## 3 Methodology
### Design of Variable Die Bending Tool  
In order to make a V-die bending process variable regarding its bending angle and thus allow the angle to be controlled, additional Degrees Of Freedom (DOF) need to be introduced in the tooling system. The concept developed for a flexible die is shown in Figure 4 in the form of a principle sketch. It shows that the die is divided into two halves, maintaining a common pivot point (MD). It has a vertical DOF and is guided horizontally in the die center. It is furthermore supported against the floor by a spring of stiffness \(\text{c}_{\text{d}}\). Moreover, each die half is guided at its other end through a floating bearing. The angle \(\beta_{\text{g}}\), in which the floating bearing is arranged, ensures that the die angle (\(a_{\text{d}}\)) is reduced when the center point is lowered (\(x_{\text{d}}>0\)). The die angle can therefore be set depending on the vertical displacement of MD. Apart from the die, the punch is designed to be flexible, too. Figure 5 shows the principle sketch of a punch with variable punch angle. As with the die, the punch is \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Sheet Thickness [mm]** & **0.5** & **0.75** & **1** \\ \hline DC01 & x & x & x \\ EN AW 6082T6 & x & - & x \\ Copper & x & - & x \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Material and sheet thickness combinations. divided into two halves with a common pivot point (MP). The ends of the punch halves (C) are connected to a floating bearing (D) via springs. It must also be ensured that D is always vertically aligned above C. The floating bearing allows for the compensation of the horizontal offset that occurs when the punch angle is reduced. Similar to the die, springs (\(\mathrm{c_{P}}\)) ensure that the appropriate coining forces can be applied for varying punch angles. Viewed individually, the punch has two DOF (\(x_{\mathrm{P1}}\), \(x_{\mathrm{P2}}\)), as the two halves of the punch can move independently. In combination with the die, these DOF are reduced to the displacement of the die center point \(x_{\mathrm{d}}\) and thus the stroke height. The tooling system is dimensioned in a way that sheet materials with a length of 160 mm and a width of 50 mm can be processed. The corresponding materials and sheet thicknesses are shown in Table 1. It should also be possible to realize angles in a range from 80 to 110\({}^{\circ}\). The required force for bending is estimated according to Equation (1) for each material and sheet thickness combination. The maximum estimated stamping force is then used to select the springs. \[F_{\mathrm{P,max}}=\mathrm{max}\Big{(}F_{\mathrm{P,req_{ij}}}\Big{)},i\in \mathbf{m} \tag{9}\]  
The vector \(\mathbf{m}\) denotes all possible combinations of material and sheet thickness. By integrating the additional DOF into the tool, the product property, i.e., the bending angle, can be controlled. Derived from [2], a cascaded controller for springback compensation is shown in Figure 6. The RL model is referred to as RL-based product-controller in the offline control loop and represents a SAC RL agent. The observable state contains the target angle \(a_{\mathrm{des}}^{\mathrm{(unl)}}\) and vector \(\mathbf{m}\). The material vector \(\mathbf{m}\) includes the elastic modulus, the Poisson ratio and the tensile strength of the selected material and sheet thickness. Based on the inputs, the agent selects a compensated angle \(a_{\mathrm{des}}^{\mathrm{(loa)}}\), which is forwarded to the online controller. The online control ensures that machine-side and environmental influences such as gear backlash and actuator inaccuracies can be eliminated. This is achieved by a camera-based inline measurement of the bending angle (50 Hz sampling frequency) and a direct feedback  
Figure 4: Principle sketch of the die. Figure 5: Principle sketch of the punch. of the measured bending angle into the machine control (PLC). Reference [20] provides a more detailed description of the similar measurement system and information flows in an air bending process.