Synchronous Measurement and Verification of Position-Independent Geometric Errors and Position-Dependent Geometric Errors of Rotary Axes on Five-Axis Machine Tools

Abstract

This paper presents a synchronous measurement methodology aimed at identifying four position-independent geometric errors (PIGEs) and six position-dependent geometric errors (PDGEs) of the rotary axis in five-axis machine tools. Previous studies and literature have emphasized the challenge of simultaneously measuring and identifying PIGEs and PDGEs of the rotary axis in five-axis machine tools. Therefore, the primary objective of this paper is to propose a measurement methodology that can identify these errors simultaneously through a single measuring process. Compared to commercially available measuring instruments, this measurement system offers several advantages: it is easy to install, cost-effective, and can be applied to various types of five-axis machine tools. These benefits enable the establishment of a fast on-machine error measurement. The initial phase of the research involves establishing a mathematical model and computing the geometric error equations based on the specific type of machine tools in use. Subsequently, the difference between the ideal and actual center positions of the calibration sphere is determined by utilizing a touch-trigger probe while positioning the machine's rotary table at various angles. Finally, the experimental data is inputted into the mathematical algorithm to obtain the PIGEs and PDGEs of the rotary table. Post-experimentation, the PIGEs and PDGEs obtained through the proposed measurement method are incorporated into the controller as compensations. The feasibility of this approach is evaluated by measuring the volumetric errors of the machine tools both with and without compensation. The results demonstrate a significant reduction in the deviation of the volumetric errors, decreasing from 11.97 to 2.31 µm after compensation. This outcome underscores the potential of the proposed method for simultaneous measurement of geometric errors in the rotary axis of machine tools across various types and scenarios.

Highlights

This paper proposes a novelly synchronous measurement methodology for identifying the four position-independent geometric errors (PIGEs) and six position-dependent geometric errors (PDGEs) of the rotary axis in five-axis machine tools.

The measurement principle, geometric errors calculation algorithm as well as simulation and experiment verification of the proposed synchronous measurement methodology are presented.

Based on experimental results demonstrating a reduction in volumetric error of up to 80.7%, it is evident that the proposed measurement method is both efficient and precise.

Introduction

Manufacturing technology is advancing rapidly in today's world [1, 2–3]. As manufacturing technology progresses, there is a growing demand for machining machines that are both precise and efficient [4, 5, 6–7]. With the incorporation of two rotary axes, five-axis machine tools exhibit enhanced efficiency and accuracy in the machining process, particularly when machining on curved and intricate workpieces [8, 9–10]. Notably, this is crucial in the production of high-precision components like turbine blades and gears [11, 12]. However, compared to three-axis machine tools, five-axis machine tools have two extra rotary axes [13]. These additional axes introduce additional error sources that can mutually influence one another, potentially reducing machining accuracy.

Building upon previous research [14, 15–16], the factors influencing the accuracy of five-axis machine tools can be classified into three primary categories: static errors, quasi-static errors, and dynamic errors. Notably, quasi-static errors emerge as the most impactful factor, constituting 60–70% of the overall machine error [17, 18]. Within the realm of quasi-static errors, geometric errors contribute to 30%, highlighting their significant influence. It is evident that geometric errors play a critical role in determining the accuracy of machine tools. Thus, ISO 230-7 categorizes the geometric errors of rotary axes into two primary groups: position-independent geometric errors (PIGEs) and position-dependent geometric errors (PDGEs). The PIGEs and PDGEs of the rotary axes are defined as the location errors of an axis' average line and the error motions of the rotation axis, respectively [19]. According to existing literature, PIGEs and PDGEs interactively affect the accuracy of rotary axes in machine tools. Consequently, achieving precise measurements of PIGEs and PDGEs for rotary axes in machine tools holds paramount significance [20].

Consequently, numerous manufacturers have undertaken research focused on measuring and compensating for geometric errors of rotary axis. The commercially available products commonly used on geometric errors measurement of rotary axis instruments in the market are R-test, Double Ballbar, XR20-W, etc. Moreover, the R-test measurement system manufactured by IBS Precision Engineering is widely adopted as commercial measurement equipment for measuring PIGEs of rotary axes on five-axis machine tools. Although it has become a common measurement instrument, setting up the measurement system can be highly challenging due to the need to avoid collisions between the three displacement transducers and the precision sphere. As a result, operating this measurement instrument requires the expertise of an experienced individual to ensure accurate and reliable measurements. In conclusion, the main drawbacks of many commercial measurement products are their high cost and the challenge of simultaneously measuring multiple error quantities. As a result, many researchers have devised distinct measurement algorithms based on existing instruments to enhance the precision of rotary axis geometric error measurement.

