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Advances in Mechanical Engineering

Volume 2013 (2013), Article ID 198487, 8 pages

http://dx.doi.org/10.1155/2013/198487

## Double Ballbar Measurement for Identifying Kinematic Errors of Rotary Axes on Five-Axis Machine Tools

School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai 200240, China

Received 28 May 2013; Revised 1 September 2013; Accepted 2 September 2013

Academic Editor: Moran Wang

Copyright © 2013 Wei Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper proposes a novel measuring method which uses double ballbar (DBB) to inspect the kinematic errors of the rotary axes of five-axis machine tool. In this study, kinematic error mathematical model is firstly established based on the analysis of the rotary axes errors which originated from five-axis machine tools. In the simulation, working conditions considering different error origins are simulated to find the relationship between the DBB measuring patterns and the kinematic errors. In the measuring experiment, the machine rotary axes move simultaneously along a specified circular path while all the linear axes are kept stationary. The original DBB measuring data are processed to draw the measuring patterns in the polar plots which can be employed to observe and identify the kinematic errors. Rotary error compensation is implemented based on the function of external machine origin shift. Both the simulation and the experiment results show the convenience and effectiveness of the proposed measuring method as well as its operability as a calibration method of five-axis machine tools.

#### 1. Introduction

High-precision and ultraprecision machining technology is becoming the critical component and future trend of modern manufacturing industry. In recent years, research work devoted to improving the machining accuracy has drawn much attention around the world. The prerequisite for reducing machining errors is to measure and identify the error origins of the machine tools, which depends on the measuring instrument and error identification methodology.

Five-axis machine tools are considered as the backbone of manufacturing industry because of their advantages such as higher metal removal rate, better tool accessibility, significantly shorter tool and fixtures adjusting time, and more flexible machining technics. However, they are more complicated and less rigid in structure compared with traditional three-axis machine tools, which often lead to lower machining accuracy. Some researches show that there are, in general, twenty-one errors in three-axis machine tools [1], while there are thirty-eight errors in five-axis machine tools [2]. So it is necessary to measure and identify error origins and then implement proper error compensation technique to improve the machining accuracy of five-axis machine tools. Some successful measuring methods specific to linear axes can be found in both laboratory experiments and industrial applications [3–5].

Double ballbar (DBB), as a simple and effective measuring instrument, is attracting increasing attention [6–9]. Bryan invented it to evaluate machine tool performance, including thermal expansion of linear axis, squareness error, backlash, and servomismatch [10, 11]. It was later adopted by ISO 230-1 to conduct circular tests on machine tools for performance evaluation [12]. Abbaszadeh-Mir et al. first proposed that a ballbar could be used to estimate eight errors related to the rotary axes of a machine tool [13]. They also proved that there were eight axis-to-axis location errors on a machine tool through mathematical analysis and numerical simulations. This has also been recognized in the new ISO 230-1:2012 standard [14]. Lei et al. designed a specified circle path resulting from the simultaneous motion of - and -axis so as to evaluate the dynamic performance of motion servosystem, including the following parameters: feed rate, position loop gains, natural frequency, and damping factor [15]. Khan and Chen introduced a method to separate the error origins caused by - and -axis and then conducted a successful error compensation experiment [16]. Zhang et al. proposed a method to measure four errors of -axis based on the homogenous transformation matrix (HTM) error model [17]. Zhu et al. designed six measuring paths to calibrate eight errors related to the rotary worktable [18].

However, there is still room to develop a more effective and high-precision measuring method based on DBB to identify the errors caused by rotary axes of five-axis machine tools. In order to avoid other error sources’ influence on the measuring result, it is better to make - and -axis move simultaneously during the measuring test, while the other three linear axes are kept stationary. In this paper, a kinematic error model related to the rotary axes is established. Within the allowable range of - and -axis, a specified measuring path is designed to conduct the DBB measuring test and error identification. In addition, the validity of the proposed method has been confirmed through experiments on a vertical five-axis machining center.

