# Design and Lateral Stability Analysis of an Attitude Adjustment Tractor for Moving on Side Slopes
## 3 Adjustment Mechanism  
One aim of this study is to propose a model that represents the relationship between the posture change of a tractor and its lateral stability. In addition, a tractor can actively adjust its posture using its adjustment mechanisms, which has been shown in Section 2. Consequently, two models are proposed: A complete adjustment mechanism and a tractor model. The former introduces the relationship between configuration and center of gravity (COG) of the adjustment mechanism. The latter expresses lateral stability parameters as a function of the COG on both sides. By combining these two models, the amount of stability is obtained for an arbitrary tractor model.  
The aim of this section is to describe the input-output relationship of the adjustment mechanism of the designed tractor in Section 2. Based on this model, the parameters of this mechanism are also designed.  
1. Mathematical model  
The mathematical model of the attitude adjustment mechanism is shown in Figure 2d-f, where the rear wheel can rotate around the rotation center to change the height of COG (center of gravity).  
As shown in Figure 2, the relationship between the height of COG (\(H\)) and rotation angle (\(\alpha\)) is:  
\[H(\alpha)=H_{0}+L_{r}*sin(\alpha), \tag{2}\]  
where \(H_{0}\) denotes the initial height of the COG, \(L_{r}\) is the rotation radius, \(\alpha\) is the rotation angle, and \(\alpha\in\left[-\frac{\pi}{2},\frac{\pi}{2}\right]\).  
Thus, the height of COG (\(H\)) is a function with the rotation angle (\(\alpha\)):  
\[H=f(\alpha), \tag{3}\]  
1. In the initial state, the height of COG is \(H_{0}\).
2. When a wheel turns upward, the height of the COG becomes lower: \[H(\alpha)=H_{0}+L_{r}*sin(\alpha),\] (4)
3. When a wheel turns downward, the height of the COG becomes higher: \[H(\alpha)=H_{0}-L_{r}*sin|\alpha|,\] (5)  
Thus, the limited positions of the height of the COG that can be achieved when a wheel turn upward/downward are:  
\[\begin{cases}H(\alpha)_{\text{min}}=H_{0}-L_{r}\\ H(\alpha)_{\text{max}}=H_{0}+L_{r}\end{cases}, \tag{6}\]
2. Mechanism design  
Unlike the ordinary tractor, the roll angle of the proposed tractor is changeable on lateral slopes. Hillside tractors are expected to level themselves on side slope angles in the range of 0-15\({}^{\circ}\). The leveled attitude of a tractor is shown in Figure 5.  
Therefore, the lateral slope angle for a reconfigurable tractor to level its attitude is:  
\[\theta_{\text{max}}=arctan\bigg{(}\frac{H_{\text{max}}-H_{\text{min}}}{B}\bigg{)}, \tag{7}\]  
where \(B\) is the track width, and \(H_{max}\) and \(H_{min}\) are the maximum and minimum values of the height of COG, respectively.  
Substituting Equation (6) into Equation (7) gives:  
\[\theta_{\text{max}}=arctan(2L_{r}/B), \tag{8}\]  
where \(B\) is the track width, and \(L_{r}\) is the rotation radius.  
The aim for designing the attitude adjustment mechanism is to level the tractor's attitude on side slope angles in the range of 0-15\({}^{\circ}\), thus:  
\[\theta_{\text{max}}\geq 15^{\circ}. \tag{9}\]  
As shown in Figure 2d-f, design the length of rotation radius \(L_{r}\):  
\[L_{r}\geq(B*\tan 15^{\circ})/2. \tag{10}\]  
Giving the tread \(\text{B}=1280\text{mm}\), therefore:  
\[L_{r}\geq 171.48\text{mm}. \tag{11}\]  
The distance between the half shaft and the drive shaft is 181 mm. In other words, with the rotation radius \(L_{r}=181\) mm, the designed parameters can satisfy the design demand. With the designed parameters, the maximum side slope angle that this tractor can level itself on is 15.79\({}^{\circ}\).