# Research on Springback Compensation Method of Roll Forming Based on Improved Fuzzy PID Control
## 2 System Modeling Based on Springback Compensation Strategy
### Kinematic Model of Rolls  
Take a horizontally adjusted three-roll circular rolling machine as an example. As shown in Figure 1, the lead screw is fixed to the output shaft of the motor (labeled \"1\"). The slider (labeled \"2\") is sleeved on the lead screw through a lead screw nut. The upper surface of the slider (2) is inclined at a certain angle. The bottom of the wedge (labeled \"3\") is in contact with the upper surface of the slider (2). Two driven rolls (labeled \"4\"), spaced a certain distance apart, are located on the top of the wedge (3). The working roll (labeled \"5\") is positioned above the middle of the two driven rolls (4) and is driven to move up and down by a hydraulic mechanism. The circular rolled sheet is placed between the driven rolls (4) and the working roll (5). A distance sensor is installed on the driven rolls (4) to measure the desired distance and the actual forming distance. The rotation of the lead screw is driven by the motor (1), enabling the slider (2) to move on the lead screw. Since the upper surface of the slider (2) is inclined at a certain angle and the bottom of the wedge (3) is in contact with the upper surface of the slider (2), when the slider (2) moves forward on the lead screw, it can drive the wedge (3) to move upward, thereby bringing the driven rolls (4) closer to the working roll (5) and shaping the radius of the circular rolled sheet. Conversely, when the slider (2) moves backward on the lead screw, it can drive the wedge (3) to move upward, thus moving the driven rolls (4) away from the working roll (5). Among them, the motor (1) controls the horizontal movement of the slider (2). Through the wedge (3), the horizontal motion of the slider (2) is transformed into the vertical movement of the driven rolls (4). By adjusting the vertical displacement of the driven rolls (4), the contact position between the sheet and the roll is altered. The rotation of the working roll (5) causes the sheet to undergo continuous elastic plastic deformation. Figure 2 presents a schematic diagram of circular roll forming. The distance sensor therein can measure the distance between the sheet and the sensor, enabling real-time monitoring of the changes in springback. Based on the feedback from the distance sensor, the rotation angle of the motor is adjusted to change the rolling angle of the sheet, thereby correcting the springback that occurs in the sheet. A kinematic model for circular rolling springback compensation is established [23; 24]. Based on the desired forming radius \(R_{q}\), the desired distance \(L_{q}\), and the sheet thickness \(D_{0}\), the relationship between the distance \(D\) from the distance sensor to the midline is obtained:  
\[D^{2}+{R_{q}}^{2}=\left(R_{q}+L_{q}+D_{0}\right)^{2} \tag{1}\]  
Based on the actual forming radius \(R_{s}\), the actual distance \(L_{s}\), and the sheet thickness \(D_{0}\), the relationship between the distance \(D\) from the distance sensor to the midline is obtained as follows:  
\[R_{s}=\frac{D^{2}}{2(L_{s}+D_{0})}-\frac{(L_{s}+D_{0})^{2}}{2} \tag{2}\]  
Figure 1: Horizontal upward adjustable three-roller rolling machine structure. 1—electric motor, 2—slide block, 3—inclined wedge, 4—driven roller, 5—working roller. Figure 2: Kinematic modeling of rolling circle rebound compensation. Then, the mathematical relationships among the desired forming radius \(R_{q}\), the radius of the driven rolls \(R_{1}\), the center-to-center distance of the driven rolls \(D_{1}\), the sheet thickness \(D_{0}\), and the upper-pressing amount \(h\) are as follows:  
\[\left(\frac{D_{1}}{2}\right)^{2}+\left(R_{q}+R_{1}+D_{0}-h\right)^{2}=\left(R_{ q}+R_{1}+D_{0}\right)^{2} \tag{3}\]  
\(D_{3}\) is the pitch on the screw, and \(D_{2}\) is the horizontal displacement of the slider. Herein, the upper-pressing amount \(h\) represents the upward displacement of the driven roll starting from the moment when the working roll, the driven roll, and the sheet come into contact.