#### 2. Kinematic Error Model of Rotary Axes

The proposed measuring method and experiments are conducted on a workpiece tilt-type five-axis machining center which is shown in Figure 1. The machine tool coordinate frame is defined as the reference frame, whose origin is located at the intersection of - and -axis. The initial position of -axis coordinate frame overlaps with the machine tool coordinate frame. The -axis coordinate frame is defined on the worktable. The workpiece coordinate frame is defined on the center of DBB measuring ball . All the coordinate frames are illustrated in Figure 2.

Generally, there are thirty-eight geometric errors in a five-axis machine tool. However, some errors have significant influence on the machining accuracy, while others contribute a little. Some researches have been conducted to investigate the relationship between the error origins and the machining accuracy [19]. This study proposes a novel method to identify four errors which are caused only by the rotary axes during the simultaneous motion of - and -axis. indicates the angular error of -axis with respect to -axis around -axis; , , and represent the linear deviations between - and -axis in the direction of -, -, and -axis, respectively, as shown in Figure 3. According to the above analysis and the symbol definition, the homogenous transformation matrix (HTM) can be deduced as follows.

denotes a HTM of the workpiece coordinate frame with respect to the -axis coordinate frame

The rotation matrix describes the motion of -axis nominal rotation by angle : where is the distance between and in -axis direction, as shown in Figure 2.

When the angular and linear errors exist between - and -axis, the error matrix is defined by the following equation:

When -axis rotates a nominal angle , HTM of -axis with respect to the machine tool coordinate frame can be defined by

Suppose the position of the measuring ball in the workpiece coordinate frame is (given in (5)); we can obtain the position of in the machine tool coordinate frame through (1)–(5): The explicit form of (6) is given as follows:

#### 3. The Measuring Path of Double Ballbar Test

During the actual machining process, the motion of the machine tool is controlled by numerical control (NC) code which is generated by computer aided manufacturing (CAM) software without considering any angular or linear errors [20]. So we can assume that all the errors in (7) are equal to zero; that is to say

Substituting (8) into (7), we can get the ideal position of measuring ball in the machine tool coordinate frame

The accessible work space of the measuring ball can be calculated through (9). As the distance between and the origin of the machine tool coordinate frame is constant when the - and -axis rotate simultaneously within their allowable range, the resulting 3D space can be seen as part of a spherical surface of radius which can be defined by .

During the measuring test, the measuring ball which is mounted on the spindle stays stationary, while the measuring ball which is mounted on the worktable moves along a horizontal circular path resulting from the simultaneous motion of both - and -axis within their allowable angular range. If no error exists during the movement, the measuring pattern should be a standard circle whose radius equals the length of the measuring bar; otherwise, deviations caused by errors can be found between the measuring pattern and the standard circle. We can identify different error origins by analyzing these measuring patterns and finding their correspondence. So it is crucial to design a proper measuring path for the DBB instrument. As illustrated in Figure 2, the center of measuring ball lies on the line passing through the center and perpendicular to the measuring path plane. In this way, the ball together with the three linear axes remains stationary during the DBB test and only the errors caused by the rotary axes appear in the measured results.

For every interpolated point of the measuring path, their position and velocity vectors can be determined by the backward kinematic transformation. As demonstrated in Figures 1 and 2, the initially relative position between the workpiece coordinate frame and the -axis coordinate frame will not exert an influence on the 3D space available for the measuring ball . In order to simplify the calculation, we can make the following assumption: where denotes the distance between the and coordinates, as shown in Figure 2.

There are usually at least two solutions resulting from the backward transformation, and the one with smallest distance to go will be selected:

During the measuring process, without loss of generality, the defined tracking angle begins with the coordinate and the tracking angle is 90° at the tracking start position. The direction of angular velocity of -axis changes when = 90°, 270°, while that of -axis changes when = 180°, 360°. According to the DBB measuring experienc, we should pay more attention to the quadrant-changing positions so as to find the influence on measuring patterns by the backlash of rotary axes. The movement of the rotary axes is documented in Table 1 with corresponding and values along the path at every 45 degrees.

