## 5 Experimental Verification of FFvI Controller
The relevant constants can be seen in Table 4. Although the values of the tuned constants are not optimal, the authors believe that they are close enough to the optimal values combining adequate speed while not exceeding our criterion of \(2^{\circ}\) overshoot on \(\alpha_{x}\) and \(\alpha_{y}\). Figure 19: Comparison between feed-forward and ANFIS controller. \begin{table}
\begin{tabular}{c c c c} \hline
**Bellows** & **Controller** & **Maximum Error [bar]** & **Mean Error [bar]** \\ \hline \multirow{2}{*}{1} & feed-forward & \(0.0959\) & \(0.0217\) \\  & ANFIS & \(0.0231\) & \(-0.0028\) \\ \hline \multirow{2}{*}{2} & feed-forward & \(0.1463\) & \(0.0219\) \\  & ANFIS & \(0.0174\) & \(-0.0011\) \\ \hline \multirow{2}{*}{2} & feed-forward & \(0.1286\) & \(0.025\) \\  & ANFIS & \(0.0294\) & \(0.0029\) \\ \hline \end{tabular}
\end{table} 
Table 3: Maximum error. The performance of both controllers can be seen in Figure 20. The reference signal is sequentially alternating between \(\alpha_{xref}=8^{\circ}\), \(\alpha_{yref}=-10^{\circ}\), \(\alpha_{xref}=-8^{\circ}\) and \(\alpha_{yref}=10^{\circ}\). These values were chosen because all bellows need to be engaged simultaneously to different degrees, they represent values in the middle part of the module workspace and it is expected that most movements will be within this area. They also demonstrate the asymmetric behavior of the platform. The performance of both controller types is shown in Table 5. It needs to be noted that, this comparison deviates from a standard comparison of step responses by not having the platform at zero tilt and testing a combined rotation around \(\alpha_{x}\) and \(\alpha_{y}\). Nevertheless, this comparison is much closer to comparing the controller performance under more realistic conditions. It can be seen, that both controllers fulfill the condition of not having more than \(2^{\circ}\) overshoot. From Table 5 it can be seen, that the FFvI controller is faster in all categories and has more overshoot only when going from \(-8^{\circ}\to 8^{\circ}\). The difference in quality between both controllers is illustrated by performance criteria, where in all cases the FFvI controller outperforms the PID controller by roughly more than 40%. The second comparison between both controllers is in following a sine reference signal with parameters written in Table 6, as can be seen in Figure 21. Here the FFvI controller can tightly and smoothly follow the sinusoidal reference signal while overshooting at the maximum and minimum of the reference signal. The overshoot, again, does not exceed \(2^{\circ}\) for either tilt angles. The PID controller, while not overshooting nearly as much, lags constantly behind the reference signal by about 0.7 s and the plot is in some parts jitter. The overall tracking error is much smaller for the FFvI than for the PID controller. It can be said that both controllers perform acceptably, while the FFvI controller is faster while still passing the overshoot criteria. \begin{table}
\begin{tabular}{c c} \hline  & **Gain Bar/deg** \\ \hline \(K_{p}\) & 40 \\ \(K_{I}\) & 240 \\ \(K_{D}\) & 5 \\ \hline \end{tabular}
\end{table} 
Table 4: PID controller gains. One ability that the FFvI controller has is to change the stiffness of the system while maintaining reference posture. Stiffness is set indirectly by inputting a number between 1-15 bar. This number represents the sum of the input pressures in all bellows and correlates with the overall stiffness of the system. The reason we did not quantify the stiffness is because stiffness is not only dependent on the input pressure, but also on the posture of the module making stiffness a three-dimensional matrix of values. This makes it impractical as an intuitive value that should be specified by the user. An algorithm to convert stiffness to aggregate pressure can be designed, but it would not change any part of the controller.