A control method with considering error coupling in milling multi-feature thin-walled parts

Multiple features of thin-walled parts have position-dependent and dimension association, which means that multi-feature machining errors can affect each other. They may be coupled in multiple processes. Therefore, multi-feature couple effect plays a significant impact on thin-walled parts machining process in some cases. The most existing error coupling analysis methods ignore the error influence between features. To overcome this problems, Multi-feature machining is investigated thoroughly, error coupling method is proposed to consider these effects in this paper. According to the influence mode of multi-feature machining, error coupling is decomposed into geometric interaction and machining interaction, and analyze individual influence mode. The geometric interaction is tolerance/datum interactions. The machining interaction is generated from machining operations. A method based on SDT (small displacement torsors) and transformation of the coordinate systems is employed and integrated to the comprehensive error model. Besides, a compensation method by adjusting the pose is proposed. Finally, a case study is performed to validate the proposed method.


Introduction
Thin walled parts are widely used in aerospace industry considering their advantages in their light weight, good stiffness, and high strength [1]. However, the machining of thin-walled parts is the most difficult problem in metal cutting. For the complex structures, such parts often have many machining features. These features are related to machining process, dimension, and tolerance. They restrict and benchmark each other. As the process going on, the errors among the features will interact and coupling. In this situation, interactive manufacturing features play important role in the analysis of thin-walled parts processing quality. It makes more difficult to predict and control the machining error of thin-walled parts.
To solve the abovementioned problems during machining multi-feature thin walled parts, many researchers have made much detailed study about this complex process problem. Pal et al. [2] analyzed the optimization algorithm needed for the interactive function of various datum/tolerances of multi-feature parts, and provided a more accurate machining strategy for NC machining process by using spatial reasoning method. According to the geometric information and boundary conditions of multiple features, Givehchi et al. [3] simplified the features to arrange the processing sequence considering the interaction between feature volume and adjacent features. Ma et al. [4] established the calculation model of instantaneous cutting point and system stiffness in multi-feature machining, a method of machining deformation homogenization was proposed to control the machining error of thin-walled parts. Dipper et al. [5] constructed a feature information model related to each other in multi-feature machining. Combined with the process information, a threedimensional information model is formed and applied to the NC machining process. Mieczyslaw and Siemiatkowski [6] presented a feature based reasoning approach for generating machining sequences in terms of part setups and the assignment of machine alternatives. The approach assumes a heavy reliance on a data input model incorporating functional requirements for parts and in particular GD&T references. Wan et al. [7] proposed a machining-feature requirement-driven workpiece-holding scheme to constrain the degrees of freedom (DOFs) of the workpiece according to the various machining features.
These models and strategies can obtain the optimized cutting method and sequence relative to the multi-feature thinwalled parts. Process optimization method try to eliminate or greatly decrease the interaction between multi-feature based on the angle of datum/tolerance and stiffness. However, it is ignored error coupling caused by the force, heat, and stress in multi-operation and multi-feature manufacturing processes.
About the influence factors of error coupling of thinwalled parts, the researchers at home and abroad mainly focuses on the error coupling of datum transformation [8,9], thermal coupling [10,11], stress coupling [12][13][14], etc. Du [15] established the change propagation model of multistage machining errors through a series of homogeneous transformations of different positioning/reference errors. Zhengshu et al. [16] modeled geometric tolerances and their interaction in truly three-dimensional context based tolerance analysis method. Liu et al. [17] proposed a method to control geometric error and thermal error simultaneously based on the relationship between geometric error of tooth surface and grinding position error. Liu et al. [18] studied a variety of error fields generated in the multi-position clamping of thinwalled parts, and built a 3D error coupling model using the state space theory. Huang et al. [19,20] followed the concept of datum flow chain (DFC), regarded each station of multi-feature machining as a series of discrete events. The transmission, coupling, and superposition of errors in the manufacturing process were analyzed. Control theory was also introduced to optimize the system, so as to improve process parameters and reduce product errors. In addition, Ge et al. [21] proposed an integrated error compensation method based on on-machine measurement (OMM) inspection for thin web parts machining. Du et al. [22] supplied a comprehensive error compensation method that includes three major error sources, which are geometric error, thermal error, and force-induced error. Considering variable stiffness structure workpieces, Li et al. [23] developed a novel systematic approach to minimize the maximum fixturing induced deformation of the surface to be machined based on elastic mechanics.
In conclusion, the existing research methods for machining quality control of thin-walled parts are mainly studied in two aspects. One is to study multi-feature NC machining and material removal considering the structural characteristics of parts. The second is to study the coupling of errors generated by different error sources of a specific single machining feature. The former mainly focuses on feature recognition and machining sequence optimization of multi-feature machining of thin-walled parts. The latter is mainly used to predict the error of multiple machining for a single feature of thin-walled parts. However, for thin-walled parts with multiple features, there is a position-dependent and dimension association, which means that multi-feature machining errors can affect each other. Both of the above ignore this correlation between features. The first machined feature affects the geometric error of the later machined feature through datum/tolerance. The later machined feature affects adjacent machined features through stiffness and cutting force/heat, resulting in a secondary change in its quality accuracy, which in turn affects the stability of the machining quality of thin-walled parts. This paper aims to present a systematic investigation on this issue by means of theoretical modeling and experimental validation.

