Position-independent geometric errors (PIGEs), caused by assembly imperfection, affect the machine tool’s accuracy. Moreover, the rotary axes of five-axis machine tools introduce more PIGEs during parts machining, hence, the need to identify them to improve the accuracy. However, the identification of rotary axes centerlines and the use of extension bars, using double ball bar (DBB), introduce additional setup errors and uncertainties during error measurements and sometimes make the process tedious. This work proposes a new method to identify the eight (8) PIGEs in the rotary axes of a five-axis machine tool with a tilting table. The homogeneous transformation matrix (HTM) was used to construct the relative motion matrix to the established coordinate systems and then deduced the error model. Three different DBB installation setups were employed for the identification of the PIGEs in each rotary axis using the planar test, the recorded DBB length data was substituted into the model to obtain the PIGEs. The results show that the proposed PIGEs identification method is accurate and can improve the accuracy of the five-axis machine tool effectively, which shows a good improvement in the manufacturing industry.
Research Article
A new method to identify the Position-Independent Geometric Errors in the Rotary axes of Five-axis Machine Tools
https://doi.org/10.21203/rs.3.rs-1868874/v1
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PIGEs
rotary axis
five-axis machine tool
Double ball bar
The Design of machine parts of diverse shapes and sizes has been one of the major goals in the manufacturing industry. Despite the complexity of the geometry of machining workpieces in various products such as engine blocks, impeller blades, etc., high-precision machine tools are required to attain a satisfactory and high accuracy, surface definition, tolerance, and repeatability, whereas programming effort, amount of fixturing, and the required motions are simplified by multi-axis machines. However, the most relevant measure of performance is accuracy; performance optimization ability by controlling or minimizing error at a maintaining cost is very important in the machine tool industry [1]–[5]. The five-axis machine tool has two rotary axes in addition to the three linear axes of the conventional three-axis machine. There should be the capability of the rotary device to support the weight of the part and/or the fixture, which is a relevant feature in considering rapid movements. However, the two rotary axes quickly introduce more errors (random or/and systematic errors) which are more difficult to identify, eliminate, or compensate for than the linear axes [6], [7].
In micro-features machining, the accuracy is affected by four main error sources: thermal errors, force-induced errors (FIEs), geometric errors, and control system errors [8], [9]. Though the thermal and geometric errors contribute about 70% of the total machine errors, however, 25–50% of the total machine errors are alone contributed by the geometric errors making it to be the principal error source during machining [10]. The geometric errors occur due to axes component shift and axes line shift, the deformation of the machine components (cutting tool, machine body, axes, spindle holder, and spindle) causes the FIEs which continuously varies as a result of the variations in the cutting conditions, and the machine tool and its ambient temperature (during operation) contribute to the thermal errors [8]–[11].
The geometric errors are classified into (i) Position-independent geometric errors (PIGEs): also known as the link errors are caused by imperfections in the machine assembly, hence, they are consistent, systematic, and significant (ii) Position-dependence geometric errors (PDGEs): motion axis or the controlled axis caused by component defects [10], [12]–[14]. There are 13 PIGEs in a five-axis machine tool consisting of three squareness errors in the three linear axes, two squareness/angular, and two-position/offset errors in each of the rotary axes, and two squareness errors in the spindle axis for machine tools with tilting head and 11 PIGEs for those with tilting rotary table [15], [16].
