# Design and Lateral Stability Analysis of an Attitude Adjustment Tractor for Moving on Side Slopes
## 4 Lateral Stability
### Mechanical Model of an Attitude Adjustment Tractor  
In this section, we propose a mechanical model of an attitude adjustment tractor on a side slope. As shown in Figure 6, the COG (center of gravity) of the tractor is located at distances \(L_{d}\) and \(L_{u}\) from the uphill and downhill wheels along the \(\mathrm{y_{r}}\) direction, respectively, and \(H_{d}\) and \(H_{u}\) from those wheels along the \(\mathrm{z_{r}}\) direction. Here, both \(H_{d}\) and \(H_{u}\) are adjusted with the input of the attitude adjustment mechanism in the rear axle, which can be obtained with Equation (2) (in Section 3), whereas both \(L_{d}\) and \(L_{u}\) are constant. Adjusting the active inputs of the rear axle on both sides together, \(H_{d}\) and \(H_{u}\) are changed. Then, the roll angle of the tractor is adjusted. Overall, the tractor can enhance its lateral stability by actively adjusting its attitude.  
The aim of this section is to establish a complete tractor model, expressing the relationship between input, configure/posture and output, and wheel-soil contact force.  
Figure 6: Tractor on a side slope: (**a**) Definition of coordinate systems; (**b**) Mechanical tractor model.  
As shown in Figure 6b, \(\theta_{l}\) denotes the roll angle of the tractor and \(\psi_{l}\) is the tilted angle of the tractor. This tractor can actively adjust its attitude at a side-slope with angle \(\theta\), thus, the roll angle of the tractor \(\theta_{l}\) is:  
\[\left\{\begin{aligned} &\text{Initial state}:\theta_{l}=\theta\\ &\text{Attitude adjustment state}:\theta_{l}=\theta-\psi_{l}\end{aligned}\right., \tag{12}\]  
As shown in Figure 6b, in the right triangle OP\({}_{1}\)P\({}_{2}\), the adjustable attitude angle of the tractor with respect to the side slope (\(\psi_{l}\)) is:  
\[\tan(\psi_{l})=\frac{H_{d}-H_{u}}{L_{d}+L_{u}}. \tag{13}\]  
As B is the track width of the tractor:  
\[L_{d}+L_{u}=\text{B}. \tag{14}\]  
Substituting Equation (14) into (13), the adjustable attitude angle of the tractor (\(\psi_{l}\)) is:  
\[\psi_{l}=\arctan\left(\frac{H_{d}-H_{u}}{B}\right). \tag{15}\]  
As shown in Figure 6b, the force analysis on the side slope for an attitude adjustment tractor begins by assessing global equilibrium. The moment static equations at the uphill and downhill contact points (P\({}_{1}\) and P\({}_{2}\)) are:  
\[\left\{\begin{aligned} & F_{u}*n-Gcos(\theta_{l})*L_{d}+Gsin( \theta_{l})*H_{d}=0\\ &-F_{d}*n+Gcos(\theta_{l})*L_{d}+Gsin(\theta_{l})*H_{u}=0\end{aligned}\right. \tag{16}\]  
where \(n=\sqrt{\left(\text{B}\right)^{2}+\left(H_{d}-H_{u}\right)^{2}}\).  
After rearranging the Equation (16), equate to:  
\[\left\{\begin{aligned} &\text{Uphill side}:F_{u}=\frac{Gcos(\theta_{l})*L_{d}- Gsin(\theta_{l})*H_{d}}{n}\\ &\text{Downhill side}:F_{d}=\frac{Gcos(\theta_{l})*L_{d}+Gsin( \theta_{l})*H_{u}}{n}\end{aligned}\right., \tag{17}\]  
where \(n=\sqrt{\left(\text{B}\right)^{2}+\left(H_{d}-H_{u}\right)^{2}}\), \(\theta_{l}=\theta-\psi_{l}\), \(L_{d}\) and \(L_{u}\) are constant. They represent the distance between the COG and downhill/uphill wheels along the \(\text{y}_{\text{r}}\) direction, respectively. \(H_{d}\) and \(H_{u}\) are the distances between COG and downhill/uphill wheels along the \(\text{z}_{\text{r}}\) direction, respectively, which both can be actively adjusted by the input of the attitude adjustment mechanism.