Interactive factors of multi-feature processing
The interactive factors in multi-feature machining can be divided into two classes as shown in Fig. 1 [24]. The first is geometric error Δ g resulting from locating and clamping, which is the error generated before machining. Locating datum plane is the reference of parts processing. If there is error in datum plane, it will affect the further processing of parts. Clamping deformation can further change the locating position and shape of the thin-walled part. The second class is machining error from machining, which is the error encountered during processing. If permanent distortions of part and fixture are ignored, the error Δ m from machining is composed of (a) the error caused by force/heat generated in the cutting process of associated features; and (b) the error related to elastic deformation due to the smaller and smaller stiffness with the removal of materials. The error variation, the tiny deviation of dimension, shape, and position for a machining feature can be expressed by some torsor parameters. Such torsor parameters are then called small displacement torsors (SDT). The SDT can be used directly in its generic form to represent potential variations along and about all three Cartesian axes. If the SDT is used to express the difference between the actual and ideal Fig. 1 Interaction of multi-feature machining quality of the key quality feature, a machining error vector, denoted as Δ tol , can be written as: Here, S actual represents the actual machined feature; S ideal is the nominal machined feature. Δx , Δy , and Δz are the translational vectors of the feature in the directions x, y, and z; Δ , Δ , and Δ are the rotating vectors around x, y, and z. According to the error sources, the vector of the resultant surface error can also be obtained as:

