Geometric error compensation method using the Laser R-test

The five-axis machine tool consists of three linear axes (X, Y, and Z axes) and two rotary axes, which has exist 43 GE terms. Based on the structure of five-axis machine tools, the composition of the two rotary axes can be divided into two categories: either a combination of the A-axis and C-axis, a pairing of the B and C axes, or an arrangement of the A-axis and B-axis. The accuracy of the rotary axis is essential in a five-axis CNC machine tool, as it significantly influences the precision of five-axis machining operations. The two rotary axes have 22 GE items including the PDGEs and PIGEs [1‐3].

The most general NC systems provide geometric error compensation functions, such as linear and angular position error compensation and thermal error compensation. Some advanced NC control systems even can provide PDGEs, PIGEs, and error volumetric compensation accessory, such as in HEIDENHAIN TNC640, Siemens 840D sl, and Fanuc 31i system.

Most of the machine tool accuracy detection equipment used at this stage is developed and sold by foreign manufacturers, such as RENISHAW, BLUM, and AGILENT. The related detection products are often expensive to purchase and maintain. Therefore, there is a certain threshold for domestic machine tool manufacturers to adjust the accuracy of their machine tools, and only some manufacturers with a certain scale can afford the abovementioned detection equipment costs.

Equipment companies such as RENISHAW CO. and H.P.Co. use an external high-precision rotating platform with a laser interferometer to detect the angle positioning error of the machine tool indexing table [4, 5], such as RENISHAW XR20-W and HP-E5290C and other products. IBS Precision Engineering developed the R-test system [6] to detect the total error of the five-axis machine tool. The IBS R-test system uses a capacitive probe. Due to the smaller sensing range of the capacitor, care must be taken during the system installation and setup process to avoid accidental collisions. Etalon developed a Laser TRACER high-speed automatic tracking laser length finder [7], which uses the GPS principle to automatically track the length data obtained by the laser to detect the three-dimensional error in the working space of the machine tool. In 2003, Tsutsumi [8] proposed a classical measuring method for PIGEs using conduct double ball bar (DBB) tests. The method has been accepted by many researchers and is included in ISO 10791–6 of test conditions for machining centers [9]. However, since the DBB is a one-dimensional sensor, multiple measurements are required to analyze a single error item, necessitating significant setup and measurement time. In 2011, Wen-Yuh Jywe and his team developed an optical non-contact detection system composed of four-quadrant sensors, lasers, and lens modules [10], based on the ISO-10791–6 testing method. This system can be used to detect errors in high-end and five-axis machine tools, such as eccentricity errors and five-axis motion trajectory errors(total error). This system can simultaneously detect errors in the X, Y, and Z directions. Therefore, it only needs to be set up once, and during a single measurement process, the X, Y, and Z trajectory errors during five-axis movement can be obtained simultaneously. This greatly reduces setup and measurement time, and the non-contact nature of the system minimizes the chance of collisions. Soichi Ibaraki’s research introduces an enhanced non-contact R-test by using three laser triangulation displacement sensors[11]. This study focuses on a new algorithm and laser sensor uncertainties to efficiently calibrate errors in five-axis machine tools, comparing its performance with traditional methods. As relevant scholars began to analyze the acquired measurement error trajectories, they often used mathematical methods to analyze PIGEs and PDGEs, aiming to improve the simultaneous five-axis trajectory errors and overall composite errors. Tran and Hsieh proposed a method for analyzing rotary axis angular positioning errors[12]. This method primarily uses the cosine theorem in a Cartesian system to address concentricity issues. Paired with the Laser R-test, it captures the optical scale X, Y, and Z coordinate signals of the linear axis. Through computation and analysis, the angular positioning errors of the C-axis or A-axis can be determined. Wang and his team [13] had introduced a method using a double ball bar (DBB) and unit dual quaternion (UDQ) to identify position independent geometric errors in five-axis machines. This approach had simultaneously pinpointed eight distinct errors, streamlining experimental processes and operations, and had promoted enhanced machining precision through regular calibrations. In 2019, Li [14] presented a cost-effective method for compensating volumetric positioning errors in five-axis machine tools. By distinguishing between linear and nonlinear errors and utilizing sag error compensation with table multiplication, the method enhanced machine accuracy, showing promise in manufacturing. Its efficacy was confirmed using the NAS and S-shape test pieces. Guo [15] proposed a method for geometric error terms model, which is established based on the multibody system theory and the method of homogeneous transformation. Based on the proposed method, the 30 PDGEs and 13 PIGEs can be measured and analyzed by using DBB and XL-80 laser interferometer. The geometric error compensation results based on test experiments of cutting impellers showed that the accuracy of the machined parts with complex curved surfaces was improved 56.22%. In 2023, Yao et al. [16] have developed an innovative method to simultaneously identify PDGEs and PIGEs in dual rotary axes. By employing a laser tracker and two retroreflectors, the method allows for precise determination of PDGEs and PIGEs with a single setup, thus enhancing measurement efficiency. An identification model, integrated with the Powell algorithm, streamlines the error decoupling process. However, the widespread application of this method is currently limited due to the high cost of laser trackers. Cheng et al. [17] present a new test for identifying geometric and thermal errors in five-axis machine tools. Utilizing a specially designed disc-shaped test piece with 12 rectangular grooves, the method employs a coordinate-measuring machine for error detection. This process successfully pinpoints four PIGEs.

