2.1 STI Tire Model
STI tire model is proposed by an American company, which named as “Systems Technologies Inc.” [30]. Compared with other tire models, the STI tire model can not only describe the tire nonlinear mechanical characteristics accurately, but also reflect the coupling relationship between the tire longitudinal force and the tire lateral force exactly [31], thus it is very suitable for the bus YSC system design in this study. To achieve the description of the STI tire model, an important variable must be defined firstly, i.e., the composite slip coefficient σ, which is expressed as [32]:
$$\sigma = \frac{{\uppi a_{p}^{2} \sqrt {k_{\alpha }^{2} \tan^{2} \alpha + k_{s}^{2} (s/1 - s)^{2} } }}{{8\mu F_{z} }},$$
(1)
where ap denotes the length of the tire contact patch, kα denotes the tire lateral stiffness, ks denotes the tire longitudinal stiffness, α denotes the tire sideslip angle, s denotes the tire longitudinal slip coefficient, μ is the road adhesion coefficient and Fz is the tire vertical load. In Eq. (1), the length of the tire contact patch can be further defined as [33]:
$$a_{p} = 0.0768\sqrt {F_{z} \cdot F_{zt} } /T_{w} (T_{p} + 5),$$
(2)
where Fzt is the tire design load, Tw is the tire width, Tp is the tire pressure. The tire lateral stiffness kα and the tire longitudinal stiffness ks in Eq. (1) can be measured by experimental tests. On this basis, to further consider the saturation effect of the tire longitudinal stiffness, a modified tire longitudinal stiffness ksm is defined as:
$$k_{sm} = k_{s} + (k_{\alpha } - k_{s} )\sqrt {\sin^{2} \alpha + \cos^{2} \alpha } ,$$
(3)
Based on the defined composite slip coefficient σ, a tire force saturation function can then be defined as:
$$f(\sigma ) = \frac{{C_{1} \sigma^{3} + C_{2} \sigma^{2} + (4/\uppi )\sigma }}{{C_{1} \sigma^{3} + C_{3} \sigma^{2} + C_{4} \sigma + 1}},$$
(4)
where C1, C2, C3 and C4 are the fixed coefficients, which can be obtained by fitting the experimental data. The function f (σ) is consistent with the mechanical properties of the tire force friction circle, thus it can reflect the tire force saturation characteristics effectively. Then, on the basis of the above equations, the standardized expressions of the tire longitudinal force and the tire lateral force of the STI tire model can be respectively described as:
$$\left\{ \begin{aligned} F_{x} & = \frac{{f(\sigma )k_{sm} s}}{{\sqrt {k_{\alpha }^{2} \tan^{2} \alpha + k_{sm}^{2} s^{2} } }}\mu F_{z} , \\ F_{y} & = \frac{{f(\sigma )k_{\alpha } \tan \alpha }}{{\sqrt {k_{\alpha }^{2} \tan^{2} \alpha + k_{sm}^{2} s^{2} } }}\mu F_{z} + Y_{\gamma } \gamma , \\ \end{aligned} \right.$$
(5)
where Fx represents the tire longitudinal force, Fy represents the tire lateral force, γ represents the tire camber angle, Yγ represents the tire camber coefficient, which is used to reflect the influence of the camber angle on the tire lateral force. According to Eq. (5), the coupling relationship between the tire longitudinal force and the tire lateral force under compound driving conditions can be reflected exactly.
To obtain the experimental data which can accurately reflect the nonlinear mechanical characteristics of the tire, the experimental tests are conducted through a flat-plate bench assisted by the KH Automotive Technologies (Guangzhou) Co., Ltd. Figure 1 shows the experimental setup of the tire mechanical characteristics tests.
