Steady-State Steering Characteristics of Mathematical Model for Semitrailer Based on Variations in Camber Parameters

To master the basic characteristics of steady-state cornering for a semitrailer, this paper summarises the current modelling methods for handling and stability and discusses their limitations. The classical linear mathematical model for a two-degree-of-freedom (DOF) handling and stability system is used to develop a new model. Analysis methods are proposed to introduce the inﬂuence of the camber angle and body roll into the model parameters. Thus, a mathematical model for the lateral stability of semitrailer with ﬁve DOFs is established. At the same time, a modiﬁed formula to calculate the stability factor of the semitrailer is developed with a MATLAB model to solve the dynamic state equation. The mathematical model, which considers the body roll and the changes in the camber angle caused by roll, compares the turning radius ratio and yaw rate as the evaluation index with the classical linear mathematical model of a two-DOF system. The vehicle parameters for three diﬀerent types of semi-tractor trailers are used to calculate and compare two mathematical models for handling and stability using real vehicle test data. The results show that the new modelling and analysis method proposed in this paper has a high calculation accuracy and fast calculation speed, is clear and concise, and is consistent with the real vehicle test data. In addition, the accuracy of the new mathematical model for handling and stability and the improved stability factor are veriﬁed.


Introduction
e handling and stability of vehicles is its ability to drive according to the driver's intention. Semitrailer and other articulated vehicles are complex dynamic systems with the coupling effects of the tractor and the trailer. e research for the handling and stability of vehicles is to maintain the vehicle trajectory and control its yaw [1][2][3]. To date, theoretical research on vehicle handling and stability is primarily through the establishment of mathematical models. Highprecision dynamics models for vehicle handling and stability are of great importance for system performance analyses and the development of stability controllers [4,5].
At present, some achievements have been made in the theoretical research on the steady-state steering characteristics of semitrailer.: Wu [6] established a five-degree-of-freedom-(DOF-) articulated vehicle using a multisteering mathematical model. e basic analytical results of the yaw/ roll performance characteristics were presented, and the effects of the roll steer about the lateral force under steadystate turning were discussed and illustrated.
e mathematical model considered the influence of roll on the steadystate steering, but it did not account for the influence of the camber angle of the tire as caused by roll, suggesting their model has some limitations. Kobayashi et al. [7] constructed a new formula of the mechanism by which direct yaw moments alter the cornering resistance and mechanical power for all wheels based on the classical mathematical model of a two-DOF system. is includes the electric loss of the motors and the inverter to study the steady-state steering characteristics of the vehicle. However, the influence of roll on the wheel deflection characteristics is not considered in the new formulation. Sankar and Surial [8] proposed a mathematical sensitivity method to study the effects of variations in parameters on the rollover stability of heavy articulated vehicles during steady turning manoeuvres. e results show that their approach provides a quick and accurate estimate of the vehicle rollover response to reasonable variations in the parameters. However, the influence of tire characteristics on the steady-state steering under roll conditions was not considered. Zhang et al. [9] proposed an analytical lateral dynamic model for skid steering with a wheeled vehicle that could be applied to its design and control. e model is described using second-order ordinary differential equations in an explicit form, and the steady-state characteristics of skid steering for wheeled vehicles were analysed based on differential equations. While the analysis results showed that the steady-state response of the skid-steered wheeled vehicle is determined from the stability factor, the influences of this factor were not given. Hussain et al. [10] established a multidegree-of-freedom, nonlinear, multibody, dynamic model for semitrailer handling and stability. e model incorporates all sources of compliance, stiffness, and damping, all of which have nonlinear characteristics. e model was used primarily to study the rollover stability of an articulated vehicle; however, under roll conditions, the rollover model does not consider the influence of changes in the tire characteristics. e steady-state steering characteristics of vehicles have been studied using a variety of methods, but the influence of changes in the camber angle of the tires as caused by roll was not considered in previous mathematical models. However, it is important to establish a model for the entire vehicle. A more complex correlation for expressions in such a model is not necessarily better, as the mechanism of the key factors should be considered. One of the key factors in the mathematical model for handling and stability is the tire characteristics, where the body roll of the vehicle can cause changes to the camber and toe angles, while the influence on the tire cornering characteristics is relatively large. is paper presents an established mathematical model for the manoeuvring stability of a vehicle based on the linear mathematical model with two DOFs. To improve the accuracy of the steady-state model for the vehicle steering stability, the influence of the camber angle caused by roll and body roll into the model parameters is proposed. A mathematical model for the lateral stability of a semitrailer with five DOFs is established. Finally, the evaluation indices of the yaw velocity and the turning radius ratio as well as experiments are used to verify the theoretical model. Compared with the traditional linear, two-DOF, mathematical model, the proposed model more accurately reflects the vehicle ground state under steady-state steering conditions. is article is divided into the following sections. Section 2 discusses the shortcomings of the traditional two-DOF mathematical model and proposes a new method to calculate the stability factor through the introduction of tire inclination parameters. Section 3 verifies the accuracy of the introduced tire inclination parameter model based on the evaluation index of the yaw velocity. Section 4 verifies the accuracy of the calculation method based on the evaluation index of the turning radius ratio. Section 5 gives the conclusions of the study.