Based on previous studies and literature, it has been observed that for the precision of a five-axis machine tool, 70% of its error originates from the rotational axis PIGEs and PDGEs [18]. Currently, certain controller parameters within PIGEs and PDGEs are available, allowing users to make adjustments and compensations, mainly focusing on angular positioning errors like ECC, EAA, and EBB, and wobble errors such as XOC, YOC, AOC, BOC, and so forth. The majority of current research primarily employs interferometers and DBB, conducting multiple experiments and samplings before being able to analyze the corresponding geometric errors. One significant limitation of methods based on previous studies is the prolonged setup time and the ability to measure only a single error vector at one time.

This paper introduces an approach utilizing the Laser R-test, aiming to develop a speedy detection path and technique to concurrently measure the five-axis machine tool’s angular positioning error, eccentricity error, and wobble error of the rotational axis. The core of this research presents a 4 × 4 matrix coordinate analysis method, combined with the Rodrigues’ rotation formula principle, to establish a wobble error model. Primarily, through the Laser R-test, the X, Y, and Z coordinate changes during the sensing movement path are measured. By incorporating these X, Y, and Z coordinate changes into the wobble error model, the corresponding PIGEs can be ascertained. This method promises a significant reduction in error measurement time and simplifies the complexity of model derivation. Lastly, this research will plan experiments to validate and compare the benefits of compensating for different PIGE parameter items concerning the trajectory errors of the five-axis machine tool. The measurement and compensation technique is adaptable for commercial CNC controllers, including HEIDENHAIN, SIEMENS, and FANUC.

This system, Laser R-test (LRT), was proposed in 2004 [10]. The original system hardware was for wired transmission and power supply. The current system has been developed to support Wi-Fi wireless transmission. The sampling frequency of the system is 1000 Hz, and it can be connected with the CNC controller. The LRT is an optical non-contact detection instrument independently researched and developed in Taiwan. The detection system includes two components as shown in Fig. 1: The first part is a 3D sensing module consisting of two sets of lasers and two quadrants of sensors. The second part includes a standard ball lens. Through the laser and photoelectric sensor modules combined with the optical glass imaging principle, the repeatability is about 1 μm, achieving non-contact detection of machine tool error technology. LRT, the five-axis measurement system, has the following characteristics:

For the software interface, this research uses C# to develop an automatic compensation technology [19], and also through this research, an error analysis technology can be established. By the detection results of the ISO-10791–6 BK1 and BK2 path, the position errors of the spindle relative to the rotation axis can be analyzed, including any eccentric errors and yaw errors of the rotation axis. The human–machine interface is shown in Fig. 2.

In this paper, the research focused on a five-axis machine tool with rotary axes A and C corresponding to the X, Y, and Z axes. Typically, five-axis machine tools exhibit eccentricity errors and gimbal errors. The eccentricity error can be calculated using a simple formula. Therefore, this study aimed to utilize the LRT system and spatial coordinate method to develop error models for AOC, BOC, COA, and BOA.

In general, two rotational axes of the five-axis machine tool often exhibit eccentricity errors and deflection errors. This study aims to utilize the LRT system and spatial coordinate methods to construct error models for AOC, BOC, COA, and BOA. The LRT system and spatial coordinate methods provide the necessary tools and techniques to accurately analyze and quantify the angular deflection errors in the rotational axes of the five-axis machine tool.

The rotation coordinate transformation matrices for a three-dimensional cartesian coordinate system based on the right-hand coordinate system [20] with X, Y, and Z axes, rotating by angles θA, θB, and θC, respectively. They are represented by Rx (θA), Ry (θ B), and Rz (θC) as shown in (1),(2), and (3).

Here, θA, θB, and θC represent the rotation angles around the X, Y, and Z axes, respectively. These matrices are used to describe the rotation transformations of an object in a three-dimensional cartesian coordinate system.

Placing the standard ball lens on the work table of the machine tool, the coordinates of point Pi (where i = 1, 2, 3, …, n) are obtained. The X, Y, and Z values of the coordinates of the point can be simultaneously measured by the LRT. Based on this method, the standard ball lens is positioned on the C-axis table to obtain the coordinates of the first point P1 \(=\left[\begin{array}{c}{P1}_{x}\\ {P1}_{y}\\ {P1}_{z}\end{array}\right]\), and then, the C-axis is rotated to the angle θC to acquire the coordinates of the second point P2(c) without any AOC or BOC angle errors, As shown in Fig. 3, The matrix formula for P2(c) is given as shown in (4).