The PIGEs contribute more than 70% of the total machine tool’s inaccuracy [46], henceforth, identifying these errors in the rotary axes of the five-axis machine tool could drastically improve the machine tool accuracy. Several measuring methods have been proposed to measure the PIGEs of machine tools; the cone test piece was machined to identify 11 PIGEs of five-axis machine tools with a tilting head, moreover, the non-cutting moves of the cone-frustum and S-shaped parts using DBB have been employed to measure machine tools’ PIGEs which simultaneously move all the motion axes, and the individual errors are decoupled computationally which sometimes takes time. Other measuring devices have been used in the measurement of PIGEs including the laser interferometer, the touch-trigger probe, and the binocular-vision-based error detection system [17]–[25]. Currently, the DBB is widely used in error measurements due to its low cost, easy installation, and measurement stability compared to other devices [26]; it is mainly used for the calibration of the rotary axes of machine tools, kinematic calibrations [27], and evaluation of feed-axes pairs [28]. The DBB circular or planar test, synchronized movements of at least a single rotary axis and with or without one or two of the linear axes, is the most widely used method to identify the PIGEs of the rotary axes [29], [30]. By simplifying the rotary axes’ PIGEs identification methods proposed in the literature, the concept of placing at least one of the DBB balls on a rotary reference axis line by first identifying the rotary axes’ centerline, and/or the use of extensions bars are mostly employed [13], [15], [31]–[42]. Methods of moving the C- and A-axes of a tilting type simultaneously, and moving them with at least one of the linear axes [42]–[45] have been proposed. The methods involving accurately identifying the rotary axes' centerline are very difficult and time-consuming, and the use of extension bars could affect the stiffness of the ball bar. Though the DBB is a high-precision measuring device, however, using the aforementioned method could introduce measurement uncertainties and additional errors like setup errors, hence, eliminating these additional error sources during measurement would facilitate the effectiveness of the error measurement processes
This work focuses on the characterization and identification of the eight PIGEs in the rotary axes of a tilting rotary table of a five-axis machine tool using a newly proposed error identification method. Three measurement patterns/setups are designed for each rotary axis, and only one rotary axis is driven in each test by changing the position of the balls; the workpiece ball is driven while the tool cup call is kept stationary, this is to decouple the PIGEs in each rotary axis. Moreover, the fixturing in this method is very simple, requiring less setup time, hence, minimizing the set-up errors. This paper is outlined as follows: Section 2 describes the machine structure and the established coordinate systems, Section 3 discusses the kinematic error model and analysis of the five-axis machine tools based on the measuring method proposed in this work, Section 4 discusses the experimental procedures, the Results is discussed in Section 5, and the work is concluded in Section 6.
2.1 Machine Structure
The geometric structure of the five-axis machine tool is shown in Fig. 1 (a). It has two kinematic chains: the tool (bed-Y-axis-X-axis-Z-axis-tool) and the workpiece chain (bed-A-axis-C-axis-workpiece). The coordinate systems established are the A-axis, C-axis, and reference coordinate systems
2.2 Coordinate Systems
Four coordinate systems are set without considering those of the linear axes since the focus is on the two rotary axes. as shown in Fig. 1 (b), they are:
- Reference Coordinate System (RCS): RCS is set in the workspace arbitrarily (0,0,0).
- The Workpiece Coordinate System (WCS): Attached to the workpiece ball.
