the distance between the rear axle and the articulation point varies from −30% to 30%, while the front part stays the same. Case C: the distance between the front axle and the articulation point is from −30% to 30%, while the distance between the rear axle and the articulation point varies from 30% to −30% at the same time.
In Figure 12, when the wheelbase of the front part increases from −30% to 30%, the sideslip angle of the tires drops with the increase of the front wheelbase and the rear tire has a bigger decreasing rate. e RMS and peak value of the vehicle sideslip vary from 9.75 to 10.4 and 4.26 to 4.5 degrees. e yaw rate of the front vehicle increases slightly while the yaw rate of the rear part and lateral force of the front and rear tire drop slightly with the variation of the front wheelbase.
In Figure 13, when the wheelbase of the rear axle increases from −30% to 30%, the front/rear tire sideslip angle has the opposite trend. e front tire's sideslip angle drops with it while the rear tire increases with it. e vehicle sideslip angle and yaw rate of the front and rear vehicle increase with the variation of the rear wheelbase at a steady rate. e lateral force of the tires varies slightly during the variation range of the rear wheelbase.
In Figure 14, when the wheelbase of the front part drops from 30% to −30% and the wheelbase of the rear part increases from −30% to 30%, the front/rear wheelbase ratio increases from 0.5 to 3. e tire sideslip angle, the yaw rate of the front and rear parts, and the lateral force of the rear tire show a decreasing trend with the decreasing rate. e lateral force of the front tire keeps still during the range. e vehicle sideslip angle increases with the front/rear wheelbase ratio, and the increasing rate becomes smaller.

Influence of Vehicle Speed.
According to the current research, the vehicle that runs at a high speed usually has poor lateral stability [38]. For the articulation vehicle, the normal operating speed range is lower than the passenger car. Based on the relevant researches, the speed range is set from 5 to 35 km/h. Figure 15 shows a clear upward trend of these vehicle performance indicators. According to Figure 15(a), the front tire sideslip angle is lower than the rear tire when the vehicle velocity is smaller than 18 km/h; after that, the angles of the front and rear tire are relatively close to each other. Based on Figures 15(c) and 15(d), the yaw rate and lateral force of the rear part are higher than the front part in the whole range. For the vehicle sideslips angle, the minimum peak value emerges at the velocity of 10 km/h instead of 5 km/h.
Because the velocity change range is larger than the other parameters, the variation range of the vehicle lateral stability criteria is far higher than the change of the tire cornering stiffness, vehicle payload, CG position, and wheelbase. During the variation range of vehicle speed, the tire sideslip angle RMS and peak value vary from 1 to 8 degrees and 4 to 18 degrees, the RMS and peak value of vehicle sideslip angle increase from 2 to 8 degrees and 4 to 18 degrees, the yaw rate of the vehicle varies from 2 to 9°/s and 7 to 18°/s, and for the lateral force, the RMS and peak range are 0.25 × 10 4 to 2 × 10 4 N and 0.5 × 10 4 to 4.5 × 10 4 N.

Sensitivity Analysis
A parametric sensitivity correlation analysis is initiated to understand the vehicle parameters sensitivity of the stability criteria. Based on the simulation result in the last part, the correlation sensitivity of the abovementioned parameters on the proposed vehicle stability criteria is shown in Figure 16. Four bars with different colors represent the variation value of the seven indicators based on the original value. Since the variation range of the speed is different from the other parameters, the influence of the speed is not shown in Figure 16.
For the front tire sideslip angle, the tire stiffness of the front wheel has the most significant negative effect on it. e rear load has the most crucial influence on the rear tire sideslip angle. For the vehicle sideslip angle, the front tire stiffness plays the most important factor in it. Both the yaw rate of the vehicle's front and rear parts have rear load and           Shock and Vibration           rear tire stiffness as the two most important factors. For the lateral force on the front and rear tire, the corresponding tire cornering stiffness is the most crucial factor. e variation range of the lateral force is from −25 to 25%, and the ranges for the yaw rate of the front and rear parts are the same, −20 to 15%. e front tire sideslip angle and vehicle sideslip angle have the minimum variation percentage, which is −15 to 10%. e rear load plays a relatively important factor in the variation of rear tire sideslip angle, vehicle sideslip angle, and yaw rate of the vehicle, which is due to its massive value of the rear load compared to the front load.
According to Figure 16, the variation trends of the RMS and peak value are similar. erefore, the sensitivity analysis is initiated based on the RMS value of the indicators. e sensitivity analysis result is shown in Figure 17. e seven lateral stability indicators have a nearly linear relationship with the vehicle parameters. For the front tire sideslip angle, the influence of the wheelbase is so small that can be neglected. e front tire stiffness harms it and the front and rear load have a positive effect on it. In Figure 17(b), the increasing of load on the front and rear axle leads the rear tire sideslip angle to increase, and the rear wheelbase has the minimum influence on it, while the other vehicle parameters have a negative influence on it. For the vehicle sideslip angle, the rear wheelbase has the minimum effect on it. e increase of the rear tire stiffness causes a minor variation of sideslip angle compared to the decrease of the tire stiffness. For the sensitivity of the yaw rate of the front and rear parts, the rear tire stiffness, front tire stiffness, and rear load have the same trend. For the lateral force, the most critical factor to the force is the corresponding tire stiffness; however, the front tire stiffness harms the rear lateral force, while the rear tire stiffness has a positive effect on the front lateral force.

Conclusion
e conflict and parametrical sensitivity of the articulated vehicle stability versus the vehicle parameter variation are analyzed based on the vehicle dynamic theory. To accomplish the goal, the single-track articulated vehicle model with 3 DOFs is established and validated with the field test result of the current research.
e snaking maneuver at the constant speed of 15 km/h is taken as the input of the simulation. e influences of the tire cornering stiffness, vehicle load, wheelbase, and speed are analyzed, and the vehicle load and wheelbase are divided into front and rear values due to the articulated vehicle's structural characteristic.
Based on the result of the analysis, the following conclusions can be drawn: (1) e tire cornering stiffness has a positive effect on the corresponding tire lateral force. e front tire cornering stiffness harms the seven indicators except for the lateral force on front tire. e rear tire cornering stiffness has a positive effect on the seven indicators except for the rear tire sideslip angle.
(2) e increase of the load on the front and rear will lead to the increase of the indicators except for the yaw rate of the front and rear parts.
(3) e load of the rear part has a comparatively high influence on the rear tire sideslip angle, vehicle sideslip angle, and the vehicle yaw rate of the front and rear part. e reason that the load of the rear part has a higher impact than the front part is that the load of the rear part is much higher than the front part.
(4) e wheelbase of the rear axle has a relatively low impact on the indicators. e wheelbase of the front axle has a relatively low effect on the indicators except for the rear tire sideslip angle and lateral force. (5) e variation range of the indicators is lower than the variation range of the vehicle parameters. e increase of the vehicle speed leads to the increase of all the indicators, which means the lateral stability gets worse.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare no conflicts of interest.