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This paper proposes a novel method for identifying carriage errors. A general mathematical model of a guideway system is developed, based on the multi-body system method. Based on the proposed model, most error sources in the guideway system can be measured. The flatness of a workpiece measured by the PGI1240 profilometer is represented by a wavelet. Cross-correlation analysis performed to identify the error source of the carriage. The error model is developed based on experimental results on the low frequency components of the signals. With the use of wavelets, the identification precision of test signals is very high.

With the rapid advances in the development of electronic and optical devices, machines need to meet high precision requirements. Component errors of machine tools are the main factor which affects the machinery accuracy. Due to some limitation reasons, some errors cannot be tracked in real-time, hence identifying the main errors which affect machinery accuracy is important.

A carriage is an oriented-device that can travel in a given trajectory. It is one of the important moving parts which can determine the surface roughness, the surface shape and the relative position. It has a direct effect on the processing results. In order to improve the accuracy of a machinery tool, analyzing and identifing the carriage error(s) is essential.

The measurement of the carriage straightness error plays an important role in metrology. Various methods are adopted in the industrial measurement field [

Some researchers have analyzed and identified the motion errors of the machine tool including the guideway errors using some direct ways. For example, Kakino

From the above, the previous work just measured or modeled the single component error of a machinery tool. These methods will introduce some errors. The analytical results are different from the actual values. Some researchers have analyzed and identified the motion errors of machine tools from several measured results, but they cannot identify the dominant error from the flatness of workpieces.

Multi-body theory is a theory developed several decades ago and used for analyzing complex mechanical systems with movement errors [

This paper proposes a model for identifying carriage errors of a multi-body system, computes the cross-correlation between the carriage errors and the machining accuracy of a workpiece, as well analyzes and identifies the error sources of the carriage. Straightness and squareness of the carriage are measured and calculated. From the fitting equations, if the errors are given, then a cylindical workpiece is machined and the flatness is measured. Hence, the dominant error sources of the carriage can be deduced by performing the cross-correlation analysis on both the simulation results obtained from the model and the actual data obtained from measuring the flatness of the workpiece. The cross-correlation results show that the impact factor of each error of the carriage on the flatness of the workpiece and the main impact factor of the error from the carriage can be identified.

In this session, the multi-body theory is used to model the errors of the carriage. In the multi-body system, topology and low-order array are used for describing the relationship of the physical body. Topology is a major area in mathematics. It is a subject that studies the preservative properties of continuous deformations of the objects, such as the deformations due to stretching without tearing or gluing. This subject has overlapped with geometry and set theory, such as space, dimension and transformation. It is used to describe a family of sets that have certain properties and are used to define a topological space which is a basic object of topology.

The total error motion of the ^{th}^{th}_{ij}_{ij}_{ij}_{ij}_{ij}_{ij}_{ij}_{ij}_{ij}_{ij}_{ij}_{ij}

A rigid solid body has six degree of freedom [_{1}, Δ_{1}, Δ_{2} and Δ_{2} are the translational errors. Δ_{1} and Δ_{2} are the roll errors of the carriage, Δ_{1} and Δ_{2} are the pitch errors. Δ_{1} and Δ_{2} are the yaw errors. Assume that the coordinates of the tool at _{x}, _{y}, _{z}):

The deviation between the actual coordinate _{A}

The cross-correlation techniques have been widely used in engineering and science, particularly in the fields of measurement and communication [_{1},….,_{n}} and _{1},….,_{n}}, respectively, the correlation coefficient is used to measure the similarity between these two random variables [

Daubechies wavelets can describe the details of signals because of their compact support and orthogonality properties. Another advantage of using compact support wavelets is they have fewer degree of freedoms than the others. Daubechies wavelets have enormous potential for the analysis of problems with local high gradients. For constructing Daubechies, the properties of Daubechies wavelets are presented below. A more detailed description can be found in [

Define _{i}_{N}_{N}

According to _{1} and Δ_{2} are the pitch error and the squareness error of the carriages, Δ_{1} is the straightness of the _{2} is the positional error of the _{x}_{y}

If the _{1}(_{1}, Δ_{2} are roll errors of carriage, Δ_{1}, Δ_{2} are the pitch errors, Δ_{1}, Δ_{2} is the yaw errors, the guideway is for turning machine tool. In this research, we turn a cylindrical flat. Since the tool fixed on the _{2}_{2}(x) is not affect by the flatness of the machinery flat workpiece. The calculated value is ignored. The pitch error Δ_{2}(_{2}(

The straightness of the

The induced yaw error:

The pitch error:

Pitch error Δ_{2}(

The machinery errors caused by the straightness:

The major causes for squareness errors are as follows:

In multi-axis machine tools, the carriages are located on the same structure. There is an angular error between the carriages and the structure of the machine tool. In

When the two axes are not perpendicular due to the upright column tilts forwards or backwards, or due to the right or left when the column base is not horizontal, the two-axis machine of a reference coordinate system will fix the machinery frame to each body in the kinematic chain. Based on the reference Cartesian coordinate systems, the _{xz}

Therefore, the machinery error caused by the straightness and the squareness of the carriage according to

The structure of a machine tool is shown in

The results obtained by measuring the data and fitting the curve are shown in ^{−4} mm, the maximum measured straightness error is 2.48 × 10^{−4} mm and the fitting error is about 0.065 μm. It is approximately at the point of 360 mm test range. Here, the measured straightness is 2 × 10^{−5} mm, and the value of the fitting curve is −4.5 × 10^{−5} mm. The actual tool displacement is from 0 to 10 mm in the X direction as shown in the blue curve of the small figure region in

To verify the effect of the errors of the carriage on the out-of-flatness of the workpiece, an aluminum workpiece with a diameter of 20 mm has been machined by the two-axis lathe. The workpiece is a cylindrical flat, and it is supported by the rotating spindle of the machine tool. The tool is fixed on the Z carriage. The displacement in the horizontal direction is controlled by the cross-slider. The

The test result of the machined workpiece includes every error of the machine tool. The errors of the carriage are geometric and the motion errors belong to the low frequency domain. The db1 wavelet is used to decompose the test result first. Then, the test result are expressed as

The first part _{1}(_{2}(t) constitutes the high-frequency signal. The signal _{1}(_{2}(

Here, the carriage errors include the straightness and the squareness. The straightness is determined by measurement. The simulation process of the carriage errors and the decomposed signal of the measured flatness of workpiece are different processes. The shapes of the simulation of the carriage errors and the decomposed low signal (_{1}(_{2}(_{1}(_{2}(_{1}(_{1}(_{2}(_{1}(_{2}(_{1}(_{1}(_{2}(_{xy}^{m} and _{x}^{1}_{y}m_{xy}m_{1}(_{2}(_{x}^{1}_{y}m_{1}(

In order to demonstrate the effect of the wavelet transform, the correlation between the carriage errors and test result is shown. The case treated by the wavelet transform and that without treating are compared. ^{4}. The longitudinal coordinate is the correlation coefficient and it is in the range of [0–1]. Both the correlation coefficient _{xy}^{1} of the squareness error _{1}(_{xy}^{2} of the straightness error _{2}(

_{1}(_{x}^{1}_{y}^{1} of the squareness error and the measured result is close to 1. It means that the squareness error of the carriage is the main error in the test results. The correlation coefficient _{x}^{1}_{y}^{2} of the straightness is similar to that shown in

In the above correlation analysis, it shows that the squareness between the cross carriage and the axis line of the spindle (that is, the normal line of the workpiece shown in ^{−6} rad:

Based on ^{−6} rad. The squareness error is Δ_{xz}^{−6} rad on the _{x}^{−3} mm. It can be verified from ^{−3} mm shown in

In [

For a large complex structure machine tool, some errors are not easy to test. With correlation analysis, the main impact error can be identified from the flatness of the workpiece of the machine tool. Some of the complicated measurements of the components of the machine tool can be eliminated. This method makes up for the drawbacks of some errors of the machine tools for which we cannot measure the component errors directly.

This paper applies the cross-correlation method and the wavelet transform to identify the main error of the carriage of a machine tool. The error model which considers the geometry and the motion errors of the carriage system provides an analytical relationship between the flatness error and the carriage system. This method can identify the correlation coefficient between the carriage errors and the flatness of the workpiece. With the wavelet analysis, the identification precision of the test signal is improved. Also, the dominant errors of the carriage and the impact on the flatness of the workpiece are identified. The method can also be applied to the error identification of other machine tool components.

This research was funded by the National Natural Science Foundation of China Grants Nos. 51105005 and 50775004, and the Fund for New Teachers from the Ministry of Education (No. 20111103120002).

The researched machine tool. (

The processing error caused by the straightness error of carriage.

Experimental setup for measurement straightness of cross guideway.

Measurement and fitting curve of straightness.

Out-of-flatness of workpiece with different feed rates.

Squareness of guideway.

Decomposed result of test by wavelet transformation.

Cross-correlation analysis before using wavelet transformation.

Cross-correlation analysis after using wavelet transformation.

Measurement of squareness between guideway and spindle rotated line.

Cross-correlation analysis of the axis errors in [