Until now, extensive research has been dedicated to measuring geometric errors in rotary axes. For example, Lasemi utilized a double ballbar to concurrently measure both PIGEs and PDGEs, while also considering the potential influence of linear axis errors on the obtained results [21]. Xia designed six distinct double ballbar measurement paths and installation positions to identified different categories of geometric errors [22]. They calculated the PIGEs of rotary axes based on the concentricity and inclination of the mean circular trajectory, additionally considering the up and down oscillations of the errors as PDGEs of rotary axes. Lee and Yang introduced four unique circular paths to measure the geometric errors of dual rotary axes using the double ballbar measuring system [23]. This study also addressed the installation errors of the double ballbar when the main axis of the ballbar and the rotary axis have varying centers. One notable limitation of the double ballbar measuring system is its ability to assess errors solely along its axial direction. Consequently, operators must reinstall the system multiple times to account for errors in different directions, resulting in difficulties in achieving automation and efficient measurement. To address this issue, Weikert proposed a solution named R-test utilizing three evenly spaced probes inclined in the horizontal plane [24]. These probes contacted a precision ceramic sphere and recorded its displacements in three directions. With these measurements, the system could calculate the geometric errors of the rotary axes. Furthermore, Hong and Ibaraki developed a non-contact R-test method to overcome the uncertainties associated with the compression or internal spring kinetics of traditional contact probes [25]. Instead of using a contact probe, non-contact R-test method employed a specular reflection type displacement sensor to collect both the measured and actual values of sphere displacements at various locations in space. This approach eliminates the need for physical contact, enhancing the accuracy and reliability of the measurements. However, the aforementioned measurement systems often lack automated calibration procedures and rely on a skilled and experienced operator.

To fulfill the requirements of an automated measuring process, the measurement instrument must be seamlessly integrated into the machine and readily accessible for use [26, 27]. Employing a touch-trigger probe can be a suitable option for identifying the geometric errors of rotary axes in machine tools, as it eliminates the need for additional manual installation. Once integrated, it can be readily used for the error measurement in machine tools, streamlining the measurement process and facilitating automation effectively. As a result, Ibaraki et al. utilized a touch-trigger-probe and the artifacts for measuring the geometric errors of rotary axes [28].

On the other hand, our previous works [29] and [30] solely focused on proposing measurement methodologies for the PIGEs and PDGEs of the rotary axis in five-axis machine tools, respectively. Additionally, our previous works targeted specific types of five-axis machine tools, including mill-turn machines and large horizontal boring and milling machine tools, to develop multiple measurement methodologies aimed at identifying the geometric errors of rotary axes [31, 32]. Specifically, our previous study [32] focused on mill-turn machine tools and developed a measurement methodology exclusively for the four PIGEs and two PDGEs of the C-axis. Nevertheless, as per ISO 230-7 standards, each rotary axis on a five-axis machine tool is characterized by four PIGEs and six PDGEs. To improve the machining accuracy of five-axis machine tools, it is essential to synchronously measure the four PIGEs and six PDGEs of rotary axes.

After reviewing the literature, it's evident that the measurement methods proposed in the aforementioned studies are limited in their ability to measure the PIGEs and PDGEs of rotary axes individually and separately, or they lack comprehensive coverage of geometric errors [32]. Nevertheless, identifying these errors separately is insufficient, potentially hindering a complete and detailed characterization of the geometric errors in the rotary axes. To address this limitation, this study proposes a novel measurement methodology for simultaneously measuring complete geometric error parameters (four PIGEs and six PDGEs) of the rotary axes. Such an integrated approach would provide more precise measurements of geometric errors, consequently enabling better optimization and compensation of five-axis machine tools. Finally, it must be emphasized that the proposed measurement method in this study is crucial for enhancing accuracy and performance in manufacturing processes.

System Structure and Measurement Principle

Definition of Geometric Errors

Geometric errors in machine tools can result in deviations in the tool's position and orientation relative to the workpiece. These errors are often caused by assembling defects and manufacturing deviations in the machine's components, leading to inaccuracies in the movement of each axis. Hence, The International Organization for Standardization (ISO) classifies geometric errors into two categories based on their causes, namely, position independent geometric errors (PIGEs) and position dependent geometric errors (PDGEs), as described in ISO 230-7 [14]. PIGEs, also known as location errors, refer to errors whose values remain constant and independent of the rotary axis position, regardless of different rotation angles. On the other hand, PDGEs, known as motion errors, have values that vary at different rotation angles due to the presence of different defects on each component. Therefore, according to the reference of ISO 230-7, each rotary axis on a five-axis machine tool has four PIGEs and six PDGEs. Table 1 lists the geometric errors of C-axis in a five-axis machine tool.

Table 1. Geometric errors of rotary axis of machine tools

Symbols used in this study		ISO
PIGEs (Rotary Axis)	Oxc (Offset error of C-axis in X direction)	Location Errors (Rotary Axis)	XOC
	Oyc (Offset error of C-axis in Y direction)		YOC
	Sxc (Squareness error of C-axis to Y-axis)		AOC
	Syc (Squareness error of C-axis to X-axis)		BOC
PDGEs (Rotary Axis)	δxc (Radial error)	Component Errors (Rotary Axis)	EXC
	δyc (Radial error)		EYC
	δzc (Axial error)		EZC
	εxc (Tilt error)		EAC
	εyc (Tilt error)		EBC
	εzc (Angular positioning error)		ECC

Machine Configuration

The study is conducted using the five-axis machine tools MF400U, manufactured by Quaser Machine tools Inc., as depicted in Fig. 1. This five-axis machine tool comprises three linear axes (X-, Y-, Z-axis) and two rotary axes (A-, C-axis). The machine is designed in an RRTTT configuration, also known as Table-tilting type.

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Fig. 1

5-axis machine tool MF400U

In this configuration, both the rotary axis and tilting axis are positioned on the worktable side, providing a wide machining range, making it a popular choice in the market. According to the previous studies and to the best of our knowledge, the geometric errors of rotary C-axis might be dominant in the machining accuracy [31, 33, 34]. These errors can have a dominant impact on the overall precision and quality of the machining process. Based on the measurement methodology, the measurement data are obtained by using the linear axis to drive the probe and make contact with the workpiece. Table 2 shows the specifications of MF400U. Moreover, Fig. 2 depicts a schematic diagram illustrating the position of each axis of the MF400U machine.

