# Simulation of the Rollover Angle of a Self-Propelled Radish Harvester for Different Load Conditions
## 2 Materials and Methods
### Mathematical Model of the Radish Harvester Stability  
The rollover of the self-propelled radish harvester was performed using a simulation program based on a mathematical model [29] involving two types of rollover angles in a stability evaluation. The first was the lateral overturning angle that occurs during a sharp turn or when the rear wheel is lifted due to driving along the contour line direction of the slope [30]. The second was the longitudinal overturning angle that occurs transverse to the slope direction line, but the field slope is the main cause of it [31]. The self-propelled radish harvester has an integrated frame and front axle. Thus, the front axle cannot swing separately, and suspension is not included. Although they may have a significant impact on the static and dynamic balance of the machine, they were not considered in this rollover test. The lateral overturning angle for the rear wheel was determined using Equations (1) and (2), which were obtained from Figure 3.  
\[R_{1Z}=0, \tag{1}\]  
\[a_{s1}=tan^{-1}\bigg{(}\frac{db}{2(lh-Ha)}\bigg{)}, \tag{2}\]  
where \(R_{1Z}\) is the reaction force of the ground (N); \(a_{s1}\) is the lateral overturning angle (\({}^{\circ}\)) for the rear wheel; \(d\) is the track (mm); \(l\) is the wheelbase (mm); \(h\) is the center of height; \(H\) is the distance (mm) from the ground to A; \(a\) is the distance (mm) from the center to rear-wheel contact point; and \(b\) is the distance (mm) from the center to front wheel contact point. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Values** \\ \hline Weight (kg) & & 1350 \\ Center of gravity & & 186.5, 710, \(-\)300 \\  & \(I_{xx}\) (kg\(\cdot\)m\({}^{2}\)) & 3040 \\ Moment of inertia & \(I_{yy}\) (kg\(\cdot\)m\({}^{2}\)) & 3902 \\  & \(I_{zz}\) (kg\(\cdot\)m\({}^{2}\)) & 1318 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The simulation parameters of the 3D model of the radish harvester used in this study. Figure 2: The 3D model of the radish harvester used in this study. The preliminary center of the radish harvester was G (\(a,\ d/2,h\)) at the line of AB. To obtain the overturning angle for the front wheel, the position of the CoG was newly assumed to be G\({}^{\prime}(x^{\prime}_{0},\ y^{\prime}_{0},z^{\prime}_{0})\) at the line BD. The front-wheel overturning angle (\(a_{sf}\)) was determined using Equation (3) [29]. \[a_{sf}=tan^{-1}\left(\frac{y^{\prime}_{0}}{x^{\prime}_{0}}\right), \tag{3}\]  
where \(a_{sf}\) is the lateral overturning angle (\({}^{\circ}\)) for the front wheel; \(x^{\prime}_{0}\) is the \(x^{\prime}\) coordinate of the CoG G\({}^{\prime}\); and \(y^{\prime}_{0}\) is the \(y^{\prime}\) coordinate of G\({}^{\prime}\). The XYZ coordinate was designed from the preliminary lateral overturning angle axis AB, which is shown in Figure 4. The \(\delta\) and \(\beta\) were determined from Equations (4) and (5), respectively. \[\delta=tan^{-1}\left(\frac{d}{2l}\right), \tag{4}\]  
\[\beta=tan^{-1}\left(\frac{H}{\sqrt{\frac{4l^{2}+d^{2}}{4}}}\right), \tag{5}\]  
where \(\delta\) is the angle of change (\({}^{\circ}\)) from \(y\)-axis to Y-axis, and \(\beta\) is the angle of change (\({}^{\circ}\)) from \(z\)-axis to Z-axis. To determine the center of gravity, the relation between \(xyz\) and XYZ coordinates are found in Equations (6)-(8) [29]. \[X=(xcos\delta-ysin\delta)cos\beta-zsin\beta, \tag{6}\]  
\[Y=ycos\delta-xsin\delta \tag{7}\]  
\[Z=xcos\beta-(xcos\delta-ysin\delta)sin\beta \tag{8}\]  
The relation between the center position of \(xyz\) (G (\(a,\ d/2,h\))) and XYZ coordinates are found in Equations (9)-(11), where the CoG is G(\(x\),\(y\),\(z\)). \[X_{0}=(acos\delta+(d/2)sin\delta)cos\beta-zsin\beta, \tag{9}\]  
Figure 2:0  
Figure 2:1 The determination of the lateral overturning of the radish harvester. Figure 2:2  
For the rear-wheel lateral overturning angle, the CoG (G) at the line of AB was converted using \(\lambda\) to the CoG G\({}^{\prime}(x_{0}^{\prime},\ y_{0}^{\prime},z_{0}^{\prime})\) at XYZ coordinate using Equations (12)-(16).