A novel error equivalence model on the kinematic error of the linear axis of high-end machine tool

The kinematic errors of the linear axis play a key role in machining precision of high-end CNC (computer numerical control) machine tool. The quantification of error relationship is still an urgent problem to be solved in the assembly process of the linear axis, especially considering the effect of the elastic deformation of rollers. In order to obtain the kinematic errors of the linear axis of machine tool, a systematic error equivalence model of slider is proposed. The linear axis contains the base, the linear guide rail, and carriage. Firstly, the geometric errors of assembly surface of linear guide rail are represented by small displacement torsor. Then, according to the theory of different motion of robots, the error equivalence model of a single slider is established, namely the geometric errors of assembly surface of linear guide rail and the pose error of slider are equivalent to the elastic deformation of roller. Based on the principle of vector summation, the kinematic error of a single slider is mapped to the carriage, and the kinematic error of the linear axis is obtained. At the same time, experiments validation of kinematic error model of the linear axis is carried out. It is indicated that the proposed model is accurate and feasible. The analysis of key design parameters shows that the proposed model can provide an accurate guidance for the manufacturing and operation performance of the linear axis in quantification, and a more effective reference for the engineers at the design and assembly stage.


Introduction
With the rapid development of precision manufacturing and unceasing enhancement demand, high-precision machining centers, namely multi-axis CNC machine tools, are widely used in all kinds of advanced manufacturing fields, such as automotive engines, aero-engines and gearboxes, valve bodies and other pivotal components. Multi-axis precision CNC machine tools are composed of the linear axis, rotation axis and other components. As an important translational component and with bearing capacity, the linear axis plays a crucial role in the machining accuracy of workpieces. Generally, the horizontal CNC machine tool with four axes consists of three linear axes (x, y and z axis) and a rotation axis. The linear axis is composed of a base, two linear guide rails, four sliders and a carriage. The kinematic error of the linear axis is mainly affected by the geometric error of the base and the kinematic error of the slider, meanwhile it also directly affects the machining accuracy of workpieces.
However, the geometric tolerances of the installing surface of the linear axis are often given according to the designers' experience, moreover the influence factors of roller deformation inside the slider are ignored in the actual working situations. Therefore, in the design stage and manufacturing process, an accurate kinematic error prediction model of the linear axis is extremely essential. In addition, the kinematic error and machining accuracy of the linear axis can be improved from the aspects of design, manufacture and assembly techniques.
In order to obtain the kinematic errors of the linear axis, the kinematic errors of a single slider moving along the guide rail need to be acquired first. Over the past few decades, a large 3 number of studies have been carried out on the kinematic errors of a single slider. There exist three categories of mainstream research. The first one is that only the geometric errors of the linear guide rail are considered, the contact analysis between the guide rail and slider is ignored. With the measuring form errors of guideway tier, He et al. [1] proposed a systematic method of motion error estimation of a linear motion bearing table. Zha et al. [2] investigated the relationship between the profile error of guide rail and vertical motion straightness of y axis slider, and provide theoretic guidance for the precision design of open hydrostatic guideways. Xue et al. [3] presented a new method to analyze the motion error of a closed hydrostatic guideway and found that the wavelength of each profile error component is the main affecting factor. Considering the geometric profile error of guideway, Fan et al. [4] introduced a systematic approach to predict the kinematic error of machine tool's guideway based on the guideway tolerance. In view of the installing base bending shape, Zhong et al. [5] established the error transfer model to obtain the accurate kinematic errors of a translational axis of machine tool. Zhang et al. [6] proposed a new approximate model to study the effect of geometric errors of guide rails and table on the motion errors of hydrostatic guideways, the result showed that the geometric error of guide rail is the main influence factor. The second one is that only the contact analysis between the guide rails and slides is performed to study the kinematic errors of linear guide rail, the geometric errors of guide rail and base are neglected. Using the finite element analysis, Chlebus et al. [7] presented a method to obtain the static properties of guideway joints with contact layers. Paweł Majda [8] introduced an analytical examination approach, aiming to study the influence of static characteristics of linear guideway on the joint kinematic errors of machine tool table. Based on Hertz contact theory, the elastic deformation of guideway and roller were considered to construct the static model to study the kinematic errors of linear guideway [9,10]. Jeong et al. [11] established the eight-spring equivalent model to link the motion components and guideways, and adopted the finite element method to analyze the static and dynamic properties of linear guideway.
