Accuracy enhancement of kinematic error model of three-axis computer numerical control machine tools

Abstract

Accurate estimation of volumetric errors is an important issue in machining operations. For this purpose, a kinematic error model is used to characterize machine tool’s related errors on its workspace. In this research, it is shown that when measuring the linear and positioning errors using a laser interferometer, part of the angular errors are converted to linear and positioning errors and their magnitudes are overestimated. These values are calculated twice in the models which use homogeneous transformation matrix since Abbe’s principle is not considered. In this article, a kinematic error model is proposed which eliminates this overestimation. This model’s methodology is based on rigid body kinematic and errors measurement by laser interferometer and can be generalized for all three-axis machine tools. A software package is developed to integrate the kinematic errors with the NC-codes. A workpiece is machined in the virtual environment and compared with a workpiece machined in real environment. It is shown that the kinematic error model developed in this research predicts the kinematic errors more accurately.

Introduction

Precision machining of components with accurate geometrical and dimensional tolerances depends on error identification, characterization and compensation. Volumetric error is the deviation of nominal tool position relative to workpiece, which has different values in machine workspace. ¹ Thermal errors, cutting force and load induced errors, control motion errors and geometric and kinematic errors are the main sources of machine tool errors.^1–4 Geometric and kinematic errors are mainly machine-dependent errors which are affected by geometric errors of machine components and relative motion of joints.

To identify the error vector in all positions of a machine workspace, a volumetric error model is needed. Leete ⁵ generalized the error model using trigonometric equations for continuous compensation of kinematic errors in a machine tool. Lee et al. ⁶ considered a three-axis milling machine and modeled the tool position by a homogeneous transformation matrix. Okafor and Ertekin ¹ developed volumetric error model with combination of thermal error and geometric/kinematic error model based on laser interferometer measurements. Vahebi Nojedeh et al. ⁷ generated a revised geometric/kinematic error model which categorized machine axes into axes carrying workpiece group and axes carrying tool group. Wu ⁸ used vector representation of errors of a machine tool to model the machine kinematic errors. Machine table and slides were considered as vectors. So the three scalar equations were derived to model the 21 error components.

Kinematic error model explains the relationship between geometric/kinematic error components and volumetric error. Laser interferometer-based method is the usual way for measuring the kinematic errors. Error measuring methods and procedures are described in ISO 230-1 and ISO 230-2, respectively.^9,10 In modeling of kinematic errors, it is assumed that the error components are independent. According to some studies, kinematic straightness errors are the integration of angular errors (AEs).^11,12 Ekinci and Mayer ¹² demonstrated that by reducing the ratio of carriage bearing spacing to guide way geometric error wave length, the joint kinematic straightness errors can be estimated by integration of joint kinematic AEs. The relationship between kinematic angular and straightness errors indicates that the deviations caused by them are along one direction and so measuring the linear and positioning errors using laser interferometer is affected by AEs. It should be noted that some studies simplified the hypothesis that AEs may be converted to straightness errors and assumed it as insignificant.^13–15 However, this simplification can affect the accuracy of the kinematic error model. ¹⁶ Bryan ¹⁷ proposed the generalized Abbe’s principle which states that the positioning and straightness measuring system should be aligned with the functional point where the displacement and straightness are to be measured. Otherwise, the AE effects should be calculated.

It should be mentioned that according to the knowledge of the authors, the models used in the research work in this area, which are described by homogeneous transformation matrix, do not comply with Abbe’s principle. The Abbe offsets in these models are not considered properly. In other words, conventional kinematic error models do not consider the measurement instruments. Therefore, to calculate the volumetric errors by classical kinematic error model, AE effects are once considered directly and measured again by linear and positioning measuring instruments, indirectly. Therefore, an overestimation is occurred in the calculation of kinematic error model.

In this study, a vectorial kinematic error model is developed for a three-axis milling machine. The calculation of AE arms is demonstrated. The kinematic error modeling considered in this research focuses on modification of AE arms and eliminating the overestimation affected by the AEs. The model is based on rigid body kinematic and is complied with Abbe’s principle. This approach is used to eliminate the linear and positioning and AE interferences. The model is validated by experimental and simulation results.

