Principle and Method of Base Station Calibration Based on a Physical Standard for Multi-Station Laser Tracking Measurement

Abstract

In the measurement of volumetric errors in CNC machine tools using multi-station laser tracking technology, the coordinate calibration accuracy of external measurement base stations is a key factor determining the system’s final accuracy. Traditional calibration approaches typically use the commanded positions of the machine tool directly to inversely determine base station coordinates, which results in strong coupling between inherent geometric errors and base station parameters. Consequently, the measurement accuracy cannot be properly evaluated, and metrological traceability of the results remains difficult to achieve. To address this issue, this paper proposes a novel calibration principle based on an independent external physical standard and develops a base station calibrator independently. This device employs a precision turntable, G5-grade precision spheres, and electromagnet groups to construct an equivalent target with four feature points at the spindle end. Verified by a high-precision coordinate measuring machine (CMM), the maximum difference in repeated calibrations of the device is 1.4 µm, indicating its excellent positioning repeatability. The calibrator was further applied to measure the positioning errors of a CNC milling machine, and comparative experiments were performed with a Renishaw XL-80 laser interferometer. The results indicate that the error variation trends obtained from the two measurement principles are highly consistent. In both the X-axis and Y-axis directions, the maximum deviations of linear errors are controlled within 3.7 µm, while the maximum deviations of angular errors remain within 4.6 µrad. Furthermore, the reliability of the system data was confirmed through an uncertainty analysis. The external physical standard developed in this study ensures that base station calibration accuracy is not affected by the inherent errors of the machine tool, providing a novel and reliable scheme for high-precision calibration and metrological traceability of machine tool spatial errors.

1. Introduction

Five-axis and multi-axis CNC machine tools act as core manufacturing equipment in aerospace and other high-end manufacturing sectors, where their spatial volumetric accuracy directly determines the machining quality of complex curved surface parts [1,2,3]. However, during actual operation, the end-effector of a machine tool inevitably produces complex spatial geometric errors due to the combined influence of multiple error sources [4,5,6,7]. To address this problem, accurate spatial error measurement followed by software compensation within the CNC system is widely regarded by both academia and industry as the most cost-effective and efficient approach to improve volumetric accuracy [8,9,10].

To efficiently obtain the volumetric errors within the workspace of machine tools, measurement technology has gradually shifted from “single-item error measurement” to “comprehensive spatial volumetric error measurement [11,12].” Traditional accuracy inspection mainly relies on laser interferometers to measure 21 geometric error components, such as straightness and yaw, individually [1]. This approach is not only time-consuming and labor-intensive but also has difficulty in accurately capturing spatially coupled errors between moving axes [13]. To overcome these limitations, indirect volumetric error measurement techniques based on the principle of multilateration have been developed [14,15]. In recent years, external large-scale non-contact measurement systems, represented by multi-station laser trackers and laser tracers, have become a major research focus in the field of precision measurement [16]. These systems provide large measurement ranges and high interferometric ranging accuracy, enabling rapid identification of six-degree-of-freedom (6-DOF) errors across the entire workspace through the deployment of multiple measurement base stations [17,18,19].

However, prior to performing spatial error detection using these external measurement systems, base station calibration must be carried out to accurately determine the spatial pose of each measurement station relative to the machine tool coordinate system. At present, common calibration methods for multilateration base stations rely on the machine tool’s own motion mechanisms to follow predefined trajectories in space, using the commanded coordinates from the CNC system as reference values to inversely determine the base station position parameters [20,21]. This calibration approach introduces an inherent logical issue: real machine tools inevitably contain initial geometric errors, causing the actual physical positions of measurement points to deviate from the CNC commanded coordinates [22]. When the mathematical model assumes these error-containing CNC values as absolute references, the resulting base station coordinates incorporate unknown systematic deviations of the machine, leading to error coupling and reduced measurement accuracy [20,23].

Such propagation and coupling of errors not only cause misalignment of subsequent measurement benchmarks but also violate the fundamental principles of metrological traceability [24]. Calibrating external base stations using CNC readings of the machine tool under test essentially forms a “self-certification trap,” where the measuring system is calibrated based on the precision of the object being measured [23].

To address these limitations, this paper proposes a novel base station calibration principle based on an independent external physical standard, highlighting a core innovation in error decoupling. By developing a dedicated base station calibrator that relies on a pre-calibrated, high-precision physical reference rather than the machine tool’s motion axes, this device achieves complete physical and mathematical decoupling of the base station coordinates from the machine’s inherent geometric errors. This approach fundamentally dismantles the self-certification trap, establishing a rigorous metrological traceability chain and providing a highly reliable theoretical basis and practical methodology for the precise acquisition of CNC machine tool spatial errors.

2. Materials and Methods

2.1. Principle of Spatial Coordinate Calculation Using Multilateration

“Multilateration” is the fundamental principle for obtaining spatial coordinates in multi-station tracking measurement technology [18]. Multi-station tracking measurement technology deploys several measurement base stations around a machine tool to synchronously track and measure the distance to a target point (e.g., the machine tool spindle). This method effectively overcomes the limitation of a single laser instrument, which can only perform one-dimensional measurements, thereby enabling simultaneous detection of multiple errors. Within this framework, the specific measurement and calculation process of multilateration is illustrated in Figure 1. Multiple external measurement base stations, denoted as $n_i$, distributed around the machine tool emit laser beams toward the measured target point $M$ to perform distance measurement, obtaining the spatial distance $l_i$ from each base station to the target point. By constructing and solving a system of spatial distance equations, the three-dimensional coordinates of the target point $M$ can be determined [12].

