# A Methodology for the Mechanical Design of Pneumatic Joints Using Artificial Neural Networks
## 4 Experimental Results
### Comparison between BNN and RSM  
The SPA with reinforcements I and III was mounted vertically, as for the tests conducted for the dataset construction, while the one with reinforcement II was mounted horizontally. This was done to assess whether the predictions could be made by neglecting the gravitational effects of the different SPA assemblies. Figure 7a-i show the deformation states of the three reinforcements at 0.4, 0.8, and 1.2 bar.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Reinforcement**} & \multirow{2}{*}{**Joint**} & \multicolumn{4}{c}{**Parameters**} \\ \cline{3-6}  & & **L (mm)** & **R (-)** & **\(\mathbf{\Theta}\) (\({}^{\circ}\))** & **N (-)** \\ \hline I & 1 & 5.0 & 0.65 & 82 & 6 \\ \hline \multirow{4}{*}{II} & 1 & 10.0 & 0.70 & 70 & 3 \\  & 2 & 10.0 & 0.60 & 70 & 2 \\  & 3 & 5.0 & 0.60 & 70 & 2 \\ \hline \multirow{4}{*}{III} & 1 & 5.2 & 0.50 & 60 & 2 \\  & 2 & 5.2 & 0.60 & 54 & 2 \\ \cline{1-1}  & 3 & 10.0 & 0.65 & 86 & 2 \\ \cline{1-1}  & 4 & 8.0 & 0.40 & 74 & 2 \\ \cline{1-1}  & 5 & 8.0 & 0.60 & 64 & 2 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Parameter values for the external reinforcements used in the experimental test.  
Figure 7: Deformation configurations for reinforcement: (**a**) I at 0.4 bar; (**b**) I at 0.8 bar; (**c**) I at 1.2 bar; (**d**) II at 0.4 bar; (**e**) II at 0.8 bar; (**f**) II at 1.2 bar; (**g**) III at 0.4 bar; (**h**); III at 0.8 bar; (**i**) III 1.2 bar.  
The same image analysis followed to construct the dataset was adopted to quantify the experimental joint bending angles (\(\alpha_{\text{exp}}\)) of each of the three reinforcements. The parameter values of each joint were used to predict the bending angle by both using BNN (\(\alpha_{\text{BNN}}\)) and a previously identified model with RSM (\(\alpha_{\text{RSM}}\)) [36]. The experimental and predicted angles of the joints are shown in Table 5. Absolute errors for BNN (\(\Delta\alpha_{\text{BNN}}\)) and RSM (\(\Delta\alpha_{\text{RSM}}\)) were calculated with Equations (4) and (5).  
\[\Delta\alpha_{BNN}=\alpha_{exp}-\alpha_{BNN} \tag{4}\]  
\[\Delta\alpha_{RSM}=\alpha_{exp}-\alpha_{RSM} \tag{5}\]  
The maximum absolute error was \(8.20^{\circ}\) for BNN and \(-8.85^{\circ}\) for RSM. However, evaluating the average error for reinforcements I, II, and III results in \(2.72^{\circ}\), \(2.80^{\circ}\), and \(2.71^{\circ}\) for BNN and \(4.55^{\circ}\), \(3.91^{\circ}\), and \(3.14^{\circ}\) for RSM, respectively. These values give further validation to the soft joint design approach by the BNN algorithm. Figure 8 plots the absolute errors as a function of pressure for each of the three reinforcements analyzed to view and compare the results of Table 5 easily. In these figures, the errors have been indicated with \(\Delta\alpha\), where the subscripts denote the reinforcement and the joint's number to which it refers, respectively. Figure 8a shows that the error with the BNN (black circles) for reinforcement I seems to have a negative linear trend. However, Figure 8b shows more variability for the absolute errors related to reinforcement II. Only for the predictions of joint 2 (blue circles) is there a negative linear trend, while for joint 1 with the RSM, there is a positive linear trend. Instead, for reinforcement III, analyzed in Figure 8c, the error obtained on joint 2 (blue circles), joint 4 (green circles), and joint 5 (purple triangles) with the BNN remains relatively constant. The error shows a positive linear trend for the RSM prediction on joint 4 (gray stars).