- The Tool Coordinate System (TCS): Attached to the DBB ball fixed on the tool cup.
- Rotary Table-axis Coordinate System (RTCS): An arbitrary position on the C-axis worktable close to the centerline (centerline is assumed, not measured or identified).
P: A point where the workpiece ball is positioned in the workspace. r: The distance, on the x-axis, from point P to the RTCS. H: The vertical distance between the RTCS and the RCS. d: The vertical distance between the RCS and TCS. z: The vertical distance between the WCS and point P.
Table 1 Parameter definition of PIGEs
3.1 Error modeling
In this study, the homogeneous transformation matrix (HTM) theory is used to develop the mathematical model of the five-axis machine tools (TTTRR).
$${\text{n}}_{{\text{T}}_{\text{i}}= {0}_{{\text{T}}_{1}} {\text{n}}_{2}\dots {\text{i}-1}_{{\text{T}}_{\text{i}}} \text{i},\text{n}=\text{0,1},2,\dots ,\text{i}.}$$
1
Where \({{n}}_{{{T}}_{{i}}}\) represents the transformation from i coordinate to n. All the linear axes are set to identity matrices since only the motions of the rotary axes (workpiece chain) are considered; Table 1 shows the parameter representation of PIGEs to be measured. Based on the kinematic and error chain of the worktable, the actual position of the workpiece ball relative to the RCS could be expressed as:
$${}_{W}{}^{R}T={}_{A}{}^{R}{T}_{actual}{}_{C}{}^{ A}{T}_{actual }{}_{P}{}^{C}T {}_{W}{}^{P}T$$
2
$${}_{A}{}^{R}{T}_{actual}=\left(\begin{array}{cccc}1& -{S}_{\text{z}\text{a}}& {S}_{\text{y}\text{a}}& 0\\ {S}_{\text{z}\text{a}}& 1& 0& {\sigma }_{\text{y}\text{a}}\\ -{S}_{\text{y}\text{a}}& 0& 1& {\sigma }_{\text{z}\text{a}}\\ 0& 0& 0& 1\end{array}\right)\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& \text{c}\text{o}\text{s}\left({\theta }_{a}\right)& -\text{s}\text{i}\text{n}\left({\theta }_{a}\right)& 0\\ 0& \text{s}\text{i}\text{n}\left({\theta }_{a}\right)& \text{c}\text{o}\text{s}\left({\theta }_{a}\right)& H\\ 0& 0& 0& 1\end{array}\right)$$
3
$${}_{C}{}^{A}{T}_{actual}=\left(\begin{array}{cccc}1& 0& {S}_{\text{y}\text{c}}& 0\\ 0& 1& -{S}_{\text{x}\text{c}}& 0\\ -{S}_{\text{y}\text{c}}& {S}_{\text{x}\text{c}}& 1& 0\\ 0& 0& 0& 1\end{array}\right)\left(\begin{array}{cccc}\text{c}\text{o}\text{s}\left({\theta }_{c}\right)& -\text{s}\text{i}\text{n}\left({\theta }_{c}\right)& 0& 0\\ \text{s}\text{i}\text{n}\left({\theta }_{c}\right)& \text{c}\text{o}\text{s}\left({\theta }_{c}\right)& 0& 0\\ 0& 0& 1& H\\ 0& 0& 0& 1\end{array}\right)$$
4
$${}_{P}{}^{C}T =\left(\begin{array}{cccc}1& 0& 0& r\\ 0& 1& 0& b\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right)$$
5
$${}_{W}{}^{P}T=\left(\begin{array}{c}0\\ 0\\ z\\ 1\end{array}\right)$$
6
$${}_{T}{}^{R}T={\left[\begin{array}{cc}0& \begin{array}{ccc}0& d& 1\end{array}\end{array}\right]}^{T}$$
7
Therefore, the recorded length of the DBB, \({L}_{BB}\) is:
$${L}_{BB}=‖{}_{W}{}^{R}T-{}_{T}{}^{R}T‖$$
$$=\sqrt{\begin{array}{c}{\left|\begin{array}{c}d-H-{\sigma }_{\text{z}\text{a}}+z\hspace{0.17em}\left({S}_{\text{y}\text{a}}\hspace{0.17em}{S}_{\text{y}\text{c}}-1\right)+{S}_{\text{y}\text{a}}\hspace{0.17em}\left({\sigma }_{\text{x}\text{c}}+H\hspace{0.17em}{S}_{\text{y}\text{c}}\right)-b\hspace{0.17em}\left({S}_{\text{x}\text{c}}\hspace{0.17em}\text{cos}\left({\theta }_{c}\right)+{S}_{\text{y}\text{a}}\hspace{0.17em}\text{sin}\left({\theta }_{c}\right)+{S}_{\text{y}\text{c}}\hspace{0.17em}\text{sin}\left({\theta }_{c}\right)\right)\\ +r\hspace{0.17em}\left({S}_{\text{y}\text{a}}\hspace{0.17em}\text{cos}\left({\theta }_{c}\right)+{S}_{\text{y}\text{c}}\hspace{0.17em}\text{cos}\left({\theta }_{c}\right)-{S}_{\text{x}\text{c}}\hspace{0.17em}\text{sin}\left({\theta }_{c}\right)\right)\end{array}\right|}^{2}\\ +{\left|\begin{array}{c}{\sigma }_{\text{x}\text{c}}-r\hspace{0.17em}\left({S}_{\text{y}\text{a}}\hspace{0.17em}\left({S}_{\text{y}\text{c}}\hspace{0.17em}\text{cos}\left({\theta }_{c}\right)-{S}_{\text{x}\text{c}}\hspace{0.17em}\text{sin}\left({\theta }_{c}\right)\right)-\text{cos}\left({\theta }_{c}\right)+{S}_{\text{z}\text{a}}\hspace{0.17em}\text{sin}\left({\theta }_{c}\right)\right)-{S}_{\text{z}\text{a}}\hspace{0.17em}\left({\sigma }_{\text{y}\text{c}}-H\hspace{0.17em}{S}_{\text{x}\text{c}}\right)\\ -b\hspace{0.17em}\left(\text{sin}\left({\theta }_{c}\right)-{S}_{\text{y}\text{a}}\hspace{0.17em}\left({S}_{\text{x}\text{c}}\hspace{0.17em}\text{cos}\left({\theta }_{c}\right)+{S}_{\text{y}\text{c}}\hspace{0.17em}\text{sin}\left({\theta }_{c}\right)\right)+{S}_{\text{z}\text{a}}\hspace{0.17em}\text{cos}\left({\theta }_{c}\right)\right)+z\hspace{0.17em}\left({S}_{\text{y}\text{a}}+{S}_{\text{y}\text{c}}+{S}_{\text{x}\text{c}}\hspace{0.17em}{S}_{\text{z}\text{a}}\right)+H\hspace{0.17em}{S}_{\text{y}\text{a}}+H\hspace{0.17em}{S}_{\text{y}\text{c}}\end{array}\right|}^{2}\\ +{|{\sigma }_{\text{y}\text{a}}+{\sigma }_{\text{y}\text{c}}+{S}_{\text{z}\text{a}}\hspace{0.17em}\left({\sigma }_{\text{x}\text{c}}+H\hspace{0.17em}{S}_{\text{y}\text{c}}\right)-z\hspace{0.17em}\left({S}_{\text{x}\text{c}}-{S}_{\text{y}\text{c}}\hspace{0.17em}{S}_{\text{z}\text{a}}\right)+b\hspace{0.17em}\left(\text{c}\text{o}\text{s}\left({\theta }_{c}\right)-{S}_{\text{z}\text{a}}\hspace{0.17em}\text{s}\text{i}\text{n}\left({\theta }_{c}\right)\right)+r\hspace{0.17em}\left(\text{s}\text{i}\text{n}\left({\theta }_{c}\right)+{S}_{\text{z}\text{a}}\hspace{0.17em}\text{c}\text{o}\text{s}\left({\theta }_{c}\right)\right)-H\hspace{0.17em}{S}_{\text{x}\text{c}}|}^{2}\end{array}}$$
(8)
3.2 Levenberg-Marquardt circle fitting algorithm
The recorded DBB length data is converted to \({(y}_{i},{z}_{i}) \text{a}\text{n}\text{d} \left({x}_{i},{y}_{i}\right), i=\text{1,2},3,\dots N,\)for A-axis and C-axis respectively. Finding the best fitting circle to observe the data point\(,\) involves the minimization of the Euclidean distance from the points to the curve [47], [48], in the C-axis:
$${d}_{i}=\sqrt{{({x}_{i}-a)}^{2}+ {({y}_{i}-b)}^{2}}-R$$
9
$$\mathcal{F}\left(a,b,r\right)=\sum _{i=1}^{N}{d}_{i}^{2}$$
10
Where (\(a,b)\) is the center and r is the radius of the circle. With these shorthand notations, the Levenberg-Marquardt runs smoothly:
$${r}_{i}=\sqrt{{({x}_{i}-a)}^{2}+ {({y}_{i}-b)}^{2}}$$
11
$${u}_{i}=\frac{-\partial {r}_{i}}{\partial a}= \frac{{({x}_{i}-a)}^{2}}{{r}_{i}}$$
12
$${y}_{i}=\frac{-\partial {r}_{i}}{\partial b}= \frac{{\left({y}_{i}-b\right)}^{2}}{{r}_{i}}$$
13
$$J= \left[\begin{array}{ccc}-{u}_{i}& -{v}_{i}& -1\\ ⋮& ⋮& ⋮\\ {-u}_{N}& -v& -1\end{array}\right]$$
14
$$\mathcal{N}=N \left[\begin{array}{ccc}\stackrel{-}{uu}& \stackrel{-}{uv}& \stackrel{-}{u}\\ \stackrel{-}{uv}& \stackrel{-}{vv}& \stackrel{-}{v}\\ \stackrel{-}{u}& \stackrel{-}{v}& 1\end{array}\right]$$
15
$${J}^{T}g =\text{N}\left[\begin{array}{c}R\stackrel{-}{u}-\stackrel{-}{ur}\\ R\stackrel{-}{v}-\stackrel{-}{vr}\\ R-\stackrel{-}{r}\end{array}\right]=\text{N}\left[\begin{array}{c}R\stackrel{-}{u}\mp a-\stackrel{-}{x}\\ R\stackrel{-}{v}+b-\stackrel{-}{y}\\ R-\stackrel{-}{r}\end{array}\right]$$
16
Where \(\stackrel{-}{uu}= \frac{1}{N}\varSigma {u}_{i}^{2},\stackrel{-}{uv}= \frac{1}{N}\varSigma \left(uv\right)\), etc.
$$g={[{r}_{1}-R,\dots ,{r}_{N}-R]}^{T}$$
17
The Levenberg-Marquardt circle fit algorithm runs as:
However, to avoid computational problems in the Cholesky factorization, a numerically stable technique that provides ultimate accuracy is recommended [48], that is,\({J}_{{\lambda }}\text{\hslash }={g}_{0}.\)
4.1 Installation of DBB and PIGEs identification
The proposed PIGEs identification method is validated on an industrial five-axis machine tool (MT) with a Siemens 840D controller. Each setup is repeated 2–3 times, and the mean errors and standard deviations are computed. The approximated time for the whole process including the experimental setup is less than 2 hours. The machine tool is warmed up for 20 minutes before beginning the test as recommended in [49].
