# Design of Unmanned Road Widths in Open-Pit Mines Based on Offset Reaction Times
## 2 Methodology
### Optimization of Road Width  
The sensor detects information when the vehicle is deviated and then transmits it to the processing center. The center subsequently sends instructions to the controller, which takes about 0.4 s to control the truck's response. In comparison, a human typically takes about 1.2 s after observing an obstacle and making a judgement [20]. As presented in Figure 2, the width of the unmanned road that is optimal can be calculated using Equation (2) based on the maximum distance of the mining dump truck's lateral offset during the sensor reaction phase (\(x_{max}\)) and the maximum distance of the truck's lateral offset during the driver reaction phase (\(x^{\prime}{}_{max}\)).  
\[x_{op}=x^{\prime}{}_{max}-x_{max} \tag{2}\]  
To obtain precise transverse offset distance measurements for mining dump trucks, it is imperative to create a compliant transverse dynamics model for these vehicles and investigate the transverse forces acting on them while steering. Due to the intricate nature of these forces, a single truck model was designed to characterize the transverse dynamics of mining dump trucks in order to simplify the algorithm. As a result, the transverse and longitudinal aerodynamic effects were overlooked.  
From the lateral dynamics model of the vehicle shown in Figure 3, it is evident that the equations of motion for the vehicle's lateral movement can be derived.  
\[F_{xf}\cos\theta+F_{xb}=m(\dot{v}_{x}+v_{y}r) \tag{3}\]  
where \(F_{xf}\) and \(F_{xb}\) represent the lateral deflection forces of the front and rear wheels, respectively; \(\theta\) denotes the angle of rotation of the front wheel; \(m\) signifies the mass of the vehicle; \(v_{x}\) indicates the lateral speed of the vehicle; \(v_{y}\) represents the longitudinal speed; and \(r\) is the transverse angular velocity.  
Figure 2: Lateral offset back to positive curve.  
The theory of the brush tire model suggests that there exists a linear relationship between the tire's side deflection force and the side deflection angle when the latter is less than \(5^{\circ}\).  
\[F_{xf},F_{xb}=\left\{\begin{array}{l}-C_{\alpha}\tan\alpha+\frac{C_{\alpha}^{2 }}{3\mu F_{z}}[\tan\alpha|\tan\alpha-\frac{C_{\alpha}^{3}}{2f\mu F_{z}}\tan^{3 }\alpha,|\alpha|<\arctan(\frac{3\mu F_{z}}{C_{\alpha}})\\ -\mu F_{z}\mathrm{sgn}\alpha,|\alpha|\geq\arctan(\frac{3\mu F_{z}}{C_{\alpha }})\\ \end{array}\right. \tag{4}\]  
where the vehicle tire's lateral deflection stiffness (\(C_{\alpha}\)), the wheel's lateral deflection angle (\(\alpha\)), the road surface adhesion coefficient (\(\mu\)), and the tire's vertical force (\(F_{z}\)) are all important factors to consider.  
When the lateral deflection angle and slip rate of the tire are low, the resulting lateral force on the tire is proportional to the lateral deflection stiffness. Therefore, it is possible to express \(F_{xf}\) and \(F_{xb}\) in Equation (4) using Equation (5):  
\[F_{xf},F_{xb}=C_{\alpha}\times\alpha \tag{5}\]  
If the distance from the front wheel of the truck to the center of gravity is known, as well as the vertical force \(F_{t}\) of the truck's tires, the truck's moment of inertia \(M\) can be calculated as follows:  
\[M_{f}=l\times F \tag{6}\]  
The relationship between the rotational moment \(M\) and the angular acceleration \(\beta\) can be described as follows:  
\[M=J\times\beta \tag{7}\]  
where the truck's moment of inertia is \(J=5757\) kg/m\({}^{3}\).  
The angular acceleration \(\beta\) is determined by the following equation:  
\[\beta=\frac{d\omega}{dt} \tag{8}\]  
\[x=\sum_{i=1}^{n-1}2v\cdot dt\cdot\tan(i\cdot\omega dt\cdot\frac{\pi}{180})+v \cdot dt\cdot\tan(n\cdot\omega dt\cdot\frac{\pi}{180})\ \ \ n=t/dt \tag{9}\]  
Based on the instantaneous turning angle \(\omega\), the sensor response time \(t_{\mathrm{s}}\), and the driver response time \(t_{\mathrm{d}}\), we can calculate the maximum distance \(x\) of the truck's lateral deflection.