When the rotary table of C-axis has AOC and BOC angle errors, the coordinate transformation matrices for \({E}_{AOC}{\mathrm{\;and\;}E}_{BOC}\) are given by (5) and (6), respectively:

When the rotary table of C-axis has AOC and BOC angle errors, the matrix representation of the spatial coordinates can be expressed as (7):

When the rotary table of C-axis has AOC and BOC angle errors, the Z-axis vectors of rotary table of C-axis can be represented as (\(\overset\rightharpoonup{k_c}\)).

According to Rodrigues’ formula method [21], in spatial coordinates, when rotating the Z-axis vector of a rotary table of C-axis with AOC and BOC errors by an angle \({\theta }_{C}\), the formula for rotation can be expressed as follows, the spatial coordinate representation \(R_{\overset\rightharpoonup{k_c}}\) (\({\theta }_{C}\)) can be calculated using the Rodrigues’ formula as (9):\(v{\theta }_{C}=1-{\text{cos}}{\theta }_{C}\), “s” represents the sine function (sin), and “c” represents the cosine function (cos).

As shown in Fig. 4, assuming P1 is located at any point on the rotary table of C-axis with AOC and BOC angle errors, and the Z-axis vector on the rotary table of C-axis is rotated by an angle \({\theta }_{C}\) to reach point \({{\text{P}}2(c)}_{\delta }\), the calculation matrix for \({{\text{P}}2(c)}_{\delta }\) is given by (10):

The spatial coordinates of \({{\text{P}}2(c)}_{\delta }\) can be expressed as shown in (11).

To obtain the \(\left[\begin{array}{c}\Delta X(c)\\ \Delta Y(c)\\ \Delta {\text{Z}}({\text{c}})\end{array}\right], \Delta X\left(c\right), \Delta Y(c)\), and \(\Delta Z(c)\) error values caused by AOC and BOC errors of rotary table of C-axis, we can subtract the coordinates of point \({{\text{P}}2(c)}_{\delta }\) (with AOC and BOC errors) from point P2(c) (without AOC and BOC errors). The resulting differences will represent the error values in each dimension, Mathematically, the error in the X-axis (\(\Delta X(c)\)), error in the Y-axis\((\Delta Y(c)\)), and error in the Z-axis (\(\Delta Z(c)\)) can be calculated as follows (12):

\(\Delta X(c)\), \(\Delta Y(c)\), and \(\Delta Z\left(c\right)\) errors can be represented in spatial coordinates form as shown in (13).

After expanding the matrix, we can obtain the calculation formulas for\(\Delta X(c)\),\(\Delta Y(c)\), and \(\Delta Z\left(c\right)\) as follows:

By solving the simultaneous equations using (14), (15), and (16), the AOC and BOC errors can be obtained.

As shown in Fig. 5, using the same principle, obtaining P2(A), \({\theta }_{A}\) is the rotation angle of the X-axis vector of the A-axis rotary table. Without any BOA or COA angle errors, point P1 is rotated by an angle \({\theta }_{A}\) to reach point P2(A). The matrix formula for P2(A) is given as shown in (17).

When the rotary table of A-axis has BOA and COA angle errors, the coordinate transformation matrices for \({E}_{BOA}\) and \({E}_{COA}\) are given by (18) and (19), respectively:

When the rotary table of A-axis has BOA and COA angle errors, the matrix representation of the spatial coordinates can be expressed as (20)

When the rotary table of A-axis has BOA and COA angle errors, the X-axis vectors of rotary table of A-axis can be represented as \(\overset\rightharpoonup{\left(l_A\right)}\).

Based on the same principle, rewriting (9) allows the matrix \(R_{\overset\rightharpoonup{l_A}}\) (\({\theta }_{A}\)) can be obtain as shown in (22).

As shown in Fig. 6, assuming P1 is located at any point on the rotary table of A-axis with BOA and COA angle errors, and the X-axis vector on the rotary table of A-axis is rotated by an angle \({\theta }_{A}\) to reach point \({{\text{P}}2({\text{A}})}_{\delta }\), the calculation matrix for \({{\text{P}}2({\text{A}})}_{\delta }\) is given by (23):

The spatial coordinates of \({{\text{P}}2({\text{A}})}_{\delta }\) can be expressed as shown in (24).

Based on the same principle as (12) and (13), the error values for \(\Delta {\text{X}}({\text{A}})\), \(\Delta {\text{Y}}({\text{A}})\), and \(\Delta {\text{Z}}\left({\text{A}}\right)\) can be obtained by (25).

