Adaptive Machining Scheme for Multi-Hole Part with Multi-Position Accuracy Requirements

: The machining of multi-hole parts often has complex correlated position accuracy requirements. When some position accuracies do not meet the requirements, several hole axes need to be adjusted. Previous methods usually correct all deviated axes to their theoretical locations. However, the correction workload is too large and inefficient. This paper proposes an efficient and adaptive hole position correction model for multi-hole part. First, the method establishes the topological relationship of the holes and faces on the part according to the position accuracy requirements of the multi-hole part. Then, the goal is to minimize the number of holes that need to be corrected. In this model, the parallelism of holes, perpendicularity, and other constraints are considered. The simulation and experimental results show that the use of this model can effectively reduce the number of holes that need to be corrected during the compensation of the position error between holes. It improves the efficiency in the subsequent compensation process significantly.


Introduction
Multi-hole part plays an important role in the mechanical connection of the assembly process and the transmission of structural parts, such as gearboxes. The position accuracy between holes matters in evaluating the machining quality of holes and it will have an important influence on assembly accuracy, part performance and life [1]. Therefore, when processing multi-hole part, how to ensure the position accuracy between holes in processing is a problem that needs to be paid attention to.
The traditional hole processing method is usually manual, and the processing quality mainly depends on the workers' experience. Manual drilling has been gradually replaced by CNC equipment due to the higher precision and stability of CNC machining equipment [2]. However, on some occasions with high precision requirements, CNC still cannot meet the machining position accuracy of the hole due to the equipment accuracy limitation. The current research on improving accuracy of holes can be divided into two categories according to different drilling equipment.
One type is based on a special drilling mechanism, such as a robot. This kind of method usually first designs the corresponding drilling fine-tuning compensation mechanism, and then designs the corresponding motion control and compensation algorithm according to the structural characteristics of the compensation mechanism, to control the action of the compensation mechanism and realize the hole position error compensation [3][4][5]. The other type is mainly processing equipment with CNC machine tools as holes. This type of equipment only needs to design the corresponding compensation algorithm to directly change the NC codes when performing hole processing, without the need to design a special hardware mechanism [6,7].
However, the limitation of previous research is that when selecting methods to improve accuracy, the focus is to study the deviation of the actual position of the single hole from its theoretical position, and to make the hole as close to its theoretical position as possible through correction. There are usually complex positional requirements between holes and holes, holes and surfaces in multi-hole parts. To ensure its position accuracy requirements, it is necessary to correct the position of each hole to make it as close to its theoretical position as possible. However, this method requires a heavy correction workload, because the position tolerance allows the position of the hole to change within a certain range. Therefore, when correcting the hole position, it is unnecessary to correct all the holes to make it meet the accuracy requirements. In order to solve this problem, the method proposed in this paper reduces the workload of correction by reducing the number of holes that need to be corrected. First, the accuracy requirements' topological relationship between hole and hole is established according to the position accuracy requirements of the hole and surface of the part. A model with the minimum number of holes to be corrected as the goal and the position accuracy requirement as the constraint is established, and the correction scheme of holes is obtained through this model. The experimental results show that when the accuracy of a certain item or several positions in the set of holes does not meet the accuracy requirements, using this method to correct the axis direction of the hole in the Set of holes can reduce the number of holes that need to be corrected, and make the position accuracy of each hole meet the requirements, reducing the workload of subsequent processing and correction.

Related work
The equipment for machining holes mainly includes two types: robot and NC machine tool. Yuan et al. used the double eccentric discs-spherical pair structure [8] as the end axis correction mechanism of the robot drilling, and designed a drilling axis correction algorithm based on this mechanism and the deviation of the drilling axis.
This algorithm can ensure that the axis deviation of the robot drilling does not exceed ±0.5°, which effectively ensures the perpendicularity of the hole to the end face [9].
Shen et al. optimized the rigidity of the robot drilling system before processing and proposed a compensation method for the hole position error during processing, which effectively improves the robot drilling accuracy [10]. In addition, position correction strategies of holes based on interpolation methods such as kriging [11], co-kriging [12], and Shepard [13] can also effectively improve hole machining accuracy.
Similar to the special drilling mechanism, corresponding improvements in hardware and software can also be made for CNC machine tools. Gao et al. added a piezoelectric actuator to the boring bar servo system to realize the online compensation for the precision boring machining error of the aspect ratio hole. The experiment proved that this method can significantly reduce the roundness error of the hole [14]. Chiu et al. designed a boring bar servo system with online compensation for machining errors. Forecasting compensatory control (FCC) was implemented in the boring bar servo system to predict machining errors, which is used for high-length-to-radius ratio precision boring machining error online compensation [15]. The last part summarizes the conclusions. According to the measurement point data showed in Fig. 1 (a), two planes can be fitted respectively to determine the plane where the center of the measurement point is located. Suppose the fitted plane equation is:

