# Lateral Stability of a Mobile Robot Utilizing an Active Adjustable Suspension
## 5 Case Studies
### Case Study for Lateral Stability
#### 5.2.2 Inputs of Uphill and Downhill Sides are Unequal  
According to the relationship between the ground clearance and the lateral stability in Section 5.2.1, the robot is most stable at its lowest ground clearance. When \(L_{ad}\
eq L_{en}\), the roll angle of the robot changes, the mode in this section is to extend the linear actuator on the downhill side. Specifically, the robot is at its lowest ground clearance in its initial state. Then, it adjusts its roll angle by expending the linear actuator on the downhill side. The roll angle of the slope was varied from \(0^{\circ}\) to \(20^{\circ}\) at \(5^{\circ}\) intervals. The adjustment angle (\(\psi_{h}\)) can be calculated using Equations (15)-(16). Figure 11 shows the relationship between the linear actuator length on the downhill side (\(L_{ad}\)) and adjustment angle (\(\psi_{h}\)). As indicated, the adjustment angle (\(\psi_{h}\)) increases with the increasing linear actuator length on the downhill side (\(L_{ad}\)). Its maximum is \(12.33^{\circ}\). In other words, the mobile robot can level its configuration when the slope angle is lower than \(12.33^{\circ}\). Figure 11: Cont. The roll angle of the robot can be obtained using Equations (13), (15) and (16). As indicated in Figure 10(b), when the slope angle is \(5^{\circ}\) or \(10^{\circ}\), the roll angle of the robot decreases to \(0^{\circ}\). Then, it increases in the inverse direction. When the slope angle is \(15^{\circ}\) or \(20^{\circ}\), the roll angle of the robot decreases with the increasing linear actuator length on the downhill side (\(L_{\text{ed}}\)). The relationship between the lateral stability criteria (\(\beta_{l}\)) and the linear actuator length on the downhill side (\(L_{\text{ed}}\)) is shown in Figure 10(c). When the slope angle is \(5^{\circ}\), \(\beta_{l}\) increases first, and then decreases. When the slope angle is \(10^{\circ}\), \(15^{\circ}\), or \(20^{\circ}\), \(\beta_{l}\) increases with \(L_{\text{ed}}\). Specifically, when the slope angle is \(5^{\circ}\), as the linear actuator length on the downhill side (\(L_{\text{ed}}\)) increases, the adjustment angle (\(\psi_{h}\)) increases, and the roll angle of the robot (\(\theta^{\prime}\)) first decreases, and then increases in the inverse direction. Referencing Equation (18), the lateral tip-over gradient on the downhill side (\(\lambda_{1}\)) increases, and the lateral tip-over gradient on the uphill side (\(\lambda_{2}\)) decreases; when \(\lambda_{1}=\lambda_{2}\), \(\beta_{l}\) obtains its maximum value (\(41^{\circ}\)), it then decreases. When the slope angle is \(10^{\circ}\), \(15^{\circ}\), or \(20^{\circ}\), as the linear actuator length on the downhill side (\(L_{\text{ed}}\)) increases, the adjustment angle (\(\psi_{h}\)) increases. Referencing Equation (18), the lateral tip-over gradient on the download side (\(\lambda_{1}\)) increases constantly, and the lateral tip-over gradient on upload side (\(\lambda_{2}\)) decreases constantly when \(L_{\text{ed}}\) is at its maximum (220 mm) and \(\beta_{l}\) reaches its maximum. By expending the linear actuator on the downhill side when the slope angle is \(5^{\circ}\), \(\beta_{l}\) increases from \(39.78^{\circ}\) to \(41^{\circ}\) (\(3.07\%\)). When the slope angle is \(10^{\circ}\), \(\beta_{l}\) increases from \(34.78^{\circ}\) to \(37^{\circ}\) (\(6.38\%\)). When the slope angle is \(15^{\circ}\), \(\beta_{l}\) increases from \(29.78^{\circ}\) to \(32^{\circ}\) (\(7.45\%\)). When the slope angle is \(20^{\circ}\), \(\beta_{l}\) increases from \(24.78^{\circ}\) to \(27^{\circ}\) (\(8.96\%\)). Overall, the lateral stability criteria \(\beta_{l}\) can be increased by expending the linear actuator on the downhill side (\(L_{\text{ed}}\)) when the initial state of the robot is in its lowest ground clearance. Then, the lateral stability of the robot increases. Then, the lowest ground clearance is the most stable with respect to the other ground clearances, while referencing Section 5.2.1.