An approach to enhancing machining accuracy of five-axis machine tools based on a new sensitivity analysis method

Identification of key geometric errors is an essential prerequisite for improving the machining accuracy of five-axis machine tools. This paper presents a new sensitivity analysis (SA) method to extract key geometric errors, and then to improve the machining performance of machine tools by compensating key geometric error components. Development of geometric error prediction model is involved to obtain geometric error values at arbitrary positions at first. Based on the multi-body system theory and flank milling theory, the machining error model is developed, which considers 37 geometric errors. Then, a new SA method is introduced by taking the machining error model as sensitivity analysis model and taking the geometric errors as analytical factors. Meanwhile, a sensitivity index, which has the characteristics of simple expression and clear physical meaning, is proposed, i.e., the peak value of the machining error caused by each geometric error. Moreover, the simulations analysis is carried out to obtain the sensitivity coefficient of each geometric error and the key error components. Finally, the validity and correctness of the proposed method are demonstrated by the experiments. Furthermore, the SA method can be extended to multi-axis machine tools.

NC machine tools, as mother machine in the industrial field, have become the embodiment of machinery manufacturing capability and development level for a country [1][2][3]. The machining accuracy of NC machine tool is an important index to measure machine tools' performance and an important symbol of the country's technological level. Thus, how to improve the machining accuracy of machine tools has become the focus of research. Until now, the accuracy optimization design and error compensation are commonly used methods for improving the performance of machine tools.
Accuracy optimization design is a way to improve the machining accuracy by reasonably distributing geometric errors to the key parts of machine tools [4][5][6][7][8]. It is widely known that the degree of influence for each geometric error on machining accuracy has significant differences. If the geometric errors are assigned based on design experience, it will cause the distortion of the optimization results. Therefore, distributing weight to each geometric error reasonably is the prerequisite for the accuracy optimization design. In addition, error compensation is another way to improve the machining accuracy by compensating the key geometric errors [9][10][11][12][13]. In economic terms, it is unreasonable to compensate all geometric errors. Only the key geometric error that has been compensated is a cost-effective manner to improve the machining accuracy. Therefore, it is an important issue to obtain the weight coefficient of each geometric error and make sure which vital geometric errors need to be compensated.
As the most effective way to quantify the impact of geometric error on the machining accuracy and extract the key error components, the sensitivity analysis (SA) has been widely applied and developed at present. In the last several decades, the abundant intensive research works about SA have been carried out.
Niu et al. [14] employed the form of sequential multiplication to establish the global SA model of error components. The influence of each error component on machining precision in a certain stroke was discussed. Wu et al. [15] selected F-value results and the Euclidean norms of parametric test results as sensitivity coefficients to identify the key geometric errors. The identified results provide the references for the accuracy design of machine tools. Based on the error mapping Jacobian matrix, a series of sensitivity indices were presented in [16] to trace 10 key error components and 10 trivial error components. Yang et al. [17] utilized the 3D geometric sensitivity analysis to obtain the influences of the 11 position-independent geometric errors on each of the measurement paths. The Monte Carlo method and kriging were integrated in [18] to develop a SA method for milling accuracy. And then, the impact of each parameter distribution on the milling accuracy was quantified. The local and global SA of reliability was introduced in [19]. The sensitivity coefficients were calculated by taking only one machining point as an example to determine the key error components. Fang et al. [20] developed a new SA model by utilizing the Lie theory. Due to the units of angular error and linear error are different, the sensitivity coefficients of two kinds errors were calculated and compared respectively. The key geometric errors were identified, which applied in accuracy design and error compensation. Zhang et al. [21] established the machining error model based on homogeneous transformation matrices to describe the impact of deformation error on the machining error. Then, the sensitivity coefficient of deformation errors is calculated based on the machining error model. The SA of 37 geometric errors was presented in [22]. The position errors and angular errors were set as fixed value, i.e., 0.1 μm and 0.1 μrad respectively. And the sensitivity coefficients were calculated by taking only three machining points as an example to determine the key error components. Yao et al. [23] taken the derivatives of volumetric error as the sensitivity coefficient of each geometric error. Simultaneity, for computational expediency, assuming all the position errors and angular errors are set as fixed value, i.e., 10 μm and 5″ respectively. The machining performance of machine tools was improved by reducing the identified key error components. Guo et al. [24] proposed the Morris global sensitivity analysis (GSA) method, and the absolute mean values and the standard deviation are utilized to calculate the sensitivity coefficients of the error parameters. The SA results can provide guidance for error compensation. Tian et al. [25] developed the probabilistic sensitivity analysis method and defined global sensitivity index to extract the key error components. Extended Fourier amplitude sensitivity test method was utilized in [26] to establish GSA model. The crucial error components were identified based on the GSA model, and the machining accuracy of CNC machine tools was improved by compensating crucial error components.
Although the above-mentioned researches have obtained some positive achievements, the existing researches are plagued with the following problems: 1. Some of these SA methods based on the assumption that all the angular errors and linear errors have the same value, which cannot reflect the key factor that the geometric error varies with machining positions and can make the SA results inaccurate or even unreliable. Thus, the SA results failed to correctly guide the accuracy optimization design and error compensation of machine tools. 2. The units of angular error and linear error are different, which need to calculate its sensitivity separately. Hence, such scenario will cause these SA methods involve too much calculation. 3. The sensitivity index of each geometric error is defined unreasonable, which lead to these SA methods with considerable computational complexity and poor universality.
To rectify the insufficiency of above-mentioned researches, this paper presents a new SA method to quantify the impact of each geometric error on the machining accuracy and extract the key error components as outlined in the logical flowchart shown in Fig. 1. First, in order to effectively figure out each geometric error at any machining position, the geometric error prediction models are established. In addition, the machining error prediction model is developed based on the multi-body system theory and flank milling theory. Then, the sensitivity analysis model is established based on the machining error prediction model. For computational expediency, the peak value of the machining error caused by each geometric error is taken as the sensitivity index, which has the characteristics of simple expression and clear physical meaning. Meanwhile, the sensitivity of each geometric error and the key error components are obtained. Finally, an experiment is carried out to prove the validity and correctness of the proposed method.
Henceforth, the structure of this paper is organized as follows. Section 2 establishes the machining error model. In Section 3, the new SA method is developed to extract the key geometric errors. The simulation verification is conducted in Section 4. For demonstrating the correctness of the proposed method, the experimental validation is carried out in Section 5. Finally, some conclusions are given in Section 6.