Mathematical Model of Vehicle Handling
and Stability e proposed mathematical model for handling and stability is introduced, which considers the continual influence of the camber angle as caused by roll and body roll. A modified formula to calculate the stability factor of a semitrailer is given.

Principle of Model Simplification.
In general, the dynamic characteristics of semitrailer are much more complex than those of a truck. ese primarily involve the relative motion between the tractor and trailer and the coupling effects between the traction seat and the traction pin of semitrailer [11][12][13]. To study the dynamic characteristics of a semitrailer, the following aspects are simplified in the semitrailer model due to combining automobile theory with multibody dynamics: (i) e tractor and trailer are regarded as rigid bodies, and the semitrailer trains are regarded as a combination of two rigid bodies connected via the coupling of a traction pin and traction seat (ii) e roll and yaw motions of the semitrailer train are considered while the vertical and pitching motions are not (iii) In the analysis of the force for the semitrailer train system, the rolling and air resistances are neglected (iv) In the analysis of the side deflection force of the tires, the side deflection angles of the left and right tires are assumed to be equal and relatively small, and the front wheel angle of the tire is also considered to be relatively small Based on the above simplifications, the semitrailer train can be simplified using a monorail and double centroid model [12,14,15] with five DOFs. e DOFs are the lateral motion y, yaw motion r, roll angle of the tractor and trailer φ, steering angle δ, and folding angles between the tractor and trailer θ. A more realistic open-loop research method is adopted, as shown in Figure 1.

Classical Mathematical Model of a Two-DOF System.
e mathematical model for the classical two-DOF system only considers the side angles of the front and rear tires. e lateral forces for each axle tire are given as where k 1 , k 2 , and k 3 are the lateral deflection angle stiffnesses for the front, middle, and rear axles of the tire, respectively (N/rad), and α 1 , α 2 , and α 3 are the tire sideslip angles of the front, middle, and rear axles of the tire, respectively (rad). e differential equations of motion for the tractor and trailer are given as follows: 2 Shock and Vibration When the vehicle is driving at a constant speed, the system enters the steady state, and the stability factors K 1 and K 2 can be obtained from the k 1 and k 2 in equations (2). e stability factor is an important parameter to characterise the steady state response of the vehicle as However, the stability factor obtained from the mathematical model for the classical two-DOF is not sufficiently accurate, and its influence factor is not comprehensive. Although this approach reflects the relationship between the most basic parameters of an automobile and the stability of its handling, it ignores the influence of changes in the tire inclination angle. Such changes significantly influence the tire force and lateral angle to vary the direction of tire movement, which greatly affects the stability of vehicle handling.