Zou and Wang [12] investigate the contact stiffness variation of linear rolling guides due to the effect of friction and wear, and indirectly analyzed the precision condition of linear rolling 4 guides by the mode shape. The third one is that the geometric error and elastic deformation of guide rail are both considered to study the kinematic errors of linear guide rail, however the research object and the key point are different. Different from the line contact form of the linear guide rail with rollers, the linear guide rail with rolling ball is the point contact form.
Ma et al. [13] proposed a method of combining the geometric error of rolling guide and the elastic deformation of rolling ball, aiming to study the motion error rule of slider under different working situations. Khim et al. [14] established the transfer function between the geometric error of the guide rail and the restoring force, and also the balance equation in the motion process to analyze the motion error of the linear guide rail. They presented a simple and systematic estimation method of motion errors of 5-DOF aerostatic linear motion stage.
In view of the difficulty in obtaining the geometric error of guide rail, Khim et al. [15] further substituted the geometric error of guide rail with the straightness error of base, and analyzed the motion error of linear guide rail. As for the aerostatic bearings, Ekinci et al. [16] proposed an equation system, aiming at investigating the relationship from the geometric errors of guideway to the motion errors of axis. Khim et al. [18] introduced a prediction model of motion accuracy of a multi-sliders motion table and constructed the transfer function between the form errors of rail and the bearing force of a bearing block.
Many researches have been carried out about the geometric errors modeling and measurement of linear guideway, also the relationship between the geometric error of guideway and the kinematic error of the linear axis. Generally, the processing position change of workpiece is mainly guaranteed by the linear axis precision, so it is very essential to obtain the kinematic error of linear axis. Furthermore, the kinematic error of the linear axis is influenced by the assembly process, including the hierarchical error transfer of base, guide rail and carriage. More importantly, the kinematic error prediction model of the linear axis of machine tool can provide an accurate guidance for precision design of new machine tools.
In order to obtain the kinematic error of linear axis, a systematic error equivalence model is proposed. Firstly, the geometric errors of assembly surface of linear guide rail are represented by small displacement torsor. Then, according to the theory of different motion of 5 robots, the error equivalence model of a single slider is established. Little work focused on the error equivalence method, the contribution point of the proposed model is that the geometric error of assembly surface of linear guide rail and the pose error of slider are equivalent to the elastic deformation of roller. Afterwards, based on the principle of vector summation, the kinematic error of a single slider is mapped to the carriage and the kinematic error of the linear axis is obtained. Finally, a measuring experiment of the kinematic errors of the linear axis is performed, aiming to verify the validity of the proposed model by comparing the theoretical results and the experimental values. The rest of this paper is organized as follows:

Geometric error representation based on MCS method
The linear axis of machine tool is mainly composed of base, guide rail, slider, roller, carriage and other parts, the carriage reciprocates along the direction of the guide rail. The kinematic error of linear axis depends on the geometric error and deformation of the parts.
The geometric error value of base and linear guide rail is unknown in the design stage, while its tolerance value is the interval value that restricts its change. The geometric tolerance is easier to obtain, but it is difficult for the precise distribution function of the geometric error.
The sampling statistical analysis method is often used to analyze its uncertain parameters under the unknown distribution function. Therefore, the interval parameters and small 6 displacement torsor method are adopted to describe the uncertain geometric error. Besides, MCS method is employed for simulation error, the geometric error distribution of linear guide rail is obtained under meeting the geometric tolerance constraints.

