# A Methodology for the Mechanical Design of Pneumatic Joints Using Artificial Neural Networks
## 5 Discussion  
Characterization conducted with a model-free approach performed better than one with RSM [36]. Both regression algorithms are data-driven and are based on experimental responses. The BNN required a dense campaign of tests to build the Bend dataset, while RSM minimized the number of trials to be conducted. It is true that with the second approach, SPA behavior can be identified with less time and few resources expended, but greater errors are made.  
The errors in the experimental tests are described in the previous section, and here the relative percentage errors are quantified. Concerning Table 5 and graphs in Figure 8, in reinforcement I, the bending angles are 20.82\({}^{\circ}\) at 0.4 bar, 45.82\({}^{\circ}\) at 0.8 bar and 80.88\({}^{\circ}\) at 1.2 bar. The absolute errors of 2.82\({}^{\circ}\), 0.40\({}^{\circ}\) and 4.94\({}^{\circ}\) with the BNN are equal to a relative error of 13.54%, 0.87% and 6.11%, respectively. With the values predicted by RSM, however, the errors rise to 23.92%, 12.61% and 3.56%.  
For reinforcement II, the maximum relative errors are for joint 1 at 0.4 bar and for joint 2 at 0.8 bar and 1.2 bar for both algorithms. At these joints and pressures, the bending angles are equal to 15.45\({}^{\circ}\), 24.60\({}^{\circ}\) and 16.14\({}^{\circ}\). Specifically, the relative errors are 18.6%, 11.91%, and 13.33% for the BNN and reach 57.28%, 31.71% and 7.79% for RSM.  
Reinforcement III, on the other hand, shows a relative error of 48.99% for joint 2 at 0.4 bar, 36.03% at 0.8 bar for joint 1, and finally 16.20% again for joint 2 at 1.2 bar for the BNN. The maximum relative errors with RSM are 59.29% at 0.4 bar for joint 5, 15.87% at 0.8 bar for joint 3, and finally 15.35% for joint 4 at 1.2 bar.  
It emerges that the largest errors occur at lower pressures when the SPA bending angles are smaller. This means that for the same absolute error, there is more influence on the relative error at low deflections. However, even when considering relative percentage errors, there were smaller errors with the use of the BNN than RSM.  
Making comparisons with other work employing ANNs for characterizing SPAs is quite complex. Indeed, a comparison of algorithm performance would be made on different types of SPAs, such as with and without reinforcement, with different geometries and different materials. In the literature, there are several studies employing ANNs, but the details of the layers or neurons used are often not provided to compare the hyperparameters employed. In addition, the training times are often not specified, and so we were not able to compare the required times either. Also, the input data or dataset responses are different, for example, some consider the spatial position of the SPA tip (X, Y, and Z coordinates) and others the bending angles, as in the present work. The ranges of input parameters can be very different, for example, in terms of actuation pressures, which depend greatly on the materials used. Therefore, a comparison in terms of absolute or relative percentage errors would not be very meaningful.  
Figure 8: Absolute errors between experimental and predicted bending angles for BNN and RSM as a function of feeding pressure for (**a**) reinforcement I, (**b**) reinforcement II, and (**c**) reinforcement III.