# Robust Design of Deep Drawing Process through In-Line Feedback Control of the Draw-In
## 2 Materials and Methods  
This study presents a methodology aimed at minimizing the influence of scattering in material properties and in lubrication conditions during a deep drawing process to achieve a zero-failure product. The adopted approach proposes the in-line optical measurement of the blank draw-in and the in-line adjustment of the blank holder force.  
The methodology is described for the T-shaped component shown in Figure 1, chosen as the case study. The part shape was purposely chosen to analyze typical defects of the drawing process, such as formability and cosmetic defects. First, a finite element (FE) model in AutoForm-Sigma R10(r) was developed for the deep drawing simulation of the part, and preliminary experimental tests were performed to calibrate the numerical model. These tests were carried out on the 3000 kN hydraulic press machine designed and manufacturing by GIGANT Italia in collaboration with industrial and academic partners of the PICO&PRO project. A passive blank holder moved by a hydraulic actuator with a maximum hydraulic force of 1000 kN and a response time of 100 ms was adopted in this press.  
The calibration of the numerical model was carried out by detecting the friction coefficient value so that FE draw-in matched the experimental one. The draw-in was evaluated at the points highlighted with the letters A, B, and C in Figure 1. Laser triangulation sensors (LK-G5000 Keyence, Osaka, Japan) were adopted near these points for experimentally acquiring the draw-in (Figure 2). These sensors had a sampling speed of 392 kHz, an accuracy of \(\pm\)0.02%, and a repeatability of 0.01 \(\upmu\)m, hence were able to perform accurate measurements during the in-line process. A 100 Mbps Ethernet communication interface was finally chosen to connect the sensors to a programmable logic controller (PLC), Siemens S7-1500 (Siemens, Munich, Germany) with CPU 1515T-2 PN, for controlling the measurements of the draw-in.  
The calibrated FE model was adopted to evaluate the influence of the drawing depth (p) and the blank holder force (BHF) on the blank thinning (formability defect) and on the curvature of the part in the zone where the surface deflection occurs due to insufficient stretching (cosmetic defect). The thinning and the curvature were chosen as quality indices. After defining the acceptance limits for these indices, the values of the blank holder force  
Figure 1: T-shaped component and points where the drw-in was monitored (A, B and C).  
Figure 2: Hydraulic press machine adopted for experimental tests and laser triangulation sensors corresponding to the points (A, B and C) where the draw-in was monitored.  
and the drawing depth that respected these limits were identified as the design configuration. For this selected combination of BHF and p parameters, stochastic FE simulations were then performed by varying the friction coefficient (\(\mu\)), yield strength (\(\sigma_{0}\)), ultimate tensile strength (\(\sigma_{R}\)), and mean anisotropy coefficient of the material (\(r_{m}\)), which were considered as noise parameters. It was evaluated how the noise parameters influence the two selected quality indices and the blank draw-in. This analysis was supported by metamodels obtained using the kriging technique. Metamodels allowed the detecting of the most sensitive sensor to the noise parameter variability and the noise parameter that most influenced the quality of the drawn component. Finally, based on the point-by-point difference between the draw-in in the design configuration and the one measured in the presence of a variation in the friction coefficient, a numerical control strategy was implemented using AutoForm and MATLAB R2022b software. Specifically, a closed-loop control was applied to the FE model so that variations in the blank holder force allowed for disturbance compensation. Using draw-in as trajectory tracking, blank holder force changes at each iteration, thanks to a proportional-integral-derivative (PID) controller, were modeled in MATLAB by means of the Ziegler-Nichols method.