# Design, Analysis, and Experiment of a Wheel-Legged Mobile Robot
## 2 Structure Design
### Further Optimization  
Steps are common obstacles in daily life and represent a complex irregular terrain that is difficult for various mobile robots to cross. The height of steps is generally 10 mm\(\sim\)20 mm. Owing to its structural advantages, the proposed hybrid wheel-legged mobile robot can quickly climb steps using a relatively simple control strategy. The following statics modeling is used to analyze the force conditions and requirements that enable step climbing. The straight-line part and circular-arc part of the straight-curved wheel-leg have different contact situations with steps during climbing and should be classified and analyzed to obtain the range of geometric parameters that allow step climbing. According to the statics analysis of wheel-leg step climbing and based on the structural characteristics of the wheel-leg, the main circumstances of the robot's step crossing are divided into several types, as shown in Figure 8:  
Figure 6: The evolution of the wheel-leg structure. (**a**) Semicircular single-spoke wheel-leg; (**b**) Curved triple-spoke wheel-leg; (**c**) Straight-curved, triple-spoke wheel-leg. Figure 7: Wheel-leg structure. (**a**) Geometric structure. (**b**) Entity structure. When the straight-line part of the wheel-leg comes into contact with a step, Formula (10) is obtained according to the geometric relationship between the wheel-leg and the step:  
\[ \begin{cases}l\sin\delta+R\cos\alpha=H\\ d=l\cos\alpha\end{cases} , \tag{10}\]  
Based on the different positions and the geometric parameters of the wheel-leg, different \(\delta\) can be obtained and compared with \(\delta_{\text{max}}\) to determine whether the steps can be crossed. When the junction between the straight-line and circular-arc part of the wheel-leg comes into contact with a step, since the position of the wheel-leg is determined, Formula (11) is obtained according to the positional relationship between the wheel-leg and the step:  
\[ \begin{cases}l\sin\delta+R\cos\alpha=H\\ \quad\eta+\delta+\alpha=\frac{\pi}{6}\\ l=0.1464R\\ \quad\eta=0.2889\end{cases} , \tag{11}\]  
The following expression is derived:  
\[\delta= \text{arcsin}\left(\frac{H}{1.0438R}\right)-1.1992, \tag{12}\]  
When the geometric parameters of the wheel-leg are determined and substituted into the above expression, the corresponding \(\delta\) is obtained. When \(\delta\leq\delta_{\text{max}}\), the wheel-leg can cross the step; when \(\delta\geq\delta_{\text{max}}\), the wheel-leg will slip. Figure 8: Wheel-legs crossing steps. (**a**) The spoke touches a step with its straight-line part. (**b**) The spoke touches a step with its junction between the straight-line and circular-arc part. (**c**) The spoke touches a step with its circular-arc part. (**d**) The spoke touches a step with its edge. When the circular-arc part of the wheel-leg comes into contact with a step, Formula (13) is obtained according to the position between the wheel-leg and the step:  
\[\left\{ \begin{array}{c}R\cos\alpha+2r\cos\left(\frac{\pi}{6}-\alpha-\beta+ \varphi\right)\sin\beta=H\\ R=\sqrt{3}r\\ \delta=\frac{\pi}{3}+\alpha+2\beta-\varphi\\ \varphi=\frac{\pi}{6}\end{array} \right., \tag{13}\]  
When the geometric parameters and initial position of the wheel-leg are determined and substituted into the above formula, the corresponding \(\delta\) is obtained. When \(\delta\leq\delta_{\max}\), the wheel-leg can cross the step; when \(\delta>\delta_{\max}\), the wheel-leg will slip at the step. When the wheel-leg climbs steps, the extreme condition of the contact between the straight part and a step is shown in Figure 9. The geometric parameters of the wheel-leg under such conditions are analyzed. According to the geometric relationship between the wheel-leg and step, Formula (14) is obtained:  
\[\left\{ \begin{array}{c}R\cos\alpha+l\sin\delta=H\\ R\sin\alpha=l\cos\delta\\ \alpha=\eta+\delta-\frac{\pi}{6}\end{array}