Synchronous Measurement System for Geometric Errors of Dual Rotary Axes in Five-Axis Machine Tools Using a Scanning Probe

1. Research Objectives

With the advancement of modern manufacturing technology and the increasing demand for high-precision and high-efficiency machining, five-axis machine tools have been widely applied in aerospace, automotive manufacturing, and other industries. However, high-precision machining also introduces complex challenges in error control, which directly affect product quality and machine tool lifespan. Therefore, the measurement of errors in five-axis machine tools has become an important research topic.

Geometric errors originate from non-ideal geometric deviations during machine tool manufacturing and assembly, such as axial, angular, and straightness errors. These errors significantly affect machining quality and tend to accumulate; thus, their measurement and compensation are crucial.

Current commercial instruments present two major limitations: (1) a single measurement usually targets only a few geometric error items, neglecting the coupling effects of other error sources and the superposition of errors during multi-axis simultaneous motion; and (2) existing instruments mostly measure installation errors or single-axis motion errors, while ignoring motion errors at different positions. Previous studies have investigated single-axis rotary errors and dual-axis installation errors, but few have considered both simultaneously. Therefore, this study aims to develop a high-precision geometric error measurement technique for five-axis machine tools and proposes a system capable of simultaneously analyzing 20 geometric error items of dual rotary axes, while considering error coupling and dynamic effects.

2. Research Methods

Experimental Equipment and Machine Tool

The experimental platform used in this study is a small vertical five-axis machining center, as shown in Fig. 1. This machine adopts a full-table rotary design and is capable of five-axis simultaneous machining. The rotational functions of the A- and C-axes increase the freedom of workpiece positioning, enabling the tool to approach the workpiece from almost any direction. It is suitable for machining aerospace components, complex automotive parts, and high-precision molds.

The measurement device used in this study is an on-machine probe system, which plays a key role in high-precision manufacturing by ensuring machining accuracy, improving production efficiency, and optimizing processes. The probe system precisely measures the positions of the workpiece and the tool, supporting the high-performance operation of five-axis machining centers.

Commonly used touch-trigger probes generate signals upon contact with the workpiece surface and provide high-precision position information. Such probes can be classified into single-point contact probes and scanning probes. Single-point probes are suitable for point-wise measurements and are commonly used for simple and precise measuring tasks. Scanning probes, by contrast, are capable of moving continuously along the workpiece surface, collecting a large number of data points and providing more detailed surface profile information. They are suitable for measuring complex curved surfaces and fine features.

In this study, an OSP60 scanning probe system (Fig. 2) is used. Compared with conventional touch-trigger probes, it can move rapidly along the workpiece surface and collect a large number of data points and directional information. It is particularly suitable for large-area or multi-point measurements, helping to more accurately capture surface contours and complex geometries. This has significant importance for precision quality control, while also shortening measurement time and improving detailed analysis and inspection capability.

Measurement System and Method

The developed measurement system uses the OSP60 scanning probe to measure a high-flatness reference block and a high-precision calibration sphere, thereby avoiding measurement uncertainty caused by sphere radius errors and block surface flatness. During the measurement process, the calibration sphere is magnetically mounted on the cradle, the block is fixed on the rotary table, and the probe is installed on the spindle, as shown in Fig. 3.

The probe system captures the center of the calibration sphere and the fixed point of the block as the initial position, then rotates the dual rotary axes and measures the sphere center position after rotation and the surface profile information of the block. All measurements are executed by an automatically designed NC-code program, and the acquired data are substituted into the mathematical model proposed in this study to calculate the errors. During the measurement process, sources of uncertainty are minimized, and all equipment is compensated using commercial measuring instruments and temperature control to obtain the most accurate results.

The measurement principle is divided into two parts: the reference sphere and the block. As shown in Fig. 4, the block is mounted on the C-axis rotary table. Due to installation inaccuracies of machine tool components, radial offsets and tilts occur in the C- and A-axes, and the coupled errors of the two axes influence the motion of the block, moving it to Position 1 and generating a volumetric error dP (PIGE). Then, the effect of motion errors is considered.