#### 4. Kinematic Error Simulation of the DBB Measuring Pattern

The DBB reading corresponds to the length of the measuring bar. In the measurement or simulation test, DBB detects the difference between the reference length (e.g., 50 mm, 100 mm, or 150 mm) and the real length of the bar. To identify the kinematic errors of the rotary axes, a relationship between the DBB readings and the kinematic errors must be established. Some key dimensions of the machine tool structure and the DBB setup as well as the assumption values of the errors are shown in Table 2.

Then four simulation tests considering different error origins are conducted as follows.(1)An angular error exists between - and -axis. The simulation pattern and the standard circle (which means no error exists in the test) are both illustrated in Figure 4. When is small, the simulation pattern is almost a circle (shown in Figure 4(a)). However, along with the increase of , it presents a pear shape gradually (shown in Figure 4(b)). In addition, when , the pattern shifts along the positive direction of -axis in the polar plot; when , the pattern shifts along the negative direction of -axis.(2)A linear error exists between - and -axis. As illustrated in Figure 5, when , the simulation pattern which still remains a circle shifts along the positive direction of -axis compared with the standard circle; when , the simulation pattern shifts along the negative direction.(3)A linear error exists between - and -axis. As illustrated in Figure 6, when , the simulation pattern shifts along the positive direction of -axis compared with the standard circle; when , the simulation pattern shifts along the negative direction. However, if the absolute values are equal, the influence of on the simulation pattern turns out to be bigger than that of .(4)A linear error exists between - and -axis. As illustrated in Figure 7, when , the simulation pattern is smaller than the standard circle and the center shifts along the negative -axis direction. In addition, the pattern presents a pear shape (small top and big bottom); when , the simulation pattern is bigger than the standard circle and the center shifts along the positive -axis direction which makes the pattern present an upside down pear shape (big top and small bottom).

From the above simulation results and the researches during the last few years, it can be concluded that the same angular or linear errors of the rotary axes produce different influences on the DBB simulation/measuring patterns compared with those of linear axes. Take the case where or exists in the motion as an example, the pattern is only -axis axisymmetric and does not distribute symmetrically with -axis. The reason is that the angle between the direction of the error vector and the sensitive DBB measuring direction (along the axial direction of the measuring bar) changes with the movement of the rotary axes.

#### 5. Experimental Validation for the Error Identification Methodology

##### 5.1. Experiments

In order to validate the effectiveness of the measuring strategy and the error identification methodology, some experiments are conducted on a high-speed five-axis machining center (model: VMC0656 by Shenyang Machine Tool Co., Ltd.) with Heidenhain iTNC530 computer numerical control (CNC) system. The structure configuration of the machining center is shown in Figure 1. This machining center adopts high-speed electronic spindle as well as linear motor driving technology for -, -, and -axis.

As the DBB measuring path in the experiment is in a specified plane in 3D space, traditional commercial DBB software is not available for this kind of data processing. New DBB software is thus developed in MATLAB language to process the original data and draw the measuring pattern in polar plot. What needs special attention here is that among the total range of tracking angle which is 0–540°, data are only captured in the period between 90° and 450° to avoid instability at the start and the end of the measuring process. The DBB measured pattern is illustrated in Figure 8.

The extraction of the values of the parameters from the real measured patterns is shown as follows.

According to (7), the distance between and of the measuring bar can be calculated:

Then we choose four points in the DBB measured circle in the diagonal line direction (four intersection points between the circle and two diagonal lines). According to Table 1, and values along the path at every 45 degrees are documented. For different values of - and -axis, with the DBB reading which corresponds to the length of DBB measuring bar, four equations can be obtained. Though more equations are helpful to the calculation result, considering the convenience of calculation, four equations are enough for extracting the values. Therefore, the error separation is implemented by calculating the four unknown variables. Then the DBB measured patterns are compared before and after error compensation; thus the compensation results are obtained.

After the calculation, the values of parameters are listed for further compensation. The linear errors , , and are 7 *μ*m, −5 *μ*m, and 6 *μ*m, respectively, while the angular error is .

