# Steering Dynamic and Hybrid Steering Control of a Novel Micro-Autonomous Railway Inspection Car
## 2 Mathematical Model of the Lateral Dynamic System
### Differential Equation of Motion  
The micro-autonomous railway inspection vehicle in this study features a magneto-rheological (MR) suspension system, which plays a crucial role in stabilizing vertical forces. Due to the MR system's ability to effectively mitigate vertical oscillations, the influence of longitudinal forces can be considered negligible in the motion analysis. As a result, the vehicle's dynamic equations are formulated primarily in the lateral direction, focusing on lateral displacement and yaw angle.  
We conducted analyses of wheel-rail interaction, particularly accounting for nonlinearities from wheel and rail profiles and adhesion constraints affecting creep force and creepage relationships [19; 20; 21; 22]. The coordinate transformation is required to convert these creep forces from the contact coordinate system to the track coordinate system, enabling a better understanding of the car's dynamics on the track, which can be expressed as follows:  
\[\left\{\begin{aligned} & F_{xcl}^{\prime}=k_{l}F_{xcl}\\ & F_{ycl}^{\prime}=k_{l}F_{xcl}\psi+k_{l}F_{ycl}cos(\beta_{l}+ \theta_{w})\\ & M_{zcl}^{\prime}=k_{l}M_{zcl}cos(\beta_{l}+\theta_{w})\\ \end{aligned}\right. \tag{10}\]  
\[\left\{\begin{aligned} & F_{xcr}^{\prime}=k_{r}F_{xcr}\\ & F_{ycr}^{\prime}=k_{r}F_{xcr}\psi+k_{r}F_{ycr}cos(\beta_{r}- \theta_{w})\\ & M_{zcr}^{\prime}=k_{r}M_{zcr}cos(\beta_{r}-\theta_{w})\\ \end{aligned}\right. \tag{11}\]  
where \(k\) denotes the coefficient to convert the model into a Shen-Hedrick-Elkins' nonlinear creep force model [14; 19].  
\[k_{i}=\left\{\begin{aligned} &\frac{1}{e_{i}},&\quad if\ e_{i}\leq 3\\ &\frac{1}{e_{i}}\left[e_{i}-\frac{e_{i}^{2}}{3}+\frac{e_{i}^{3}}{2 ^{\prime}}\right],&\quad if\ e_{i}>3\\ \end{aligned}\right. \tag{12}\]  
The unlimited resultant creep force was calculated using [PERSON]'s linear theory, expressed as follows [19]:  
\[F_{ri}=\sqrt{F_{xcl}^{2}+F_{ycl}^{2}} \tag{13}\]  
\[e_{i}=\frac{F_{ri}}{\mu N_{i}} \tag{14}\]  
The total force and moment acting on a single wheeleste were calculated as follows:  
\[F_{yw}=F_{yml}+F_{ymr}+F_{ycl}^{\prime}+F_{ycr}^{\prime} \tag{15}\]  
\[M_{zw}=M_{zcr}^{\prime}+M_{zcl}^{\prime}+\frac{\left(F_{xcr}^{\prime}-F_{ xcl}^{\prime}\right)d}{2}+\left(F_{ycr}^{\prime}-F_{ycl}^{\prime}+F_{ymr}-F_{ yml}\right)d\psi \tag{16}\]  
where \(d\) denotes track width and \(w\) denotes the wheelbase, given that the wheels were directly connected to the bogie frame without any primary suspension. \(F_{ce}\) denotes the centrifugal force acting on the vehicle during cornering, which is determined by the bend radius of the track, for which \(F_{ce}=\frac{m\pi^{2}}{R}\). The total force and moment of the bogie with two wheelsets (\(f\) = front and \(b\) = back) were obtained using rigid body dynamics and calculated using the following:  
\[F_{ywf}+F_{ywb}-\ F_{ce}=m\ddot{y} \tag{17}\]  
\[M_{zwf}+M_{zwb}+\left(F_{ywf}-F_{ywb}\right)\frac{w}{2}=l\ddot{\psi} \tag{18}\]  
The lateral displacement on the front and rear wheelsets (\(y_{f},\ y_{b}\)) was calculated as follows:  
\[y_{f}=y-\frac{\mathit{wtan}(\psi)}{2} \tag{19}\]  
\[y_{b}=y+\frac{w\,tan(\psi)}{2} \tag{20}\]