Table 2. MF400U Specifications

Type	MF400U	Unit
Tool holder	BBT 40
Stroke
A-axis	+ 30° ~ − 120°/0.0033°	degrees
C-axis	360°/0.005556°	degrees
X-axis	410 (16.14’’)	mm
Y-axis	610 (24.02’’)	mm
Z-axis	510 (20.08’’)	mm
Machine size
Length Width Height	3070 (120.87’’) 2020 (79.53’’) 2820 (111.02’’)	mm mm mm

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Fig. 2

Schematic diagram of MF400U

Measurement System Set-Up

The verification experiment of the proposed measurement system required three calibration spheres and a Renishaw OMP40-2 touch-trigger probe, as shown in Table 3, which provides the specifications of the touch-trigger probe. The touch-trigger probe is utilized to capture the coordinates of the three calibration spheres, enabling the calculation of the geometric errors of the rotary axis. In order to facilitate the explanation of the measurement system installation, Fig. 3 depicts the method and the holder used to connect the touch-trigger probe to the spindle of the machine tools. Furthermore, the calibration spheres, with a radius of approximately 18.999 mm, are securely fixed to the worktable using a magnetic base. For more accurate measurements of the tilt errors (ε_xc, ε_yc) of the rotary C-axis, the calibration spheres were positioned as close to the edge of the workpiece table as possible, as shown in Fig. 4. This setup was employed to accentuate the characteristics of the angular errors, thereby ensuring more precise data calculations. In addition, to increase the effect of the angular error (ε_zc) in the Z direction, the three calibration spheres were set at different heights.

Table 3. Specifications of touch-trigger probe Renishaw OMP40-2

Type	Renishaw OMP40-2
Protection class	IPX8
Approach direction	$\pm$ X, $\pm$ Y, -Z
Measuring force in X and Y directions Measuring force in Z direction	0.5N 5.85N
Max deflection in X and Y directions Max deflection in Z direction	$\pm$ 12.5° 6 mm
Repeatability	1 μm 2σ
Max probing speed	3 m / min
Mass	250 g
Signal transmission Range	Infrared $\pm$ 60° in Z, 360° in X / Y
Storage temperature Operating temperature	-25°C… + 70°C  + 5°C… + 55°C

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Fig. 3

Schematic diagram for connection between touch-trigger probe OMP40-2 and BBT40 tool holder

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Fig. 4

Measurement experiment by using three standard calibration spheres

Measurement Objectives and Principle

The main objective of this paper is to measure the PIGEs and PDGEs of the rotary axis simultaneously and analyze total 10 geometric error items at once. Theoretically, a rotary axis has only one degree of freedom in its direction of rotation. However, due to the existing of geometric errors, it may exhibit six degrees of freedom during rotation. For measuring the PIGEs and PDGEs of the rotary axis simultaneously, three calibration spheres were mounted on the rotary axis. With reference to ISO 230-1 and ISO 230-7, the measurement path involves rotating only the C-axis, and it is designed and executed accordingly [19, 35]. Furthermore, by referring to the R-test measurement system, the positions of the three calibration spheres at each measurement position are obtained. By calculating the position deviations of the three calibration spheres at each measurement position during the operation of the measurement path, four PIGEs and six PDGEs of the C-axis are identified. As illustrated in Fig. 5, the deviation (dp) between the ideal position and the actual position of the spheres is also referred to as the volumetric errors of the machine tools. Among them, θn represents the C-axis angular positions at the n-th measurement position. By analyzing the relationship and difference between the actual and ideal position of the spheres, the PIGEs and PDGEs of the rotary axis are calculated simultaneously.

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Fig. 5

Deviation in the position of the rotary axis

Establishment of Measurement Algorithm

Machine Tools Kinematic Chain

Based on the construction of the five-axis machine tools utilized in this paper, the relationship between the tip of the touch-trigger probe and the calibration spheres was established through kinematic modeling. To accomplish this, two kinematic chains were developed: the probe kinematic chain and the sphere kinematic chain. The sequencing of each machine tool component in the two kinematic chains is determined by the designed diagram of the five-axis machine tool. The probe kinematic chain adheres to the following sequence: reference coordinate system → Z-axis → probe. Meanwhile, the sphere kinematic chain follows this path: reference coordinate system → Y-axis → X-axis → A-axis → C-axis → sphere. To facilitate the construction of the measurement equations, coordinate systems are placed on each machine tool component. Firstly, the assumption was made that the reference coordinate system aligns with the machine coordinate system, and the relationship between each machine tool component coordinate system is established using a homogeneous transformation matrix (HTM). Secondly, the coordinate systems of both the probe and the calibration spheres are transformed into a unified coordinate system. Furthermore, to reduce the complexity of measurement and calculation, it is necessary to simplify the kinematic model and eliminate unnecessary distance parameters.

As depicted in Fig. 6, the alignment of the C-axis coordinate system (CCS), A-axis coordinate system (ACS), X-axis coordinate system (XCS), and Y-axis coordinate system (YCS) was assumed. Additionally, the probe coordinate system (PCS) was assumed to be aligned with the Z-axis coordinate system (ZCS). Notably, parameter z₄ signifies the distance parameter in the z-direction between the reference coordinate system and the Z-axis coordinate system. Parameters x_w, y_w and z_w represent the distance between the center of the calibration sphere and the C-axis coordinate system.

