# Steering Dynamic and Hybrid Steering Control of a Novel Micro-Autonomous Railway Inspection Car
## 2 Mathematical Model of the Lateral Dynamic System
### Nonlinear Creep Forces During Wheel and Rail Contact  
The equations presented in this section are derived from established fundamental theories in wheel-rail interaction modeling, specifically Hertzian contact theory and [PERSON]'s linear theory [19]. Hertzian theory defines the normal contact pressure distribution between the wheel and rail, while [PERSON]'s theory provides a linearized approximation of creep forces under small creepage conditions. These theories have been widely validated and serve as the theoretical foundation for analyzing creep forces, as presented in the following equations.  
Creep forces in the x/y direction \(\left(F_{xx},\ F_{yc}\right)\) and moment in the z-axis \(\left(M_{zc}\ \right)\) at the position of contact between the rail and wheels were determined using [PERSON]'s linear theory, which is expressed as follows [18]:  
\[\left\{\begin{aligned} & F_{xcl}=-f_{111}\xi_{xl}\\ & F_{ycl}=-f_{221}\xi_{yl}-f_{23l}\xi_{sl}\\ & M_{2 cl}=f_{23L}\xi_{yl}-f_{33l}\xi_{sl}\end{aligned}\right. \tag{1}\]  
\[\left\{\begin{aligned} & F_{xcr}=-f_{11r}\xi_{xr}\\ & F_{ycr}=-f_{22r}\xi_{yr}-f_{23r}\xi_{sr}\\ & M_{zcr}=f_{23r}\xi_{yr}-f_{33r}\xi_{sr}\end{aligned}\right. \tag{2}\]  
where \(f_{11}\), \(f_{22}\), \(f_{23}\), and \(f_{33}\) denote the creep coefficients reliant on the geometry at the contact point between each wheel and the rail. The Kalker coefficients \(C_{11}\), \(C_{22}\), \(C_{23}\), and \(C_{33}\) were calculated using lookup table interpolation based on the SIMPACK dynamics model.  
Figure 3: (**a**) Contact geometry between wheel and rail; (**b**) components of the forces at the wheel and rail contact position.  
\(\xi_{x}\), \(\xi_{y}\), and \(\xi_{s}\) denote the longitudinal, lateral, and spin creepages, respectively. Subscript i is used to represent the left (I) and right (r) wheels.  
\[\left\{\begin{array}{l}f_{11i}=G\big{(}\frac{A_{i}}{\pi}\big{)}C_{11i}\\ f_{22i}=G\big{(}\frac{A_{i}}{\pi}\big{)}C_{22i}\\ f_{23i}=G\big{(}\frac{A_{i}}{\pi}\big{)}^{3/2}C_{23i}\\ f_{33i}=G\big{(}\frac{A_{i}}{\pi}\big{)}^{2}C_{33i}\end{array}\right. \tag{3}\]  
where \(G\) is the rigidity modulus of the wheel and rail material. \(A\) is the contact area between the wheel and rail at the point of contact.  
Creep forces primarily depend on creepages (\(\xi\)), which represent vehicle speed and wheelset movements on the rail [14,19].  
\[\left\{\begin{array}{l}\xi_{xl}=1-\frac{r_{l}}{r_{0}}-\frac{d\dot{\psi}}{v} \\ \xi_{yrl}=\Big{(}\frac{\dot{\psi}_{w}}{v}+\frac{r_{l}}{v}\dot{\theta}_{w}-\psi \Big{)}cos(\beta_{l}+\theta_{w})\\ \dot{\zeta}_{sl}=-\frac{sin\beta_{l}}{r_{0}}+\frac{\dot{\psi}}{v}cos(\beta_{l }+\theta_{w})\\ \end{array}\right. \tag{4}\]  
\[\left\{\begin{array}{l}\xi_{xr}=1-\frac{r_{l}}{r_{0}}+\frac{d\dot{\psi}}{v} \\ \xi_{yr}=\Big{(}\frac{\dot{\psi}_{w}}{v}+\frac{r_{l}}{v}\dot{\theta}_{w}-\psi \Big{)}cos(\beta_{r}-\theta_{w})\\ \dot{\xi}_{sr}=\frac{sin\beta_{r}}{r_{0}}+\frac{\dot{\psi}}{v}cos(\beta_{r}- \theta_{w})\end{array}\right. \tag{5}\]  
where \(r_{l},r_{r}\) are the radius of the left and right wheel at the point of contact with the rail. \(r_{0}\) is the nominal radius of the wheel. \(\beta_{l},\beta_{r}\) are the contact angles between the left and right wheel, respectively, and the rail. \(v\) is the velocity of the vehicle. Following the Hertzian contact theory, \(\theta_{w}\) denotes the wheelset rolling angle, which was calculated from the difference between the rolling radius of the left and right wheels [19]. \(d\) is the track width.