In the multilateration calculation model, the coordinates of the base stations themselves are necessary known parameters for determining the position of the target point. This implies that the measurement accuracy of the target point $M$ depends on the precision of the base stations’ own coordinates. Therefore, before conducting formal machine tool error measurements, the accurate spatial positions of all base stations within the machine tool reference coordinate system must be established. This procedure is referred to as base station calibration. If deviations exist in the initial calibration of a base station position, such errors will be directly transmitted and amplified in the final coordinate calculation results [1]. To enhance measurement reliability and reduce random noise, a redundant measurement strategy is generally introduced in practice, involving the deployment of four or more base stations to form redundant equations [24]. This strategy not only allows rapid recovery of measurements after laser beam interruption but also supports self-calibration and correction of base station position parameters.

In summary, measurements using external measurement base stations involve two main steps: first, the calibration of the base stations, and second, the measurement of target points. Since the measurement principles and algorithms for target points are already relatively mature, achieving high-precision base station calibration has become a critical technical challenge that must be addressed to realize high-precision measurement.

2.2. Principles of Base Station Calibration and Machine Tool Error Separation

Based on the aforementioned multilateration principle, when the coordinates $P\left(x,y,z\right)$ of the external measurement base station itself need to be inversely calibrated, assuming that the coordinate points of four fixed target mirror positions in space $T_i\left(x_i,y_i,z_i\right)\left(i=1,2,3,4\right)$ are known, together with the measured distances $L_1,L_2,L_3,L_4$ between the external measurement base station and these four positions. By applying the distance formula between two points, a system of three-variable high-order nonlinear redundant equations can be established.

$$\left\{\begin{array}{c}{\left(x-x_1\right)}^2+{\left(y-y_1\right)}^2+{\left(z-z_1\right)}^2=L_1^2\\ {\left(x-x_2\right)}^2+{\left(y-y_2\right)}^2+{\left(z-z_2\right)}^2=L_2^2\\ {\left(x-x_3\right)}^2+{\left(y-y_3\right)}^2+{\left(z-z_3\right)}^2=L_3^2\\ {\left(x-x_4\right)}^2+{\left(y-y_4\right)}^2+{\left(z-z_4\right)}^2=L_4^2\end{array}\right.$$

(1)

By solving this equation system, the actual coordinates $P\left(x,y,z\right)$ of the external measurement base station can be obtained.

Mathematically, determining a three-dimensional spatial coordinate inherently requires at least three independent distance measurements. However, solving a system with only three spherical distance equations typically yields two possible intersection points in space. Introducing a fourth fixed target mirror not only uniquely resolves this spatial ambiguity but also constructs an overdetermined system of redundant equations. This redundancy is crucial, as it allows for the application of least-squares optimization algorithms to effectively suppress random measurement noise and significantly enhance the robustness and accuracy of the base station coordinate calibration.

After completing base station calibration, the machine tool is moved to drive the base station calibrator to the position to be measured. At this stage, the actual coordinates $P\left(x,y,z\right)$ of the base station are known. Provided that the relative positional relationships among the four target mirrors are known, the true coordinate values $T_i^{\prime }\left(x_i^{\prime },y_i^{\prime },z_i^{\prime }\right)$ of the measured target mirrors at this position can be determined through a similar distance equation by measuring the length readings $L_i^{\prime }$ from the external measurement base station to the target mirrors:

$${\left(x_i^{\prime }-x\right)}^2+{\left(y_i^{\prime }-y\right)}^2+{\left(z_i^{\prime }-z\right)}^2={\left(L_i^{\prime }\right)}^2$$

(2)

Furthermore, based on the fixed spatial positional relationship between the target mirrors and the measured target point on the machine tool, the actual coordinates of the measured target point at the current position can be derived. After obtaining these actual coordinates, the comprehensive volumetric error of the CNC machine tool at the current position can be determined by comparing them with the theoretical commanded coordinate values. Finally, by applying an error separation algorithm for CNC machine tool geometric errors, each independent geometric error component can be effectively decoupled.

The base station calibration and the measurement process of the measured target points are illustrated in Figure 2. First, a system of redundant equations is established using the coordinates of the four fixed target mirror points $T_i$ and their measured distances to the external measurement base station (laser head) to analytically determine the actual coordinates of the external measurement base station. After completing the base station calibration, the machine tool moves and drives the base station calibrator to the positions to be measured (e.g., “Test point to be tested 1” and “Test point to be tested 2” in Figure 2). At this stage, the distances from the fixed target mirror points to the external measurement base station are measured again. By using the known base station coordinates and the positional relationships among the four fixed points, the coordinates of the fixed target mirrors at the current position are solved, thereby enabling accurate calculation of the spatial position of the measured target point.

3. Working Principle and Structural Design of the Base Station Calibrator

To satisfy the prerequisite of “knowing the coordinates of four fixed target mirrors in space” in the aforementioned principle, directly arranging four non-interfering fixed target mirrors within the limited workspace of a machine tool would inevitably result in spatial conflicts and line-of-sight occlusions. Moreover, accurately determining the discrete spatial positions of these four target mirrors presents a considerable challenge. Therefore, this paper designs a base station calibrator that utilizes a single physical target mirror. A precision rotary mechanism is employed to drive the target mirror, switching its position at $90°$ intervals in space. This approach not only resolves the line-of-sight occlusion issue but also equivalently provides four target mirrors.