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Fig. 6

Coordinate system of each axis of 5-axis machine tool

Forward Kinematics

The positions of the probe and the calibration sphere with respect to the reference coordinate system (RCS) are obtained by multiplying each of the two kinematic chains. Equation (1) illustrates the relationship between the actual transformation matrix,${{}^{A}T}_C^e$, which represents the transfer matrix from the C coordinate system to the A coordinate system in the actual case and the ideal transformation matrix ${{}^{A}T}_C^i$, which represents the transfer matrix from the C coordinate system to the A coordinate system in the ideal case. Among these, the PIGEs matrix and PDGEs matrix are denoted as $T_{\mathit{PIGES}}$ and$T_{\mathit{PDGES}}$, respectively, and also represented by Eqs. (2) and (3). Additionally, the transfer matrix from the C coordinate system to the A coordinate system in the ideal case can be expressed as Eq. (4).

1

$${{}^{A}T}_C^e=T_{\mathit{PIGES}}{T_{\mathit{PDGES}}{}^{A}T}_C^i$$

2

$$T_{\mathit{PIGES}}=\left[\begin{array}{cccc}1& 0& 0& O_{\mathit{xc}}\\ 0& 1& 0& O_{\mathit{yc}}\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{cccc}1& 0& S_{\mathit{yc}}& 0\\ 0& 1& -S_{\mathit{xc}}& 0\\ -S_{\mathit{yc}}& S_{\mathit{xc}}& 1& 0\\ 0& 0& 0& 1\end{array}\right]$$

3

$$T_{\mathit{PDGES}}=\left[\begin{array}{cccc}1& 0& 0& {\delta }_{\mathit{xc}}\\ 0& 1& 0& {\delta }_{\mathit{yc}}\\ 0& 0& 1& {\delta }_{\mathit{zc}}\\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{cccc}1& {-\epsilon }_{\mathit{zc}}& {\epsilon }_{\mathit{yc}}& 0\\ {\epsilon }_{\mathit{zc}}& 1& -{\epsilon }_{\mathit{xc}}& 0\\ -{\epsilon }_{\mathit{yc}}& {\epsilon }_{\mathit{xc}}& 1& 0\\ 0& 0& 0& 1\end{array}\right]$$

4

$${{}^{A}T}_C^i=\left[\begin{array}{cccc}\mathit{cos}({\theta }_C)& \mathit{sin}({\theta }_C)& 0& 0\\ -\mathit{sin}({\theta }_C)& \mathit{cos}({\theta }_C)& 0& 0\\ 0& 0& 1& z_3\\ 0& 0& 0& 1\end{array}\right]$$

Here, ${\theta }_C$ represents the C-axis angular positions, and the parameter $z_3$ denotes the distance between ACS and CCS in the z-direction. Considering the minor impact of the quadratic term associated with geometric errors of rotary axis, we have decided to exclude it from consideration due to its negligible value.

Additionally, ${{}^{R}T}_S^e$ and ${{}^{R}T}_P$ represent the motion chain of the calibration sphere and the probe, respectively. The outcomes are presented in Eqs. (12) and (13), correspondingly.

5

$${{}^{R}T}_S^e={{}^{R}T}_Y{{}^{Y}T}_X{{}^{X}T}_A{{}^{A}T}_C^e{{}^{C}T}_S$$

6

$${{}^{R}T}_P={{}^{R}T}_Z{{}^{Z}T}_P$$where ${{}^{U}T}_V$ denotes the transfer matrix from the V coordinate system to the U coordinate system.

Inverse Kinematics

Following the equations mentioned above, the pose matrix of the probe and the calibration sphere relative to reference coordinate system are obtained. Furthermore, Fig. 7. illustrates the relationship between forward kinematics and inverse kinematics. During the measurement process, the probe contacts the calibration sphere. At this moment, the elements of the homogeneous coordinate transformation matrix that represent the coordinate positions of the probe and the calibration spheres are equal. Consequently, the inverse kinematics are employed to solve for the initially unknown command values of each axis in ideal case.

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Fig. 7

Relationship between forward kinematics and inverse kinematics

Under ideal conditions, the transformation matrix of the calibration sphere is referred to as${{}^{R}T}_S^i$, while the transformation matrix reflecting the actual situation is denoted as${{}^{R}T}_S^e$, considering the influence of the geometric errors of the rotary axes. Among the homogeneous coordinate transformation matrix${{}^{R}T}_P$, the vector ${[\begin{array}{ccc}{t}_x& t_y& t_z\end{array}]}^T$ represents the position vector of the origin o_p of the PCS with respect to the RCS. Similarly, the vector ${[\begin{array}{ccc}{t}_x& t_y& t_z\end{array}]}^T$ in the homogeneous coordinate transformation matrix ${{}^{R}T}_S$ represents the position vector of the origin o_s of the SCS with respect to the RCS. Additionally, in the homogeneous coordinate transformation matrix ${{}^{R}T}_P$ and${{}^{R}T}_S$, the vector ${[\begin{array}{ccc}{I}_x& I_y& I_z\end{array}]}^T$, ${[\begin{array}{ccc}{J}_x& J_y& J_z\end{array}]}^T$ and ${[\begin{array}{ccc}{K}_x& K_y& K_z\end{array}]}^T$ describe the orientation of the three-unit vectors of the PCS and RCS with respect to the RCS.