Modeling of volumetric errors considering kinematic errors

Each motion axis has six degrees of freedom in a machine tool. So for each axis six error components may exist. These errors are concerned with geometric errors of the machine structures and their relative motions. Six error components of horizontal X-axis are shown in Figure 1. These errors are measured relative to machine coordinate system. As a result, for a three-axis milling machine, adding the three squareness errors there are 21 error components.^1,18,19 The combination of these errors depends on the machine structural loop.^3,20

Figure 1.

Error components of a horizontal X-axis. ¹

Each error component is measured relative to machine coordinate system and can be considered as a vector in a machine workspace. A kinematic error model represents the deviation of tool relative to the workpiece, considering 21 kinematic error components of a three-axis machine tool.^1,19,21 Based on rigid body kinematic, a vectorial kinematic error model is explained as follows:

Straightness and positioning errors

A linear X-axis positioning error creates a vector in X-direction and two straightness error vectors normal to movement in Y- and Z-directions. A functional point is considered on the moving component and lateral displacements of its trajectory are measured. The functional point is the tool position in tool carrying axes as well as the center of table in workpiece carrying axes. So in order to measure the straightness error by a laser interferometer, the straightness reflector should be placed on the functional point. Equation (1) represents the linear and positioning errors in X-axis

$$\{\begin{array}{c}\hfill e_x={\delta }_x(x)\hfill \\ \multicolumn{1}{c}{e_y=-{\delta }_y(x)}\\ \multicolumn{1}{c}{e_z=-{\delta }_z(x)}\end{array}$$

(1)

where e_x, e_y and e_z are error magnitudes along X-, Y- and Z-directions, respectively.

AEs

Every AE vector can be resolved into two components. The magnitudes of these components depend on error angle and the distance between the center of rotation (COR) and the tool position. In this article, these distances are called AE arms. In a kinematic error model, angular and linear and positioning errors are independent; however, during the measurement of linear and positioning errors, AEs have an effect on the linear and positioning errors.^3,16,17,22 So an overestimation is occurred in the calculation of kinematic error model. To distinguish the effects of AEs from linear and positioning errors, AE arms have to be identified correctly. For this purpose, the machine axes are categorized in three groups: a single axis carrying the tool, multiple linear axes carrying the tool and multiple linear axes carrying the workpiece. Therefore, the kinematic error model of all three-axis machine tool configurations can be developed using the integration of these groups.

Single axis carrying tool

The AE arms of a tool carrying axis are measured from the center of joint to tool tip. These arms never change during the movement of axis, along its direction. These arms are called Δx, Δy and Δz.^4,7 When linear and positioning errors of the axis are measured, all of the AE effects are also measured since the measuring system is always aligned with the functional point where linear error is to be measured. This condition complies with Abbe’s principle. So under the assumption of repeatability conditions, AE vectors have a constant value in every position and are measured by linear and positioning reflectors. So measuring the AEs and considering them in the overall volumetric error model causes an overestimation of this error. The errors of a Z single axis carrying tool are obtained as equation (2)

$$\{\begin{array}{c}\hfill e_x={\delta }_x(z)\hfill \\ \multicolumn{1}{c}{e_y={\delta }_y(z)}\\ \multicolumn{1}{c}{e_z={\delta }_z(z)}\end{array}$$

(2)

Multiple axes carrying tool

To illustrate the kinematic AEs in multiple axes carrying tool, two X- and Z-axes are considered (Figure 2). All effects of Z-axis AEs are measured by linear and positioning reflectors similar to single axis carrying tool, since under repeatability conditions, the AE arms and error angles do not change in each position. Therefore, all AE effects are constant in each position and can be measured by linear and positioning reflectors.

Figure 2.

The influence of X-axis pitch error in X-direction with different Z heights.

