Collision Detection Algorithm of Belt Grinding of the Blisk Based on Improved Octree Segmentation Method

In the process of belt grinding aero-engine Blisk(Bladed Disk), the abrasive belt can easily interfere with the Blisk, which will damage the valuable Blisk. Therefore, it is indispensable and significant to study the collision detection of belt grinding the Blisk. However, the application of traditional collision detection algorithms in this complicated realistic scene is difficult to obtain satisfactory results. In order to improve the accuracy and efficiency of the collision detection of grinding the Blisk, a collision detection algorithm based on the improved octree segmentation method is proposed in this paper. Firstly, the Oriented Bounding Box (OBB) is applied to establish the collision detection model for the abrasive belt. Secondly, the traditional octree segmentation method is optimized based on the k-means clustering algorithm, and an improved octree segmentation method is presented, in addition, the flow chart of the collision detection algorithm for belt grinding of the Bliskis given. Finally, algorithm verification and experimental verification are carried out based on a certain type of the Blisk. The results suggest that compared with the traditional method, the method in this paper not only promotes the accuracy of collision detection, but also promotes the efficiency of collision detection, and meets the requirements of object collision detection in this tanglesome scene with both accuracy and speed.


Introduction
The Blisk plays a decisive role in the manufacture and production of aero engines [1]. Bliskis one of the key parts of aero engine, its profile is thin and twisted, and its flow channel is deep and narrow, making it difficult to machine. Therefore, its precision machining has become the key in the field of aero engine manufacturing. Belt grinding is one of the commonly used methods for the precision machining of the Blisk of the engine

OBB bounding box model of abrasive belt
Bounding box is an algorithm for solving the optimal enclosing space of a set of discrete points.
Its basic idea is to use simple geometric bodies (called bounding box) to approximately replace complex geometric objects, so as to achieve the purpose of rapid collision detection [15]. OBB is defined as the smallest regular hexahedron that contains the object and is arbitrarily oriented relative to the coordinate axis. Its biggest feature is that the direction of the bounding box is arbitrary, which makes it possible to tightly where, Finally, the OBB bounding box model of the abrasive belt is established, as shown in Figure 2.

Improved octree segmentation method based on k-means clustering algorithm
In order to solve the many shortcomings caused by the above-mentioned traditional methods, this paper adopts the idea of "divide and conquer, overall optimization", and introduces the k-means clustering algorithm [16] to improve it, that is, firstly, perform appropriate point cloud classification of the individual blades of the Blisk, then perform the octree segmentation after classification respectively, finally, the result is stored in the corresponding data structure for subsequent collision detection.
In this paper, the Euclidean distance between data objects [17] is used as the basis of k-means algorithm clustering, where k represents the number of clusters, and means represents the mean value of the data objects in the clusters. It should be pointed out that the change of k value will affect the result of clustering. In order to find the best k value of k-means clustering algorithm, the range of k value is 2~8 in this paper.
For a given data set . Now we have to use the k-means clustering algorithm to divide the data set into k clusters, and we use C to represent the set of k clusters, that is: , there must be a center point the Euclidean distance is selected as the criterion for judging the similarity and distance of each data in the cluster. Assuming that the sum of the squares of the distances from each point in the cluster to its center point Obviously, the purpose of using the clustering algorithm is to minimize the formula (5).
The steps for applying the algorithm in this paper are as follows: Step 1. Enter the number of clusters k and the data set , and randomly select k initial center points 1 ,, k  L ; Step 2. Calculate the distances from all data points ( 1, 2, , ) in the data set to these k initial center points respectively. If any data point ; if the distances to multiple initial center points are equal, they can be divided into arbitrary clusters. Any data point Step 3. After classifying all the data points, calculate the mean value in each cluster as the new center point of the cluster. This process is also called updating the cluster mean; Step 4. Repeat steps 2 and 3 until the new center point of the cluster is equal to the initial center point, and output a data set of k clusters.
Step 5. Find the optimal k value of the data set

Collision detection algorithm
The definition of the separation axis theorem is as follows [18,19]: if an axis can be found in the three-dimensional space of the target object, and the two target objects are projected on the axis separately, and no overlapping projection part can be found on the axis, then this axis is defined as the separation axis.  Fig. 6 The principle diagram of the collision detection of the separation axis method [20] According to Fig. 6 Combining equations (9) and (10) into equation (8), we can get: Finally, the flow chart of the collision detection algorithm in this paper is shown in Figure 7.  Table 1, and the split time comparison between the traditional octree split method and the improved octree split method is shown in Table 2. The comparison chart is shown in Figure   9 and Figure 10 respectively. It is worth pointing out that according to the clustering results obtained in the above experiment, the mean value of all data points in each cluster is calculated, so as to ensure that the initial center point of the cluster in the collision detection algorithm in this paper is not randomly selected.    In terms of algorithm verification, this paper develops a set of abrasive belt grinding and polishing the Blisk collision detection software based on the above theoretical methods. The collision detection process is shown in Figure   11(a). In terms of experimental verification, this paper has realized the experimental verification of the collision detection of the Blisk abrasive belt grinding and polishing processing by a robot, and the collision detection process diagram is shown in Figure 11(b). The experimental results obtained from algorithm verification and experimental verification are shown in Table 3 and Table 4, respectively, the visualization results are shown in Figure 12(a) and Figure 12(b) respectively.
(a) (b)  Due to the introduction of clustering algorithm in the algorithm in this paper, the Blisk and the point cloud of the blade are divided into multiple parts from a whole, and each part is divided into different clusters. As a result, the collision space required by the re-divided point cloud area is more compact, which improves the accuracy of collision detection. From Table 3 and Table 4, it can be concluded that the accuracy of the collision detection of the algorithm in this paper is about 45% higher than that of the traditional algorithm on average. In terms of collision detection time, because the algorithm in this paper can eliminate the space that does not participate in collision detection, this greatly reduces the number of nodes divided by the octree, thereby reducing the traversal time required for collision detection. It can also be concluded from Table 3 and Table 4 that the collision detection efficiency of the algorithm in this paper is 18.60% and 18.44% higher than the traditional algorithm respectively.     (2) The feasibility of the collision detection algorithm proposed in this paper is verified   The principle diagram of the collision detection of the separation axis method [20] Figure 7 Flow chart of collision detection algorithm in this paper Comparison of the total number of split nodes between the traditional octree partition method and the improved octree partition method Figure 10 Comparison of the split time between the traditional octree segmentation method and the improved octree segmentation method Figure 11 The algorithm veri cation of the collision detection of the Blisk of the belt grinding (a), and the experimental veri cation based on the robot processing (b) Figure 12 The visualization result of algorithm veri cation (a) and the visualization result of experimental veri cation (b)