Geometric interaction of multi-feature machining
In the process of machining thin-walled parts, the tolerance/ datum between features will affect each other. However, the tolerance/datum plane is the reference for machining parts. If there are errors in the datum plane, the further machining of parts will be affected. As shown in Fig. 2, the complex thin-walled part is clamped twice with different datum planes to mill different features. Its geometric features such as planes, holes, and cylindrical surfaces are labeled in Fig. 2a. Figure 2c shows the clamping and positioning method when machining different features. Different machining features can share the same datum, and the machined features can also be the machining datum of other quality features. Therefore, different machining features of the same part can influence each other through datum. The features of complex thin-walled parts interact through positioning and datum. The relationship between the 10 features of the sample part is illustrated in Fig. 3. According to Fig. 3, the relationship between datum and machining features is defined as Where N is the number of clamping. H i is the relationship between datum and machining feature in the ith installation.
In Eq. (3), j = 1, 2, 3 is the number of machining features, and k = 1, 2, 3 is the locating surface. D i1 , D i2 , D i3 is the first positioning/datum, the second positioning/datum plane, and the third positioning/datum plane respectively. A ij denotes machining the jth feature in the ith clamping. ijk is the attribute of machining feature and positioning/datum plane. When ijk = 0 , the machining feature is parallel to the positioning/datum plane. When ijk = 90 , the locating surface is perpendicular to the machining feature; when ijk = , there is an angle relationship between machining features and location/datum; when ijk = 1 , there is only dimension relationship between machining features and locati.on/datum.
There are two kinds of errors between machining feature and positioning/datum. That is, the error Δ pv caused by the incorrect position of the positioning part (unideal fixture position and manufacturing error of fixture), and the variation value Δ wv belongs to the size of the part. The error between the machining feature and the positioning/reference plane can be expressed as When the position of the positioning element is deviated, the position and direction of the part will change, and the machining error will also be affected by these changes. Compared with the ideal positioning, this change shows the change of datum. After the positioning datum changes, any point on the machining feature of the part will also change. Considering the uneven datum plane and clamping deformation, the fitting surface is obtained by fitting the changed discrete points to express the noni deal surface features. The three locating surfaces are expressed by fitting plane equations. The locating surface is expressed by plane equation. (I) When the machining feature is parallel to the positioning/datum plane, that is, when ijk = 0 , the equal error between the positioning surface and the machining feature is shown in Fig. 4.
In the unified coordinate system, it is assumed that the normal vector n j of the datum plane needs to rotate around the x -axis, around the y-axis, and γ around the z-axis to coincide with the normal vector n d of the positioning plane. Namely The variation of positioning/datum plane will lead to the position change of machining features. Suppose that the ideal coordinate of any point on the machining feature is M(x, y, z) , and it actually becomes M ′ x ′ , y ′ , z ′ after positioning. If point M ′ simply rotates angle anticlockwise around the x ′ axis, then M ′ and x ′ coincide, then The matrix form is as follows That is where Similarly, the rotation matrices of simple rotation around the y ′ -axis (rotation angle is ) and simple rotation around the z ′ -axis (rotation angle is ) are as follows Suppose that the translation of M(x, y, z) and M ′ x ′ , y ′ , z ′ along the three axes is (Δx, Δy, Δz) , the translation matrix can be written as Then, the relation between M(x, y, z) and M ′ x ′ , y ′ , z ′ is It can be expressed as The general error produces a small rotation angle, so the above equation can be simplified as follows Then, the error of any point of the part caused by the positioning/datum plane can be computed as (II) When the machining feature is perpendicular to the positioning/datum plane, that is, ijk = 90 , the vertical error between the positioning plane and the machining feature is shown in Fig. 5. As shown in Fig. 6, when the machining feature is parallel or vertical to the positioning/datum plane, according to the geometric characteristics of the part and positioning element, the rotation angle ΔR of the machining parallel feature and vertical feature caused by the positioning/datum plane error is the same. The analysis process is consistent with the above parallel machining features. Therefore, when machining vertical features such as holes and cylinders, the relationship between the ideal position of any point on the feature and the actual position is as follows The error is the same as Eq. (19).
The traditional measurement method of positioning error can use laser interferometer. With the emergence of computer aided design software, it is possible to analyze and measure the positioning error in the computer simulated virtual environment, which provides great convenience for

Interaction between features caused by machining
Milling process is a very complex process. The problem involved is not simple structural analysis or thermal analysis, but the coupling effect of various physical fields. Therefore, in this kind of analysis, the function and interaction of each physical quantity need to be considered. Among them, the deformation caused by clamping is an important factor affecting the machining accuracy in the multi-feature machining of thin-walled parts. It can make the adjacent machining feature produce elastic deformation by clamping force, which leads to the machining error of the adjacent feature in the cutting process. At the same time, the cutting force and heat in cutting metal are a common physical phenomenon. It may influence the features connected with it through force, heat, and stress. The calculation of machining deformation under these coupling effects involves many related theories, including elastic-plastic mechanics theory, heat transfer theory, advanced dynamics theory, and vibration mechanics theory. This section mainly analyzes the influence of clamping force and cutting force on multi-feature machining of thin-walled parts.

Influence caused by clamping force
For a part, under a certain positioning scheme, as shown in Fig. 7, the clamping deformation of feature surface can be expressed as the displacement of feature surface in x, y, and z directions of spatial coordinate system, and the displacement is related to the clamping force in the three directions. Suppose n i is the inner normal unit vector of the contact between the part surface and the ith positioning element. i and i are two orthogonal unit vectors of the tangent of the part surface at the contact point as shown in Fig. 8. Then, the three directions of force generated by the positioning element on the part at the contact point are as follows In which, f i = f ni , f i , f i T is the contact force at the contact point in the normal and two orthogonal tangential directions. The tangential forces are the friction forces at the contact point. When the part is in contact with the positioning element, the contact deformation will occur at the contact point due to the clamping force, which will make the part deviate from the original position. It is assumed that K i is the contact stiffness of the ith positioning clamping element and the part. According to the Hertz contact theory, the relationship between the contact force f i and the displacement Δd i of the locating point is as follows On the other hand, the simulation software is used to analyze the clamping deformation of thin-walled parts, and the results are shown in Fig. 9. The normal component of generating points on machined surface adjacent to (24) the clamping surface is the machining error. Suppose that i is the actual coordinate value of the surface at a certain time, and the corresponding ideal coordinate value is q i x i , y i , z i . It can be seen from Fig. 10 that the machining error of the surface caused by the clamping force is: In which, i is the number of nodes.

