## 3 Results and Discussion  
In this section, all obtained results are displayed and discussed. To obtain the results, two NN were implemented on the basis of the parameters reported in the following Table 1: the first NN was used to calculate the angle \(\theta_{\mathrm{s}}\) (\(\theta_{\mathrm{c}}\), \(\theta_{\mathrm{p}}\)) solving a non linear equation, while the second one was used to estimate muscular forces optimizing the cost function described above. The \(\theta_{\mathrm{s}}\) angle estimated signals, for one of the subject, obtained using deterministic algorithm optimization and using neural network are both shown in Figure 5.  
In the Table 2 the correlation of the two signals, for three different subjects, is shown in terms of Root Mean Square Error (RMSE). Neural network, for all three subjects, makes a RMS Error well below 1% and a SD Error below 10\({}^{-3}\), showing a good convergence in the research of the optimal solution. Regarding the second step, 1180 random samples out of 118,000 have been taken for the analysis of the results. In the Figure 6, the Bland-Altman plot for the Rectus Femoris is shown. While the 1.96 \(\sigma_{\mathrm{diff}}\) boundaries are not negligibly narrow, they can be reasonably attributed to the intrinsic noise affecting the deterministic algorithm (more on this matter will be discussed below). The distribution is acceptably symmetric around the mean axis, excluding the possibility of a systematic measurement error. There is no apparent pattern in the error distribution apart from a border effect, \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Output** & **NN topology** & \begin{tabular}{c} **Number of** \\ **inputs** \\ \end{tabular} & \begin{tabular}{c} **Number of neurons** \\ **in the hidden layer** \\ \end{tabular} & \
\begin{tabular}{c} **Training set** \\ **(Samples)** \\ \end{tabular} &
\begin{tabular}{c} **Validation set** \\ **(Samples)** \\ \end{tabular} \\ \hline \(\theta_{\mathrm{s}}\) angle & 1 MISO & 2 & 4 & 7,050 & 118,000 \\ Muscle Forces & 9 MISO & 3 & 15 & 2,360 & 118,000 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of Neural Networks (NN) used to obtain the results. Figure 5: Angle \(\theta_{\mathrm{s}}\) signals, obtained by the deterministic optimization algorithm (solid gray) and by the neural network (dashed black). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Subject** & **RMS Error \%** & **Standard Deviation Error** \\ \hline
1 & 0.085\% & \(2.68\times 10^{-4}\) \\
2 & 0.134\% & \(7.35\times 10^{-4}\) \\
3 & 0.077\% & \(4.53\times 10^{-4}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Results of RMS Error percentage and SD Error for the calculation of \(\theta_{\mathrm{s}}\) angle. consisting in a clustering of samples on the left side around the axis origin. This is due to boundary conditions used in the model: muscular forces cannot have negative value, because of obvious physiological reasons. In Table 3, both the upper and lower 1.96\({}_{\mathrm{odiff}}\) boundaries are shown, together with the width of the boundary divided by the relative muscle mean force (normalized boundary). Aside from the error, a consideration can be made on the networks performance predicting muscular forces. Indeed, the number of unknowns in (2) is higher than the number of equations (9 unknowns for 3 equations), so the numerical optimization of such problem is multimodal, yielding a very noisy solution. The NN however performs an average of the different solutions, smoothing effectively the results. This effect is highlighted in Figure 7: the NN completely filters unnatural \"high\" frequency components in the muscular activity signals. In terms of computational complexity, the computational time required to elaborate a full sample, thus to compute the nine muscular forces starting from the pedals data, is about 26 ms on a Core i7 Machine, against the 2,200 ms required to compute the deterministic original approach on the same machine, yielding a considerable speed up.