Geometric error modeling
Geometric errors, which are caused by the manufacturing errors and assembly errors of machine tools' key parts, are one of the key influence factors for machine tools' machining accuracy. According to the ref. [27], a five-axis machine tool with three linear axes and two rotary axes has 37 geometric errors, which consisted of position-independent geometric errors (PIGEs) and position-dependent geometric errors (PDGEs), as listed in Table 1. Figures 2 and 3 display the schematic diagram of PDGEs for X-axis and C-axis respectively. Figure 4a shows the schematic diagram of PIGEs for linear axis, and Fig. 4b represents PIGEs for rotary axis.
Since the geometric error is different along with the change of machining positions, assuming all the angular errors and linear errors have the same value will lead to the sensitivity analysis results with poor universality. Therefore, obtaining geometric error values at any position in the entire workspace is an essential step to carried out the sensitivity analysis. First, the geometric errors of the linear axis are identified by the laser interferometer (XL-80 by Renishaw), and the geometric errors of the rotary axis are identified by double ball bar (QC20-W by Renishaw). Then, the Fourier curve fitting method is utilized to establish the geometric error model based on the identified results. Due to limited space, only the modeling results of geometric errors for X-axis are finally displayed in Eq. (15) in Appendix 1.

Volumetric error modeling
To date, the multi-body system (MBS) theory is the most famous and widely used volumetric error modeling method because of its higher applicability and generality [27,28]. In this research, a gantry type five-axis NC milling machine is taken as research objective to establish the volumetric error modeling, which consists of X-, Y-, Z-, B-, and C-axis, as displayed in Fig. 5. Based on MBS theory, the simplified analysis of gantry type five-axis NC milling machine is carried out. And the topological structure of the machine tool is established, which consists of tool branch and workpiece branch, as shown in Fig. 6. Then, the static and motion transformation matrices between the adjacent bodies of machine tool are established based on the homogeneous coordinate transformation, as shown in Tables 6 and 7 in Appendix 2, respectively. The position vector of the body reference coordinate system origin of each body is listed in Table 8 in Appendix 2.

Conclusions
Fourier curve fitting method In the actual machining process, the position vector of the tool center point in the workpiece coordinate system can be obtained, as shown in Eq. (1).
where 1-6 denote the key parts of gantry type five-axis NC milling machine, as displayed in Fig. 5; w denotes the workpiece; t denotes the tool; i j T p and i j T pe denote the ideal and actual static transformation matrices between the adjacent bodies of machine tool; i j T s and i j T se denote the ideal and actual motion transformation matrices between the adjacent bodies of machine tool; and r t denotes position vector of tool center point in tool coordinate system, i.e., r t = (0, 0, −l, 1) T , where l denotes the tool length.
In the ideal machining process, i.e., each geometric error value is zero, the position vector of the tool center point in the workpiece coordinate system can be obtained, as shown in Eq. (2).
Ultimately, based on MBS theory, the volumetric error model can be calculated by subtracting Eq. (2) from Eq. (1), as shown in Eq. (3).
By ignoring the high-order terms, the volumetric error model of gantry type five-axis NC milling machine is obtained.