Small displacement torsor description
The kinematic error of the linear axis of machine tool usually depends on the manufacturing error, bearing deformation and assembly error of the parts, but the geometric error is uncertain, so it is obviously difficult to study. Hence, some characteristic parameters are employed to describe the geometric elements, and then are transformed into the study of the uncertain geometric error.
Torsor theory is an important mathematical tool for studying space motion. Torsor is composed of two three-dimensional vectors, which respectively represent the direction and position of vectors [19]. Furthermore, small displacement torsor is commonly used, which is a vector composed of weeny displacement generated by a rigid body with six motion components. In 1996, Bourdet et al. [20] first introduced small displacement torsor into the tolerance modeling field, and proposed to replace the variation of the actual tolerance surface in tolerance zone with the variation of small displacement torsor parameters. In the three-dimensional space, any physical quantity has three translational degrees of freedom along the coordinate axis and three rotational degrees of freedom around the coordinate axis, u v w    is the small displacement torsor parameters.
In three-dimensional space, any physical object can be simply abstracted into the basic elements, namely points, lines and planes. Similarly, the geometric elements of the geometric error can be expressed by their respective small displacement torsors [21]. In theory, there are six torsor parameters for any three-dimensional characteristic surface, namely three 7 translations and three rotations. However, in practical product processing, due to the existence of constraints, some torsor parameters of geometric elements with different types of geometric errors are taken as zero. The rule of setting the parameter as zero is : when the geometric element moves along a certain direction, its motion trajectory does not generate new error sweep entity. Ref. [22] listed several common torsor expressions corresponding to form and position errrors.

Geometric error constraints of guide rail
The kinematic error of the linear axis of machine tool is mainly manifested as the deviation between the actual and ideal position of rolling guide rail, which can be expressed by torsor parameters ( ) , ,0, , , uv    . According to the design requirements, neglecting the dimension error, the pose error of rolling guide rail is determined by the planeness error of the assembly main datum surface, the perpendicularity error and the parallelism error of the assembly subordinate datum surface. In this paper, they are called the geometric error of linear guide rail, as shown in Fig. 1.
The variation range formulas of main assembly datum surface A are where x, z value is the four limit positions of surface A, that is four corners on the assembly main datum surface A in the figure, namely the setting range of x value is ( ) / 2, / 2 aa − , and the setting range of z value is ( )

(2) parallelism error
According to the design requirements, the assembly subordinate datum surface C of the left guide rail is required to have a parallelism error c T with the assembly main datum surface A, furthermore the corresponding small displacement torsor model is

(3) perpendicularity error
Similar to the subordinate datum surface C of left guide rail, according to the design requirements, the assembly subordinate datum surface B of the right guide rail is required to have a perpendicularity error c T with the assembly main datum surface A, moreover the corresponding small displacement torsor model is ( ) Similar to the constraints form of parallelism error, the torsor parameters constraints of perpendicularity error are where y, z value is the four limit positions of surface B, that is four corners on the assembly subordinate datum surface B in the figure, namely the setting range of y value is

MCS method
MCS is a method to solve approximate solutions of problems through statistical tests and stochastic simulation of random variables [24]. It has many advantages such as strong adaptability, simple calculation method, and solving errors independent of problem dimensions. With the MCS method, small displacement torsor parameters of geometric error are sampled, the sampling flow chart of MCS simulation is shown in Fig. 3. The generated corresponding random parameters are utilized to simulate flatness and parallelism and perpendicularity error, and then these random numbers meeting the constraint conditions are retained, and in contrast unsatisfactory ones are removed, finally above process are regarded as a basis of solving the kinematic error of the linear axis of machine tool. Taking the flatness error of linear guide rail's main assembly datum surface A as an example, the steps to simulate the actual variation range of flatness error by MCS are as follows: (1) Determine an ideal probability distribution model of geometric error torsor parameters. In general, the geometric error distribution of parts manufactured by mechanical processing conforms to the law of normal distribution [25], and its probability density function is assumed to be 12 (2) Determine the mean value and variance of ideal probability distribution for torsor parameters of geometric error. The calculation results show that the probability of the value x ranging from -3 u  to +3 u  is 99.73%. From the perspective of engineering, it is generally considered that the distribution range of the normal distribution is 3  . According to the torsor variation in equation (1), the mean and variance of the ideal distribution of torsor parameters , , (3) According to the constraint requirements, the torsor parameters of geometric errors are sampled, and the total samples number in this paper is 10000.