The DBB was set up on a five-axis machine tool worktable. As shown in Fig. 2, the workpiece ball is put in the pivot tool cup fixed on the rotary table, and the spindle tool cup holds the spindle ball of the DBB. A dial indicator was used to minimize the misalignment of the two balls from their respective cups introduced by the setup errors. The XY and YZ planar tests were conducted to obtain the DBB length variation data of the C- and A-axis respectively, and further analyses were done to obtain their respective PIGEs; Table 2 shows the DBB setup specifications.
| Parameters | Values |
|---|---|
| DBB Type | Renishaw QC20-W |
| DBB Nominal length (mm) | 100.00 |
| DBB Calibrated length (mm) | 99.9957 |
| Testing feed rate (mm/min) | 500 |
| Dial indicator tolerance (µm) | 1.00 |
4.2 Experimental Procedure
Only the PIGEs of the rotary axes (A and C) are measured in this work. Hence, the linear axes errors are ignored or assumed compensated for. There are three (3) different setups for the identification of PIGEs in each rotary as shown in Fig. 3 and Fig. 4; each setup for the A-axis and C-axis was repeated three and two times respectively, and data were taken at every 5o step angle.
A-axis: The C-axis is kept stationary whiles the A-axis was rotated from − 30 to 30o. Setup 1: The RCS is set, and the workpiece ball is fixed in the workspace at point P at a distance, r (on the X-axis). The spindle/tool ball is fixed on the tool cup as shown in Fig. 2 (a). Setup 2: The workpiece ball system is moved to position (r,b). Setup 3: The workpiece ball system is placed on a platform to elevate the workpiece ball (increasing z, the distance from point P to the WCS), and the spindle ball is moved vertically to distance d, hence, the spindle ball position is (0,0,d).
C-axis: The A-axis is kept stationary whiles the C-axis is rotated from 0o to 360o. Setup 1: The RCS is set, and the workpiece ball is fixed in the workspace at point P at a distance, r (on the X-axis). Setup 2: The workpiece ball and spindle ball are aligned vertically by moving the spindle ball up at a distance of d (on the Z-axis), making the spindle position (0,0, d). Setup 3: The workpiece ball system is moved at distance r (on the X-axis) whiles the spindle ball is kept at d (on the Z-axis) keeping the spindle ball at position (0,0, d). Table 3 shows the parameter values set for all setups for both rotary axes as shown in Fig. 2 (b).
The two positional PIGEs in each of the rotary axes were computed using the Levenberg-Marquardt circle fitting algorithm and their orientational PIGEs were computed by using an optimization technique to solve Eq. (8), all mathematical computations were done using MATLAB software.
The presence of the PIGEs affects the rotary axes of the five-axis machine tool, hence, the actual motion of the axes deviates from the ideal. Therefore, the actual DBB length data recorded for each setup (black color) varies from the nominal length (green color) for both rotary axes as shown in Fig. 5 and Fig. 6 with the fitted circles (red color).
| A-axis | |||
|---|---|---|---|
| Setup Parameter | Setup 1 (mm) | Setup 2 (mm) | Setup 3 (mm) |
| r | -100 | -60 | -100 |
| H | -70.33 | -70.33 | -70.33 |
| b | 0 | -80 | 0 |
| z | -70.33 | 70.33 | 124.33 |
| d | 0 | 0 | 54 |
| C-axis | |||
| Setup Parameter | Setup 1 (mm) | Setup 2 (mm) | Setup 3 (mm) |
| r | -100 | 0 | -60 |
| H | -70.33 | -70.33 | -70.33 |
| b | 0 | 0 | 0 |
| z | 70.33 | 68.33 | 70.33 |
| d | 0 | 98 | 80 |
The DBB length data recorded for the A-axis rotary table show low variabilities from the ideal/nominal length and were almost the same for the three setups. However, the variabilities were high in the setups for the C-axis, and this could be due to the wear of the C-axis rotary table.
