# Steering Dynamic and Hybrid Steering Control of a Novel Micro-Autonomous Railway Inspection Car
## 2 Mathematical Model of the Lateral Dynamic System
### Wheel and Rail Contact  
The Hertzian normal contact theory and [PERSON]'s linear theory were used to calculate the movement of the vehicle's center while in motion by analyzing wheel axle movement during wheel and rail contact [12; 13; 14]. A mathematical model of the vehicle was then derived using these equations. To assess lateral dynamics, the vehicle was assumed to proceed down the track at a constant speed of v = 30 km/h. The MR mount detection system was employed to swiftly reduce the vehicle's vertical vibrations, ensuring stability in the vertical direction [11]. Given the effectiveness of this system in mitigating vertical oscillations, we assumed that vertical vibrations were nonsignificant in our analysis. Consequently, this study focused exclusively on the lateral dynamics of our micro-autonomous railway inspection vehicle, incorporating its unique design and hybrid steering control system. By leveraging the MR suspension to stabilize vertical forces, we were able to concentrate on analyzing lateral forces, the wheelset's self-steering behavior, and the hybrid control system's performance in maintaining track alignment.  
The vehicle coordinate systems of this study are presented in Figure 2. Oxyz denotes the wheelset coordinate system at the center of the track during movement at a constant speed relative to the fixed reference frame of the track. O\({}_{\text{c}}\)x\({}_{\text{c}}\)y\({}_{\text{c}}\)z\({}_{\text{c}}\) denotes the coordinate system at the center of the car. O\({}_{\text{w}}\)x\({}_{\text{w}}\)y\({}_{\text{w}}\)z\({}_{\text{w}}\) denotes the coordinate system at the center of the wheelset axle. O\({}_{\text{w}}\)x\({}_{\text{l}}\)y\({}_{\text{l}}\)z\({}_{\text{l}}\) and O\({}_{\text{r}}\)x\({}_{\text{r}}\)y\({}_{\text{r}}\)z\({}_{\text{r}}\) denote the respective coordinate systems attached to the instantaneous contact points of the left and right rails and their wheels. \(\beta\) denotes the contact angle between the wheels and rail. \(\theta\) denotes the roll angle of the wheelset axle rotation around the x-axis during motion. \(\psi\) denotes the yaw angle of vehicle rotation around the z-axis during motion. Wheel/rail normal and tangential contacts, which control the vehicle dynamics, change as a result of the geometry at the point of contact [15; 16].  
The main forces ensuring a train remains in place on a track when stopped or traveling at a steady pace include creep force and normal force, as shown in Figure 3[17; 18]. The term normal force (\(F_{n}\)) refers to the force exerted by the rail upward onto the train's wheel and is perpendicular to the contact surface. Creep force (\(F_{c}\)) at the wheel-rail contact refers to the small amount of relative motion or slippage that can occur between a train's wheel and the rail's contact surfaces. The term lateral creep force (\(F_{yc}\)) refers to the lateral movement of the wheel on the rail, which is particularly common when the train navigates  
Figure 2: **(a)** Wheelset coordinate systems; **(b)** diagram of contact geometry between wheel and rail.  
curves, due to the differences in radii between the curvature of the track and the wheel profile. Lateral creep force enables the wheels to align with the curve while maintaining proper contact.  
The interaction between the wheel and rail involves both normal forces and creep forces, each playing a crucial role in the vehicle's lateral dynamics. Due to the complexity of their behavior, these forces are analyzed separately before being incorporated into the overall motion equations.