## 2 Materials and Methods
### Theoretical Analysis of the Miniature Model's Static Falling down Sidelong Angle  
The lateral overturning of the tractor is analyzed theoretically by literature review, and the relationship between the original and miniature models is derived based on the analysis result. The main assumptions made to simplify the theoretical analysis are as follows:  
* The subject of the analysis is a four-wheel-drive tractor; hence, the weight of the front axle and front wheels cannot be ignored. \begin{table}
\begin{tabular}{c c} \hline \hline
**Items** & **Specification** \\ \hline Model & RT135 \\ Manufacturer, Nation & DaeHo, Korea \\ Length/Width/Height (mm) & 4080/2345/2810 \\ Weight (N) & 46,300 \\ Engine & Rated power (kW)/Speed (rpm) & 95.6/2250 \\ Minimum ground clearance (mm) & 374 \\ Traveling speed & Minimum (km/h) & 1.86 \\ Maximum (km/h) & 48.81 \\ Minimum turning radius (mm) & 3.06 \\ Static falling down sidelong angle (\({}^{\circ}\)) & 45.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Specifications of the tractor used. Figure 1: Picture of the tractor used. Figure 2: Pictures of the performance tests: (**a**) Minimum turning radius test; (**b**) Static falling down sidelong angle test. The slope angle increases quasi-statically (ignoring the dynamic effect), and the tire makes point contact with the ground. The track widths of the front wheels and rear wheels are the same. Slip does not occur between the tire and the ground, and the tire stiffness is ignored. Figure 3a illustrates a simplified version of the tractor. In the static falling down sidelong angle experiment, the tractor is in a stationary state. Accordingly, the force exerted in the tractor's moving direction (z) is zero. The free-body diagram of the tractor is illustrated in Figure 3b. When the tractor is placed in the direction of the contour of the slope, the ground contact forces of the rear and front wheels located in the upper part of the slope are expressed by Equations (1) and (2), respectively [10]. \[F_{R1}=\frac{m_{R}g}{w}\bigg{[}\frac{L_{1}}{L}\Big{(}h_{1}sin\theta-\frac{w}{2 }cos\theta\Big{)}-\big{(}h_{1}+k_{Ry}\big{)}sin\theta+\Big{(}\frac{w}{2}+k_{Rx }\Big{)}cos\theta\bigg{]} \tag{1}\]  
Here,  
\(F_{R1}\) = Rear upslope tire contact force  
\(m_{R}\) = Mass of a component body of the tractor at rear (kg)  
\(g\) = Gravitational acceleration (m/s\({}^{2}\))  
\(w\) = Track width (m)  
\(L\) = Length of wheel base (m)  
\(L_{1}\) = Dimension from rear axle to rear center of gravity (m)  
\(h_{1}\) = Height of front-axle pivot (m)  
\(\theta\) = Slope angle  
\(k_{Rx}\) = Downslope displacement of rear body center of gravity with respect to midplane (m)  
\(k_{Ry}\) = Height of rear body center of gravity above height of pivot (m)  
\[F_{F1}=\frac{m_{R}g}{w}\bigg{(}\frac{L_{1}}{L}\bigg{)}\Big{(}\frac{w}{2}cos \theta-h_{1}sin\theta\Big{)}+\frac{m_{F}g}{w}\big{[}\Big{(}\frac{w}{2}+k_{Rx }\Big{)}cos\theta-\big{(}h_{1}+k_{Fy}\big{)}sin\theta \tag{2}\]  
Here,  
\(F_{F1}\) = Front upslope tire contact force  
\(m_{F}\) = Mass of a component body of the tractor at front (kg)  
\(k_{Rx}\) = Downslope displacement of front body center of gravity with respect to midplane (m)  
Figure 3: Tractor model for theoretical analysis: (**a**) Simplification of the general tractor model; (**b**) Free body diagram. \(k_{Fy}\) = Height of front body center of gravity above the height of pivot (m)  
When the inclination angle of the slope increases, either the front or rear wheel located in the upper part of the slope is lifted off the ground first, causing a Phase I rollover [8]. Afterward, the other wheel is lifted off the ground if the inclination angle continues to increase, and the tractor quickly overturns sideways. The moment the wheel begins to lift off the ground, the ground contact force of the wheel becomes zero. If \(\theta_{R1}\) is the inclination angle at which the rear wheel starts to lift off the ground, it is the inclination angle when \(F_{R1}\) is zero; hence, \(\theta_{R1}\) can be derived from Equation (1) as expressed below.