The length data recorded for setups 1 and 3 show much deviation from the ideal length compared to setup 2, and this is due to the workpiece ball being positioned at a distance from the axis’ centerline, hence, the closer the setup (the workpiece ball) to the C-axis centerline as shown in Fig. 4 (b), the less the variation in length from the ideal and vice versa; the results are shown in polar plots in Fig. 5. The C-axis offset errors are magnified when the workpiece ball is far from the axis centerline, but the orientation errors are higher when close to the axis centerline. Hence, the orientation errors in the C-axis are higher for setup 2 whiles the offset errors are higher for setups 1 and 3 as shown in Table 4.
The proximity of the workpiece ball to the A-axis centerline showed the same effect on the DBB length variations in each setup as shown in Fig. 6, however, both the offset and orientational PIGEs are higher in setup 3 and relatively lower in setup 1 as shown in Table 4
This shows that some of the PIGEs in the rotary axes are magnified or abased based on the proximity of the setup (workpiece ball) to the axes' centerline. Hence, this method could be used to identify the PIGEs in the full workspace of the rotary axes of five-axis machine tools with a tilting table. Moreover, the newly proposed method for identifying the PIGEs in the rotary axes follows the experimental guidelines recommended by ISO [28], [50], hence, the results obtained in this work are accurate and valid. The results are summarized in Table 4.
| Error Parameter | Setup 1 | Setup 2 | Setup 3 | Mean | Standard Deviation |
|---|---|---|---|---|---|
| Eya (mm) | -0.0347 | -0.0380 | -0.0390 | -0.0372 | 0.0022 |
| Eza (mm) | 0.0028 | 0.0029 | 0.0032 | 0.0030 | 0.0002 |
| Tya (o) | -0.00004 | -0.00005 | 0.00053 | 0.0002 | 0.0003 |
| Tza (o) | -0.0004 | -0.0002 | -0.0220 | 0.0030 | 0.0125 |
| Exc (mm) | -0.0065 | -0.0067 | -0.0243 | -0.0125 | 0.0102 |
| Eyc (mm) | 0.3562 | 0.0035 | 0.2142 | 0.1913 | 0.1774 |
| Txc (o) | -0.0201 | -0.4280 | -0.0005 | -0.1493 | 0.2423 |
| Tyc (o) | -0.0010 | -0.4642 | -0.0001 | -0.1551 | 0.2677 |
This work presents a new measuring pattern to identify the eight (8) PIGEs in the rotary axes of a five-axis machine tool using the DBB planar test; the method was validated on a five-axis machine tool with Siemens 840D controller (TTTRR configuration). Three different measuring patterns were proposed to identify the PIGEs in each rotary axis. The A-axis was first measured since the C-axis rotary table is mounted on the C- axis tilting table. The recorded DBB length data collected were analyzed and the PIGEs errors were computed using MATLAB software.
In conclusion, the benefits of the new identification method proposed in this study are:
-
Decoupling the PIGEs in each rotary axis is easy in this method since a single rotary axis is controlled during the test.
-
This new method is easy and eliminates additional errors and uncertainties since no extension bar and rotary axis centerline are required.
-
Moreover, this method could be used to explore the full workspace of the rotary axes since PIGEs identification could be affected by the setup (the workpiece ball) proximity to the axis’ centerline.
-
Other types of five-axis machine tools with rotary axes in the workpiece chain can employ this new method as well.
Therefore, this newly proposed method can be sufficiently used as an identification method to detect the PIGEs in the rotary axes of five-axis machine tools with tilting tables, and it is also applicable to the standard accuracy testing of the five-axis machine tools in the manufacturing industry due to its effectiveness.
Acknowledgments The author would like to thank the National Natural Science Foundation of China 52175456, CAEP research Project S22H0086, Sichuan Science and Technology Project 2021YFG0052, and Central University Fundamental Research Project ZYGX2019J032 as the funders.
Author contribution All authors contributed to the ideas in this study. Osei Seth: methodology, software, writing the original draft. Wang Wei (the corresponding author): funding acquisition, supervision, project administration, and editing. Qicheng Ding: Experimentation, Machine operation, and editing.
Data availability All data supporting this work are available upon reasonable request.
Author contribution All authors contributed to the idea and method
Data availability All data supporting this work are available upon reasonable request.
Consent for publication All authors have read and agreed to publish the manuscript.
Competing interests The authors declare no competing interests.
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