# Research on Three-Dimensional Scanning Path Planning of Casing Parts Based on Industrial Robot
## 3 Scan Path Planning
### Mathematical Modeling of the Circular Scan Path  
To facilitate the 3D scanner to scan the inside and outside of the casing, the casing was fixed on the six degrees of freedom parallel tooling support platform in the direction of the radius. In order to avoid the interference collision between the 3D scanner and the six degrees of freedom support platform, bottom-up layer-by-layer circular scanning was adopted. The whole scanning process was divided into two parts: one part is a circular rotating scan around the inner and outer walls of the magazine, and the other part is a straight translation scan along the outer wall of the engine magazine. In the first part of the circular scanning process, the scanner performs a linear translation motion along the  
Figure 5: Metra SCAN 3D scanning acquisition area. Figure 6: Aero engine casing part. circumcircle tangent of the engine casing. When the scanner moved beyond the acquisition distance, the attitude of the scanner was adjusted. That is, the equidistant offset rotation scanning trajectory was fitted by a multi-segment linear translation motion. Therefore, the total length of the scanner's linear movement and the number of pose adjustment times were important indicators to evaluate the efficiency of the scanning path-planning strategy. The scanning starting point of the scanner was set as point A, and the movement mode of the robot within the effective measurement range of the scanner was linear translation. \(AW\) is the optimal measurement distance of 30 cm, \(\theta\) is the angle of the longitudinal direction of the scanning sector, where \(\theta=50^{\circ}\), \(A_{i}B_{i}\), \(A_{i}C_{i}\) is the distance from the laser line to the surface of the workpiece, where \(A_{i}B_{i}\), \(A_{i}C_{i}\in[20\text{ cm},40\text{ cm}]\), H is the height of the casing, \(R_{0}\) is the radius of the casing bottom and \(\beta\) is the cone of the casing. When the robot reaches the maximum capacity of the scanning sector near point A\({}_{1}\), it completes a linear translation. When the scanner moves to A\({}_{1}\), the scanning area changes, and the intersection points of the vertex, the boundary and the center \(O\) currently are, respectively, represented by red dotted lines in Figure 7a. Scanning along points A, A\({}_{1}\), A\({}_{2}\),..., A\({}_{n}\), and completing the first scan, the scanning trajectory is an external polygon, as shown in Figure 7a. The scanner moves points A, E, and F, upward along the outer wall of the casing, and completes the scanning of the A-F ring until the scanning was completed. The Establishment of the Mathematical Model of Circular Scanning  
1. Circular scanning process  
It is assumed that the starting point of the scanner is A and \(OA\) is the distance from the scanning area to the center of the casing. The translational scanning motion reaches the pose adjustment point A\({}_{1}\), \(OA_{1}\) is the distance from the scanning area to the center of the casing and then there is \(OA_{i}=R_{i}+AW\). The deflection angle between \(OA\) and \(OA_{1}\) was \(\gamma\), and the distance between point A and point A\({}_{1}\) was \(L_{1}\). According to the geometric position relationship of the three, the length of the scanning trajectory in the measurement range of the scanner is as follows:  
Figure 7: Circular scanning trajectory diagram. (**a**) Circular scanning top view; (**b**) Circular scanning main view. \[L=n\sum_{i=1}^{m}\tan\gamma(R_{i}+AW) \tag{2}\]  
where \(i\) is the number of circles scanning; \(R_{i}\) is the scanning radius of circles scanning.