In order to measure the X-axis linear and positioning errors, Z-axis should be placed on a constant Z height. During the measurement, a part of AE effects are directly measured by linear and positioning reflectors. Therefore, when Z-axis moves in a constant X position, the X-axis linear and positioning error values are changed. The condition where tool tip and reflector are placed in different positions has been shown in Figure 2. The effect of ε_y(x) AE is obtained as equation (3)

$$E_{\mathrm{real}}={\epsilon }_y(x)\times z_0$$

(3)

where z₀ is the distance between tool tip and joint center. The magnitude of E_m is already measured during the measurement of positioning errors. Therefore, the AE arm should be considered from positioning reflector position to tool tip (z_xx). E_c is the part of ε_y(x) AE which should be calculated as AE magnitude and is obtained as equation (4)

$$E_c={\epsilon }_y(x)\times z_{\mathrm{xx}}$$

(4)

So E_real is sum of E_m (that is measured from positioning reflector) and E_c (that is obtained from equation (4)). When AE arm is considered from the joint center, E_m magnitude is measured two times.

So E_c indicates the pitch error magnitude. Consequently, kinematic errors of X- and Z-axes are obtained as equation (5)

$$\{\begin{array}{c}\hfill e_x={\delta }_x(x)-{\epsilon }_z(x)\times \Delta Y+{\epsilon }_y(x)\times z_{\mathrm{xx}}+{\delta }_x(z)\hfill \\ \multicolumn{1}{c}{e_y={\delta }_y(x)+{\epsilon }_z(x)\times \Delta X-{\epsilon }_x(x)\times z_{\mathrm{yx}}+{\delta }_y(z)}\\ \multicolumn{1}{c}{e_z={\delta }_z(x)-{\epsilon }_y(x)\times \Delta X+{\epsilon }_x(x)\times \Delta Y+{\delta }_z(z)}\end{array}$$

(5)

where ΔX and ΔY are AE arms of Z-axis which are the distance between joint center and tool tip along X- and Y-directions, respectively, and z_xx and z_yx are AE arms which are calculated from tool tip.

Multiple linear axes carrying the workpiece

In multiple linear axes carrying the workpiece by moving an axis, the instantaneous COR errors is changed during the time. However, it can be described by considering a linear and rotational movement about the functional point. The location of linear and positioning reflectors can determine the position of the functional point. So the AE arms are the distances between tool position and reflector position along effective direction.

For example to calculate the pitch error (ε_y(x)) component along X-direction, a virtual axis is assumed as the COR. Therefore, Z height between virtual axis and nominal tool position (z₀) is considered as the pitch error arm. As it is shown in Figure 3(a), E_real represents the real magnitude of pitch error along Y-direction (equation (6))

$$E_{\mathrm{real}}={\epsilon }_y(x)\times z_0$$

(6)

Figure 3.

Effects of X-axis: (a) pitch error, (b) roll error and (c) yaw error considering modified AE arms.

Since part of this error is detected by straightness reflector (E_m), the remaining error (E_c) can be calculated as pitch error component along X-direction. So the distance between tool position and straightness reflector position (z_xx) indicates the modified AE arms and E_c is obtained from equation (7)

$$E_c={\epsilon }_y(x)\times z_{\mathrm{xx}}$$

(7)

As shown in Figure 3, E_real is the sum of errors measured by straightness reflector (E_m) and the calculated roll error (E_c). The roll and yaw error magnitudes can also be calculated in the same manner (Figure 3(b) and (c)).

Consequently, the distance between specific linear or positioning reflector position and tool position can be considered as modified AE arms. These modified AE arms comply with Abbe’s offsets. Other AE arms can be calculated in the same manner described. The error magnitudes of roll, pitch and yaw errors are given as equations (8), (9) and (10)

$$\{\begin{array}{c}\hfill e_x=0\hfill \\ \multicolumn{1}{c}{e_y={\epsilon }_x(x)\times z_{\mathrm{yx}}}\\ \multicolumn{1}{c}{e_z=-{\epsilon }_x(x)\times y_{\mathrm{zx}}}\end{array}$$

(8)

$$\{\begin{array}{c}\hfill e_x=-{\epsilon }_y(x)\times z_{\mathrm{xx}}\hfill \\ \multicolumn{1}{c}{e_y=0}\\ \multicolumn{1}{c}{e_z={\epsilon }_y(x)\times x_{\mathrm{zx}}}\end{array}$$