3.1. Overall Structural Design of the Base Station Calibrator

The overall structure of the base station calibrator is illustrated in Figure 3 and Figure 4. Its main components include a precision turntable, a target mirror, precision spheres together with a target mirror adjustment device, electromagnet groups, electromagnet adjustment brackets, and a spindle connection device. Four electromagnet adjustment brackets are rigidly mounted on the base of the turntable; they are circumferentially distributed with a $90°$ interval between adjacent brackets. The target mirror and precision spheres are mounted on the target mirror adjustment device, which is firmly attached to the turntable, allowing synchronous rotation with the turntable. When rotated to a designated position, the precision spheres engage with the electromagnet group through kinematic coupling, thereby accurately determining the spatial pose of the target mirror within the local coordinate system of the base station calibrator. The spindle connection device at the bottom of the calibrator is clamped by the machine tool spindle, serving to establish the spatial transformation relationship between the base station calibrator and the machine tool coordinate system. It is worth noting that the spindle connection device is machined from high-rigidity alloy steel. This design ensures that the clamping force exerted by the CNC machine tool spindle does not induce micro-strain or structural deformation in the base station calibrator, thereby preserving the spatial integrity of the target mirrors.

The calibration principle of the base station calibrator is illustrated in Figure 5. First, the coordinate point of the target mirror is determined. The motorized turntable rotates to drive the precision spheres and the target mirror to electromagnet group A. Upon energization, the electromagnets generate magnetic force, attracting the precision spheres to engage with the non-magnetic spherical sockets on the electromagnets. This action fixes the position of the target mirror, thereby defining its first position within the base station calibrator. Subsequently, the turntable rotates by $90°$ to electromagnet group B, and the aforementioned kinematic coupling process between the precision spheres and the non-magnetic sockets is repeated to determine the second position of the target mirror. By analogy, the four positions of the target mirror are determined, achieving high-precision positioning of the four feature points $T_1,T_2,T_3,T_4$. The four positions of the target mirror are shown in Figure 5.

As described above, the positions of the target mirror and its adjustment device depend on the precision spherical kinematic pairs formed by the engagement of three precision spheres with the non-magnetic spherical sockets on the electromagnets. To enhance overall positioning accuracy, the precision spheres adopt international G5-grade standard spheres, with a sphericity error less than 0.13 µm, a surface roughness value less than 0.014 µm, and a batch diameter variation less than 0.25 µm. One end of the electromagnet adopts a conical structure to mate with the standard sphere; this configuration is widely used in precision instruments such as the double ballbar. The positioning repeatability after engagement between the precision spheres and the electromagnets can reach better than 0.3 µm. The high-precision repeatable positioning at each location of the base station calibrator is ensured by these three precision spherical kinematic pairs.

3.2. Target Mirror Adjustment Device and Electromagnet Adjustment Bracket

After completing the measurement at target mirror position $T_1$, the precision spheres and the target mirror must return to a neutral state before being driven by the turntable to the next position for subsequent kinematic coupling. To enable repeated calibration at four different positions, the precision spheres and the target mirror adjustment device must support 3-DOF fine adjustment. This ensures that the precision spheres can firmly engage with the electromagnets through magnetic attraction and subsequently return to equilibrium positions after each calibration step.

Figure 6 illustrates the structural design of the target mirror adjustment device. The target mirror is rigidly mounted on this device, and its spatial pose is uniquely defined by the three precision spheres.

The precision sphere mounting plate is housed within the adjustment plate. As shown in Figure 7, leaf springs are employed to accommodate fine adjustment of the precision spheres and the target mirror along the X- and Z-axes. In addition to facilitating initial fine adjustment, the low-stiffness design of these leaf springs plays a crucial role in stress release during dynamic operation. If the motorized turntable exhibits a slight angular positioning error (such as stepping motor overshoot or undershoot), the flexible leaf springs provide necessary mechanical compliance. This allows the precision spheres to be pulled into perfect kinematic seating by the electromagnetic attraction without inducing significant residual bending stress on the adjustment device, thereby ensuring the stability of the micrometer-level positioning accuracy.”

Because the precision spheres and electromagnets engage magnetically, a relatively large adjustment stroke is required along the Y-axis. Therefore, cylindrical guide rails and linear sliders are utilized to reduce friction and ensure smooth kinematic motion, as illustrated in Figure 8, balancing springs are installed on both sides of the slider to ensure that the precision spheres and the target mirror consistently return to their equilibrium state. This mechanism effectively satisfies the requirements for 3-DOF spatial adjustment of the precision spheres and the target mirror.

To ensure accurate alignment between the electromagnets and the precision spheres, an electromagnet adjustment bracket was designed, as shown in Figure 9. A horizontal slot is machined in the electromagnet bracket to allow translation along the X-axis, while a vertical slot is incorporated in the electromagnet base to enable translation along the Y-axis. This configuration provides the electromagnets with 2-DOF mobility, enabling precise positional adjustment to match the locations of the precision spheres.

After completing the mechanical structural design of the base station calibrator, a Programmable Logic Controller (PLC) was programmed to achieve automated control. This allows the calibrator to autonomously perform measurement operations during volumetric error measurement of CNC machine tools. Furthermore, to mitigate potential thermal expansion caused by the electromagnets, the PLC employs a strict short-duration energization strategy. The electromagnets only operate at full power during the brief moment of kinematic seating. Given that the automated calibration cycle for all four positions takes only about 90 s, the temperature rise in the electromagnet brackets is negligible (less than 0.2 °C), fully ensuring that the 1.4 µm micrometer-level positioning accuracy is not compromised by thermal effects.