Mathematical Model considering the Camber Angle.
Changes in the camber and toe angles of the tire are affected by body roll, which manifest as the yaw rate response of the semitrailer train and affect the steering stability of the entire vehicle [2,[16][17][18]. e influence of the body roll under steering conditions is considered here. Variations in the camber angle under the same lateral force are introduced into the model parameter as the influence factor, as shown in Figure 2. e force analysis in Figure 2 is based on Newton's Second Law and assumes that the velocity along the x-axis remains unchanged while the longitudinal velocity of the centre of mass is equal; that is, u � u 1 . e differential equations of motion for the tractor and trailer are obtained as follows [19]: e lateral forces at traction pin A from the tractor and the trailer are equal; that is, f YA1 � f YA2 , and the angle of the front wheel of the tire is small (the simplification cos δ � 1 is made). ese two conditions are brought into equations (1) to obtain where m 1 and m 2 are the qualities for the tractor and trailer, respectively (kg); u is the longitudinal velocity of the vehicle (m/s); I z 1 and I z 2 are the yaw moments of inertia for the tractor and trailer, respectively (kg·m 2 ); a is the distance from the tractor centroid to the front axle (m); b is the distance from the tractor centroid to the middle axle (m); a 1 is the distance from the trailer centroid to the hinged point (m); b 1 is the distance from the trailer centroid to rear axle (m); c is the distance from the tractor centroid to the hinged point (m); L 1 and L 2 are the wheelbases for the tractor and Shock and Vibration trailer, respectively (m); ω r 1 and ω r 2 are the yaw rates of the tractor and trailer centroids, respectively (rad/s); β 1 and β 2 are the sideslip angles of the tractor and trailer centroids, respectively (degree); and F Y 1 , F Y 2 , and F Y 3 are the lateral forces of the front, middle, and rear axle of the tire, respectively (N). From Figure 2, the equations for tire lateral deflection angle on each axle are where δ is the angle of the front wheel of the tractor (rad) and θ is the relative angle between the tractor and the trailer (rad).
In the classical two-DOF model, the lateral force of the tire only considers the force caused by lateral deviations. However, the proposed mathematical model also considers the lateral force caused by changes in the camber angle for the force analysis of the tires. e influence of body roll on the tire force and changes in the camber angle are introduced into the model parameters. When the wheel is steering, the outer wheel camber angle changes in the positive direction, and the inner wheel camber angle changes in the negative direction. is weakens the ability of the tire to carry the lateral force and affects the understeering of the vehicle [20], as shown in Figure 3. e equations of tire lateral forces for each axle are where C 1 , C 2 , and C 3 are the camber angle stiffnesses of the front, middle, and rear axle tires, respectively (N/rad), and c 1 , c 2 , and c 3 are the camber angles for the front, middle, and rear axle as caused by body roll, respectively (rad). Under steady steering conditions, the body tilting moment and the lateral force of each wheel can be calculated using where M ϕ 1 and M ϕ 2 are the roll torques of the tractor and trailer, respectively; h 1 and h 2 are the heights of the tractor and trailer centroids, respectively (m); and a Y 1 and a Y 2 are the lateral accelerations for the tractor and trailer, respectively (m/s 2 ). In this case, the variations in the camber angle caused by body roll and lateral forces can be calculated using [21] where φ 1 and φ 2 are the roll stiffnesses of the tractor and trailer, respectively (N·m/rad); Γ 11 , Γ 21 , and Γ 31 are the variation ratios of the camber angle of the front, middle, and rear suspensions under roll conditions, respectively (rad/ rad); and Γ 12 , Γ 22 , and Γ 32 are the variation ratios for the camber angle of the front, middle, and rear suspensions under lateral force conditions, respectively (rad/N). By simultaneously solving equations (8) and (9), the following is obtained: e relative rotational resistance moment at the hinge of the tractor and trailer can be calculated with equation (8) as where M A is the relative rotational resistance moment at the hinge of the tractor and trailer and C A is the relative rotation damping coefficient. By simultaneously solving equations (5)- (7), (10), and (11), the following is obtained: where a 11 � (m 1 + m 2 )u, a 12 � − m 2 c, a 13 � − m 2 a 1 , In equation (12), [δ] is the input vector and [X] � β 1 ω r 1 ω r 2 θ is the output vector. Equation (12) can be expressed in matrix form as where A, B, and C are given by the matrices in equations (14)-(16), respectively: [A] � a 11 a 12 a 13 a 14 a 21 a 22 a 23 0 a 31 0 a 33 a 34 0 0 0 1 [B] � Shock and Vibration 5 When the vehicle system is under steady-state steering conditions, it is considered that ω r 1 � ω r 2 � ω r , β 1 � β 2 � β, and θ are all approximately constant. In the differential equation system of equation (12), ω r 1 · � ω r 2 · � 0, β · � 0, θ · � 0. en, the matrix form in equation (13) can be simplified as [22] BX � Cδ. (17) e yaw rate gain of the tractor and trailer and the relative rotation angle gain between them are obtained from solving the matrix equation as

Improved Method to Calculate the Stability Factor.
A new method to calculate the stability factor can be obtained from equations (12) and (18) as follows. It is seen from equations (19) and (20) below that the roll torque and camber angle impact the stability factor, which affects the understeer of the entire vehicle [23]: where A � ((a 1 h 1 /φ 1 )(C 2 /k 2 )Γ 21 ) − ((bh 1 /φ 1 )(C 1 /k 1 )Γ 11 ), e A and C (E and G) values are the influence from changes in the tire camber angle of the tractor (trailer) as caused by body roll. e B and D (F and H) values are the influence from changes in the tire camber angle change of the tractor (trailer) caused by the lateral forces. Variations in the camber angle reduce the tire cornering stiffness and increase the tire sideslip angle and their differences (α 1 − α 2 ), which increases the vehicle's understeer.