Geometric error simulation of guide rail
The structure of the linear axis contains the base, linear guide rail and carriage in Fig. 4.
The kinematic error of the linear axis is determined by the combined pose error, composed by the assembly main datum surface and subordinate datum surface of left and right guide rails.
According to the master-slave relation of the assembly guide rails, the right guide rail is taken 14 as the datum one, the torsor parameters of pose error for left and right guide rail are respectively ( )

Error equivalence model of a single slider
In order to obtain the kinematic error of the linear axis of machine tool, the kinematic error of a single slider is firstly calculated. Anytime, the linear guide rail keep balance under the load namely itself gravity and the elastic restoring force of rollers. Moreover, the geometric error of guide rail and the position error of slider will influence the contact condition among the slider, guide rail and rollers in Fig. 8. Consequently, the elastic deformation of rollers can be effected. Hence, in this paper, the error equivalence model is around x, y and z axis, as is shown in Fig. 9. Furthermore, the kinematic errors of these five degrees of freedom of the machine tool mainly derive from the straightness of the guide rail.
Hence, in this paper, five items of kinematic errors, namely the straightness along x and y axis ( y  and x  ), and angular errors of pitch, yaw and roll ( , , u v w    ) , the physical meaning of each error is listed in Table 1.
When the slider is under the loads, the elastic deformation occurs between rollers and raceway. Since the width of the contact region between roller and raceway is far less than the curvature radius of contact point, it can be equivalent to the contact problem between an elastic cylindrical and a rigid plane [26]. Therefore, as for the motion of a finite length roller in the raceway, the relationship between the elastic deformation and load can be obtained by the Palmgren empirical formula [27], as shown in Eq.
where e n denotes the number of effective rollers subject to load in each column.  Hence, the offset of the center distance of guide rail raceway and slider raceway is equivalent to the elastic deformation of roller. Ideally, the contact model of rollers before and after elastic deformation is shown in Fig. 11.
(a) 26 (b) Fig. 11 (a) Contact model of linear guide rail before deformation; (b) Error equivalence model after deformation In Fig. 11 (a), o is the geometric center of roller,  is the angle between the center line of roller and raceway and horizontal direction before deformation.  14), respectively. In Fig. 11 (b),  is the angle between the center line of roller and raceway and horizontal direction after deformation.

Kinematic error model of the linear axis
The geometric error of base and the kinematic error of a single slider are studied in above sections in this paper, then the kinematic error of the linear axis is the main research object in this section. In addition, the kinematic error of the linear axis is the same as that of a single slider, and there are five categories of errors: straightness along x and y axis, pitch, yaw, and roll angular error. In this section, the kinematic error of a single slider is transformed to derive and obtain the kinematic error of the linear axis. As shown in Fig. 12

Experiment setup
Some approaches have been utilizied to measure the kinematic error of the linear axis [30], in this paper, the horizontal machine tool of a certain company is regarded as a case to carry out the verification experiments. The Renishaw multi-laser interferometer XL-80 is employed to measure angular errors ( pitch and yaw) of z-axis of machine tool, the instrument resolution is 0.01″, the angle measurement range is ±10°, the angle accuracy is(±0.2%± 0.5±0.1M) μm/m, M is the measurement distance，the unit is m. The dial indicator and marble square are adopted to measure the straightness along the plane xoz and yoz, the electronic level meter is used to measure the angular error (roll), as shown in Fig. 16. Besides , the type NSK RA55 of the linear guide rail is used in the machien tool, some parameters is listed in Table 2. All guide rail, roller and slider of linear guide rail use the material GCr15, the elasticity modulus is 206 GPa, and the Poisson's ratio is 0.3. Therfore, all five kinematic errors of z-axis of horizontal machine tool have been measured [31], as shown in Fig. 17. 36 Fig. 16 Renishaw equipment of multi-laser interferometer