4. Target Mirror Position Calibration Experiment of the Base Station Calibrator

According to the principle of base station calibration and machine tool error separation, to calibrate the external measurement base station, the accurate coordinate values of the four fixed target mirror points on the base station calibrator must be known. To obtain the true coordinates of the measured target point, the relative positional relationships among these four fixed target mirror points must be determined; that is, the spatial coordinates of the first target mirror position should be able to mathematically describe the spatial coordinates of the other three positions. Therefore, a target mirror position calibration experiment is required to determine the accurate coordinates and relative positional relationships of the four fixed points.

During the target mirror calibration, the coordinate measuring machine (CMM, COORD3 ARES 10.7.5, Italy, with a nominal volumetric accuracy of 1.8 + L/333 µm and a resolution of 0.1 µm) was first initialized and prepared. Subsequently, the adjusted base station calibrator was placed and rigidly fixed on the granite measuring table of the CMM to conduct the calibration experiment, as illustrated in Figure 10. When the target mirror was positioned at electromagnet adjustment device A, the CMM probed points on the spherical surface of the target mirror from multiple angular directions to calculate and determine its geometric center. To ensure that the relative positions among the four target mirrors remain consistent during both base station calibration and measured target point measurement, the base station calibrator must exhibit high positioning repeatability. More than 20 repeated measurements were conducted for each target mirror position to verify its positioning repeatability and determine its relative coordinates. The average value of the 20 sets of coordinate data was taken as the spatial position coordinates for each target mirror location. Furthermore, the standard deviation was calculated as a dispersion indicator to represent the fluctuation of the measurement results under identical conditions.

The specific coordinate data for electromagnet position adjustment devices A, B, C and D are presented in

Table 1. Coordinate values at electromagnet group A.

Position A	Mean Value (mm)	Max (mm)	Min (mm)	Difference ($\mathbf{\mu }\mathbf{m}$)	Standard Deviation (mm)
X	383.3369	383.3375	383.3363	1.2	0.00189
Y	335.6237	335.6242	335.6231	1.1	0.00148
Z	112.3574	112.3581	112.3568	1.3	0.00153

,

Table 2. Coordinate values at electromagnet group B.

Position B	Mean Value (mm)	Max (mm)	Min (mm)	Difference ($\mathbf{\mu }\mathbf{m}$)	Standard Deviation (mm)
X	356.8247	356.8253	356.8242	1.1	0.00169
Y	361.7155	361.7163	361.7149	1.4	0.00170
Z	112.3388	112.3394	112.3381	1.3	0.00134

,

Table 3. Coordinate values at electromagnet group C.

Position C	Mean Value (mm)	Max (mm)	Min (mm)	Difference ($\mathbf{\mu }\mathbf{m}$)	Standard Deviation (mm)
X	331.7053	331.7061	331.7047	1.4	0.00156
Y	334.5298	334.5304	334.5291	1.3	0.00161
Z	112.3804	112.3811	112.3798	1.3	0.00142

and

Table 4. Coordinate values at electromagnet group D.

Position D	Mean Value (mm)	Max (mm)	Min (mm)	Difference ($\mathbf{\mu }\mathbf{m}$)	Standard Deviation (mm)
X	357.5912	357.5916	357.5903	1.3	0.00117
Y	308.3845	308.3851	308.3839	1.2	0.00152
Z	112.3383	112.3389	112.3378	1.1	0.00134

. The spatial position coordinates at electromagnet adjustment device A are $T_1\left(383.3369,335.6237,112.3574\right)$; at device B, $T_2\left(356.8247,361.7155,112.3388\right)$; at device C, $T_3\left(331.7053,334.5298,112.3804\right)$; and at device D, $T_4\left(357.5912,308.3845,112.3383\right)$.

By analyzing the calibration experiment data of the base station calibrator, it can be concluded that the maximum difference in repeated calibrations across the four electromagnet position adjustment devices is 1.4 µm, while the maximum standard deviation is 0.00189 mm. The small variations in repeated calibrations and the low standard deviations indicate that the measurement data are highly concentrated, demonstrating excellent positioning repeatability. It is important to note that this 1.4 µm maximum difference represents the internal relative positioning repeatability of the four target mirrors within the rigid framework of the calibrator. Because the calibrator acts as an integral, high-rigidity solid body, these internal relative geometric relationships remain invariant even when the device is unmounted from the CMM and remounted onto the CNC machine tool spindle, ensuring the pre-calibrated precision is perfectly preserved for actual measurements.

When measuring machine tool errors, assuming that the coordinates of $T_1$ within the machine tool coordinate system are $T_1\left(x,y,z\right)$, the spatial positions of the other three target mirrors can be respectively expressed as: $T_2\left(x-26.5122,y+26.0918,z-0.0186\right)$, $T_3\left(x-51.6316,y-1.0939,z+0.0230\right)$, and $T_4\left(x-25.7457,y-27.2392,z-0.0191\right)$. Once the relative positional relationships among the target mirrors are mathematically established, the base station calibrator can be used to perform base station calibration for the external measurement base station (e.g., laser trackers or laser tracers), measure the target points, and ultimately calculate the machine tool errors.

