## 5 Experimental Verification of FFvI Controller  
This section will focus on the experimental comparison between the FFvI controller and controllers designed according to established algorithms. First of all, the performance of the feed-forward part of the FFvI algorithm will be compared with an ANFIS controller designed using the same data-set as our feed-forward controller. In the second part, the complete FFvI controller will be compared to a constant gain PID controller. The ANFIS controller was designed using the MATLAB neuro-fuzzy designer. Three controllers for each bellows separately were created. The input data are tilt angles \(\alpha_{x}\) and \(\alpha_{y}\) and the output is the corresponding pressure. The teaching data are picked from the same data, that is used to design the feed-forward controller, but are limited to having an aggregate pressure of \(9\pm 1.5\) bar. The feed-forward controller is also set to the same level of aggregate pressure. This will decrease the teaching time and ensure more reliable results. The neural network used is a Sugeno-type network [39] with 10 linear generalized bell-shaped membership functions for each input. The minimum achieved teaching error is in Table 2. \begin{table}
\begin{tabular}{c c} \hline
**Bellows** & **Teaching Error [bar]** \\ \hline
1 & 0.0205 \\
2 & 0.0227 \\
3 & 0.0272 \\ \hline \end{tabular}
\end{table} 
Table 2: Minimum teaching error. Figure 18: Comparison between constant and variable gain I-controller under different conditions. (a) high feed-forward error; (b) low feed-forward error; (c) reference angle change \(\alpha_{y}\)= 8\({}^{\circ}\)\(\rightarrow\)10\({}^{\circ}\). Both controllers were supplied with sets of reference postures and their outputs were compared to input pressures, that produced these postures during measurement. The resulting comparison between the outputs of the feed-forward controller and the ANFIS controller is shown in Figure 19. The minimum error and mean error are in Table 3.  
The ANFIS controller has a consistently lower error and the mean error is also lower. Nevertheless, as can be seen from the above results, both controllers perform well, with the maximum error of the feed-forward controller not exceeding \(0.15^{\circ}\). The magnitude of this error is still well within what would be considered acceptable for this application. Considering that the feed-forward controller can approximate the whole pressure augmented workspace, while the ANFIS controller is specifically designed to work in the tested aggregate pressure region, these results are encouraging. When assessing the complete FFvI controller, one must keep in mind that the physical module is meant to be part of a robot. Therefore, we have specified that the overshoot, when performing a movement, should not exceed \(2^{\circ}\) on \(\alpha_{x}\) and \(\alpha_{y}\) separately. This number, although arbitrary, should be a good controller benchmark for this case and the controller is expected to be further tuned after being installed in the complete robot control system. As a comparison to our FFvI controller, a PID-based controller was used. The controller consists of three identical constant gain PID controllers controlling all the bellows separately. The PID controllers were tuned using the P-I-D tuning approach described in [40]. It is an experimental method that starts by experimentally tuning the P gain to a satisfactory level while keeping the I gain and D gain at zero. After that, the I gain is calculated according to the step response of the system and the P gain is reduced to retain stability. The D gain can by calculated from the I gain. In this case, the I gain was later increased to speed up the controller, as this method leaves room for improvement in this regard. The D gain was lowered to decrease the effects of noise.