Motion errors originate from manufacturing defects of machine tool components, causing relative motion offsets and tilts between components. This produces another volumetric error dP (PDGE) and moves the block to Position 2. The deviation between Position 2 and the ideal position, i.e., the difference between the actual and ideal positions, constitutes the total volumetric error dP. By establishing the relationship between volumetric errors and the geometric errors of the rotary axes, both installation and motion errors of the dual rotary axes can be identified.

The system determines the plane equation by fitting the sharp-point position of the block, measuring the plane normal vector and one point on the plane. Then, three plane equations are solved simultaneously to calculate the machine coordinates of the sharp point, thereby avoiding measurement errors. Subsequently, the relationship between the deflection of the plane normal vector and the geometric errors of the rotary axes is considered.

Fig. 4 shows that geometric errors cause volumetric errors at the sharp point of the block, leading to directional volumetric deflection dV of the plane normal vector. Similarly, this includes dV(PIGE) caused by installation errors and dV(PDGE) caused by motion errors, where only angular errors affect the direction of the normal vector. By establishing the relationship between the angular geometric errors of the rotary axes and the normal vector, the deviations of the plane normal vectors of the block can be measured to obtain volumetric error information and thus identify the geometric errors of the rotary axes.

Fig. 5 illustrates the deviations of the calibration sphere under the influence of geometric errors. In this system, the calibration sphere is placed on the A-axis cradle mechanism. Therefore, when the rotary axes rotate, the sphere center position is first affected only by the A-axis installation error and moves to Position 1, and then is affected by motion errors and moves to Position 2, generating a volumetric error. This design allows the separation of the influence of A-axis geometric errors while simultaneously analyzing the volumetric error of the block sharp point on the rotary table, facilitating measurement path design.

The detailed experimental procedure is as follows: the scanning probe is installed on the spindle, the calibration sphere is placed on the A-axis, and the block is fixed on the C-axis. After eliminating probe errors, the coordinates of the calibration sphere and block are measured at the initial position with A- and C-axis angles set to 0°. The automatic measurement program is then executed to perform measurements at various angular positions, record geometric information, and calculate the error terms.

The probe measurement program is developed using NC code. Through circular path scanning, linear path scanning, and single-point triggering, automatic measurements are performed, and the sphere center coordinate deviation at different angles (XYZ center error), the start-point normal vector coordinate deviation of the linear path (Start point error), and the path angle deviation (Best fit angle) are obtained.

Table 1 summarizes the relationship between the 12 groups of rotary axis positions and measurements. Excluding the initial position, a total of 110 error terms are identified in the experiment, including A-axis installation errors (4 items) and motion errors (6 DOF × 6 positions), as well as C-axis installation errors (4 items) and motion errors (6 DOF × 11 positions). The A-axis performs sphere center XYZ error measurements at the initial position and positions 7 to 12. For the latter six positions, transformed sphere coordinates are used. For each of the five planes of the block, two orthogonal scanning paths are generated, and the normal vectors are obtained by calculating the cross product of the path vectors. The Start point error and Best fit angle of each plane are measured, and these data are used to convert volumetric error offsets and tilts to identify geometric errors. Due to posture interference at certain rotary angles, some plane measurements are omitted.

3. Kinematics

To identify geometric errors, two kinematic chains between the probe and the measured objects are established in this study:

1. Probe P → Z-axis → Y-axis → X-axis → A-axis → C-axis → Block B

2. Probe P → Z-axis → Y-axis → X-axis → A-axis → Reference Sphere S

Homogeneous coordinate transformation matrices are used to derive the relative relationships between the block and the calibration sphere with respect to the probe.

Multiple coordinate systems are established between the machine coordinate system, motion axes, and measured objects, such as the machine coordinate system, probe coordinate system, and reference sphere coordinate system. Distance parameters between coordinate systems, such as the distances between axes and the distances from the rotary axis centers to the reference origin, are considered. Parameters that do not affect the results are then eliminated to simplify subsequent calculations and avoid unnecessarily complex equations. Figures 6(a) and 6(b) show the kinematic chains before and after simplification.

In the derivation of the mathematical model, each coordinate system is expressed using homogeneous transformation matrices. Based on the relative geometric relationships, the matrices are multiplied to obtain the geometric relationships of each kinematic chain relative to the machine origin. Both ideal and actual cases (including errors) are calculated. The results include the probe center position, the sphere center point, the block sharp point coordinates, and the plane normal vectors of the block.