According to the proposed measurement methodology in this paper, the geometric errors associated with the rotary axis are addressed by analyzing the deviation between the ideal and actual transformation matrix. As demonstrated in Eqs. (7)–(9), the fourth column elements of the matrices ${{}^{R}T}_P$, ${{}^{R}T}_S^i$ and ${{}^{R}T}_S^e$ are extracted, and equations of equality are established among them. Based on this approach, it becomes feasible to determine the present command values for the X-, Y-, and Z-axes of the five-axis machine tools during measurement process. Among these elements, the values a, b, and c of the ${{}^{R}T}_S^e$ matrix are elaborated in detail within Eqs. (10)–(12).

7

$${{}^{R}P}_{\mathit{tcp}}=\left[\begin{array}{c}0\\ 0\\ z_{\mathit{cmd}}+z_4\\ 1\end{array}\right]$$

8

$${{}^{R}P}_{\mathit{scp}}^i=\left[\begin{array}{c}{x}_{\mathit{cmd}}+x_w\mathit{cos}{\theta }_C+y_w\mathit{sin}{\theta }_C\\ y_{\mathit{cmd}}+y_w\mathit{cos}{\theta }_C-x_w\mathit{sin}{\theta }_C\\ z_w\\ 1\end{array}\right]$$

9

$${{}^{R}P}_{\mathit{scp}}^e=\left[\begin{array}{c}a\\ b\\ c\\ 1\end{array}\right]$$

10

$$a={\delta }_{\mathit{xc}}+x_{\mathit{cmd}}+O_{\mathit{xc}}cos({\theta }_C)+O_{\mathit{yc}}sin({\theta }_C)+x_w(cos({\theta }_C)+{\epsilon }_{\mathit{zc}}sin({\theta }_C))+y_w(sin({\theta }_C)-{\epsilon }_{\mathit{zc}}cos({\theta }_C))+z_w(S_{\mathit{yc}}+{\epsilon }_{\mathit{yc}})$$

11

$$b={\delta }_{\mathit{yc}}+y_{\mathit{cmd}}-O_{\mathit{xc}}sin({\theta }_C)+O_{\mathit{yc}}cos({\theta }_C)-x_w(sin({\theta }_C)-{\epsilon }_{\mathit{zc}}cos({\theta }_C))+y_w(cos({\theta }_C)+{\epsilon }_{\mathit{zc}}sin({\theta }_C))-z_w(S_{\mathit{xc}}+{\epsilon }_{\mathit{xc}})$$

12

$$c={\delta }_{\mathit{zc}}+z_w+sin({\theta }_C)(S_{\mathit{xc}}+{\epsilon }_{\mathit{xc}})-sin({\theta }_C)(S_{\mathit{yc}}+{\epsilon }_{\mathit{yc}})-x_w(cos({\theta }_C)(S_{\mathit{yc}}+{\epsilon }_{\mathit{yc}})+sin({\theta }_C)(S_{\mathit{xc}}+{\epsilon }_{\mathit{xc}}))+y_w(cos({\theta }_C)(S_{\mathit{xc}}+{\epsilon }_{\mathit{xc}})-sin({\theta }_C)(S_{\mathit{yc}}+{\epsilon }_{\mathit{yc}}))$$

Geometric Errors Calculation

Based on our prior research, we previously focused solely on considering the volumetric errors that stem from the difference between the expected and actual positions of calibration spheres [29, 30]. This consideration was specifically aimed at respectively identifying the PIGEs and PDGEs of the rotary axes in five-axis machine tools. However, it is not possible in principle to simultaneously identify the PIGEs and PDGEs of rotary axes by only analyzing the difference between the expected and actual positions of calibration spheres. Hence, this study focuses on measuring and identifying the PIGEs and PDGEs of rotary axes in five-axis machine tools simultaneously. Since the comprehensive mathematical derivation of the volumetric errors can be reviewed in our prior publication [29, 30–31], this subsection provides a concise overview and highlights the distinctions. Based on the forward kinematics equations mentioned above, the correlation between the geometric errors and the servo command values of each axis are utilized to identify and characterize the PIGEs and PDGEs of rotary axes in five-axis machine tools. Exactly, $P^r$ denotes the position indicating the actual position of the calibration spheres, while $P^i$ represents the position symbolizing the ideal position of the calibration spheres, as illustrated in Eqs. (13) and (14), respectively:

13

$$P^r=\left[\begin{array}{c}{x}_{\mathit{cmd}}^r\\ y_{\mathit{cmd}}^r\\ z_{\mathit{cmd}}^r\end{array}\right]=\left[\begin{array}{c}{-\delta }_{\mathit{xc}}-O_{\mathit{xc}}\text{cos}({\theta }_C)-O_{\mathit{yc}}\text{sin}({\theta }_C)-x_w\text{cos}({\theta }_C)-{\epsilon }_{\mathit{zc}}\text{sin}({\theta }_C)-y_w(\text{sin}({\theta }_C)-{\epsilon }_{\mathit{zc}}\text{cos}({\theta }_C))-z_w(S_{\mathit{yc}}+{\epsilon }_{\mathit{yc}})\\ {-\delta }_{\mathit{yc}}+O_{\mathit{xc}}\text{sin}({\theta }_C)-O_{\mathit{yc}}\text{cos}({\theta }_C)+x_w(\text{sin}({\theta }_C)-{\epsilon }_{\mathit{zc}}\text{cos}({\theta }_C))-y_w(\text{cos}({\theta }_C)+{\epsilon }_{\mathit{zc}}\text{sin}({\theta }_C))+z_w(S_{\mathit{xc}}+{\epsilon }_{\mathit{xc}})\\ {\delta }_{\mathit{zc}}+z_w-x_w(\text{sin}({\theta }_C)(S_{\mathit{xc}}+{\epsilon }_{\mathit{xc}})+(\text{cos}({\theta }_C)(S_{\mathit{yc}}+{\epsilon }_{\mathit{yc}}))+y_w(\text{cos}({\theta }_C)(S_{\mathit{xc}}+{\epsilon }_{\mathit{xc}})-\text{sin}({\theta }_C)(S_{\mathit{yc}}+{\epsilon }_{\mathit{yc}}))-z_4\end{array}\right]$$