Number of each row of contact rollers 18
After the dial indicator and marble square are adjusted, and the linear axis of machine tool is moved along z axis, then the straightness is measured. The top and side surface of the marble square is measured with the dial indicator, respectively, namely the straightness of surface yoz and xoz along z axis in Fig. 17 (a) and (b). Secondly, after the laser interferometer is adjusted, and the linear axis of machine tool is also moved along z axis, the angular error of pitch and yaw are measured in Fig. 17 (c) and (d). Finally, after the electronic level is adjusted, and the linear axis of machine tool is also moved along z axis, the angular error of roll is measured in Fig. 17 (e). In addition, the z axis moving stroke of the machine tool is 610 mm.
The above moving along the z axis of machine tool are set up 10 measuring points each stroke and repeated for 10 times. A total of 100 groups of measurement data are obtained.

Verification
With measured straightness and angular error of z axis of machine tool, the corresponding kinematic errors of the linear axis are all calculated by the proposed model in this paper, as shown in Fig. 18. In order to verify the validity of the proposed model, the 39 theoretical model results are compared with the experiment data, as shown in Fig. 19.
Five groups of measured kinematic error parameters were plotted as the frequency histogram, and then Gaussian fitting was performed to obtain the Gaussian curve. Afterwards, five kinds of kinematic error parameters calculated by the proposed theoretical model were plotted as the frequency vs error curves, moreover the peak lines of the frequency curves of the experiment data were compared and analyzed with the theoretical model. The comparison results show that for five kinds of kinematic errors, the frequency value of the theoretical model curve is much smaller than that measured by the experiment in Table 3. It is mainly due to the larger sample size of the theoretical model than the experimental model.
Furthermore, the comparison mainly focuses on the frequency peak value, namely the average value of kinematic error.
When machine tool moves along z axis, on one hand, as for the straightness of two surfaces, and surface xoz along z axis, as shown in Table 3, the error of the maximum frequency value between the theoretical model and experiment results is 0.18% in Fig. 18 (a) and Fig. 19 (a). Similarly, surface yoz along z axis, the error of the maximum frequency value between the theoretical model and experiment results is 0.26% in Fig. 18 (b) and Fig. 19 (b).
On the other hand, as for the angular error around x, y and z axis, the pitch error around x axis between the theoretical model and experiment results is 6.46% in Fig. 18 (c) and Fig. 19 (c), and the yaw error around y axis between the theoretical model and experiment results is 8.24% in Fig. 18 (d) and Fig. 19 (d), and the roll error around z axis between the theoretical model and experiment results is 8.54% in Fig. 18 (e) and Fig. 19 (e).
It can be seen from the above results that the errors of five kinematic error values calculated by the theoretical model are all within 10%, thus it belongs to the acceptable accuracy range. Therefore, the results show that the theoretical model proposed in this paper is accurate and feasible. In addition, the theoretical model proposed in this paper can also provide a scientific and reasonable guidance for the design, manufacture and assembly of the linear axis of the machine tools.

Discussions
In the manufacturing and assembly process of the linear axis, the geometric error of parts and the external load play a key role in making a precision linear axis of the machine tool.
According to above verification results in section 4.2, some effect factors on the kinematic error of the linear axis is researched in the proposed model, including the preload of guide rail, the geometric error of the assembly surface of linear guide rail, and the external load and moment.

Effect of preload
According to literatures [13,32], as for a single slider, the preload played an important role in the straightness of the assembly main datum surface A and the subordinate datum

Effect of parallelism
It can be seen from Fig. 22

Effect of perpendicularity
As shown in Fig. 23 Table 5. As can be seen from the following Table,    In future study, we will add the study of the surface topography effect of linear guide rail on the kinematic error of the linear axis, and further obtain the accuracy retention rules of the linear axis, aiming to predict the operation life of machine tool.
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