Influence caused by cutting force
For complex thin-walled parts, they are more sensitive to the force, heat, and stress in the machining process, so the interaction of their features is more prominent. The influence mode is shown in Fig. 11. Under the interaction of tool and part, machined feature ① is affected by post machined feature ②. The characteristic plane is extruded in normal and tangential directions, resulting in normal warping deformation and tangential torsion deformation. The deformation is shown in Fig. 11. The influence of tangential torsion deformation on the flatness of planar features is relatively small, which can be ignored generally, while the influence of normal deformation is relatively large. where Δx im , Δy im , and Δz im are the deformation of point i in three directions under the action of cutting force.F ix , F iy , and F iz is the cutting force. K ix , K iy , and K iz are the stiffness of the thin-walled part in three directions positions, respectively.

The comprehensive error model based on SDT
According to the analysis and modeling of the two error sources in Sects. 3 and 4, the comprehensive error model can be expressed based on SDT in the key feature coordinate system. The key feature is the important machining quality that must be guaranteed for the final machining of parts. However, the geometric error is modeled in the workpiece coordinate system while the force-induced error is calculated in the key feature coordinate system. To realize the comprehensive error compensation, the error models need to be unified into a coordinate system. In order to facilitate the adjustment of part pose, the workpiece coordinate system is selected here. Then the comprehensive error can be obtained by rotation and translation between coordinate systems as: where Δ i is the comprehensive error; d ix , d iy , d iz , Δ , Δ , and Δ are the translation and rotation values between the two coordinates.

Posture adjustment based on comprehensive error model
As shown in Fig. 12, there is a posture error between the theoretical coordinate and the actual coordinate in the workpiece coordinate system (WCS). Posture error includes coordinate value and direction. According to the above comprehensive error model, the angle error is ΔR={Δ , Δ , Δ } ; the displacement error is ΔP={Δx, Δy, Δz} . That is, there are rotation error ΔR and displacement ΔP error between the actual coordinate system and the theoretical coordinate system of key machining feature.
It is assumed that the theoretical coordinate system of machining feature i is recorded as PCS 0 i x 0 i , y 0 i , z 0 i and the actual coordinate system is PCS i x i , y i , z i . Then, the relationship between PCS 0 where cos(Δ ) , sin(Δ ) , cos(Δ ) , sin(Δ ) , cos(Δ ) , and sin(Δ ) are high-order small quantities and can be ignored. Therefore, the above equation can be simplified to It can be deduced from the equation that the posture to be adjusted for machining key feature i is: Considering that the absolute position accuracy and direction accuracy of milling features meet the design requirements, an error compensation strategy is proposed. According to the posture error, the NC code of machining feature is modified based on the WCS to realize the posture error compensation of the milling system under the target pose. The specific methods are as follows: (1) the milling target pose is imported into the post processor in the form of NC code as the system input; (2) according to the above error prediction value, the error amount to be compensated for the target pose is determined; (3) after the compensation is completed, the motion command is input to the motion control system in the form of NC code to control the motion of the machine tool to obtain the right pose.

Case study
To validate the feasibility of the proposed comprehensive compensation method, an example is given to show the application. The object of which is an aerospace thin-walled shell part with multiple typical machining features shown in Fig. 2.

The interaction of geometric errors in multi-feature machining
The relationship between machining features and locating/ datum in clamping planning is shown in Table 1. The first clamping and the second clamping share the positioning/ reference ②, and the features ④ and ⑨ processed in the first clamping are the positioning/reference plane of the second clamping. After the first clamping process, there will be machining error. When the second clamping takes the feature with the error as the reference/locating plane, the machining error caused by the first clamping will be transferred and superimposed into the second clamping machining. According to the datum/locating error analysis in Sect. 3, the positioning/datum plane affects the parallelism error of features ④, ⑤, and ⑩, and also affects the verticality of features ⑥, ⑧, and ⑨ in the first clamping of the sample part. Due to the difference in the distance between the feature and the datum, there is a proportional difference in the error caused by the positioning/datum error. In addition, different layouts of positioning elements will be different. Tables 2  and 3 respectively show the error analysis results caused by different features of complex thin-walled parts in datum conversion. Here, all positioning/datum error analysis is carried out in the workpiece coordinate system (WCS).