14

$$P^i=\left[\begin{array}{c}{x}_{\mathit{cmd}}^i\\ y_{\mathit{cmd}}^i\\ z_{\mathit{cmd}}^i\end{array}\right]=\left[\begin{array}{c}-x_w\mathit{cos}{\theta }_C-y_w\mathit{sin}{\theta }_C\\ -y_w\mathit{cos}{\theta }_C+x_w\mathit{sin}{\theta }_C\\ z_w-z_4\end{array}\right],$$

During the measurement process, a total of 12 unique rotation angles are measured, with each angle being separated by intervals of 30 degrees. Consequently, the difference between the actual and ideal position of three calibration spheres is given by Eq. (15):$$d{P}_k={\left[\begin{array}{c}⋮\\ d{P}_x\\ d{P}_y\\ d{P}_z\\ ⋮\end{array}\right]}_{9n}=A_{9n\times 10}E$$$$=\left[\begin{array}{cccc}⋮& ⋮& ⋮& ⋮\\ 1-\text{cos}{\theta }_c^n& -\text{sin}{\theta }_c^n& 0& 0\\ \text{sin}{\theta }_c^n& 1-\text{cos}{\theta }_c^n& 0& 0\\ 0& 0& {-y}_w+y_w\text{cos}{\theta }_c^n-x_w\text{sin}{\theta }_c^n& x_w-x_w\text{cos}{\theta }_c^n-y_w\text{sin}{\theta }_c^n\\ ⋮& ⋮& ⋮& ⋮\end{array}\right)$$

15

$${\left(\begin{array}{cccc}⋮& ⋮& ⋮& ⋮\\ -1& 0& 0& 0\\ 0& -1& 0& z_w\\ 0& 0& 1& y_w\text{cos}{\theta }_c^n-x_w\text{sin}{\theta }_c^n\\ ⋮& ⋮& ⋮& ⋮\end{array},\begin{array}{cccc}⋮& ⋮& & \\ -z_w& y_w\text{sin}{\theta }_c^n-x_w\text{cos}{\theta }_c^n& & \\ 0& {-x}_w\text{cos}{\theta }_c^n-y_w\text{sin}{\theta }_c^n& & \\ x_w\text{cos}{\theta }_c^n-y_w\text{sin}{\theta }_c^n& 0& & \\ ⋮& ⋮& \end{array}\right]}_{9n\times 10}\left[\begin{array}{c}{O}_{\mathit{xc}}\\ O_{\mathit{yc}}\\ S_{\mathit{xc}}\\ \begin{array}{c}{S}_{\mathit{yc}}\\ {\delta }_{\mathit{xc}}^n\\ \begin{array}{c}{\delta }_{\mathit{yc}}^n\\ \begin{array}{c}{\delta }_{\mathit{zc}}^n\\ {\epsilon }_{\mathit{xc}}^n\\ \begin{array}{c}{\epsilon }_{\mathit{yc}}^n\\ {\epsilon }_{\mathit{zc}}^n\end{array}\end{array}\end{array}\end{array}\end{array}\right]$$where ${\theta }_c^n$ represents the n-th measurement rotation angle. $A_{9n\times 10}$ denotes the coefficient matrix of the equation, and $E$ signifies the set of the 4 PIGEs and 6 PDGEs of the rotary axis which are depicted above.

Furthermore, the equations concerning the actual positions of the calibration spheres were also employed to identify the PIGEs and PDGEs within the rotary axes simultaneously, as shown in the Eq. (16):$${\left[\begin{array}{c}⋮\\ x_{\mathit{cmd}}^r\\ y_{\mathit{cmd}}^r\\ z_{\mathit{cmd}}^r\\ ⋮\end{array}\right]}_{3n}=B_{9n\times 10}E$$$$=\left[\begin{array}{cccc}⋮& ⋮& ⋮& ⋮\\ -\text{cos}{\theta }_c^n& -\text{sin}{\theta }_c^n& 0& -z_w\\ \text{sin}{\theta }_c^n& -\text{cos}{\theta }_c^n& z_w& 0\\ 0& 0& y_w\text{cos}{\theta }_c^n-x_w\text{sin}{\theta }_c^n& x_w\text{cos}{\theta }_c^n-y_w\text{sin}{\theta }_c^n\\ ⋮& ⋮& ⋮& ⋮\end{array}\right)$$