(9)

$$\{\begin{array}{c}\hfill e_x={\epsilon }_z(x)\times y_{\mathrm{xx}}\hfill \\ \multicolumn{1}{c}{e_y=-{\epsilon }_z(x)\times x_{\mathrm{yx}}}\\ \multicolumn{1}{c}{e_z=0}\end{array}$$

(10)

y_xx, z_xx, x_yx, z_yx, x_zx, y_zx are AE arms, where first subscript indicates the measuring direction of linear and positioning errors and second subscript denotes the direction of motion. These arms are defined as equation (11)

$$\{\begin{array}{c}\hfill x_{\mathrm{yx}}=x_w+x_t-x_{s2x}\hfill \\ \multicolumn{1}{c}{x_{\mathrm{zx}}=x_w+x_t-x_{s3x}}\\ \multicolumn{1}{c}{y_{\mathrm{xx}}=y_w+y_t-y_{\mathrm{px}}}\\ \multicolumn{1}{c}{y_{\mathrm{zx}}=y_w+y_t-y_{s3x}}\\ \multicolumn{1}{c}{z_{\mathrm{xx}}=z_w+z_t-z_{\mathrm{px}}}\\ \multicolumn{1}{c}{z_{\mathrm{yx}}=z_w+z_t-z_{s2x}}\end{array}$$

(11)

where x_w, y_w and z_w are workpiece coordinate with respect to machine coordinate system; x_t, y_t and z_t are nominal tool position with respect to workpiece coordinate system; and third term represents the linear and positioning reflector coordinate systems, where in first subscript p, s2 and s3 represent the positioning reflector, straightness reflector in Y- and Z-directions, respectively; and the second subscript represents the direction of motion.

The values of X-axis kinematic errors are obtained from equation (12)

$$\{\begin{array}{c}\hfill e_x={\delta }_x(x)+{\epsilon }_z(x)\times y_{\mathrm{xx}}-{\epsilon }_y(x)\times z_{\mathrm{xx}}\hfill \\ \multicolumn{1}{c}{e_y=-{\delta }_y(x)-{\epsilon }_z(x)\times x_{\mathrm{yx}}+{\epsilon }_x(x)\times z_{\mathrm{yx}}}\\ \multicolumn{1}{c}{e_z=-{\delta }_z(x)+{\epsilon }_y(x)\times x_{\mathrm{zx}}-{\epsilon }_x(x)\times y_{\mathrm{zx}}}\end{array}$$

(12)

As a result, the AE arms are the distances between the specific linear reflector location and nominal tool position. The specific linear and positioning reflector is a reflector which measures the effect of AE. Using this approach, AE arms are equal to Abbe’s offsets. Whereas in the model based on homogeneous transformation matrix, the AE arms are measured from the machine coordinate system and are not equal to Abbe’s offsets. The values of Y- and Z-axes errors are also calculated in the same manner.

Derivation of kinematic error model

To derivate the kinematic error model, the positioning, straightness, squareness and AE effects are integrated along X-, Y- and Z-directions according to the described method in sections “Straightness and positioning errors” and “AEs.” Three squareness errors can be calculated as equation (13)

$$\{\begin{array}{c}\hfill e_x={\alpha }_{\mathrm{xy}}\times y_w-{\alpha }_{\mathrm{xz}}\times z_w\hfill \\ \multicolumn{1}{c}{e_y={\alpha }_{\mathrm{zy}}\times z_w-{\alpha }_{\mathrm{xy}}\times x_w}\\ \multicolumn{1}{c}{e_z={\alpha }_{\mathrm{xz}}\times x_w-{\alpha }_{\mathrm{zy}}\times y_w}\end{array}$$

(13)

where α_xy is squareness error between Y- and X-axes; α_xz is squareness error between X- and Z-axes and α_zy is squareness error between Z- and Y-axes. ¹

