# Lateral Stability of a Mobile Robot Utilizing an Active Adjustable Suspension
## 3 Active Suspension's Model
### Mechanical Model of a Reconfigurable Mobile Robot  
In this section, we propose a mechanical model of a reconfigurable mobile robot on a slope. As shown in Figure 8, we assume that the COG of the mobile robot is located at distances \(L_{d}\) and \(L_{u}\) from the downhill and uphill wheels along the \(y^{(r)}\) axis, respectively, and at distances \(H_{d}\) and \(H_{u}\) from those wheels along the \(z^{(r)}\) axis. Here, both \(L_{d}\) and \(L_{u}\) are constant, whereas \(H_{d}\) and \(H_{u}\) vary with the input of the active suspension, which is obtained with Equation (10). We can actively adjust the inputs of the two active suspensions in both sides together. Then, both \(H_{d}\) and \(H_{u}\) are changed, and the roll angle of the robot adjusts. Overall, the mobile robot increases its lateral stability with the inputs of the two suspensions together.  
As shown in Figure 8, \(\theta^{\prime}\) denotes the roll angle of the mobile robot, and \(\psi_{h}\) is the adjusted angle with regard to the slope. The mobile robot actively adjusts its roll on the slope. The expression of \(\theta^{\prime}\) is:  
\[\left\{\begin{array}{c}\text{Initial state: }\theta^{\prime}=\theta\\ \text{Reconfigure state: }\theta^{\prime}=\theta-\psi_{h}\end{array}\right. \tag{13}\]  
Figure 6: Robot tip-over angles.  
Figure 7: Lateral stability margins for the robot.  
Therefore, the expressions of the lateral tip-over gradients \(\gamma_{i}(i=1,2)\) are:  
\[\left\{\begin{array}{l}Downhill\;side:\tan(\gamma_{1}+\theta^{\prime})=\
icefrac{{ \iota_{d}}}{{H_{d}}}\\ Uplill\;side:\tan(\gamma_{2}-\theta^{\prime})=\
icefrac{{\iota_{u}}}{{H_{u}}} \end{array}\right. \tag{14}\]  
As shown in Figure 8, in the right-angled triangle, OP1P2, the adjustable angle of the mobile robot with regard to slope \(\psi_{h}\), is:  
\[\tan(\psi_{h})=\frac{H_{d}-H_{u}}{L_{d}+L_{u}} \tag{15}\]  
where \(H_{d}\) varies with the input of the downhill-side suspension, and \(H_{u}\) varies with the input of the uphill-side suspension.  
As referenced in Section 3, we denote \(L_{eu}\) and \(L_{eu}\) as the inputs of the active suspensions in the uphill and downhill sides, respectively. Putting \(L_{eu}\) and \(L_{eu}\) into Equation (10), we get:  
\[\left\{\begin{array}{l}Downhill\;side:H_{d}=f(L_{eu})\\ Uplill\;side:H_{u}=f(L_{eu})\end{array}\right. \tag{16}\]  
By substituting Equations (13), (15) and (16) into Equation (14), solutions for lateral tip-over gradients are obtained employing the inputs, \(L_{eu}\) and \(L_{eu}\), which are the lengths of downhill-side and uphill-side linear actuators, as variables.  
* Initial state: \[\left\{\begin{array}{l}Downhill\;side:\lambda_{1}=\arctan(\
icefrac{{L_{d} }}{{H_{d}}})-\theta\\ Uplill\;side:\lambda_{2}=\arctan(\
icefrac{{L_{u}}}{{H_{u}}})+\theta\end{array}\right.\] (17)
* Reconfigure state: \[\left\{\begin{array}{l}Downhill\;side:\lambda_{1}=\arctan(\
icefrac{{L_{d} }}{{f(L_{eu})}})-\left(\theta-\frac{f(L_{eu})-f(L_{eu})}{L_{u}+L_{d}}\right)\\ Uplill\;side:\lambda_{2}=\arctan(\
icefrac{{L_{u}}}{{f(L_{eu})}})+\left( \theta-\frac{f(L_{eu})-f(L_{eu})}{L_{u}+L_{d}}\right)\end{array}\right.\] (18) Therefore, the lateral stability criteria, \(\beta_{l}\), is: \[\beta_{l}=\min(\lambda_{1},\lambda_{2})=\
u(L_{eu},L_{ed})\] (19) where \(L_{ed}\) and \(L_{eu}\) are the linear actuator lengths on the downhill and uphill sides, respectively, and \(\theta\) is the slope angle.  
Therefore, the lateral stability criteria, \(\beta_{l}\), can be obtained by employing the inputs \(L_{eu}\) and \(L_{ed}\) as variables.  
Figure 8: Mechanical mobile robot on a side slope.