## 4 Experiments
### Experimental Results  
A total of 36 stair-climbing experiments were conducted, repeating the low-, medium-, and high-speed scenarios three times each, as outlined in Table 5. Figure 13 shows representative images from the basic and high-k configurations (S1 and S4). Below, Figure 14 shows the wheel's CoR trajectory and its velocity profile for each scenario at different operating speeds. Each scenario was tested three times, and representative graphs are presented here. For PWM duty ratios of 20%, 40%, and 60%, the measured average angular velocities during a single-step climb were 3.2, 8.5, and 13 rad/s, respectively.  
Figure 12: Trajectory comparison of the wheel CoR between simulation and marker tracking data.  
In the simulation, the DCS appeared as an abrupt velocity drop, but the experimental data in Figure 14 show--consistent with Figure 5--that it unfolds over a finite time and maintains a relatively steady slope. This observation supports defining dynamic stability  
Figure 14: Representative CoR trajectories and linear velocity profiles for four Scenarios (S1–S4). Each was repeated three times. (Low speed: \(\omega\) = 3.2 rad/s, middle speed: \(\omega\) = 8.5 rad/s, high speed: \(\omega\) = 13 rad/s, low k: \(k_{t}\) = 0.206 N-m/rad, middle k: \(k_{t}\) = 0.547 N-m/rad, high k: \(k_{t}\) = 0.909 N-m/rad).  
Figure 13: Stair-climbing experiments for scenarios S1 (**a–e**) and S4 (**f–j**).  
in the DCS phase by using the acceleration \(a\), as it directly reflects how rapidly the velocity changes under inertial forces, providing a clear indicator of the system's ability to cope with sudden transitions.  
For each DCS interval, one value each of \(\Delta t_{i},\ \Delta v_{i},\ a_{i}\) is obtained. Because the robot passes through three steps (steps 1, 2, and 3), three sets of these parameters are collected, one for each DCS period. The average value is then defined as the mean of the three corresponding values from steps 1, 2, and 3. Tables 6-8 summarize the time interval, velocity gap magnitude, and acceleration for all four scenarios (basic, low-k, middle-k, and high-k suspensions) under low-speed, medium-speed, and high-speed conditions, respectively. Figure 15 presents bar charts of the average values for \(\Delta t_{i},\ \Delta v_{i}\), and \(a_{i}\) across all four stiffness levels under each speed condition.  
Table 9 summarizes the average velocity \(\overline{v}\) and standard deviation \(s\) across all speed conditions and scenarios. Figures 16 and 17 present these results in bar-chart form.  
\begin{table}
\begin{tabular}{c} **Middle** \\ **k** \\ \end{tabular} & **High k** \\ \hline
**Low speed** & 178.8 & 169.3 & 168.6 & 165.4 & 52.89 & 72.86 & 98.40 & 113.2 \\ \hline
**Middle speed** & 416.0 & 400.0 & 413.6 & 416.2 & 111.7 & 133.1 & 133.8 & 119.4 \\ \hline
**High speed** & 592.6 & 643.4 & 641.7 & 661.8 & 189.3 & 237.4 & 186.7 & 156.1 \\ \hline \end{tabular}
\end{table}
Table 9: Values of mean velocity and standard deviation of velocity for three drive speeds and scenarios S1–S4.  
Figure 15: Average values of time interval, velocity gap magnitude, and acceleration during contact transition in DCS for scenarios S1–S4. (**a**) Low-speed (\(\omega=3.2\) rad/s) results. (**b**) Middle-speed (\(\omega=8.5\) rad/s) results. (**c**) High-speed (\(\omega=13\) rad/s) results.