## 2 Experimental and Methodology
### Background
Furthermore, advanced sealing and guiding technologies, describing the initial state of this research, result in less pronounced frictional dynamics across the entire velocity range of the hydraulic cylinder [41]. In the case of the forming process, experimental analysis emphasizes that the movement of the hydraulic cylinder \(X_{C}\) exhibits significant changes in the response error \(\Delta X_{C}\), as shown on the \(X_{C}(t)\) graph in the red circle area, due to specimen deformation, which this study aims to detect, address, and minimize. Here, sheet metal bending is selected as the simplest forming process to be integrated into the hydraulic press, with the aim of investigating the action-reaction relationship between the forming process and the behavior of the hydraulic press. Moreover, the hydraulic press cycle consists of three phases, where the first phase involves rapid forward and backward movements, which are necessary for the hydraulic cylinder to perform idle motions, thereby extending the cycle time. The second phase, bending, occurs when the die tool makes contact with the specimen and deforms it into the desired shape. The final phase, the leveling phase, involves holding the forming tool stationary while the static effects of the forming process on the hydraulic press are examined. The actual response of the observable parameters when a condition change occurs can be observed from the data collected by the sensors \(S\)\(=\)\(\{s_{1},s_{2},s_{3},s_{4},s_{5}\}\), with a sampling frequency of 50 Hz, through actual experiments in the physical environment. A displacement sensor (Messotron DLH300 (Messotron, Radeberg, Germany), \(\pm\)150 mm range, \(\pm\)1% accuracy) monitors the hydraulic cylinder displacement. A pressure sensor (Turck PT40 OR-2004-U1-H1143 (Turck, Mulheim an der Ruhr, Germany), 0-40 MPa range, \(\pm\)0.3% accuracy) monitors the pressure in cylinder chambers A and B. A force sensor (ME KM30z (ME-Messysteme GmbH, Hennigsdorf, Germany), 0-20 kN range, \(\pm\)0.25% accuracy) tracks the cylinder's force, while an LVDT sensor (Messotron, Radeberg, Germany) with accuracy \(\pm\) 0.5% monitors the hydraulic valve opening. In this context, the variables denote real-time measurements of the displacement of the hydraulic cylinder \(X_{C}\), the pressure of the hydraulic cylinder in the chambers \(p_{A}\) and \(p_{B}\), the force of the hydraulic cylinder \(F_{C}\), and the displacement of the hydraulic valve \(X_{V}\). The parameters significantly impact experimental results, as sensor accuracy and range affect data reliability, influencing the understanding and control of the interaction between the forming process and the hydraulic press. Experimental studies were carried out on a hydraulic press to determine the hydraulic press response error \(\Delta X_{c}\)\(=\)\(\{\Delta X_{c,1},\ \ldots,\Delta X_{c,n}\}\), which is the difference between the actual and referenced hydraulic cylinder displacement \(X_{C}\) for the entire range of the sampling time sequence \(t=\)\(\{t_{1},\ \ldots,\ t_{n}\}\) as follows:  
\[\Delta X_{C}(t)=X_{C,ref}(t)-X_{C,meas}(t); \tag{1}\]  
Meanwhile, it has been shown that the most relevant features that are most relevant in the studied research problem are the valve spool position \(X_{V}\) along with either the forming force \(F_{C}\) or the pressure difference \(\Delta p\) in the bending phase, the hydraulic cylinder velocity \(v_{\mathbb{C}}\) and the valve spool position \(X_{V}\) in the leveling phase, and the hydraulic cylinder velocity \(v_{\mathbb{C}}\) in the movement phase [42]. Furthermore, the hydraulic cylinder velocity \(v_{\mathbb{C}}\) is described as a vector quantity that measures the hydraulic cylinder displacement \(X_{\text{C}}\) over the time interval \(dt\) as follows:  
\[\text{v}_{\text{C}}(t)=\frac{X_{\text{C},meas}(t)}{dt}; \tag{2}\]  
Hydraulic presses are intentionally built to generate a compressive force and allow movement during the forming process through the energy of the hydraulic fluid.