# Simple Degree-of-Freedom Modeling of the Random Fluctuation Arising in Human-Bicycle Balance
## 5 Identification Results
### Identification Results  
Table 2 lists the identified vector components of \(\mathbf{p}^{(s)}\) by PSO for all participants \(s=1,\cdots,8\) and the corresponding KS cost value \(E(\mathbf{p}^{(s)})\). The seventh column indicates the cost value as a PDF fitness value of the form  
\[\text{Fitness}:=(1-E)\times 100\,\%, \tag{21}\]  
which indicates the accuracy of our human-bicycle fluctuation model (9) in terms of the reproducibility of PDFs. The best and worst results are indicated by \" ** \" and \" \", respectively. The second-last and the last columns show the cost value and the corresponding fitness value, respectively, between the measured \(P_{hum}^{(s)}(x_{1},x_{2})\) and the mathematical two-dimensional Gaussian PDF \(P_{Gauss}^{(s)}(x_{1},x_{2})\) with the same mean vector and covariance matrix as those of the measured \(P_{hum}^{(s)}(x_{1},x_{2})\).  
The results clearly show that our proposed model (9) successfully achieved over 98.7% PDF fitness, even in the worst case (\(s=5\)). This implies that it provides much better fitness than conventional Gaussian models.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & \multicolumn{6}{c}{**Our Proposed Fitting**} & \multicolumn{3}{c}{**Gaussian Fitting**} \\ \hline \(\mathbf{s}\) & \(\mathbf{p_{1}}\) & \(\mathbf{p_{2}}\) & \(\mathbf{p_{3}}\) & \(\mathbf{p_{4}}\) & \(\mathbf{E(\mathbf{p}^{(s)})}\) & **Fitness** & \(\mathbf{E}\) & **Fitness** \\ \hline
1 & \(-32.6272\) & \(3.35867\) & \(-5.84823\) & \(2.74614\) & \(9.611\times 10^{-3}\) & \(99.04\%\) & \(1.134\times 10^{-1}\) & \(88.66\%\) \\
2 & \(-39.6902\) & \(1.21198\) & \(-1.72440\) & \(1.53507\) & \(7.189\times 10^{-3}\) & \(99.28\%\) & \(1.684\times 10^{-1}\) & \(83.16\%\) \\
3 & \(-44.5088\) & \(1.08744\) & \(-5.97971\) & \(3.70913\) & \(7.056\times 10^{-3}\) & \(99.29\%\) & \(1.234\times 10^{-1}\) & \(87.66\%\) \\
4 & \(-46.6330\) & \(1.17335\) & \(-6.31916\) & \(2.58083\) & \(7.478\times 10^{-3}\) & \(99.25\%\) & \(1.652\times 10^{-1}\) & \(83.48\%\) \\
5 & \(-37.5965\) & \(1.61197\) & \(-1.77080\) & \(1.54042\) & \(1.228\times 10^{-2}\) & \(98.77\%\) * & \(1.640\times 10^{-1}\) & \(83.60\%\) \\
6 & \(-34.6138\) & \(0.993204\) & \(-7.01156\) & \(2.96107\) & \(5.689\times 10^{-3}\) & \(99.43\%\) ** & \(1.351\times 10^{-1}\) & \(86.49\%\) \\
7 & \(-38.5872\) & \(1.21683\) & \(-7.31069\) & \(3.07624\) & \(6.011\times 10^{-3}\) & \(99.40\%\) & \(9.134\times 10^{-2}\) & \(90.87\%\) \\
8 & \(-32.5560\) & \(2.55268\) & \(-2.15329\) & \(1.63554\) & \(8.500\times 10^{-3}\) & \(99.15\%\) & \(1.225\times 10^{-1}\) & \(87.75\%\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Identified \(\mathbf{p}^{(s)}=(p_{1},p_{2},p_{3},p_{4})\) and its cost value for the \(s\)th participant. “ ** \" denotes the best result and “ \", the worst. The last two columns show the corresponding Gaussian results.  
Next, the left column of Figure 5 shows the difference of our simulated \(P_{sim}^{(s)}(x_{1},x_{2})\) from the measured \(P_{hum}^{(s)}(x_{1},x_{2})\), and the right column shows that for the Gaussian PDF \(P_{Gauss}^{(s)}(x_{1},x_{2})\), for all \(s\). The plot ranges are the same as those of all plots.  
As already indicated by the fitness values in Table 2, our simulated PDFs agree well with the measured PDFs, compared to the Gaussian PDFs. As the shapes of Gaussian PDFs are symmetric by definition, the right column's results imply that our measured PDFs have asymmetric shapes, as indicated by the deep-colored peaks and troughs.  
Therefore, our proposed model properly reproduced such asymmetric shapes, which were barely reproduced by the Gaussian distribution.