Based on the experimental data, the STI tire model parameters can then be fitted. According to the expressions of the tire forces, it is obvious that the fitting of the tire model parameters is actually to determine the fixed coefficients C1, C2, C3 and C4 in Eq. (4). Before achieving the fitting of the fixed coefficients, the tire lateral stiffness and the tire longitudinal stiffness need to be firstly measured by experimental tests. According to the test results, the tire lateral stiffness is finally determined as − 66463 N/rad and the tire longitudinal stiffness is finally determined as 84000 N/rad. On this basis, the value of the composite slip coefficient σ can then be calculated for different tire vertical loads, tire sideslip angles and tire longitudinal slip coefficients. Meanwhile, according to Eq. (5) and the experimental results of the tire forces, the value of the tire force saturation function can also be determined. Therefore, by combining the value of the composite slip coefficient and the value of the tire force saturation function, the fixed coefficients C1, C2, C3 and C4 can then be determined by using the curve fitting function in the Origin software. Take the case of the tire bench test under low road adhesion coefficient, the values of the four fixed coefficients C1, C2, C3 and C4 are finally determined as 10, 8.98, 10 and 0 respectively.
To verify the fitting accuracy, the comparison between the experimental results and the simulation results of the tire force saturation function is shown in Figure 2. It can be seen that the experimental results of the tire force saturation function are all distributed around the fitting curve, which shows that the fitting results are highly consistent with the experimental results. Figure 3 further shows the fitting residual of the tire force saturation function. It can be seen that most of the absolute values of the fitting residual are less than 0.15, which indicates that the fitting of the tire force saturation function in this work is trustworthy. Note that there are two other parameters that can also show the fitting effect, i.e., the R-squared and the adjusted R-squared. However, because of the residual diagram can show the fitting effect more intuitively, thus we only provide the fitting residual results in this paper.
2.2 Bus Dynamics Model
In this section, a 7-DOF bus dynamics model is established based on previous literatures [34–36], which includes the longitudinal motion, lateral motion, yaw motion and the rotational dynamics of four wheels. According to Newton’s theorem, the dynamic equations of the bus longitudinal, lateral and yaw motions can be established as follows [37]:
$$\left\{ \begin{aligned} m(\dot{v}_{x} - r \cdot \dot{v}_{y} ) & = (F_{xfl} + F_{xfr} )\cos \delta \\ & \quad - (F_{yfl} + F_{yfr} )\sin \delta + F_{xrl} + F_{xrr} , \\ m(\dot{v}_{y} + r \cdot v_{x} ) & = (F_{xfl} + F_{xfr} )\sin \delta \\ & \quad + (F_{yfl} + F_{yfr} )\cos \delta + F_{yrl} + F_{yrr} , \\ I_{z} \cdot \dot{r} & = [(F_{xfl} + F_{xfr} )\sin \delta + (F_{yfl} + F_{yfr} )\cos \delta ]a \\ & \quad + [(F_{xfr} - F_{xfl} )\cos \delta + (F_{yfl} - F_{yfr} )\cos \delta ]\frac{{t_{w1} }}{2} \\ & \quad + (F_{xrr} - F_{xrl} )\frac{{t_{w2} }}{2} - (F_{yrl} + F_{yrr} )b, \\ \end{aligned} \right.$$
(6)
where Fxij and Fyij are the tire longitudinal and lateral forces respectively, the first subscript i = f or r represent front and rear axles respectively, and the second j = l or r denote left and right wheels. m is the vehicle weight, vx and vy are the longitudinal and lateral velocities of the vehicle, δ is the front wheel steering angle, a is the distance from the center of the vehicle to the front axle, b is the distance from the center to the rear axle, tw1 and tw2 are the wheel bases of the front and rear axles, r is the vehicle yaw rate. Note that the tire longitudinal and lateral forces in Eq. (6) are calculated by the STI tire model which has been established in the previous section. To represent the lumped disturbance which includes the system uncertainties and external disturbance, a variable D(t) which is usually assumed to be a bounded function is defined as follows:
$$D(t) \le \delta_{0} ,$$
(7)
where δ0 is the upper bound of D(t).
In addition, the dynamic equations which represent the rotation of the four wheels are given by:
$$J \cdot \dot{\omega }_{k} = T - F_{xij} \cdot R,$$
(8)
where J is the moment of inertia of the wheel, ωk (k = 1, 2, 3, 4) represent the angular velocity of the four wheels, R is the wheel radius, Tk (k = 1, 2, 3, 4) represent the braking torques acted on the four wheels.