The interaction of machining errors in multi-feature machining
The three holes (H1 ~ 3) of the sample part are clamped and machined at one time through five-axis end milling, as shown in Fig. 13. The diameter d 1 , d 2 , and d 3 of the three holes are Ф 40 mm,Ф 40 mm, and Ф 25 mm. Using a cemented carbide tool with diameter Ф 12 mm, the three holes are machined with three axial milling forces of 50 N, 100 N, and 150n respectively. The influence of three holes on the coaxiality of existing features (H4 and H5) under different milling forces and machining    Fig. 13. Two circular sections are examined for each hole, and each circular section is fitted by four observation points.
The connecting line of the two center of H 4 is regarded as the reference axis. According to the definition of coaxiality, 2 times (2 × max (t 1 , t 2 )) of the maximum distance from the center of two cross-section circles of H 5 to the reference axis is the coaxiality. The least square method is used to fit the circle.
Assumptions are as follows: (a) no other deformation on the machined feature except clamping and milling; (b) fixtures, machine tools, and cutting tools have good rigidity, that is, the deformation of fixtures, machine tools, and cutting tools is not considered; and (c) the errors are generated by the current processing, and the previous processing quality is regarded as the ideal value. The coaxiality error caused by clamping and cutting deformation is predicted by FEA and verified by experiment using CMMs. The analysis results are shown in Fig. 14.
According to the simulation results shown in Fig. 14, the hole machining sequence of "213" has the least impact on the coaxiality of holes 4 and 5 under the action of milling forces 50 N and 100 N, both of which are reduced by 0.2%. When the milling force is 150 N, the coaxiality of holes 4 and 5 can be reduced by 0.63% in the machining sequence of "312" and "321." It can be seen that the milling force directly affects the machining sequence of features. Different milling forces and reasonable feature machining sequence can reduce the mutual machining influence between features.

Error reduction based on posture adjustment
In this section, the coaxiality of H 4 and H 5 is mainly investigated; the tolerance is 0.2 mm. Since the adjacent feature H 1 , H 2 , and H 3 affect the coaxiality of features H4 and H 5 , the machining analysis is carried out with the machining sequence of 321 and the milling force of 150 N. In the coordinate system shown in Fig. 15, the posture matrix of H 1 ∽H 5 is listed in Table 4. According to the calculation and analysis in Sects. 5.1 and 5.2 and Eq. (29), the final error value after machining the characteristic holes is shown in Table 5. Therefore, the comprehensive coaxiality error of H 4 and H 5 is Thus, it can be seen, the accuracy requirement of the coaxiality cannot meet the tolerance requirements. According to the posture error, the NC code of H5 is modified based on the WCS to realize the posture error compensation of the milling system under the target pose. The coaxiality of the thin-walled part is tested by the equipment three-coordinate measuring instrument. The test results are shown in Table 6. Through the coaxiality test value, it can be seen that the coaxiality prediction of H4 and H5 is relatively close to the experimental value, which proves the effectiveness of the prediction algorithm. The coaxiality error after compensation is significantly reduced, and the accuracy is improved by 53.69%, which can meet the machining requirements, indicating that the compensation algorithm is also effective.

Conclusions
In this study, a comprehensive error model is established to predicting coupling effects between features caused by machining, mainly on the final machining error of features. By analyzing the influencing factors between features, two coupling modes are determined: geometric interaction and machining interaction. On this basis, a comprehensive error model is developed by combing the theory of SDT and homogeneous coordinate conversion. The control actions were determined using the method of adjusting posture according to the comprehensive error model.
In order to verify the accuracy and effectiveness of the error model and control method, a case of a thin-walled part was presented. In the case study report, the geometric interaction and machining interaction in the multi-feature machining process of thin-walled parts are analyzed respectively, and the coupling error prediction value is given. According to the error prediction value, the corresponding error control method is implemented. The results show that the method in this paper can effectively improve the machining accuracy of features. The experiment results show that the machining accuracy of the feature match well with the really measured ones, and the error is reduced by 47.21% with the help of the proposed control method.
Author contribution Chengjun Zhang contributes to investigation, data curation, formal analysis, and writing original draft. Xiaoyan Zuo contributes to conceptualization, methodology, and writing-review and editing. Chi Zhang contributes to investigation. Xiangyang Zhou contributes to supervision. Yin Zhou and Tao Xiong particularly contribute to reviewing the simulation works in Sect. 4.