# A Stair-Climbing Wheelchair with Novel Spoke Wheels for Smooth Motion
## 3 Theoretical Analysis
### Theoretical Validation  
The number of spokes n plays a crucial role in influencing the wheelchair's overall stability and stair-climbing capability. To investigate this relationship, theoretical calculations were conducted to assess the range of the body height during Flexwheel rotation, as well as the maximum achievable stair-climbing height (the detailed calculation process is described in Section 3.3). An even number of spokes is adopted to ensure symmetrical force distribution during rotation. Accordingly, a comparative analysis was performed on Flexwheel configurations with 4, 6, and 8 spokes. A schematic diagram of the maximum stair-climbing height is presented in Figure 4, and the corresponding results are summarized in Table 1. While increasing the number of spokes could enhance the continuity of ground contact, it simultaneously reduces the stair-climbing performance. In contrast, fewer spokes would undermine overall stability. After balancing these factors, six spokes were selected as the optimal configuration, achieving a balance between stability, climbing capability, and lightweight design. Figure 4: Schematic diagram of the maximum stair-climbing height of Flexwheels with different numbers of spokes. In the simulation environment, the friction coefficient was set to \(\mu_{0}=0\). Based on the parameters in Table 2, the spring constant selection method outlined in Section 3.1.2 is employed. The results of each step are illustrated in Figure 5. According to Equation (6), the allowable range of the body height ratio \(j\) was calculated as \(0.4<j<0.866\). As shown in Figure 5a, when \(j=0.47\) (corresponding to \(h_{\text{B}}=70.5\text{mm}\)), the variance of the spring constant \(k\) reaches its minimum. Therefore, \(h_{\text{BBEST}}\) is determined to be 70.5 mm. In Step 2, the variation of \(k\) with respect to the spoke-ground angle \(\theta\) was analyzed. Figure 5b reveals that the spring constant \(k\) varies within \(61.41\leq k\leq 61.86\). Substituting these values into Equation (13), the variance of \(h_{\text{B}}\) across the full range of \(\theta\) was further calculated. The results indicate that the minimum variance occurs at \(k=61.52\text{N}/\text{m}\), determining the optimal spring constant \(k_{\text{BEST}}\) to be 61.52 \(\text{N}/\text{m}\). Figure 5d demonstrates the variation of \(h_{\text{B}}\) with \(\theta\), showing a range of \(70.17\text{mm}\leq h_{\text{B}}\leq 70.63\text{mm}\), with a total variation of 0.46 mm. These results confirm that the proposed method for selecting the spring constant \(k\) effectively stabilizes the body's CoM height under frictionless conditions. Finally, by substituting \(h_{\text{I}}=70.17\text{mm}\) into Equation (14), the required pre-compression displacement \(l_{\text{I}}\) was calculated as 69 mm. In real-world conditions, friction arises due to the relative motion between the upper and lower spokes. Given that the friction coefficient between 304 stainless steel is \(t\mu_{0}=0.5\), the selection process for the optimal spring constant \(k\) was repeated under these conditions. The results are illustrated in Figure 6. From the analysis of the spring constant range in Figure 6b, it is evident that the variance of \(h_{\text{B}}\) decreases monotonically as \(k\) increases. To further pinpoint the value of \(k\) that minimizes the variance of \(h_{\text{B}}\), an expanded parameter space was employed for exhaustive iteration. This optimization yielded an optimal spring constant \(k_{\text{BEST}}=252\text{N}/\text{m}\). Under these conditions, the body's CoM height varies within \(111.6\text{mm}\leq h_{\text{B}}\leq 131\text{mm}\), and the range is 19.4 mm. The presence of friction leads to a substantial increase in \(k_{\text{BEST}}\), while the oscillation amplitude and variation trend of the body closely resemble those of traditional spoke wheels.