# Design, Analysis, and Experiment of a Wheel-Legged Mobile Robot
## 2 Structure Design
### Geometry Definition and Design  
The wheel-leg structure is a key component of a wheel-legged mobile robot and directly determines the stability, mobility, and obstacle-crossing capability. A hybrid wheel-leg can achieve smooth movement with strong obstacle-crossing capability and simple control. Next, the structural design of the wheel-leg is proposed. First, the number of wheel-leg spokes is determined. The wheel-leg structure of existing wheel-legged mobile robots can be roughly classified into four types according to the number of spokes: single-spoke, triple-spoke, four-spoke, and five-spoke. The single-spoke wheel-leg structure has poor stability and complex gait planning [30; 41]. Therefore, the hybrid wheel-leg structure with three spokes or more is analyzed herein. The obstacle-crossing capability of the wheel-leg structure varies in forward and reverse rotation. During reverse rotation of the wheel-leg structure, for ease of analysis, it is assumed that the radius of curvature of spoke \(\mathbf{r}=\infty\), namely, it is assumed that spoke is linear. As shown in Figure 3, according to the geometric relationship, the expression for the crossing height \(H_{s}\) is obtained as:  
\[H_{s}=R\left(\sin\frac{\varphi_{s}}{2}+\cos\theta_{s}\right), \tag{9}\]  
where \(s\) represents the number of spokes; \(R\) represents the cord length; \(\varphi_{s}\) represents the angle between spokes; and \(\theta_{s}\) represents the angle between the wheel-leg spoke that contacts the ground and the vertical plane. The following results are derived: \(H_{3}=1.732R\), \(H_{4}=1.414R\), \(H_{5}=1.539R\), \(H_{6}=1.5R\), \(H_{7}=1.409R\), and \(H_{8}=1.307R\). Figure 3: Obstacle crossing by a reverse rotating wheel-leg. When the wheel-leg contacts obstacles while rotating forwardly, a similar obstacle-crossing result is desired for the analysis of the wheel-leg's obstacle-crossing capability. It is assumed that the force of the robot body exerted on each wheel-leg is the same. Under the condition of the robot's force balance, the relationship between the spoke's radius of curvature \(r\) and chord length \(R\) is set as \(r=0.67R\), and the coefficient of friction is 0.5. The constraint \(\delta_{\text{1 max}}=53.15^{\circ}\)[42] is obtained. That is, under the corresponding conditions, as long as the angle between the tangent of the wheel-leg at the contact point with the step and the horizontal line is smaller than \(53.15^{\circ}\), the wheel-leg structure is capable of crossing the obstacle, as shown in Figure 4. For a wheel-leg structure with \(s\) spokes, the forward crossing heights \(h_{s}\) are derived as \(h_{3}=1.44R\), \(h_{4}=1.12R\), \(h_{5}=1.17R\), \(h_{6}=1.15R\), \(h_{7}=1.10R\), and \(h_{8}=1.02R\). Based on the above analysis, the forward and reverse obstacle-crossing capability of wheel-legs with different numbers of spokes can be obtained, as shown in Figure 5. It can be seen that the obstacle-crossing capability of a wheel-leg declines with an increase in the number of spokes \(s\). Therefore, to achieve the strongest obstacle-crossing capability, we use \(s=3\). After the number of spokes is determined, further optimization of the spoke shape is desired. The curved triple-spoke wheel-leg shown in Figure 6b evolved from the semicircular single-spoke wheel-leg shown in Figure 6a. Its spokes can touch the ground alternately, which improves the space utilization rate. However, during step climbing, there is a certain interference among spokes and the impact of curved structure on the stiffness of the wheel-leg. Since robots usually operate on flat ground or slopes, their structure is optimized while  
Figure 4: Obstacle crossing by a forward rotating wheel-leg. Figure 5: Wheel-leg obstacle-crossing capability. retaining the alternating movement of curved triple-spoke wheel-legs.