5. Experimental Measurement of Machine Tool Errors Based on the Base Station Calibrator

Following the structural design of the base station calibrator and the determination of the relative positional relationships among its internal target mirrors, an experimental investigation was carried out regarding its application in base station calibration and measured target point measurement. By employing the base station calibrator together with a custom-developed 3D laser telescoping ballbar measurement device, the geometric errors of the X- and Y-axes of a CNC milling machine were evaluated. The obtained results were subsequently compared with those measured using a Renishaw XL-80 laser interferometer (Renishaw plc, Wotton-under-Edge, Gloucestershire, UK) to verify the feasibility and accuracy of the base station calibrator as an external physical standard.

The Renishaw XL-80 laser interferometer, used for measuring linear displacements, has a resolution of 0.001 µm and an instrument calibration uncertainty of ±0.5 ppm (k = 2). Consequently, the accuracy of the base station calibrator as an external physical standard was validated through comparative analysis with the machine tool errors measured by the laser interferometer. The experimental platform selected was a Nanjing Sikai GD650 CNC milling machine, with an X-axis travel of 650 mm, a Y-axis travel of 400 mm, a Z-axis travel of 450 mm, and a spindle speed range from 40 to 4000 rpm. The ambient laboratory temperature was strictly controlled at 20 ± 1 °C. Considering that laser wavelengths are sensitive to atmospheric pressure variations, the atmospheric pressure was continuously monitored during the experiments and maintained at 1013 ± 2 hPa.

According to the ISO 230-1 standard [25], the geometric errors of a machine tool are classified into two main categories: linear motion errors and rotary motion errors. The Renishaw XL-80 laser interferometer is primarily used for detecting the five geometric errors associated with linear motion; the measurement of rotary angular errors requires additional specialized equipment, such as a rotary encoder or an XR20-W. Therefore, this study only presents comparative analyses for these five linear motion-related geometric errors. Initially, the laser interferometer was employed to measure the positioning error of the machine tool X-axis, as shown in the experimental setup in Figure 11. The machine tool spindle was programmed to drive the reflector along the X-axis with a measurement interval of 50 mm over a range of [0 mm, 300 mm]. The measurement procedure for the Y-axis positioning error was identical to that of the X-axis. The positioning error data for both axes, as recorded by the laser interferometer, are provided in

Table 5. Measurement results of X/Y-axis positioning errors using the laser interferometer.

Position (mm)		0	50	100	150	200	250	300
Laser interferometer ($\mathrm{\mu }\mathrm{m}$)	X axis	0	4.2	4.7	6.5	4.7	8.5	9.8
	Y axis	0	6.5	8.4	17.9	15.5	19.6	19.2

. The measurement procedures for the remaining geometric errors followed a similar approach. The main difference was the need to reconfigure the optical path of the laser interferometer for different error components; the specific details of these adjustments are omitted here for brevity. The corresponding data for the remaining four geometric errors are presented in

Table 6. Measurement results of X-axis straightness errors using the laser interferometer.

Position (mm)		0	50	100	150	200	250	300
Laser interferometer ($\mathrm{\mu }\mathrm{m}$)	Y-directions	0	2.5	7.1	8.9	11.3	12.6	16.2
	Z-directions	0	−10.3	−8.6	−9.1	−3.2	−1.4	−9.5

,

Table 7. Measurement results of X-axis pitch/yaw errors using the laser interferometer.

Position (mm)		0	50	100	150	200	250	300
Laser interferometer $\mathrm{\mu }\mathrm{m}$	pitch	0	4.7	19.7	25.6	18.4	29.8	32.1
	yaw	0	−7.6	−3.4	7.6	11.5	18.2	11.6

,

Table 8. Measurement results of Y-axis straightness errors using the laser interferometer.

Position (mm)		0	50	100	150	200	250	300
Laser interferometer ($\mathrm{\mu }\mathrm{m}$)	X-directions	0	7.8	13.5	16.4	22.1	28.3	22.4
	Z-directions	0	1.8	−2.1	−8.4	−17.1	−15.8	−18.4

and

Table 9. Measurement results of Y-axis pitch/yaw errors using the laser interferometer.

Position (mm)		0	50	100	150	200	250	300
Laser interferometer ($\mathrm{\mu }\mathrm{m}$)	pitch	0	14.1	9.4	15.3	27.8	23.1	28.6
	yaw	0	−8.7	−6.1	−18.7	−21.5	−13.9	11.6

.

After completing the laser interferometer measurement experiments, a setup similar to a laser tracer was constructed using the base station calibrator and a custom-developed 3D laser telescoping ballbar measurement device to obtain the actual coordinates of the measured target points [26] A nine-line error separation algorithm was applied to process the data and separate the geometric errors in the X- and Y-directions [27]. The overall layout of the measurement system is illustrated in Figure 12. The base is rigidly mounted on the machine tool worktable. One end of the spindle connection device is connected to the base station calibrator, while the other end is clamped by the machine tool spindle. As the machine tool spindle or worktable moves, it drives the custom laser tracking ballbar to perform dynamic tracking measurements.

The experimental measurement setup is shown in Figure 13. With the base station calibrator connected to the CNC machine tool spindle and the 3D laser telescoping ballbar securely mounted on the worktable, five independent measurements were conducted at each designated position to evaluate the repeatability of the measurement system. During this process, the time consumption was also recorded. The automated calibration of the four feature points took approximately 90 s, and the comprehensive error measurement for a single axis was completed within 1.5 h. The mean values and standard deviations were then calculated.

Table 10. Measurement results of X-axis geometric errors obtained by the base station calibrator.