Detection of hole's position error
The least square method is used to construct the objective function as the minimum value of the sum of the distances from the measured multiple points to the fitting surface, the minimum value of the formula (2): Then the values of A, B, C, and D in the fitted plane equation can be obtained,  (1), and the fitted space circle can be expressed: The coordinates of the center of the circle and the value of the radius R in (3) can be obtained by taking the minimum value of formula (4) by using the least square method: Calculate the coordinates of the centers of the two measuring points circles respectively, and connect the two centers to form the measured vector value of the axis of the hole. The focus of this paper is on the requirements of parallelism and perpendicularity. Therefore, only the characteristics of holes related to the parallelism and perpendicularity of the holes can be considered, the hole is represented by a vector whose mold is equal to the length of the hole and the direction is the same as the axis of the hole. Then the parallelism error f of hole 1 l relative to hole 2 l or plane 1 p can be expressed as: Equation (7) shows whether the i-th hole needs to be corrected. If it needs to be corrected, it is recorded as 1, and if it does not need to be corrected, it is recorded as 0.
The value obtained by accumulating i x is the number of holes that need to be corrected for the entire part. The minimum number of holes to be corrected is used as the optimization target, then the optimization target AimN can be expressed as: Equation (9) establishes the relationship between whether the axis needs to be corrected and the change of the axis before and after correction. The left side of the formula (9) represents the difference between the axis vectors before and after correction, and M is a large value. This formula ensures that when the vector difference is 0, when the axis does not need to be corrected, the value of i x can be 0.
In optimizing the axis vector of the hole, the modulus of the axis vector of the hole represents the length of the hole. Therefore, the length of the hole after correction and before correction should be slightly different. the modulus difference of the axis vector is small, and this needs to be restricted. Then there are: In formula (10),  is a small value, which limits the change of the length of the shaft hole before and after correction.
The shape and position accuracy requirements of the i-th hole for the j-th hole or the j-th surface are expressed as ij LB , and The mathematical expression of the accuracy requirements of parallelism and perpendicularity shown in equations (11) and (12) can be further obtained from equations (5) and (6) respectively. When formula (11) is the parallelism of hole-to-hole and hole-to-face, the value of sin can be obtained by formulas (13) and (14) respectively. When formula (12) is the perpendicularity of hole to hole and hole to face, the value of cos  can be obtained by formula (15) and (16) respectively. Where n represents the normal vector of the plane.
In the process of adaptive processing correction, we hope that the number of correction holes is the smallest and the deviation of the corrected hole relative to the theoretical axis does not exceed the deviation of the hole before the correction to the theoretical axis. Therefore, the following constraints should be added: In summary, the target optimization model can be constructed as shown in equation (18):

Experiments:
In this section, several sets of holes data are generated to simulate the method proposed in this paper to verify the effectiveness, and then a set of inspection data after actual box processing is used to verify the feasibility of the method under actual conditions. The tool for solving the model is LINGO11, which is commonly used to solve optimization problems [19,20].

Exp 1: Parallelism verification
In the experiment, the parameters Only the axial direction of some holes needs to be corrected, and the other holes can be processed according to the original processing technology, reducing the correction workload. When the accuracy requirements are high, the number of holes that need to be corrected will increase, but the number of holes that need to be corrected can still be reduced to a certain extent. This phenomenon is of great significance to the actual correction, and an accuracy index higher than the actual machining accuracy requirement can be set during optimization to ensure the effect of the correction.  The experimental results are shown in Table 2. The number of holes that need to be corrected in the three tests is 2, 3, and 4, respectively. The experimental results show that, similar to the parallelism experiment, this method is also adaptable to the verticality error of the holes.

Exp 3: Comprehensive test of actual processing multi-hole part
The actual processing of multi-hole part is taken as an example to verify the effectiveness of the method in the actual holes processing and correction. As shown in Figure 4, the part is a certain type of gearbox part of the actual production process.
The left picture is the surface where the key holes are distributed, and the right picture is the other side of the multi-hole part. The most important processing of this part is the processing of holes, and the cooperation of multiple axes of the machine tool is required to realize the processing of holes on multiple sides during the processing.
During the machining process, because of the existence of errors, there will be a certain deflection between the actual machined holes and the theoretical position.
When this deflection reaches a certain level, the shape and position accuracy of some holes does not meet the accuracy requirements. In the figure, the holes and faces that have positional accuracy correlation are numbered for the convenience of further analysis.  There is a parallelism requirement of 0.06 between the hole and the hole in the figure, and a perpendicularity requirement of 0.06 between the hole and the surface. It can be seen that the complexity of this part is mainly manifested in: (1) The complexity of the accuracy index: there are requirements for both parallelism and perpendicularity.
As shown in the figure, there is a perpendicularity requirement between the hole and the surface, and there is a parallelism requirement between the hole and the hole. (2) The complexity of the relationship: one hole may have requirements for shape and position accuracy between multiple holes and surfaces. As shown in the figure, there is a parallelism requirement between hole 1 and hole 2 and hole 4, and there is a perpendicularity requirement between face 1, face 4, and face 9 at the same time.
According to the content of Section 2, the hole and the surface are detected, and the corresponding fitting calculation is performed to obtain the value of the axis vector of the hole and the normal vector of the plane. The shape and position accuracy between them is calculated, as shown in Table 3. It can be seen from Table 3 that the shape and position accuracy calculated for the verticality number 1, 2, 3, 4, 5, 7 exceeds the allowable range, so the workpiece needs to be corrected and optimized accordingly.