A novel scheme to measure 2D error motions of linear axes by regulating the direction of a laser interferometer☆
Introduction
A linear axis has translational error motions in three directions, namely the linear positioning and straightness error motions, and angular error motions around three axes, namely yaw, pitch and roll error motions [1]. In the standardized accuracy test procedures for machine tools, described in ISO 230-1 [1], each error motion is separately measured in a different setup with a different measuring instrument. For example, the linear positioning deviations is typically measured by using a laser interferometer. The straightness error motion is typically measured by a straight edge and a linear displacement sensor. The squareness error is measured by using a square. Angular error motions are measured by an autocollimator or a precision level. Typical machine tools have three or more linear axes. Full evaluation of all the error motions requires significant time and a lot of measuring instruments.
The multilateration measurement, the term in [1], enables a user to identify all the error motions by using a tracking interferometer (the term in [1]), or a laser tracker. Fig. 1 illustrates its 2D version. The distance from the tracking interferometer to the retroreflector, installed on a machine spindle, is measured. By the distances from three or more tracking interferometers, based on the trilateration principle, the position of the retroreflector can be calculated (the detailed formulation will be presented in Section 2.1). By using the machine’s kinematic model, all the error motions of linear axes can be estimated. This “indirect” error calibration has been long studied [2], [3], [4], [5]. Its commercial product is available (Etalon [6], [7]). Unlike more typical commercial laser trackers, where the target’s 3D position is estimated by the distance to it and the orientation of the laser beam, the multilateration is based on the measured distances, and does not use the angular orientation of the laser beam in its calculation, which significantly reduces its position measurement uncertainty. All the error motions can be evaluated by using a single measuring instrument only — this is a significant practical advantage.
When four tracking interferometers are available (for the 3D case), the multilateration measurement can be done by a single setup. In practice, due to the instrument’s higher cost, many users have only one. In such a case, the same test must be repeated four times or more with different tracking interferometer positions, assuming the machine’s positioning repeatability [6]. It typically takes several hours. Furthermore, more importantly, a single tracking test does not give any useful information to a user. When all the four tests are finished, all the error motions can be calculated. It is, in a sense, a “black-box” test, which outputs the results only when all the four tests are input. This can potentially limit its applications:
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The multilateration measurement can be used only for the final inspection or the numerical compensation of a completely assembled machine. During an assembly process, a machine tool builder may want to check error motions of each axis and modify the assembly accordingly. It is not possible to apply the multilateration measurement to such a quick check.
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Even if one of the tests fails or has abnormally large measurement error due to e.g. some setup error or external disturbance, it will likely not be noticed until all the tests are finished.
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Due to long measurement time, it cannot be applied to, for example, the evaluation of the thermal influence on the volumetric accuracy [8], [9].
This paper proposes a novel test scheme consisting of the following tests: (1) direct measurement of the linear positioning deviation of two linear axes, and (2) the distance measurement to the retroreflector, positioned on a rectangular path, by continuously regulating the laser beam direction of a laser interferometer. It requires a laser interferometer only. Its objective is to identify all the 2D error motions of two linear axes, i.e. the straightness deviation and the angular deviations of each linear axis, as well as the squareness error between them.
In the proposed scheme, a laser interferometer is installed on a machine spindle with a rotary axis. As the two linear axes position it on a rectangular path, the laser beam orientation is regulated by the rotary axis to the prescribed retroreflector position. The distance from the laser interferometer to the retroreflector is measured. This measurement is analogous to the laser tracker measurement. The difference is that (1) the proposed scheme requires a laser interferometer only. Lower implementation cost is its major practical advantage. (2) While the conventional multilateration requires “black-box” calculation based on all of the four tracking tests, as discussed above, the proposed scheme separately identifies a part of the error motions by each test. It can be potentially applied to a quick check or axis-to-axis accuracy check in a machine assembly or accuracy inspection.
The proposed scheme targets error motions of two linear axes on a 2D plane only. To evaluate all the 3D error motions of three linear axes (X-, Y- and Z-axes), it can be applied to all the planes, i.e. XY, YZ and ZX planes.
The conventional laser interferometer measurement is applied to a linear axis moving along a straight line. The proposed test extends it to a rectangular trajectory. In this sense, the proposed test can be seen analogous to the diagonal test [10], the step-diagonal test [11] and the multi-line tests [12], [13]. In the diagonal test, the laser beam is fixed on a diagonal of the measured volume and the machine moves on this diagonal. In the step diagonal test, the laser beam is fixed on the diagonal, and the machine moves on a step-shaped zigzag path on the diagonal. These tests are proposed to indirectly identify error motions of linear axes by a laser interferometer only, similarly as the proposed test. The diagonal test is, however, effective only to estimate the squareness error between the two linear axes [14]. It has been clarified in [15], [16], [17], [18] that the step diagonal test fails when angular error motions are significant. In the multi-line tests, many laser beam paths, e.g. 15 [13], 21 [12] or 33–55 [19] lines, are measured. In all these tests, the laser beam is fixed. This paper’s scheme enables a user to identify all the error motions of two linear axes by three tests only, by regulating the laser beam direction to follow the machine’s command trajectory.
Unlike a commercial tracking interferometer, the proposed scheme does not regulate the laser beam to automatically follow the retroreflector; its orientation is regulated to the prescribed retroreflector position, i.e. in a “open-loop” control manner. The same concept, “open-loop” tracking interferometer, has been proposed by Ibaraki et al. [20], [21], [22]. In these works, it is applied to the conventional multilateration, i.e. at least four tracking tests are needed to identify all the 3D error motions of three linear axes. An original contribution of this paper is on the proposal of the test scheme requiring a single tracking test only, in addition to direct measurement of the linear positioning deviation of each axis.
Section snippets
Review: conventional multilateration algorithm
As the background, the conventional multilateration algorithm is briefly reviewed (see [5], [6] for further details). To compare with the proposed scheme, this section presents its 2D version. The X-Y plane is taken as an example.
This paper considers a machine configuration where the -axis is mounted on the -axis. The objective is to estimate error motions of X- and Y-axes, shown in Table 1, for all the command positions, and . See Fig. 1. Suppose that the retroreflector,
Experimental setup
Fig. 4a shows the experimental setup (Step 3 in Section 2.2). Fig. 4b shows the laser interferometer attached to the spindle face plate. Its position in the plane vertical to the -axis can be adjusted such that the laser beam approximately intersects with the -axis (see Remark #4 in Section 2.2). Major specifications of the laser interferometer and the cat’s eye retroreflector are shown respectively in Table 2, Table 3.
Identified linear axis error motions
Fig. 5 shows the command trajectory of spindle’s reference point, as well
Objective of uncertainty analysis
The proposed test procedure has uncertainty contributors that are in principle negligible in conventional automated tracking interferometers. For example, since the laser beam direction is regulated by Eq. (4), when there exists the machine tool’s positioning error, i.e. an error with in Eq. (4), the laser beam would not be directed exactly to the center of the retroreflector. The angular positioning error of -axis, as well as an error in the initial estimate of the retroreflector
Conclusion
The conventional multilateration enables a user to evaluate all the error motions of three linear axes by using a single measuring instrument only, which is its strong advantage. On the other hand, unlike conventional standardized measurement procedures, where individual error motion is independently measured in a different setup, all the error motions can be calculated only when all the four tests are finished. It is, in a sense, a “black-box” test, where a single tracking test does not give
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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This paper was recommended by Associate editor Masanori Kunieda.