Measurement Location (mm)	0	50	100	150	200	250	300
Positioning error $\left(\mathrm{\mu }\mathrm{m}\right)$	0	3.6	5.1	7.2	5.8	7.9	9.1
Standard deviation $\left(\mathrm{\mu }\mathrm{m}\right)$	0	0.93	1.03	1.51	1.11	1.06	0.98
Straightness error Y $\left(\mathrm{\mu }\mathrm{m}\right)$	0	4.1	6.3	10.1	13.2	15.1	17.5
Standard deviation $\left(\mathrm{\mu }\mathrm{m}\right)$	0	0.94	1.38	1.67	0.80	1.31	2.32
Straightness error Z $\left(\mathrm{\mu }\mathrm{m}\right)$	0	−11.9	−5.7	−11.5	−4.7	−3.6	−10.8
Standard deviation $\left(\mathrm{\mu }\mathrm{m}\right)$	0	1.39	1.66	1.55	0.74	0.99	0.87
Pitch error $\left(\mathrm{\mu }\mathrm{r}\mathrm{a}\mathrm{d}\right)$	0	7.6	21.2	27.9	15.1	26.9	34.8
Standard deviation $\left(\mathrm{\mu }\mathrm{r}\mathrm{a}\mathrm{d}\right)$	0	0.88	1.28	1.38	1.14	1.36	1.45
Yaw error $\left(\mathrm{\mu }\mathrm{r}\mathrm{a}\mathrm{d}\right)$	0	−9.8	−5.1	10.3	15.1	16.9	9.7
Standard deviation $\left(\mathrm{\mu }\mathrm{r}\mathrm{a}\mathrm{d}\right)$	0	1.13	0.78	1.57	1.09	0.93	1.16

and

Table 11. Measurement results of Y-axis geometric errors obtained by the base station calibrator.

Measurement Location (mm)	0	50	100	150	200	250	300
Positioning error $\left(\mathrm{\mu }\mathrm{m}\right)$	0	5.6	8.6	16.5	14.6	18.3	17.7
Standard deviation $\left(\mathrm{\mu }\mathrm{m}\right)$	0	1.5	1.11	1.34	1.16	1.36	1.25
Straightness error Y $\left(\mathrm{\mu }\mathrm{m}\right)$	0	9.4	10.1	19.8	25.3	24.9	20.1
Standard deviation $\left(\mathrm{\mu }\mathrm{m}\right)$	0	1.22	1.88	1.15	1.43	1.36	1.25
Straightness error Z $\left(\mathrm{\mu }\mathrm{m}\right)$	0	3.5	−4.8	−11.5	−15.5	−12.1	−15.1
Standard deviation $\left(\mathrm{\mu }\mathrm{m}\right)$	0	1.13	1.41	1.66	1.3	2.11	1.03
Pitch error $\left(\mathrm{\mu }\mathrm{r}\mathrm{a}\mathrm{d}\right)$	0	17.2	12.8	18.7	30.4	19.1	31.5
Standard deviation $\left(\mathrm{\mu }\mathrm{r}\mathrm{a}\mathrm{d}\right)$	0	1.28	1.26	2.22	1.60	2.15	2.61
Yaw error $\left(\mathrm{\mu }\mathrm{r}\mathrm{a}\mathrm{d}\right)$	0	−11.1	−4.1	−15.7	−26.5	−11.2	8.9
Standard deviation $\left(\mathrm{\mu }\mathrm{r}\mathrm{a}\mathrm{d}\right)$	0	1.54	1.43	1.32	1.70	1.48	1.53

present the geometric error measurement results for the X- and Y-axes of the CNC machine tool, respectively.

Figure 14 presents a comparison of the machine tool positioning errors obtained using the two different measurement methods, where the red and blue curves represent the positioning error values in the X- and Y-directions, respectively. Figure 15 shows the comparative analysis of the X-axis straightness errors in the Y- and Z-directions, with the red and blue curves corresponding to the straightness errors in the Y- and Z-directions, respectively. The pitch and yaw errors of the X-axis are compared in Figure 16, where the red and blue curves denote the pitch and yaw error components, respectively. For the Y-axis, Figure 17 shows the comparison of its straightness errors in the X- and Z-directions, with the red curve representing the X-direction straightness error and the blue curve representing the Z-direction straightness error. Finally, Figure 18 presents the comparison of the pitch and yaw errors of the Y-axis, where the red and blue curves indicate the pitch and yaw errors, respectively.

An analysis of the comparative results for the above five geometric errors indicates that, after introducing the external physical standard for base station calibration, the measurement results show high agreement with the reference values obtained from the laser interferometer. The experimental results show that for the X-axis, the maximum deviations are 1.8 µm for positioning, 2.9 µm for straightness, and 3.6 µrad for angular errors. For the Y-axis, the maximum deviations are 2.3 µm for positioning, 3.7 µm for straightness, and 4.6 µrad for angular errors. Considering the inherent precision limitations of the machine tool itself, as well as unavoidable thermal deformation and vibration disturbances during dynamic measurement, the measurement principle based on the base station calibrator is demonstrated to be feasible and accurate.

Furthermore, the development and application of the base station calibrator effectively address the long-standing problem of independently calibrating base station coordinates in multi-station laser measurement systems. By introducing a high-precision physical standard, the calibration process for external measurement base stations no longer depends on the machine tool CNC commanded coordinates, thereby isolating the influence of inherent machine errors on calibration accuracy. This improvement ensures that the measurement data are traceable to a high-precision external physical reference, enabling reliable metrological traceability and providing strong technical support for evaluating CNC machine tool spatial accuracy.