The \(\Delta {\text{X}}({\text{A}})\), \(\Delta {\text{Y}}({\text{A}})\), and \(\Delta {\text{Z}}\left({\text{A}}\right)\) errors can be obtained by calculation formulas as follows after expanding the matrix.

The BOA and COA errors can be obtained by solving the simultaneous equations represented by (26), (27), and (28). The definitions and explanations for the relevant symbols can be found in list of Method Symbols.

In this study, coordinate positions on the under different trajectory paths sare sampled using a LRT. Coordinates such as G0, G1, G2, G3, and G4 are acquired, as illustrated in Table 2, with G0 typically designated as the initial coordinate position. These coordinates are then fed into the human–machine interface developed in this research, enabling the analysis of PIGEs for the A-axis or C-axis. The four PIGE parameters for the C-axis are derived by calculating the coordinates of G0, G1, and G2. Similarly, for the A-axis, its four PIGE parameters are ascertained by calculating the coordinates of G0, G3, and G4, as depicted in Table 3.

The first step: Using the X, Y, and C axes to move two points arbitrarily, and by using the LRT, it will obtain the coordinate positions of G0 and G1 points as shown in the Table 2. The initial coordinate position of the system installation is represented by the G0 point. Inserting them into the formula and the human–machine interface (as shown in Fig. 2) developed in this research, the C-axis eccentric position parameter XOC YOC can be obtained.

The second step: Using the X, Y, and C axes to move three points arbitrarily, and by using the LRT, it will obtain the coordinate positions of G0, G1, and G2 points as shown in the Table 2 below by using the LRT. Inserting them into the formula and the human–machine interface developed in this research, the C-axis AOC and BOC errors can be obtained.

The third step: using the Y, Z, and A axes to move two points arbitrarily, and by using the LRT, it will obtain the coordinate positions of G0 and G3 points as shown in the Table 2. Inserting them into the formula and the human–machine interface developed in this research, the A-axis eccentric position parameters YOA and ZOA can be obtained. The fourth step: Using the Y, Z, and A axes to move three points arbitrarily, and by using the LRT, it will obtain the coordinate positions of G0, G3, and G4 points as shown in the Table 2. Inserting into the formula and the human–machine interface developed in this research, the A-axis BOA and COA errors can be obtained.

Using ISO-10791–6 BK1:YZA synchronous motion and BK2:XYC synchronous motion path to compare eccentric error before and after compensation, it is found that from the result (as shown in Fig. 12) before BK2 compensation that there are sin and cos waves in the X-axis and Y-axis directions; the errors are 12 µm and 30 µm, respectively, and there is a jump error of about 8 µm in the Z-axis direction. After compensating for the eccentricity, the error of the X-axis and Y-axis can be significantly improved and reduced to 8 µm, but the direction error of the Z-axis still cannot be changed, so it can be understood that there may be a yaw error in the C-axis turntable.

Moreover, it is found that from the result (as shown in Fig. 13) before BK1 compensation, the direction error of the X-axis is particularly large. Therefore, after compensating the eccentric ZOA and YOA, the error compensation results can be effectively improved, and the errors in the Y and Z directions can be reduced to less than 5 µm, while the main wobble error of the rotating axis displayed in the X direction is approximately 16 µm of error, and cannot be eliminated or compensated.

The comparison between the results of the 4-parameter (XOC, YOC, ZOA, and YOA) compensation and the 8-parameter (XOC, YOC, ZOA, YOA, AOC, BOC, COA, and BOA) compensation will be performed using the BK1 and BK2 trajectory paths, employing the passive voice construction. It can be seen from Fig. 14A that the trajectory errors of the X, Y, and Z axes of BK1 can be reduced to within \(\pm\) 5 µm. The errors caused by COA and BOA have been effectively reduced in the synchronous trajectory.

Furthermore, from Fig. 14B, it can be observed that after compensation, the Z-direction trajectory error in BK2 has been reduced from the original 5 µm to within 2 µm. The significant influence of AOC and BOC on the synchronous trajectory error of BK2 has been effectively reduced.

Finaly, the ISO 10791–6 BK4 simultaneous motion path was utilized for verification, wherein the synchronous trajectory errors before and after compensating for the 4 and 8 parameters were compared. The results shown in Fig. 15 indicate that the errors in the five-axis synchronous trajectory have been reduced from approximately \(\pm\) 80 μm to within \(\pm\) 10 μm. Comparing the results of compensating for 4 parameters and 8 parameters, it was found that the error could be effectively reduced from \(\pm\) 30 to \(\pm\) 10 μm. The feasibility of the system and the derived method has been confirmed through the verification of the experimental results mentioned above. Through this method, the accuracy of multi-axis simultaneous motion has been effectively and significantly enhanced.