Consequently by adding X-, Y- and Z-axes kinematic errors, kinematic errors of a three-axis milling machine configuration of which is shown in Figure 4 is obtained as equation (14)

$$\begin{array}{cc}\multicolumn{1}{c}{e_x=}& {\delta }_x(x)+{\epsilon }_z(x)\times y_{\mathrm{xx}}-{\epsilon }_y(x)\times z_{\mathrm{xx}}-{\delta }_x(y)\hfill \\ \multicolumn{1}{c}{}& +{\epsilon }_z(y)\times y_{\mathrm{xy}}-{\epsilon }_y(y)\times z_{\mathrm{xy}}+{\delta }_x(z)\hfill \\ \multicolumn{1}{c}{}& +{\alpha }_{\mathrm{xy}}\times y_w-{\alpha }_{\mathrm{xz}}\times z_w\hfill \\ \multicolumn{1}{c}{e_y=}& -{\delta }_y(x)-{\epsilon }_z(x)\times x_{\mathrm{yx}}+{\epsilon }_x(x)\times z_{\mathrm{yx}}\hfill \\ \multicolumn{1}{c}{}& +{\delta }_y(y)-{\epsilon }_z(y)\times x_{\mathrm{xy}}+{\epsilon }_x(y)\times z_{\mathrm{yy}}\hfill \\ \multicolumn{1}{c}{}& +{\delta }_y(z)+{\alpha }_{\mathrm{zy}}\times z_w-{\alpha }_{\mathrm{xy}}\times x_w\hfill \\ \multicolumn{1}{c}{e_z=}& -{\delta }_z(x)+{\epsilon }_y(x)\times x_{\mathrm{zx}}-{\epsilon }_x(x)\times y_{\mathrm{zx}}\hfill \\ \multicolumn{1}{c}{}& -{\delta }_z(y)+{\epsilon }_y(y)\times x_{\mathrm{zy}}-{\epsilon }_x(y)\times y_{\mathrm{zy}}\hfill \\ \multicolumn{1}{c}{}& +{\delta }_z(z)+{\alpha }_{\mathrm{xz}}\times x_w-{\alpha }_{\mathrm{zy}}\times y_w\hfill \end{array}$$

(14)

Figure 4.

The configuration of three-axis milling machine and laser interferometer used in this study.

Results

To evaluate the proposed kinematic error model, a three-axis Droop & Rein computer numerical control (CNC) machine tool is used, a schematic view of which is shown in Figure 4. The kinematic error components of the machine tool in forward and backward directions are measured. The roll errors are measured using electronic level and the other errors are measured by ML10 laser interferometer. The volumetric error of machine workspace is calculated taking into account the kinematic error model based on homogeneous transformation matrix (conventional models) and kinematic error model presented in this study.

The results of calculations, in machine workspace for kinematic error model based on homogeneous transformation matrix and kinematic error model presented in this study, are mapped in Figure 5. Analyzing the graphs, it can be shown that the mean error estimated by the presented model is 24.7% smaller than the homogeneous transformation matrix model. This discrepancy is due to differences in AE arms, which causes the overestimation in homogeneous transformation matrix model magnitude.

Figure 5.

Volumetric error magnitude estimated by (a) kinematic error model based on homogeneous transformation matrix and (b) kinematic error model presented in this research.

In order to validate the presented kinematic error model, an enhanced virtual machining is used to compare the error models. Enhanced virtual machining is defined as the combination of volumetric error model and NC simulation. ²³ For this purpose, a software package based on conventional and presented error model is developed. The NC-codes and 21 error components are defined as the software input. After the fragmentation of NC-codes to small linear path, the kinematic error model translates the nominal points to real points. Figure 6 illustrates the process of kinematic error identification at each tool position using the kinematic error model based on conventional and presented model.

Figure 6.

Flowchart of the software package.

The workpieces are then machined in a NC simulator environment, using the NC-codes altered by applying the kinematic error models (presented in this article and conventional models). The virtual machined workpieces are then used as an input to a CAD software and their dimensions are compared with the nominal workpiece. The workpieces are then machined by the CNC machine tool and the coordinate of a number of points on them are measured and used as an input to the CAD software. The validation procedure is carried out by comparing the differences between the values of data obtained from the real machined and nominal workpieces as well as virtual machined and nominal workpieces.

Experimental validation

For experimental validation two free form Spline-shaped slots are machined by the CNC machine tool. One of them is machined on the center of table and the other on the corner of table (Figure 7). Since the results are influenced by the effects of dynamic properties of the machine tool, it is important to minimize these effects as much as possible. For this reason, the slots are pre-machined under roughing and semi-finishing conditions. An amount of 0.3 mm material is left on the walls of the slots. The slot walls are finished by down milling and under same environmental conditions. Table 1 shows the machining conditions.