To more accurately reflect the vehicle load transfer effect during vehicle driving process, the tire vertical load can be calculated by combining the static tire vertical load and the load transfer effect caused by longitudinal and lateral acceleration. Thus, the related equations are expressed as Ref. [38]:
$$\left\{ \begin{aligned} F_{zfl} & = mg\frac{b}{2l} - m\dot{v}_{x} \frac{{h_{g} }}{2l} - m\dot{v}_{y} \frac{{h_{g} }}{{t_{w1} }} \cdot \frac{b}{l}, \\ F_{zfr} & = mg\frac{b}{2l} - m\dot{v}_{x} \frac{{h_{g} }}{2l} + m\dot{v}_{y} \frac{{h_{g} }}{{t_{w1} }} \cdot \frac{b}{l}, \\ F_{zrl} & = mg\frac{a}{2l} + m\dot{v}_{x} \frac{{h_{g} }}{2l} - m\dot{v}_{y} \frac{{h_{g} }}{{t_{w1} }} \cdot \frac{a}{l}, \\ F_{zrr} & = mg\frac{a}{2l} + m\dot{v}_{x} \frac{{h_{g} }}{2l} + m\dot{v}_{y} \frac{{h_{g} }}{{t_{w1} }} \cdot \frac{a}{l}, \\ \end{aligned} \right.$$
(9)
where hg is the height of the center of the bus, l is the length of the bus. In addition, the calculation of the tire sideslip angle of each wheel is given as follows:
$$\left\{ \begin{aligned} \alpha_{fl} & = \delta - \arctan \left( {\frac{{v_{y} + ar}}{{v_{x} - {{t_{w1} r} \mathord{\left/ {\vphantom {{t_{w1} r} 2}} \right. \kern-\nulldelimiterspace} 2}}}} \right), \\ \alpha_{rl} & = \delta - \arctan \left( {\frac{{v_{y} + ar}}{{v_{x} + {{t_{w1} r} \mathord{\left/ {\vphantom {{t_{w1} r} 2}} \right. \kern-\nulldelimiterspace} 2}}}} \right), \\ \alpha_{rl} & = - \arctan \left( {\frac{{v_{y} - br}}{{v_{x} - {{t_{w2} r} \mathord{\left/ {\vphantom {{t_{w2} r} 2}} \right. \kern-\nulldelimiterspace} 2}}}} \right), \\ \alpha_{rl} & = - \arctan \left( {\frac{{v_{y} - br}}{{v_{x} + {{t_{w2} r} \mathord{\left/ {\vphantom {{t_{w2} r} 2}} \right. \kern-\nulldelimiterspace} 2}}}} \right). \\ \end{aligned} \right.$$
(10)
Based on the above calculations for bus driving states, the wheel center speed of each wheel can then be obtained as follows [39, 40]:
$$\left\{ \begin{aligned} v_{t\_fl} & = \left( {v_{x} - \frac{{t_{w1} }}{2}r} \right)\cos \delta + (v_{y} + ar)\sin \delta , \\ v_{t\_fr} & = \left( {v_{x} + \frac{{t_{w1} }}{2}r} \right)\cos \delta + (v_{y} + ar)\sin \delta , \\ v_{t\_rl} & = v_{x} - \frac{{t_{w2} }}{2}r{\kern 1pt} ;\;v_{t\_rr} = v_{x} + \frac{{t_{w2} }}{2}r. \\ \end{aligned} \right.$$
(11)
Since the angular velocity and the wheel center speed of the four wheels have been calculated, the tire longitudinal slip coefficient can then be given by:
$$\left\{ \begin{aligned} s_{fl} & = \frac{{\omega_{fl} R - v_{t\_fl} }}{{v_{t\_fl} }},\;s_{fr} = \frac{{\omega_{fr} R - v_{t\_fr} }}{{v_{t\_fr} }}, \\ s_{rl} & = \frac{{\omega_{rl} R - v_{t\_rl} }}{{v_{t\_rl} }},\;s_{rr} = \frac{{\omega_{rr} R - v_{t\_rr} }}{{v_{t\_rr} }}. \\ \end{aligned} \right.$$
(12)
According to Eqs. (9)‒(12), it can be seen that the tire vertical load, the tire sideslip angle and the tire longitudinal slip coefficient can be calculated by the vehicle model in real time, thus on this basis, the tire longitudinal and lateral forces can be obtained through the established STI tire model, which lays an important foundation for the following YSC system design