# A Stair-Climbing Wheelchair with Novel Spoke Wheels for Smooth Motion
## 3 Theoretical Analysis
### Theoretical Validation
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Number of Spokes** & **The Value of \(k_{\text{BEST}}\)** & **The Range of the Body Height** & **Maximum** \\ \(n\) & & **Climbing Height** \\ \hline
4 & 56.88 \(\text{N}/\text{m}\) & 2.3 mm & 120.27 mm \\
6 & 61.52 \(\text{N}/\text{m}\) & 0.5 mm & 110.85 mm \\
8 & 62.82 \(\text{N}/\text{m}\) & 0.1 mm & 103.00 mm \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The performance of Flexwheels with different spoke numbers. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Description** & **Symbol** & **Value** \\ \hline Length & \(l_{\text{L}}\) & 380 mm \\ Width & \(l_{\text{W}}\) & 200 mm \\ Number of spokes & \(n\) & 6 \\ Original spoke length & \(l\) & 150 mm \\ Spring free length & \(l_{\text{S}}\) & 100 mm \\ Rigid segment length & \(l_{\text{R}}\) & 50 mm \\ Sliding groove length & \(l_{\text{C}}\) & 10 mm \\ Body mass & \(4m\) & 2 kg \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Model parameters of the physical prototype. by Spoke B, leading to a subsequent increase in height. By substituting \(l_{\rm A}=h_{\rm B}/\sin\theta\) into Equation (13), the body height during Phases (a) and (b) can be expressed as:  
\[ \begin{cases}l_{\rm A}=max\Big{(}l_{\rm AMIN},\ l-\frac{mg(\sin\theta-\mu_{0} \text{cos}\theta)}{k}\Big{)}\\ h_{\rm B}=l_{\rm A}\cdot\sin\theta\end{cases}  \tag{15}\]  
At the critical angle \(\theta_{\rm C1}\), the following condition is satisfied:  
\[l-\frac{mg(\sin\theta_{\rm C1}-\mu_{0}\text{cos}\theta_{\rm C1})}{k}=l_{\rm AMIN} \tag{16}\]  
For \(\theta_{\rm C1}<\theta<\theta_{\rm C2}\), the sliding groove restricts Spoke A's minimum length to \(l_{\rm AMIN}\), resulting in an initial rise in body height followed by a decline. During this descent, Spoke B contacts the ground prematurely. Based on the sine law, the critical angle \(\theta_{\rm C2}\) satisfies:  
\[\frac{l_{\rm AMIN}}{\sin(\theta_{\rm C2}-\pi/3)}=\frac{l-l_{\rm I}}{\sin(\pi- \theta_{\rm C2})} \tag{17}\]  
For \(\theta<\theta_{\rm C2}\), the body height is primarily determined by the length of Spoke A (\(l_{\rm A}\)). Conversely, for \(\theta>\theta_{\rm C2}\), the height is governed by the length of Spoke B (\(l_{\rm B}\)). When Spoke B contacts the ground, its spoke-ground angle is \(\theta-\pi/3\). This smaller angle leads to reduced spring force and deformation. Although theoretical predictions suggest that \(l_{\rm B}\) should elongate, the pre-compression displacement \(l_{\rm I}\) prevents excessive extension. Thus, the body height in Phase (c) is reformulated as:  
\[ \begin{cases}l_{\rm B}=min\Big{(}l-\frac{mg(\sin(\theta-\pi/3)-l_{0}\text{cos} (\theta-\pi/3))}{k},l-l_{\rm I}\Big{)}\\ h_{\rm B}=l_{\rm B}\cdot sin(\theta-\pi/3)\end{cases}  \tag{18}\]  
As illustrated in the figure, when Spoke A disengages from the ground, the body height seamlessly transitions its initial contact value, ensuring smooth motion without sudden discontinuities, in alignment with mechanical principles. During Phase (b), when Spoke A is perpendicular to the ground (\(\theta=90^{\circ}\)), the body height reaches its peak value:  
\[h_{\rm AMAX}=l_{\rm AMIN}\cdot\sin(90^{\circ})=l-l_{\rm I}-l_{\rm C} \tag{19}\]  
To minimize body height fluctuation, the peak height \(h_{\rm AMAX}\) must match the initial body height \(h_{\rm I}\):  
\[h_{\rm AMAX}=h_{\rm I} \tag{20}\]  
where \(h_{\rm I}\) represents the body height at the moment Spoke A first makes contact with the ground. From Equation (14), it can be expressed as:  
\[h_{\rm I}=(l-l_{\rm I})\cdot\sin\Big{(}\frac{\pi}{2}-\frac{\pi}{n}\Big{)} \tag{21}\]  
By substituting parameters from Table 2 into the above equations, the pre-compression displacement is calculated as \(l_{\rm I}\approx 75\)mm. Consequently, after installation, the effective spoke length without ground contact is \(l-l_{\rm I}=75\)mm.