# Design and Lateral Stability Analysis of an Attitude Adjustment Tractor for Moving on Side Slopes
## 5 Case Study and Experiment Variation  
### Case Study  
One purpose of this article is to investigate how the attitude adjustment technology prevents lateral rollover. A general model is presented in Section 4, representing the relationship between a tractor's posture change/configuration and stability evaluation index on transverse slopes. In this section, taking the described tractor in Section 2 as example, the effects of attitude/configuration on tractor lateral stability are studied. Calculations were performed using MATLAB software whose version number is 2021. * Attitude adjustment mechanism This study employs the simplest method: turning downward the downhill side wheels and turning upward the uphill side wheels by the same amount. There are three advantages of this method. First, it reduces the amount of change in the rotation angle on one side necessary for obtaining the desired attitude, and therefore it enables the tractor to actively adapt promptly to the target terrain and continue traversing. Second, continuous roll change can be obtained around the nominal configuration. Third, during the attitude adjustment process, the wheelbases for each side are the same. Thus, based on Equations (4) and (5), the relationship between configuration and the center of gravity (COG) of the adjustment mechanism is:  
\[\left\{ \begin{aligned} &\text{Downhill side}:H_{d}=f(\alpha)=H_{0}+L_{r}* \sin\lvert\alpha\rvert\\ &\text{Uphill side}:H_{u}=f(\alpha)=H_{0}-L_{r}*\sin\lvert\alpha \rvert\end{aligned} \right., \tag{25}\]  
where \(\alpha\) is the rotation angle of the adjustment mechanism of the rear axle and \(\alpha\in[-\pi/2,\pi/2]\), \(H_{0}\) is the initial height of the COG, and \(L_{r}\) is the rotation radius; \(\alpha\) can be actively adjusted, whereas both \(H_{0}\) and \(L_{r}\) are constant values. * Lateral stability This section investigates the relationship between active input and lateral stability in straight-line mode. The input-output model of the adjustment mechanism is established in Section 3. A complete tractor model representing the relationship between a tractor's configuration/attitude and its lateral stability is introduced in Section 4. By combining these two models, the relationship between the active inputs of the tractor and its lateral stability can be estimated. Numerical simulation is carried out in this section with the structure parameter parameters listed in Table 1.  
The adjustment attitude angle of a tractor, \(\psi_{l}\), is obtained by substituting Equation (25) into Equation (15):  
\[\psi_{l}=\arctan\biggl{(}\frac{H_{d}-H_{u}}{B}\biggr{)}=\arctan\biggl{(}\frac {2L_{r}*\sin\lvert\alpha\rvert}{\text{B}}\biggr{)}, \tag{26}\]  
The lateral instability limit angle can be deduced as putting Equation (25) into Equation (24):  
1. Initial state \[\theta_{lim}=\arctan\bigl{(}L_{d}/H_{0}\bigr{)}.\] (27)
2. Attitude adjustment state \[\theta_{lim}=\arctan\bigl{(}L_{d}/H_{d}\bigr{)}+\arctan\bigl{(}\frac{H_{d}- H_{u}}{B}\bigr{)}=\arctan\Bigl{(}\frac{L_{d}}{H_{0}+L*\sin\lvert\alpha\rvert} \Bigr{)}+\] (28) \[\arctan\Bigl{(}\frac{2L_{r}*\sin\lvert\alpha\rvert}{B}\Bigr{)}.\]  
Thus, based on Equation (28), the lateral instability limit angle of a tractor, \(\theta_{lim}\), is a function of rotation angle of rear wheel, \(\alpha\), that is:  
\[\theta_{lim}=o(\alpha), \tag{29}\]  
where \(\alpha\) is the rotation angle of the adjustment mechanism in the rear axle. The effect of active input on the value of the attitude adjustment angle is shown in Figure 7a. The tractor can actively adjust its roll angle with attitude adjustment. The attitude adjustment angle of the tractor, \(\psi_{l}\), is obtained with Equation (26). As indicated in Figure 7a, the adjustment angle \(\psi_{l}\) increases with increasing active rotation angle of the rear axle. The maximum value of \(\psi_{l}\) is \(15.79^{\circ}\). In other words, the designed tractor can level its attitude when the slope angle is lower than \(15.79^{\circ}\).