To evaluate the influence of thermal variations during prolonged operation, changes in both the ambient temperature and the localized temperature near the spindle of the Nanjing GD650 machine tool were monitored, as listed in

Table 12. Temperature variations recorded during the machine tool operation.

Time ($\mathbf{h}$)	Ambient Temperature ($°\mathbf{C}$)	Near-Spindle Temperature ($°\mathbf{C}$)
0	19.0	19.3
0.5	19.2	19.6
1.0	19.5	19.9
1.5	19.7	20.3
2.0	20.0	20.5
2.5	20.4	21.0

. Over a continuous 2.5 h measurement period, the ambient temperature showed a slight increase from 19.0 °C to 20.4 °C, whereas the temperature near the spindle increased from 19.3 to 21.0 °C. These observations indicate that internal heat generation during machine operation results in gradual localized temperature rise. Although such thermal changes appear small, in high-precision metrology they can induce slight structural thermal deformation, which may affect the stability and accuracy of the measurement process.

Furthermore, to systematically examine the effect of structural vibrations on measurement accuracy, acceleration sensors were used to collect vibration signals from the machine bed and spindle regions at a sampling frequency of 10 kHz. The recorded time-domain signals were processed using the Fast Fourier Transform (FFT) to obtain the vibration spectrum distribution of the machine tool, as summarized in

Table 13. Vibration spectrum analysis of the machine tool.

Frequency ($\mathbf{H}\mathbf{z}$)	Acceleration Amplitude ($\mathbf{m}\mathbf{m}/{\mathbf{s}}^2$)	Illustrate
50	0.6	Power supply low frequency interference
120	1.2	Spindle motor harmonics
260	2.0	Tool cycle vibration peak value
400–700	0.3–0.5	Stray mechanical noise

.

The spectral analysis shows that the dominant vibration frequencies are mainly concentrated within the 100–300 Hz range. As indicated by the data in Table 13, the acceleration amplitudes in both the low-frequency (50 Hz) and high-frequency (400–700 Hz) ranges remain relatively low, measuring 0.6 $\mathrm{m}\mathrm{m}/{\mathrm{s}}^2$ and 0.3–0.5 $\mathrm{m}\mathrm{m}/{\mathrm{s}}^2$, respectively. In contrast, the vibration amplitudes show a noticeable increase within the 100–300 Hz frequency band. In particular, a subharmonic peak at 120 Hz, associated with the rotational speed of the spindle motor, reaches an amplitude of 1.2 $\mathrm{m}\mathrm{m}/{\mathrm{s}}^2$, indicating certain variability in the vibration behavior of the machine tool. In addition, a clear peak is observed near 260 Hz, corresponding to the mechanical excitation generated during the cutting process. These pronounced vibration components can cause read-out fluctuations in the measurement system, potentially introducing micrometer-level measurement errors.

To quantitatively evaluate the reliability and accuracy of the measurement results, a comprehensive uncertainty analysis was conducted for the obtained values. A unified method was applied to assess the uncertainties of all geometric error parameters, including positioning, straightness, pitch, and yaw errors. Based on the Guide to the Expression of Uncertainty in Measurement framework, this analysis incorporated multiple influencing factors, including system resolution, repeatability deviations, optical path calibration accuracy, and environmental disturbances affecting the custom 3D laser telescoping ballbar measurement system. Based on five independent measurements conducted at each spatial position, the calculated uncertainties for the respective error components are summarized below.

Table 14. Uncertainty analysis of the positioning error measurement results for the X-axis.

Measurement Position (mm)	Standard Deviation ($\mathbf{\mu }\mathbf{m}$)	Combined Standard Uncertainty ($\mathbf{\mu }\mathbf{m}$)	Expanded Uncertainty (k = 2) ($\mathbf{\mu }\mathbf{m}$)
0	0.00	0.56	1.28
50	1.00	1.22	2.44
100	1.05	1.26	2.52
150	1.59	1.73	3.46
200	1.22	1.40	2.80
250	1.21	1.39	2.78
300	1.03	1.24	2.48

,

Table 15. Uncertainty analysis of the straightness error (Y-direction) measurement results for the X-axis.

Measurement Position (mm)	Standard Deviation ($\mathbf{\mu }\mathbf{m}$)	Combined Standard Uncertainty ($\mathbf{\mu }\mathbf{m}$)	Expanded Uncertainty (k = 2) ($\mathbf{\mu }\mathbf{m}$)
0	0.00	0.57	1.32
50	0.93	1.15	2.30
100	1.54	1.70	3.40
150	1.76	1.88	3.76
200	1.16	1.36	2.72
250	1.34	1.50	3.00
300	1.55	1.71	3.42

,

Table 16. Uncertainty analysis of the straightness error (Z-direction) measurement results for the X-axis.

Measurement Position (mm)	Standard Deviation ($\mathbf{\mu }\mathbf{m}$)	Combined Standard Uncertainty ($\mathbf{\mu }\mathbf{m}$)	Expanded Uncertainty (k = 2) ($\mathbf{\mu }\mathbf{m}$)
0	0.00	0.64	1.45
50	1.18	1.36	2.72
100	1.71	1.85	3.70
150	1.68	1.83	3.66
200	0.88	1.12	2.24
250	1.06	1.26	2.52
300	1.05	1.25	2.50

,

Table 17. Uncertainty analysis of the pitch error measurement results for the X-axis.