Compared with the simulation patterns, the measured pattern has the following two main characteristics (neglecting the noise comes from the ballbar itself).(1)A periodic component (saw tooth shape) is observed in the measured pattern. This is mainly due to the motor servosystem and drive mechanism of the rotary axes. During the measuring process, the CNC system firstly sends the position and velocity command to the motor servosystem of the rotary axes, and then the worktable is driven by the transmission mechanism connected with the motors. Angular displacement and velocity are detected by the sensors where some noises exist inevitably. These noises will be sent back to the servosystem as feedback signal in the position loop and velocity loop. So the measured pattern has some saw teeth due to these noises brought by the servosystem of the rotary axes.(2)Outward steps appear at quadrant changes. When = 90°, 270°, the angular velocity direction of -axis changes, resulting in the steps in -axis of the polar plot; when = 180°, 360°, the angular velocity direction of -axis changes, resulting in the steps in -axis. It can also be found that backlash in -axis seems bigger than that -axis. One reason is that -axis bears more weight than -axis during the simultaneous motion, which is determined by the structure of the machine tool. If backlash compensation is implemented, these outward steps will disappear in the measured pattern.

##### 5.2. Error Compensation and Result

Another key technology in the error compensation system is to find a proper method to add compensation values obtained from the error model to the motion of the rotary axes. Considering the Heidenhain iTNC530 CNC system, this study proposes an implementation technique based on the function of external machine origin shift. This function can be employed in our developed system to compensate for the rotary axes simultaneously. Compensation values obtained from error compensation software are stored in W576 to W584 address in programmable logic controller (PLC). CNC system can read these data within a cyclical scan cycle through which the compensation values can be added to the feed axes controller within one PLC scan cycle. In addition, the software and CNC system are connected through the Ethernet connection under the extended data interface with LSV2 protocol. The well-acknowledged Ethernet data transmission technology can insure the stable transmission of data without electromagnetic interference under complicated workshop environments. The flowchart of error compensation strategy for rotary axes and the data exchanging method are illustrated in Figure 9.

After error identification, the kinematic errors have been compensated by moving the origin of the work coordinate frame, and the backlash compensation value of either axis has been added to the CNC motion control system. Figure 10 shows the measured pattern after error compensation. The linear errors , , and are compensated from 7 *μ*m, −5 *μ*m, and 6 *μ*m to 2 *μ*m, 1 *μ*m, and −2 *μ*m. The angular error is compensated from to . It can be said that the machining accuracy has been improved and it is accurate enough for conventional machining.

#### 6. Discussions and Conclusions

In this paper, the kinematic error model based on the DBB method is proposed in order to measure and identify the kinematic errors of the rotary axes during the simultaneous motion of - and -axis. The proposed method has the advantage of driving only two rotary axes to move simultaneously and keeping the other three linear axes stationary so as to avoid other errors’ influence on the measured results. Though the kinematic formula derivation under ideal condition is similar to [15], the aims of the two studies are different. Reference [15] focused on detection of the servo mismatch of the rotary axes, and the influence on the trace pattern, such as parameters of the proportional gain , the natural frequency , the damping factor , and no error compensation, was made. In this paper, we mainly focus on the kinematic errors between two rotary axes that result in the different trace patterns of DBB measurements. Furthermore, we implement the error compensation experiment so the kinematic errors are substantially reduced.

In addition, this method reduces the multiple times installation of measuring instrument and improves the efficiency of error measurement compared to the traditional method. The relationship between the measuring patterns and the kinematic errors has been established through the simulation under different kinematic error assumptions. This is useful for the diagnosis of error origins. The measuring method is applied to a five-axis machining center, and, furthermore, error identification and compensation experiments are conducted. From the experiment results, it can be concluded that the proposed measuring method and identification methodology can be conveniently and successfully utilized as a calibration method of five-axis machine tools.

#### Conflict of Interests

The authors do not have any conflict of interests with the content of the paper.

#### Acknowledgments

This research was sponsored by Chinese National Science and Technology Key Special Projects “Top Grade CNC Machine Tools and Basic Manufacturing Equipment” (no. 2011ZX04015-031) and The National Science Foundation Project of China (nos. 51175343, 51275305).

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