When the probe tip contacts the measured object, the condition that the positions are identical can be regarded as the overlap of the probe tip with the sphere center or the overlap of the probe tip with the block sharp point. By solving for the commanded axis positions under ideal and actual conditions, the relationships between geometric information and rotary axis geometric errors can be obtained. The equation variables include the measurement angles θ_C and θ_A and the coordinates x_w, y_w, z_w of the block and sphere relative to the rotary axis coordinate system. The optimal solutions of unknowns are obtained by minimizing the sum of squared residuals. In nonlinear cases, iterative methods are used to approximate the solutions, and the system is linearized to simplify the solution process.

According to the measurement path designed in this experiment, 180 equations can be obtained from a single measurement. After consolidating all equation relationships, the system of geometric error equations for the A- and C-axes is obtained and used to simultaneously solve the geometric errors of the dual rotary axes.

4. Experimental Results

Five groups of measurement results were averaged, and the repeatability among the groups was compared. The measurement results are shown in Figs. 7 to 9.

From the results in Fig. 7, the A-axis installation errors are −23.1704 μm, −72.8639 μm, −29.0082 arcsec, and 5.8683 arcsec, respectively, while the C-axis installation errors are 7.0554 μm, 5.0641 μm, 3.6991 arcsec, and 51.7812 arcsec, respectively. The offset error of the A-axis and the angular error of the C-axis are relatively large, which is speculated to be caused by the coupling of installation errors from other components, such as the squareness error of the linear axes. In the system analysis, it is assumed that the linear axes have no errors, but in reality, linear axis errors exist in the machine tool and influence the installation error values of the rotary axes.

Since the experimental machine does not have a compensation function and there are limited related studies domestically, this study adopts a numerical comparison method to verify the measurement errors. Currently, machine tool manufacturers often use the reference sphere positioning method to roughly estimate rotary axis deviations. The specific method is to locate the sphere center coordinates at the initial position, rotate the rotary axis to 180°, and then locate the second set of coordinates. The average of the two is taken to obtain the offset of the rotary center. If measurements are taken every 90° and circle-fitting is performed, the rotary axis installation error and tilt can be roughly estimated. Although this method cannot accurately measure installation errors, it can roughly estimate the rotation center and motion behavior.

For the A-axis, due to travel limitations, a complete circular path could not be covered, so four coordinates at 30° intervals were used to fit the best rotation center. The results were then compared with those from a dual rotary axis installation error measurement system developed in previous research, and the error analysis results were found to be generally consistent. Although there were slight differences between the methods, which may have been caused by uncertainties such as temperature or vibration, the differences were within an acceptable range. The method developed by previous research was verified by compensation tests and proved to be reliable, and the circle-fitting method also has certain accuracy; therefore, the correctness of the error calculation results in this study was validated.

5. Conclusions and Future Work

This study proposes a measurement system for five-axis machine tools that can simultaneously measure 20 geometric error items of dual rotary axes. The system adopts a scanning probe with dynamic measurement capability and can perform fast, fully automated measurements after simple calibration. This method not only enables the identification of more geometric errors, but also offers advantages such as low cost, easy installation and operation, and short measurement time.

Scanning probe technology is an innovative approach that can continuously collect data during motion. Compared with traditional touch-trigger probes, it can acquire dense coordinate points on the workpiece surface more rapidly and is suitable for comprehensive measurement of complex workpieces on five-axis machining centers. In addition, the system can flexibly adapt to different configurations of five-axis machine tools. By modifying the mathematical model and programs, the geometric errors of different configurations can be analyzed, providing accurate solutions.

Future development directions include system intelligence, automation, and multifunctional measurement. Through artificial intelligence technologies, the probe can achieve automatic calibration and error compensation, reducing human intervention and improving measurement accuracy and efficiency. The measurement parameters can be further expanded to include workpiece dimensions, shape, and surface roughness to achieve comprehensive quality inspection. In addition, real-time temperature compensation technology can be combined to automatically correct measurement results based on temperature sensor data or to adjust using thermal expansion coefficient models. At the same time, active vibration control technology can suppress vibrations through counter-vibration and combine vibration compensation algorithms to correct measurement results and reduce error influence.

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