Measurement Position (mm)	Standard Deviation ($\mathbf{\mu }\mathbf{rad}$)	Combined Standard Uncertainty ($\mathbf{\mu }\mathbf{r}\mathbf{a}\mathbf{d}$)	Expanded Uncertainty (k = 2) ($\mathbf{\mu }\mathbf{r}\mathbf{a}\mathbf{d}$)
0	0.00	0.47	1.21
50	1.62	1.69	3.38
100	1.32	1.39	2.78
150	1.87	1.94	3.88
200	1.43	1.50	3.00
250	1.49	1.57	3.14
300	1.18	1.26	2.52

and

Table 18. Uncertainty analysis of the yaw error measurement results for the X-axis.

Measurement Position (mm)	Standard Deviation ($\mathbf{\mu }\mathbf{r}\mathbf{a}\mathbf{d}$)	Combined Standard Uncertainty ($\mathbf{\mu }\mathbf{r}\mathbf{a}\mathbf{d}$)	Expanded Uncertainty (k = 2) ($\mathbf{\mu }\mathbf{r}\mathbf{a}\mathbf{d}$)
0	0.00	0.63	1.12
50	1.24	1.33	2.66
100	1.27	1.35	2.70
150	1.55	1.63	3.26
200	1.13	1.22	2.44
250	1.16	1.25	2.50
300	1.22	1.31	2.62

present the uncertainty estimations for the five geometric errors of the X-axis. The estimation procedures for the Y-axis are consistent with those for the X-axis, and the final uncertainty results for the five Y-axis geometric errors are summarized in

Table 19. Uncertainty analysis of the geometric error measurement results for the Y-axis.

Error Term	Positioning Error	Straightness Error (X)	Straightness Error (Z)	Pitch Error
Measuring range	0–300 mm	0–300 mm	0–300 mm	0–300 mm
Standard deviation	1.11–1.5 $\left(\mathrm{\mu }\mathrm{m}\right)$	1.15–1.88 $\left(\mathrm{\mu }\mathrm{m}\right)$	1.03–2.11 $\left(\mathrm{\mu }\mathrm{m}\right)$	1.26–2.60 $\left(\mathrm{\mu }\mathrm{r}\mathrm{a}\mathrm{d}\right)$
Combined standard uncertainty	1.57–1.94 $\left(\mathrm{\mu }\mathrm{m}\right)$	1.44–2.20 $\left(\mathrm{\mu }\mathrm{m}\right)$	1.40–2.37 $\left(\mathrm{\mu }\mathrm{m}\right)$	1.39–2.67 $\left(\mathrm{\mu }\mathrm{r}\mathrm{a}\mathrm{d}\right)$
Expanded uncertainty (k = 2)	3.14–3.88 $\left(\mathrm{\mu }\mathrm{m}\right)$	2.88–4.40 $\left(\mathrm{\mu }\mathrm{m}\right)$	2.8–4.74 $\left(\mathrm{\mu }\mathrm{m}\right)$	2.78–5.34 $\left(\mathrm{\mu }\mathrm{r}\mathrm{a}\mathrm{d}\right)$

.

In conclusion, this systematic uncertainty analysis comprehensively assesses the effects of various error sources on the measurement results. The calculated expanded uncertainties satisfy the experimental precision requirements, thereby ensuring the reliability and repeatability of the measurement data. This evaluation provides a quantitative basis for accurate characterization of machine tool geometric errors and establishes a solid theoretical foundation for subsequent error compensation and structural optimization.

6. Conclusions

To address the technical bottlenecks associated with base station calibration in multi-station laser tracking measurement systems, this study proposes a novel calibration method based on an external physical standard. Furthermore, a base station calibrator capable of achieving high-precision repetitive positioning across four fixed feature points was independently developed. This research aims to provide a high-precision, traceable, and cost-effective approach for measuring the geometric errors of CNC machine tools. The primary conclusions are summarized as follows:

A base station calibrator integrating a precision turntable and an electromagnetically coupled fine-tuning mechanism was successfully developed. By employing precision spherical kinematic pairs (kinematic coupling), this device realizes physical position switching among four spatial feature points with excellent positioning repeatability (a maximum calibration difference of only 1.4 μm). From a hardware perspective, it provides an absolute geometric reference—fully independent of the machine tool motion—for the accurate inverse calculation of external measurement base station coordinates.

By pre-calibrating the feature point coordinates using a high-precision coordinate measuring machine (CMM), the logical paradox of “calibrating the measuring instrument using the precision of the measured object itself” was effectively eliminated. This approach successfully decouples the inherent geometric errors of the machine tool from the coordinate parameters of the external measurement base stations, thereby enabling authentic metrological traceability for the volumetric error measurement results.

The high precision and reliability of the proposed method were verified through experimental validation. Comparative results with the Renishaw XL-80 laser interferometer show that, after integrating the base station calibrator, the system measurement results exhibit high consistency across five geometric errors, including positioning, straightness, pitch, and yaw. A comprehensive uncertainty analysis further confirms that the obtained expanded uncertainties satisfy the strict verification requirements for precision CNC machine tools.

In summary, the base station calibrator proposed in this paper provides a feasible and novel technical scheme for the calibration of external measurement base stations, characterized by high precision, excellent repeatability, and low cost. This device not only overcomes the engineering challenges of limited calibration accuracy and untraceable measurement data but also establishes a solid foundation for the broader industrial application of large-scale measurement equipment (e.g., laser tracers and laser trackers) in the spatial error evaluation of CNC machine tools.