## 3 Design of Passive Torsional Compliant Suspension  
### Dynamic Stability Metrics for Curved-Spoke Wheel  
As introduced in Section 2.2 (Figure 3), the theoretical analysis suggests that the wheel's center of rotation (CoR) can drop from its maximum to minimum velocity instantaneously during the DCS interval, resulting in an infinite deceleration. However, in real-world conditions, inertia and friction cause the velocity to decrease over a finite duration, producing a finite deceleration (Figure 5). This deceleration imposes inertial forces on the robot's main body and payload, which in turn can undermine dynamic stability.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**C1** & \multicolumn{2}{c}{**C2**} \\ \hline \(\mathbf{\omega}\) **(rad/s)** & \(\mathbf{\Delta v}\) **(mm/s)** & \(\mathbf{H}\) **(mm)** & \(\mathbf{\Delta v}\) **(mm/s)** \\ \hline
0.10 & 12.11 & 120 & 15.74 \\
0.12 & 17.41 & 160 & 17.41 \\
0.14 & 19.96 & 200 & 19.11 \\ \hline \hline \end{tabular}
\end{table}
Table 2: The velocity decrease \(\Delta v\) during DCS obtained from C1 and C2.  
Figure 4: Simulation results of CoR velocity. (**a**) Velocity profiles at different drive angular velocities (\(\omega\) = 0.10, 0.12, and 0.16 rad/s) with a fixed stair dimension (\(S\times H\) = 300 mm \(\times\) 160 mm). (**b**) Velocity profiles for different stair heights (\(H\) = 120, 160, and 200 mm) under a fixed drive angular velocity (\(\omega\) = 0.12 rad/s). C1 refers to changing the wheel’s angular velocity while fixing the stair dimensions, and C2 refers to varying the stair height under a constant angular velocity.  
Because acceleration (or deceleration) naturally captures the rate at which velocity changes--and has long served as a standard performance metric for dynamic stability in robots and vehicles [26; 27; 28]--we adopt it here as our primary measure of stability during the DCS phase. This straightforward approach aligns well with how abrupt deceleration spikes can undermine control and increase the risk of slip or rollover, especially on uneven terrain. Thus, we define the deceleration in the DCS interval to quantify how severely the CoR velocity is forced to drop, which we regard as a direct indicator of dynamic instability.  
\[Acceleration\ a_{i}=\frac{\Delta\mathbf{v_{i}}}{\Delta\mathbf{t_{i}}}=\frac{\mathbf{v_{max }}_{i}(i-1)-\mathbf{v_{min}}}{\Delta\mathbf{t_{i}}} \tag{1}\]  
The slope of the velocity profile during the DCS phase, reflecting how quickly the velocity changes.  
Where \(\Delta\mathbf{t_{i}}\) and \(\Delta\mathbf{v_{i}}\) are defined as follows:  
\[Time\ interval\ \Delta t_{i} \tag{2}\]  
The duration over which the CoR velocity rapidly decreases during the DCS phase.  
\[Velocity\ gap\ magnitude\ \Delta v_{i}=v_{max,(i-1)}-v_{min,i} \tag{3}\]  
The total velocity drop from the previous step(i - 1)'s maximum to the current step(i)'s minimum.  
In addition, while the robot ascends N steps, we track the CoR's mean velocity and standard deviation as supplementary metrics to compare overall climbing performance. A higher mean velocity with lower variability implies faster and more stable stair-climbing behavior.  
\[Mean\ Velocity\overline{v}=\frac{\sum_{j=1}^{N}v_{j}}{N} \tag{4}\]  
Average CoR velocity during N-step climbing.  
\[Standard\ deviation\ of\ velocity\ s=\frac{\sum_{j=1}^{N}\bigl{(}v_{j}-\overline{v}\bigr{)}^{2}}{N} \tag{5}\]  
The degree of how tightly the CoR velocity is clustered around the mean velocity \(\overline{v}\) during N steps.  
Figure 5: Illustrative depiction of a real-world CoR velocity profile showing DCS interval and key stability metrics. The red-shaded region highlights the DCS phase, where \(\Delta t_{i}\) indicates the duration, \(\Delta v_{i}\) the velocity drop, and \(a_{i}\) the deceleration magnitude.