Yaw Rate Verification
A MATLAB model was established to solve the dynamic state equation of the semitrailer system. e mathematical model considers body roll and changes in the camber angle as caused by roll and was used to compare the yaw rate as the evaluation index with the classical two-DOF linear model. Based on the vehicle parameters for three different types of semitrailer, two mathematical models for the handling and stability are calculated and compared with the real vehicle test data.

MATLAB Model.
e MATLAB calculation model was established based on two mathematical models, as shown in appendix Figure 4. e MATLAB model was used to verify the mathematical model, and the calculation results for the two mathematical models are further compared. e model is verified based on the vehicle parameters for three different types of semitrailer, as shown in Table 1.

Experimental Verification.
e yaw rate is an important indicator to describe the vehicle's transient response. e yaw rate evaluation index mainly evaluates the steady-state steering from the frequency domain. Lateral transient response tests were carried out according to the ISO 7401-2003 standard to further verify the accuracy of the proposed mathematical model for the entire vehicle [24]. e ISO 7401-2003 test method is more reasonable for equipment accuracy and test site and road requirements; the test method is more convenient and easy to operate, so this paper uses ISO 7401-2003 to verify the model. e vehicle was driven at a test speed of 80 km/h in a straight line, where the initial speed should not deviate by more than 2 km/h. Starting from a 0 ± 5°/s yaw velocity equilibrium condition, a steering input is applied as rapidly as possible to a preselected value and maintained for several seconds after the measured vehicle motion variables have reached a steady state. is ensures the steering input is short relative to the vehicle response time. e time between 10% and 90% of the steering input should not be greater than 15 s. e steeringwheel angle amplitude is determined from the steady-state driving on a circle with a radius that provides the preselected steady-state lateral acceleration (typically 6 m/s 2 ) at the required test speed. All test runs are performed at least three times. e steering wheel force and angle measuring instrument are shown in Figure 5(a), the gyroscope is shown in Figure 5(b), the GPS is shown in Figure 5(c), and the test road is shown in Figure 5(d).

Experimental Results.
e lateral transient response tests indicate the simulation results using the mathematical model for the three different types of semitrailer are compared to the experimental yaw rate. To facilitate the research, this paper takes the calculation results for the tractor as an example for further comparisons and explanations, as shown in Table 2.
e results show that, for the yaw rat, the steady-state error is controlled to within 0.23°/s, the peak error is controlled to within 0.28°/s, the overshoot error is controlled to within 1.9%, and the response time error is controlled within 0.02 s. e calculation results from the mathematical model,  In1  In2  In3  In4  In5  In6  In7  In8  In9  In10  In11  In12  In13  In14  In15  In16  In17  In18  In19  In20  In21  In22  In23  In24  In25  In26   which considers body roll and the changes in the camber angle as caused by roll, agree well with the experimental data. However, the calculation result for the classical two-DOF model deviated greatly from the test data. A comparison of the yaw rates between the simulation and test data curves for the three different types of semitrailer is shown in Figure 6. Figures 7(a)-7(f ) are plots of the yaw rates for the tractor and trailer for semitrailers I, II, and III, respectively. e following conclusions are based on Figure 6.
(i) e calculation curves of the mathematical model, which considers body roll and changes in the camber angle as caused by roll, are more consistent with the test curve with a higher calculation accuracy than the classical two-DOF model.
(ii) When the wheel is turning, the vehicle body generates a roll, causing the inner and outer suspension springs to compress and extend, which is equivalent to the tire bouncing up and down. e camber angle of the tire changes in the negative direction, which is reflected in the yaw angular velocity response. e steady-state yaw rate calculated using the classical two-DOF model is generally higher than the test value due to the influence of the camber angle not being considered. (iii) e response time, peak, and overshoot of the yaw rate calculated using the proposed model agree well with the experimental results, while the classical two-DOF model shows large distortions.
Force and angle measuring instrument

Turning Radius Ratio Verification
e relationship between the stability factor of the semitrailer and the turning radius ratio is derived. e proposed model was used to compare the turning radius ratio as the evaluation index with the classical two-DOF model. e vehicle parameters for three different types of semitrailer were used in the two mathematical models to calculate the  handling and stability as compared with the real vehicle test data.

