# Design, Analysis, and Experiment of a Wheel-Legged Mobile Robot
## 2 Structure Design
### Statics Modeling  
First, the force condition of a single wheel-leg during step climbing is analyzed. As shown in Figure 1, to meet practical needs, it is assumed that the width and height of a single step in a staircase are W = 25 mm and H = 15 mm, respectively. \(F\) and \(P\) are the component force of the robot body on the wheel-leg in horizontal and vertical directions; \(N\) and \(f\) are the support force and friction force of the steps on the wheel-leg; \(M_{N}\) and \(M_{f}\) are the torque generated by \(N\) and \(f\) at the geometric center of the wheel-leg; \(T_{m}\) is the driving torque generated by the motor on the wheel-leg; \(\delta\) is the angle between the tangent of the wheel-leg at its contact point with the step and the horizontal line. It is assumed that \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Category** & **Previous Works** & \multicolumn{2}{c}{**Features**} \\ \hline Track-legged & \begin{tabular}{c} TALBOT Robot \\ ([PERSON] et al., 2022) \\ \end{tabular} & \begin{tabular}{c} Advantages: \\ Great obstacle avoidance; \\ Reliable control. \\ \end{tabular} & \begin{tabular}{c} Advantages: \\ Great obstacle avoidance; \\ Reliable control. \\ \end{tabular} \\ \cline{2-4}  & \begin{tabular}{c} Black Walker \\ ([PERSON] et al., 2017) \\ \end{tabular} & \
\begin{tabular}{c} Weakness: \\ Complex structure. \\ \end{tabular} \\ \hline \hline \end{tabular}
\begin{tabular}{c} Wheel-track-legged \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Cont. the motor speed is slow and that the driving torque provided by the motor is sufficient. Then, Formula (1) can be obtained based on the force balance of the wheel-leg:  
\[\left\{ \begin{array}{l}P+\mu N\text{cos}\,\delta-N\text{sin}\,\delta=0\\ F-\mu N\text{sin}\,\delta-N\text{cos}\,\delta=0\\ T_{m}-M_{\mathcal{O}_{1}(N)}-M_{\mathcal{O}_{2}(f)}=0\end{array} \right., \tag{1}\]  
It is assumed that the maximum static friction force is approximately equal to the sliding friction force and that the dynamic friction coefficient between the wheel-leg and the step is \(\mu\). Then, when the wheel-leg is about to slip during step climbing, we have \(f=\mu N\). Thus, Formula (2) is obtained:  
\[\frac{F}{P}=\frac{\mu\text{tan}\,\delta{+}1}{\text{tan}\,\delta-\mu}, \tag{2}\]  
Next, the statics analysis of the entire robot during climbing steps is conducted. As shown in Figure 2,\(f_{1}\), \(N_{1}\),\(f_{4}\), and \(N_{4}\) are the friction and pressure of the robot when its front and rear wheel-legs come into contact with a step; \(P_{1}\), \(F_{1}\), \(P_{4}\), and \(F_{4}\) are the component force of the robot body on the wheel-leg in the horizontal and vertical directions; \(P_{1}\), \(F_{1}\), \(P_{4}\), and \(F_{4}\) are the reaction forces of \(P_{1}\), \(F_{1}\), \(P_{4}\), and \(F_{4}\), respectively; \(T_{\text{m1}}\) and \(T_{\text{m4}}\) are the driving torque of the front and rear drive motors on the wheel-leg, respectively; \(T_{\text{m1}}\) and \(T_{\text{m4}}\) are the reaction torques of \(T_{\text{m1}}\) and \(T_{\text{m4}}\), respectively; \(G\) is the gravity of the robot; \(M_{\mathcal{O}1(f_{1})}\) and \(M_{\mathcal{O}1(N\text{I})}\) are the torque of \(f_{1}\) and \(N_{1}\) at the geometric center \(O_{1}\) of the front wheel-leg, respectively; \(M_{\mathcal{O}4(\mu)}\) and \(M_{\mathcal{O}4(N\text{4})}\) are the torque of \(f_{4}\) and \(N_{4}\) at the geometric center \(O_{4}\) of the rear wheel-leg, respectively; \(l_{1}\) and \(l_{2}\) are the distance from point \(O\) to \(O_{1}\) and \(O_{2}\), respectively; and \(\gamma\) is the angle between the robot body and the horizontal line. Ignoring the gravity of the wheel-leg, it is assumed that the force condition of the front and rear wheel-legs is the same. Then, Formula (3) can be obtained from the force balance of the robot:  
\[\left\{