Machining error modeling
Flank milling, which processing the machined surface with the tool side edge, is a kind of processing method to process complex curved surface. The schematic diagram of flank milling is shown in Fig. 7. The machining error is determined by the normal distance between the ideal cutting surface and the actual cutting surface, as shown in Fig. 8.
According to Eq. (1) and Fig. 7, the actual position vector of the tool center point corresponding to the kth cutting point in the workpiece coordinate system can be obtained, as shown in Eq. (4).
(1)  where d represents the tool radius and n sjk represents unit normal vector of the kth cutting point.
According to Eq. (5) and Fig. 8, ideal cutting point of machined workpiece can be obtained by taking the point of ideal tool axis trajectory plane offset the tool radius along the unit normal vector of the kth cutting point, as shown in Eq. (7).
According to Fig. 9, the machining error can be calculated, as shown in Eq. (8).
where e v (v = x, y, z) denotes the component of machining error in the v-direction. Therefore, the machining error model of gantry type five-axis NC milling machine is obtained, as shown in Eq. (16) in Appendix 3.

Sensitivity analysis of geometric error components
The impact of different geometric errors on machining errors is widely different. To quantify the impact of each geometric error on machining error and trace the key geometric errors, the sensitivity analysis (SA) of geometric error was carried  (9) e = e x , e y , e z , 0 T out. However, the existing SA methods, as described in Sect. 1, are plagued with the above-mentioned problems. In order to tackle the above deficiencies, this paper has developed a new SA method, which by taking the machining error model as sensitivity analysis model and taking the geometric errors as analytical factors, to trace the key geometric errors effectively.
In this study, the peak value of the machining error, which can truly reflect the impact of each geometric error on machining error of machine tools, is firstly proposed to represent the sensitivity index of geometric error. The greater the sensitivity of each geometric error, the greater the impact of each geometric error on machining error.
According to Eq. (9), the general form of machining error model can be established, as shown in Eq. (10).
where G denotes the geometric error vector composed of n-term geometric errors of NC machine tools, G = g 1 , g 2 , ..., g n T ; g i denotes the ith geometric error of   where i represents the ith geometric error, i.e., i = 1, 2, 3, ..., 37.
Therefore, according to Eq. (11), the sensitivity of each geometric error can be defined as: In order to quantify the impact of geometric error on the machining accuracy and extract the key geometric errors more intuitively, the sensitivity coefficient of each geometric error is obtained by normalizing the sensitivity of each geometric error, which is shown as: Therefore, the sum of the normalized sensitivity coefficient of all the error components is 1, as shown in Eq. (14).

Simulations
To obtain the SA results, a simulation is carried out on the researched gantry type five-axis NC milling machine. First, the impact of each geometric error on machining error can be obtained by employing the machining error model (i.e., Eq. (11)). Then, the SA is carried out to obtain the sensitivity coefficient of each geometric error and extract the key geometric errors.

The impact of each geometric error on machining error
The "S" shaped test specimen, which possesses the characteristics of the twist angle, open-close angle conversion, and variable curvature, is widely applied for characterization of complex curved surface. And its profile error can really The trajectory surface of actual tool axis The trajectory surface of ideal tool axis The actual cutting surface The key parameters of machine tool and the cutting parameters of test specimen The identified geometric errors results represent the machining error of machine tools. Therefore, the "S" shaped test specimen is selected as the machined workpiece in this research.
To qualitatively analyze the impact of each geometric error on machining error for the researched machine tool, simulation analysis is conducted by MATLAB R2016b. The f lowchar t of the simulation is outlined in Fig. 10. And the procedures to fulf ill machining er rors prediction are shown as follows: Fig. 11 The flow chart of NC instruction code generation for "S" shaped test specimen

3D model
The post processor

Cutter-axes vector Tool tip trajectory
The cutter location file A B C Table 2 The key parameters of the researched machine tool First, the 3D model of "S" shaped test specimen is built by UG NX10.0, as shown in Fig. 11a. Then, the cutter location file is attained through the UG NX CAM processing module, as shown in Fig. 11b. Finally, on the basic of the cutter location file, the NC machining instruction of "S" shaped test specimen is obtained by the post processor of UG NX 10.0. Part of NC machining instruction codes is displayed in Fig. 11c.

On the basic of geometric error model established in
Section 2.1, the value of geometric errors at arbitrary positions is calculated.
3. The profile errors of "S" shaped test specimen are predicted by utilizing the machining error model and the NC machining instruction.
The key parameters of the researched machine tool are listed in Table 2. Based on the above-mentioned analysis, the influence law of each geometric error on machining error for the researched machine tool can be obtained. Due to limited space, only the influence law of X-axis geometric errors on  Fig. 14 The flowchart of the experiment machining errors is displayed in Fig. 12. From Fig. 12, it can be clearly seen that the influence law changes with different geometric errors and machining positions. Thus, assuming all the angular errors and linear errors have the same value can make the SA results inaccurate or even unreliable.

