# Vision-Based Path Guidance to Achieve Dies-Free Roller Hemming Process
## 2 Experimental Apparatus and Methodology
### 3D Surface Reconstruction and Coordinate Calibration Methodology  
In order to estimate the 3D coordinates from the sensor module, the poses, including positions and orientations of the image receiver and the laser emitter, are calibrated relative to the hemming tool using the hand-eye calibration method [13; 14]. As shown in Figure 5, the hand-eye coordinate system includes the world frame (also called the user frame), the sensor frame, the tool frame, and the robot frame. The location of the sensor frame is placed at the matching point of the two rollers with its \(y\)-axis pointing along the roller's circumferential direction, with the \(z\)-axis being perpendicular to the lower roller's surface and the \(x\)-axis being perpendicular to the \(y\)-axis and \(z\)-axis. In order to transform the measured coordinates from the sensor frame into the world frame, a multi-frame transformation is performed to solve the rotation and translation relation between the robot and the sensor  
Figure 4: Schematic illustration of the quick-change sensor module and the quick-change interface. Figure 3: The design of the proposed quick-change sensor module: (**a**) schematic illustration; (**b**) photograph of the module on a robot arm. the conversion relationship \(\overset{O}{H}\mathbf{T}\) between the end frame and the robot frame. Then, the conversion relationship \(\overset{W}{Q}\mathbf{T}\) between the world frame and the robot frame is calibrated by the known \(\overset{O}{H}\mathbf{T}\). Finally, through \(\overset{O}{H}\mathbf{T}\) and \(\overset{W}{Q}\mathbf{T}\), the hand-eye relationship can be calibrated and established to obtain the conversion relationship \(\overset{H}{L}\mathbf{T}\) between the sensor frame, the eye, the end-effector frame, and the hand of the robot arm. In order to perform the above calculation and proceed with the calibration, a calibration device as shown in Figure 8 was designed and used for establishing the references of the sensor. This device is composed of three modules, namely an optic calibration plate, the reference ball used to establish the robot reference, and the scale block used to identify the XYZ position of the robot TCP. The conversion relationship \(\overset{O}{H}\mathbf{T}\) from the robot arm frame \(\{O\}\) to the robot arm end frame \(\{H\}\) is calibrated using the following steps. First, the calibration jig is installed to the original TCP of the robot arm as a hemming tool. Since the designed fixture size is the same as the actual hemming tool, the end-effector frame \(\{H\}\) can be calibrated by aligning the hemming tool frame. Secondly, the robot arm is manipulated with the calibration jig to the origin of the calibration module at three or more different angles as illustrated in Figure 9. In this step, one needs to let the spherical device on the front of the jig lightly touch the meter module to make the meter module produce displacement readings. According to the changes of the readings, the endpoint of the calibration fixture with different postures can be adjusted to the same position, thereby reducing the influence of the gap within the robot itself on the calibration accuracy. When multiple postures are recognized, the last step is to calculate the conversion relationship between the end frame and the original robot arm TCP frame to obtain the transformation matrix \(\overset{O}{H}\mathbf{T}\) from the robot arm frame \(\{O\}\) to the end-effector frame \(\{H\}\). Figure 8: (**a**) A schematic drawing and (**b**) photograph showing the hand–eye calibration device used in this study. Figure 7: The calibration procedure. The calibration procedure for the world frame is similar to the above-mentioned TCP calibration process. After installing the calibration fixture on the end of the robot, one needs to manipulate the robot to touch the origin's X-direction point and Y-direction point of the world frame \(\{W\}\) with the same posture as shown in Figure 9b.