Figure 7.

(a) Machining the slots and (b) measuring the slots with a coordinate measuring machine.

Table 1.

Finish machining conditions of test pieces.

Parameter	Magnitude
Spindle speed in side cutting operation (r/min)	650
Feed rate in side cutting operation (mm/min)	90
Radial depth of cut (mm)	0.3
Air temperature (°C)	19–22

The coordinates of a number of points on the slots are recorded by a DEA coordinate measuring machine (CMM; Figure 7) and are used as input to a CAD software. Finally, some curves are passed through the points and the differences between the nominal and measured curves are obtained. The free form Spline-shaped slots are then machined in the NC simulator environment applying the NC-codes generated by software package using conventional kinematic error model as well as the developed kinematic error model in this article. The slots are machined in NC simulator in both corner and center of the table. The geometry of virtual machined slots applying the current and conventional kinematic error models and also the real machined slots are compared with nominal slots and the results are shown in Figures 8 and 9. Figure 8 shows the results of the slot machined in the center of machine table and Figure 9 shows the results of slot which is machined in the corner of table.

Figure 8.

Discrepancies between nominal slot and (a) virtual machined slot using kinematic error model based on homogeneous transformation matrix, (b) virtual machined slot using presented kinematic error model, (c) real machined slot and (d) a, b, c together (graph analysis), in the center of table.

Figure 9.

Discrepancies between nominal slot and (a) virtual machined slot using kinematic error model based on homogeneous transformation matrix, (b) virtual machined slot using presented kinematic error model, (c) real machined slot and (d) a, b, c together (graph analysis), in the corner of table.

As shown in Figures 8 and 9, the accuracy of presented kinematic error model is better than the model based on the homogeneous transformation matrix. This issue is more obvious in the Spline-shaped slot which is machined in the corner of the table (Figure 9) since the AEs and AE arms in the corner of table are larger than these values at the center of the table. Therefore, the effects of AE arms calculated by presented kinematic error model and the conventional kinematic error model are more obvious in the corner of the table.

As mentioned earlier, laser interferometer measurement technique is the typical way to measure the kinematic errors. However, the conventional kinematic error models do not consider the measurement instrument. Since the linear reflectors measure part of AEs (Figures 2 and 3), some parts of AEs are measured twice. These are once calculated directly and measured again by linear and positioning measuring instruments, indirectly. Hence, the volumetric error predicted by conventional methods is overestimated (Figures 8 and 9).

The presented kinematic error model contributes to a more accurate prediction of the volumetric error in machine workspace. Moreover, the presented model can be applied for all three-axis machine tool configurations. It can also be integrated with a CAM software where the generated NC-codes are modified to produce more accurate parts. The presented kinematic error model can also be integrated with NC simulators to assess machine tool ability to produce workpieces with certain tolerances.

Conclusion

The main objective of a kinematic error model is the prediction and compensation of machine kinematic errors on its workspace. In this article, the effects of AE arms on the kinematic error model are analyzed. It is demonstrated that in a single axis carrying tool, the effects of all AEs are measured by linear and positioning reflectors. So, calculating the AEs, which are measured by linear reflectors, results in an overestimation of the errors. In addition, the specific reflector coordinate system is the measurement reference to calculate the AE arms, in the workpiece carrying axes. Also, in multiple axes carrying tool, an axis is treated as a single axis and in the other axes, specific linear and positioning reflectors are the measurement references for calculating the AE arms. A kinematic error model is developed based on the modified AE arms. The presented error model is consistent with Abbe’s principle; therefore, the overestimations are eliminated. The results indicate that the presented error model enhances the accuracy of the volumetric error for about 25% in the machine tool used in this study.

To validate these characteristics, a software package is developed based on the kinematic error model. Applying this software, two similar parts are machined in real and virtual environments. A comparison of these two parts shows the effectiveness of the error model. The presented kinematic error model can improve the workpiece accuracy by modifying the NC-codes. It can also be integrated with a NC simulator to assess machine tool’s ability to produce workpieces with certain tolerances.