Identification of key geometric errors
On the basic of the influence law of each geometric error on machining error displayed in Section 4.1 and Eq. (13), the sensitivity coefficient of each geometric error can be obtained, as listed in Table 3. In addition, in order to reflect the SA results more intuitively, the sensitivity coefficient of each geometric error is sorted, as shown in Fig. 13. From Fig. 13 and Table 3, the sensitivity coefficients of the 10th, 17th, 22nd, 24th, 37th, 6th, and 21st geometric error are relatively large (i.e., 0.1527, 0.1510, 0.1392, 0.1300, 0.0498, 0.0388, and 0.0379, respectively), accounting for approximately 70% of total. Therefore, it can be concluded that the 10th, 17th, 22nd, 24th, 37th, 6th, and 21st geometric error is considered as the key geometric error.

Experimental validation
To demonstrate the correctness and validity of the SA method presented in this study, the machining and measurement experiment of "S" shaped test specimen are carried out. The machining experiment is implemented on the researched machine tool. The measurement experiment is conducted on the coordinate measuring machine (CMM) named ZEISS PRISMO navigator. The flowchart of the experiment is shown in Fig. 14.
Compensating the key geometric errors that were extracted by utilizing the SA method proposed in this paper was defined as the compensation scheme 1. Compensating all geometric errors was defined as the compensation scheme 2. Before machining, the corrected NC machining instruction codes based on two different Fig. 17 The detection points of the "S" shaped test specimens compensation schemes are generated by employing the actual inverse kinematics method proposed in [29]. Additionally, the semifinished products of "S" shaped test specimens are the same. A Φ20-mm cylindrical milling cutter is selected to finish milling. Then, based on abovementioned conditions, the machining experiment is conducted, as displayed in Fig. 15. Subsequently, two "S" shaped test specimens are obtained. Finally, the profile errors of two "S" shaped test specimens are measured by the CMM. The measurement scene is shown in Fig. 16. According to Ref. [30], for the sake of simplicity, three detection lines (i.e., L1, L2, and L3) along the Z-direction of the "S" shaped edge strip are selected to test the profile errors of "S" shaped test specimen, as shown in Fig. 17. Twenty-five detection points are distributed uniformly in each detection line. In addition, the indoor ambient temperature is set at 20℃ ± 2℃ [31]. For improving the stability of measurement results, the measurement experiment is repeated by 5 times. The average values of 5 times measurement results are taken as the final profile errors of the machined "S" shaped test specimen, as displayed in Table 4. Figure 18 depicts residual errors of profile errors between two different compensation schemes. From Fig. 18, it can be clearly seen that the vast majority of residual errors are less than 0.01 mm, which are considerably small.
In order to verify the correctness of the proposed SA method more intuitively, the comparisons of the average profile errors for two "S" shaped test specimens are listed in Table 5. From Table 5, in contrast to compensation scheme 1, the average profile error obtained by compensation scheme 2 is reduced from 0.037 to 0.034 mm, 0.033 to 0.030 mm, and 0.042 to 0.037 mm at L1, L2, and L3, respectively. In other words, the average profile error obtained by compensation scheme 2 has decreased by 0.003 mm, 0.003 mm, and 0.005 mm at L1, L2, and L3, respectively, which is considerably small. Summarized above, the comparison results show that there is no significant difference between

Conclusions
This paper proposes a new SA method to quantify the impact of geometric error on the machining accuracy and extract the key geometric errors. By comparison with the existing SA methods [14][15][16][17][18][19][20][21][22][23][24][25][26], the superiorities of the new SA method are listed as follows: (1) A novel sensitivity index is proposed to solve the problem that the SA results of angular error and linear error need compared separately, which significantly simplifies the SA procedures and improves the computational efficiency. (2) The proposed SA model takes into account the characteristic of geometric error changes with different machining positions, which solves the problem that the universality of SA results obtained by assuming all the angular errors and linear errors have the same value are poor. To verify the practicability and validity of the presented SA method, experiments on a gantry type five-axis NC milling machine are conducted. The experimental results show that residual errors are considerably small and can be neglect. And the average profile errors obtained by compensation scheme 2 have decreased by 0.003 mm, 0.003 mm, and 0.005 mm at L1, L2, and L3 respectively when compared with compensation scheme 1. The comparison results show that there is no significant difference between the profile errors obtained by two compensation schemes. Thus, the key geometric errors identified by the proposed SA method have significant effect on the machining accuracy of the researched machine tool. It is therefore reasonable to conclude that the presented SA method is correct and effective. This method is universal and provides a reference for key geometric error identification of multi-axis NC machine tool. Fig. 18 The residual errors of profile errors between two different compensation schemes     1-2 (X-axis)