# Regression Approach to a Novel Lateral Flatness Leveling System for Smart Manufacturing
## 2 Machine Application Background  
The schematic drawing of the model leveling machine can be seen in Figure 2. As illustrated in Figure 2a, the sheet metal coil is fed through between the top layer and bottom layer of the leveling machine. The leveling machine consists of work rolls, intermediate rolls, and backup rolls. The work rolls are in direct contact with the sheet metal. The intermediate rolls act as a pressure distribution buffer between the backup rolls and work rolls. As is shown in Figure 2b, five backup roll sets, which are placed evenly along the sheet's lateral direction but with rolls in each set packed similar to the work rolls and intermediate rolls along the sheet's traveling direction, are used to support the work roll through the intermediate roll. In addition, the five backup roll sets are used to prevent the deflection of the work rolls while leveling the coil sheet material. The other main purpose of the backup roll sets is shape correction, particularly for the coil sheet's lateral leveling process. Therefore, backup roll adjustments were our primary focus for this paper.  
In the neutral setting, the work rolls and backup rolls are straight and parallel. This setting is used for leveling coil set and crossbow symptoms as shown in Figure 1, as those two cases do not have uneven lateral coil problems. However, lateral (widthwise) leveling requires adjusting the height of each of the roll in the backup set; that is, it requires adjusting the force pointing down toward the work rolls through the intermediate rolls for each of the five backup sets as shown in Figures 2 and 3. For the leveling machine, it is common to place the multi-roll along the sheet's traveling direction (i.e., the lengthwise direction), since the sheet inevitably exhibits a waveform in its shape from its initial coil set [26]. However, in such a design, an assumption is made of an ideal or perfect state along the sheet's lateral direction and that no waviness is allowed to appear in the sheet's widthwise portion. In reality, due to manufacturing imperfections and handling damages, this is not actually true. For example, for wider sheet metal coils, cross-width imperfections in the form of edge wave and center buckle can be easily observed. As such, the leveling or adjustment of the waviness of the sheet becomes important. In such cases, as illustrated in Figure 3, due to edge wave imperfection, the sheet has a longer length in its two edges than its center part, whereas for the center buckle case, it has a longer length in the center than at the two edges. Since the mentioned backup sets in present industry machines can only make adjustments to the so-called non-flat sheet span along its lateral direction (i.e., making the sheet thinner), this adjustment will need to adjust where the fibers are shorter to equalize the length and make the sheet flat.  
In the case of leveling center buckle or the sheet retaining a long center along its lateral direction, such as in the case shown in Figure 3b, it is required to stretch the edges until they are the same length as the center [27]. In this scenario, the backup rollers are adjusted to be tight on the center and loose in the edges. On the other hand, in the case of edge waves or long edges, as shown in Figure 3a, it is required to stretch the center until it is the same length as the edges [28]. In this scenario, the backup rollers are adjusted to be tight on the edges and loose in the center. This means that we will stretch the edges in a controlled manner. There are examples in the literature that have addressed backup roll design [28; 29]. However, backup roll adjustments for lateral leveling have been limited. This is a common practice in the coil leveling industry from machine operators with years of experience. Therefore, utilizing the regression method for parameter prediction for backup adjustments is the motivation of this research.