## 4 Simulation of Running Motion Generation
### Simulation Results  
Table 2 summarizes the initial values used in the simulation. Table 3 displays the value of the base gain \(K_{*\_\_\_\_}\)base and the minimum values of \(E_{\_}{\text{ref}}\) for each tactic. Figure 2(a) presents the results of Participant 1's running simulation with the base gain, serving as a representative example. This simulation achieved a stable running state after approximately 10 s. Figure 2(b) presents the results that most closely matched Participant 1's experimental data by adjusting the combination of tactics through varying weight assignments. Figure 4 plots all the results obtained for stride and cadence across various gain weights and applied energy levels. Stride was calculated as the distance between consecutive landing points, while cadence was determined as the inverse of the time taken for one complete cycle of running. The amplitude was calculated using the difference between the highest and lowest COG points in one step. Table 4 provides corresponding weight values \(w\) and energy \(E_{\_}{\text{ref}}\) applied to the model, whereas Table 5 summarizes the running parameters from both the experiment and the simulation results for comparison. We calculated the 95% confidence intervals for stride, cadence, and amplitude and conducted statistical tests to determine whether the mean values of the simulation data and experimental data were consistent. The results are summarized in Table 5.  
Figure 3: (**a**) is a simulation result with the base gain. The first 20 s of the simulation with the base gain were extracted. (**b**) is the result with weighted gains to reproduce the participant’s running. Both figures are the results from Participant 1, which was selected as a representative example. Each figure contains 4 graphs. The first graph plotted the height of the mass coordinate, while the second graph showed the average and reference velocity per step in blue and red lines, respectively. The third graph illustrates the total energy state (purple) alongside the reference energy (red). The fourth graph presents the attitude angle (green) on the first axis and stiffness (blue) on the second axis.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Participant** & \(\mathbf{w_{k}}\) & \(\mathbf{w_{\theta}}\) & \(\mathbf{w_{f}}\) & \(\mathbf{E_{\mathrm{ref}}}\) \\ \hline P.1 & 0.6 & 1.2 & 0.6 & 936 \\ P.2 & 0.6 & 1.0 & 2.0 & 926 \\ P.3 & 0.4 & 1.0 & 0.4 & 804 \\ P.4 & 1.0 & 0.8 & 1.8 & 710 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Explored weights  
Figure 4: Cadence and stride. The plots illustrate the combinations of stride and cadence that the proposed model reproduced. Each simulated value is plotted as a point, and the color map indicates the reference energy value \(E_{\mathrm{ref}}\). The black line represents the relationship where Cadence = 2.22/Stride (C = 2.22/S). The red cross in the plots represents the experimental results. The mean of the experimental values is the intersection of the crosses, and the standard deviation is represented by the length of the line.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Participant** & \(\mathbf{K_{k\_base}}\) & \(\mathbf{K_{\theta\_base}}\) & \(\mathbf{K_{f\_base}}\) & \(\mathbf{E_{\mathrm{min}}}\) \\ \hline P.1 & \(-1000\) & 0.15 & 11 & 900 \\ P.2 & \(-1200\) & 0.16 & 10 & 890 \\ P.3 & \(-2000\) & 0.16 & 14 & 770 \\ P.4 & \(-600\) & 0.15 & 8 & 660 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Base gain.