16

$${\left(\begin{array}{cccc}⋮& ⋮& ⋮& ⋮\\ -1& 0& 0& 0\\ 0& -1& 0& z_w\\ 0& 0& 1& y_w\text{cos}{\theta }_c^n-x_w\text{sin}{\theta }_c^n\\ ⋮& ⋮& ⋮& ⋮\end{array},\begin{array}{cccc}⋮& ⋮& & \\ -z_w& y_w\text{sin}{\theta }_c^n-x_w\text{cos}{\theta }_c^n& & \\ 0& {-x}_w\text{cos}{\theta }_c^n-y_w\text{sin}{\theta }_c^n& & \\ x_w\text{cos}{\theta }_c^n-y_w\text{sin}{\theta }_c^n& 0& & \\ ⋮& ⋮& \end{array}\right]}_{9n\times 10}\left[\begin{array}{c}{O}_{\mathit{xc}}\\ O_{\mathit{yc}}\\ S_{\mathit{xc}}\\ \begin{array}{c}{S}_{\mathit{yc}}\\ {\delta }_{\mathit{xc}}^n\\ \begin{array}{c}{\delta }_{\mathit{yc}}^n\\ \begin{array}{c}{\delta }_{\mathit{zc}}^n\\ {\epsilon }_{\mathit{xc}}^n\\ \begin{array}{c}{\epsilon }_{\mathit{yc}}^n\\ {\epsilon }_{\mathit{zc}}^n\end{array}\end{array}\end{array}\end{array}\end{array}\right]$$

Similarly, $B_{9n\times 10}$ represents the coefficient matrix of the equation.

Since the primary objective of this study is to simultaneously identify the 4 PIGEs and 6 PDGEs of the rotary axis, the Eqs. (15) and (16) are combined for the solving process. Furthermore, the least squares method is applied to identify the 4 PIGEs and 6 PDGEs of the rotary axes on a five-axis CNC machine tool in this study. The set of 4 PIGEs and 6 PDGEs, denoted as E, of the rotary axis is calculated by minimizing the squared residuals $(dP-CE{)}^2$ within the equations. The equations for the least squares method are given by Eq. (17):

17

$$Q\left(E\right)=\underset{E}{\mathit{min}}\sum _{s=1}^{9n}{R}_s^2=\underset{E}{\mathit{min}}{∥\left[\begin{array}{c}⋮\\ d{P}_x\\ d{P}_y\\ d{P}_z\\ x_{\mathit{cmd}}^r\\ y_{\mathit{cmd}}^r\\ z_{\mathit{cmd}}^r\\ ⋮\end{array}\right],-,\left[\begin{array}{c}{A}_{9n\times 10}\\ B_{9n\times 10}\end{array}\right],\left[\begin{array}{c}{O}_{\mathit{xc}}\\ O_{\mathit{yc}}\\ S_{\mathit{xc}}\\ \begin{array}{c}{S}_{\mathit{yc}}\\ {\delta }_{\mathit{xc}}^n\\ \begin{array}{c}{\delta }_{\mathit{yc}}^n\\ \begin{array}{c}{\delta }_{\mathit{zc}}^n\\ {\epsilon }_{\mathit{xc}}^n\\ \begin{array}{c}{\epsilon }_{\mathit{yc}}^n\\ {\epsilon }_{\mathit{zc}}^n\end{array}\end{array}\end{array}\end{array}\end{array}\right]∥}^2={\underset{E}{\mathit{min}}\Vert dP-CE\Vert }^2$$

Simulation Verification

In this subsection, the 4 PIGEs and 6 PDGEs of the rotary C-axis solved by the least square method of the equation are verified. In the simulated condition, the initial values for 4 PIGEs and 6 PDGEs of the rotary C-axis are randomly assigned. Based on the results mentioned in Sects. 3.1, 3.2, 3.3, 3.4 and 3.5, the simulated $d{P}_k$ and position of the calibration spheres ($x_{\mathit{cmd}}^r$, $y_{\mathit{cmd}}^r$ and $z_{\mathit{cmd}}^r)$ are obtained. Additionally, the simulated values for the 4 PIGEs and 6 PDGEs of the rotary C-axis are calculated using the simulated $d{P}_k$ and position of the calibration spheres. In order to achieve more precise simulation verification results, dozens of simulation experiments are conducted. In addition, the magnitude for each impact factor of the PIGEs and PDGEs of the rotary C-axis are random in each simulation. After the simulation verification, it was observed that the differences between the given and simulated values of the PIGEs and PDGEs were less than 10^–12 percent. This indicates that the proposed measurement algorithm is mathematically accurate and the proposed measurement method is robust.

Experiment Verification

Three precise calibration spheres were used as the measuring workpieces because the distance from any point on the sphere's surface to its center is the same. Furthermore, through a specially designed measurement procedure mentioned above, the center of the calibration sphere can be correctly obtained at each measurement position. As shown in Fig. 8, the calibration spheres were approached in five different directions (± x, ± y, and -z directions) by using the touch-trigger probe. By calculating these five obtained coordinate values, the center positions of the calibration spheres were calculated and identified.

[See PDF for image]

Fig. 8

Measurement procedure of center coordinates of the calibration sphere [30]

According to the experimental requirements, the angle and number of measurement positions were set. Based on our experience, reducing the interval between rotation angles for measurements can improve measurement accuracy. Nevertheless, augmenting the number of measurement points will inevitably lead to an increase in the experimental time needed. Therefore, a total of 12 different rotation angles with 30 degrees intervals are taken for the measurement points in this paper, as shown in Fig. 9. Before conducting the experiment, it is essential to calibrate the touch-trigger probe to ensure that its errors fall within the acceptable range specified by the probe's manufacturer. Moreover, the geometric errors of the three linear axes are calibrated to minimize their impact on the measurement results. Following a series of preliminary calibration procedures, the measurement experiment can be conducted. Initially, the C-axis is set to 0 degrees, after which three calibration spheres are securely positioned on the workpieces table of C-axis. The three linear axes are controlled by the machine controller to approach the three calibration spheres from five directions, allowing us to acquire their respective center coordinates. Measurements of the three calibration spheres are conducted sequentially at each measurement point. By following this measurement procedure, a set of experiments was completed and took approximately 32 min.