Calculation Model.
e steady-state steering of the semitrailer can be characterised with the turning radius ratio. erefore, the accuracy of the proposed model was verified by deriving the relationship between the stability factor and turning radius ratio.
As shown in Figure 7, the loads for each axle of the tractor and the trailer are calculated as where G 1 and G 2 are the front and middle axle loads (N) and f ZA and G 3 are the saddle and rear axle loads (N). e calculation equations for K 1 and K 2 from Table 2 and equation (6) are expressed as When the vehicle system enters steady-state steering, the lateral acceleration of each shaft is a y , and the following equations can be obtained from equation (3) with F Yi � G i a yi : where M y � a y /g and g is the acceleration due to gravity (m/s 2 ). e distance from the traction pin of the semitrailer to the middle axle is relatively small, generally 50-200 mm [25]. e model is simplified so that the traction pin coincides with the middle axle. When there is no lateral deviation, R 01 ≈ L 1 /δ and R 02 ≈ L 2 /θ; when there is a lateral deviation, e theoretical equations to calculate the turning radius ratio are obtained by combining equation (8) with the above relationship as

Experimental
Verification. e understeer, neutral, and oversteer characteristics of an automobile are commonly referred to as the steady-state steering characteristics. e turning radius ratio R/R 0 is an important index used to characterize the steady-state steering characteristics of automobile.
e tests for the steady-state circular driving behavior constant steering-wheel angle were performed based on the ISO 4138-2012 standard [26] test methods. e vehicle travels along a circle with a radius of 40 m (R � 40 m); the steering-wheel angle is fixed while the speed is increased continuously at a slow rate up to the limit of control. e test curve for semitrailer I and the turning radius ratio fit curve are shown in Figure 8. e test curve and turning radius ratio of the tractor unit and the semitrailer units are shown in Figures 8(a)-8(d), respectively. As shown in Figure 8, the fit curve is consistent with the test data for 6.5 m/s 2 , indicating the test data are reasonable. erefore, the curves for the test and simulation calculations are compared and analysed within the data for lateral accelerations from 0 to 6.5 m/s 2 .
e test data of the other vehicle models are fit and processed using the same method.

Experimental Result.
e curves for the test data after fitting are compared with the curves calculated using the proposed model, as shown in Figure 9. Figures 4(a)-4(f ) are the turning radius ratio curve of the tractor and trailer of the semitrailers I, II, and III, respectively. Figure 9 provides the following conclusions.
(i) As shown in Figures 4(a)-4(d), the curve for the turning radius ratio of the proposed model is more consistent with the test curve than the classical two-DOF model. e trend of the understeer is also more consistent with the test curve, which indicates that changes in the camber angle weaken the lateral stiffness of the tire, increase the lateral deflection angle of the tire, and increase the difference (α 1 − α 2 ), thus enhancing the understeering of vehicles.   (ii) As shown in Figure 9(e), when the lateral acceleration of the test curve for the tractor is a y � 5.25 m/s 2 , the vehicle begins to enter an understeer. When the lateral acceleration is a y � 5.21 m/s 2 , the proposed model considering roll begins to enter an understeer, which indicates that the calculation curve of the proposed model agrees well with the test curve.

Conclusions
is paper summarised the situations and shortcoming of modelling methods for handling and stability. Furthermore, the new modeling method is proposed. Body roll and the influence of the change of camber angle are introduced into the parameters of the mathematical model, and the dynamic equation of state is solved by MATLAB, and the results are compared with the classical linear mathematical model of the two DOF and the experimental results data. e conclusions are as follows: (1) e calculation results for the proposed model, which considers body roll and changes in the camber angle caused by roll, are closer to the experimental values with a higher calculation accuracy than the classical two-DOF model. inclination angle, or the influence of the K&C characteristic parameters of the suspension.
us, further studies can address the influence of these characteristic suspension parameters and further establish a more accurate dynamic equation for semitrailer trains.