## 7 Discussion  
In this paper, a new type of controller for the control of the posture of a pneumatic bellows actuated module of the cascade robot PneuTrunk was presented. The controller consists of a feed-forward controller designed using experimental data and a error dependent variable gain I-controller. The feed-forward controller is created by fitting the data at a certain pressure level using a polynomial function and subsequently, again fitting the resulting set of polynomial function constants by another set of polynomials. This allows for a simple and fast controller, that not only allows to control the posture of the module, but also its stiffness. The variable gain I-controller supplements the feed-forward controller by adding a feedback loop, hence facilitating disturbance rejection and correcting for feed-forward controller imperfections. The variable gain allows for fast error correction while limiting overshoot and windup at the same time. This hybridisation approach allows for a simple controller design for a complex MIMO systems that can be easily adjusted and updated. This controller was compared to other established controllers. On the feed-forward level, the controller was compared with an ANFIS controller, delivering comparable results. Comparing the complete controller to a tuned PID controller showed that our FFvI controller is faster, and can reliably follow a harmonic reference signal while maintaining required performance parameters. This controller was also tested under dynamic load with satisfactory results. The maximum FFvI controller error during the positioning of the module with consideration of dynamic disturbance was only \(1.52^{\circ}\). The aforementioned tests show a robust behaviour of the FFvI controller in the face of both internal feed-forward model imperfections, as shown by quick error rejection in the step response tests and external dynamic disturbance forces, as shown by the dynamic tests. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Velocity; Pressure** & **I on [\({}^{\circ}\)]** & **I off [\({}^{\circ}\)]** \\ \hline
10 mm/s; 6 bar & 0.93 & 3.57 \\
30 mm/s; 6 bar & 0.66 & 4.29 \\
50 mm/s; 6 bar & 1.52 & 3.48 \\
70 mm/s; 6 bar & 1.51 & 6.18 \\
10 mm/s; 9 bar & 0.31 & 3.84 \\
30 mm/s; 9 bar & 0.62 & 3.75 \\
50 mm/s; 9 bar & 1.02 & 3.66 \\
70 mm/s; 9 bar & 1.05 & 6.82 \\
10 mm/s; 12 bar & 0.28 & 2.78 \\
30 mm/s; 12 bar & 0.60 & 2.58 \\
50 mm/s; 12 bar & 0.88 & 3.17 \\
70 mm/s; 12 bar & 1.15 & 5.42 \\
10 mm/s; 15 bar & 0.52 & 2.74 \\
30 mm/s; 15 bar & 0.63 & 2.68 \\
50 mm/s; 15 bar & 0.87 & 3.24 \\
70 mm/s; 15 bar & 1.14 & 3.44 \\ mean & 0.86 & 3.85 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Maximum total error under dynamic load. To further improve the performance of the controller, it is necessary to create a comprehensive mathematical model of the module, mainly to combat the detrimental effects of the hysteretic behavior of the bellows. It is also appropriate to compare different types of functions driving the variable gain of the I-controller. In the future, this controller will be applied as a part of a larger control system controlling the pneumatic cascade robot PneuTrunk. M.V.--Investigation and Formal analysis; I.V.--Project administration; M.K.--Supervision; L.M.--Methodology; Z.B.--Validation; P.J.S. and T.M.--Software. All authors have read and agreed to the published version of the manuscript. This research was funded by Slovak Grant Agency VEGA 1/0436/22 Research on modelling methods and control algorithms of kinematically redundant mechanisms and VEGA 1/0201/21 Mobile mechatronic assistant. This research has been also elaborated under the support of the project Research Centre of Advanced Mechatronic Systems, reg. No. CZ.02.1.01/0.0/0.0/16019/0000867 in the frame of the Operational Program Research, Development and Education. Not applicable. The data presented in this study are available on request from the corresponding author. The data are not publicly available due to funding project restrictions.