[See PDF for image]

Fig. 9

Measurement path

Measurement Result

In this experiment, a total of 11 sets of different rotation angles ranging from 30 to 330 degrees of the C-axis were measured. Based on the equations mentioned in Sect. 3 and the measurement strategy mentioned in Sect. 4, total 4 PIGEs and 6 PDGEs of the C-axis on the five-axis machine tool were obtained simultaneously. From the experimental results, the offset error O_xc and O_yc of the C-axis are − 1.945 μm and − 0.023 μm, respectively. Similarly, the squareness error S_xc and S_yc of the C-axis are − 0.02 arcsec and 2.17 arcsec, respectively. For determining the repeatability of the proposed measurement method, it is important to emphasize that each set of measurement data was averaged from five consecutive sets of data. Furthermore, Figs. 10 and 11 show the measurement results of 4 PIGEs and 6 PDGEs, respectively, of the C-axis in five-axis CNC machine tool.

[See PDF for image]

Fig. 10

Identified PIGEs of C-axis with compensation a offset error and b squareness error

[See PDF for image]

Fig. 11

Identified PDGEs of C-axis a δ_xc, b δ_yc, c δ_zc, d ε_xc, e ε_yc, and f ε_zc

Verification Experiment

For the precision manufacturing industry, assessing volumetric errors is paramount importance as it serves as a critical indicator of the precision of the machine tools. Volumetric errors are defined as the three-dimensional offset errors between the ideal and actual cutting or measurement locations in the working space of the machine tools. As a result, compensating for the volumetric errors obtained from the proposed measurement method can demonstrate an enhancement and effectiveness in the accuracy of the measurement system.

The verification procedure of the volumetric error compensation is shown in Fig. 12. As shown in Fig. 12, the volumetric errors compensated values are obtained through the measurement equations mentioned above and the calculated PDGEs of C-axis. In other words, the coordinate positions of the calibration spheres ($x_{\mathit{cmd}}^r,y_{\mathit{cmd}}^r,z_{\mathit{cmd}}^r$) with and without the PDGEs compensation are obtained. The verification results of volumetric errors, with and without PDGEs compensation for the C-axis, are illustrated in Fig. 13a, b respectively. As depicted in Fig. 13, the volumetric errors of the three calibration spheres at each measurement position have decreased from a maximum of 11.97–2.31 µm. This substantial reduction in volumetric errors serves to validate the accuracy of the geometric error measurement methodology presented in this paper.

[See PDF for image]

Fig. 12

The verification process of volumetric errors

[See PDF for image]

Fig. 13

Volumetric errors a without compensation and b with compensation

Conclusions

A measurement method for simultaneously identifying 4 PIGEs and 6 PDGEs of the rotary axis in five-axis machine tools has been proposed in this paper. Traditionally, PIGEs and PDGEs of the rotary axes are measured separately using different measurement systems. However, it is important to note that PIGEs and PDGEs interactively and simultaneously affect the machining accuracy of five-axis machine tools. Thus, separately measuring the PIGEs and PDGEs of the rotary axes essentially disregards another type of geometric errors. With the proposed measurement methodology, it supplements the lacking measurement capabilities of commercially available measurement equipment for measuring 4 PIGEs and 6 PDGEs of the rotary axis in five-axis machine tools simultaneously. By establishing forward and inverse kinematics, applying homogeneous transformation matrices, and employing the least squares method, the measurement equations were formulated to simultaneously identify four PIGEs and six PDGEs of the rotary axis. Additionally, experimental verification was conducted on the Quaser Machinery MF400U five-axis machine tool, a commonly used small to medium-sized machine in general machining facilities. Furthermore, the experimental results clearly demonstrate that the volumetric error has been reduced by up to 80.7% after executing the proposed measurement methodology and compensating for the PIGEs and PDGEs of the rotary axes. Based on the results of experimental verification, it is evident that the proposed measurement method is efficient, precise, and reliable. Conclusively, it is imperative to emphasize that this paper successfully accomplishes the simultaneous measurement of both the PIGEs and the PDGEs of the rotary axis. However, similar to many other measurement methods that use touch-trigger probes described in previous literature for calibrating the PIGEs or PDGEs of rotary axes, the proposed measurement method also requires intermittent physical contact. Therefore, designing a novel and continuous measurement methodology to calibrate the geometric errors of rotary and/or linear axes by using scanning probes will be studied in our future research.

Author Contributions

YTC: writing- original draft preparation and methodology. YJY: methodology, conceptualization, software, and validation. CSL: writing- reviewing and editing, supervision and project administration.

Funding

The authors gratefully acknowledge the financial support provided to this study by the National Science and Technology Council of Taiwan under grant numbers NSTC 110-2218-E-002-038, 111-2222-E-150 -002 and 111-2218-E-002-032.

Availability of Data and Materials

Not applicable.

Declarations

Competing interests

The authors declare that there is no conflict of interest regarding the publication of this study.

Ethical Approval and Consent to Participate

Not applicable.

Consent to Publish

Not applicable.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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