Measurement of thermal influence on a two-dimensional motion trajectory using a tracking interferometer

CIRP Annals - Manufacturing Technology 65 (2016) 483–486

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Measurement of thermal influence on a two-dimensional motion trajectory using a tracking interferometer S. Ibaraki a,*, P. Blaser b, M. Shimoike c, N. Takayama c, M. Nakaminami c, Y. Ido c a

Department of Micro Engineering, Kyoto University, Nishikyo-ku, Kyoto 615-8530, Japan Institute of Machine Tools and Manufacturing (IWF), ETH Zurich, Technoparkstrasse 1, PFA E81, 8005 Zurich, Switzerland c DMG Mori, Co., Ltd., Meieki 2-35-16, Nakamura-ku, Nagoya 450-0002, Japan Submitted by S. Shimada (1), Osaka, Japan b

A R T I C L E I N F O

A B S T R A C T

Keywords: Metrology Machine tool Thermal error

Conventional thermal deformation tests in ISO 230-3:2007 only measure the tool displacement when it is positioned at a single point. This paper proposes the application of a tracking interferometer to the evaluation of the thermal influence on two-dimensional motion trajectory. The full multilateration algorithm requires at least four tests repeated at different tracking interferometer positions, when only one tracking interferometer is available. This paper proposes the identification of 2D geometric errors of linear axes by single-setup tests. Since the measurement time is significantly reduced, it can be applied within a thermal test. The uncertainty analysis is also presented. ß 2016 CIRP.

1. Introduction Thermal deformation of machine structure, caused typically by the heat generation in spindle and feed drive motors or environmental temperature change, is clearly among major error sources for any machine tools. As reviewed in [1–3], numerous efforts have been reported on the measurement, modelling and compensation of thermal errors. Thermal tests described in ISO 230-3:2007 [4] are widely adopted by machine tool builders. The tests in [4] measure the displacement of the tool centre point (TCP), as well as the orientation of the tool, when it is positioned at a single point (or two points at most), as the heat is generated by the spindle rotation or the reciprocal motion of linear axes. When the heat only causes simple linear expansion of machine structure to some direction, the TCP displacement would be the same at any points in the workspace. However, the heat often affects straightness and angular error motions of linear axes, as illustrated in Fig. 1. In such a case, the tool’s displacement/ orientation can be significantly different depending on the location in the workspace. In other words, the heat may influence the machine’s volumetric accuracy [1]. Conventional thermal tests in [4] cannot evaluate such influence at all. Thermal influence on a trajectory, not a point, has been studied in few publications. Florussen et al. [5] studied thermal influence on ball bar tests. The ball bar can be applied only to a circular path in a small region. Gebhardt et al. [6] applied the R-test to evaluate thermal influence on a rotary axis’ rotating trajectory. The term ‘‘volumetric accuracy’’ is defined in ISO 230-1:2012 [7]. ISO/TR 16907:2015 [8] describes the numerical compensation of

* Corresponding author. E-mail address: [email protected] (S. Ibaraki). http://dx.doi.org/10.1016/j.cirp.2016.04.067 0007-8506/ß 2016 CIRP.

volumetric errors. The evaluation of the volumetric accuracy requires the measurement of the TCP’s 3D position for any command positions in the workspace. Such a 3D position measurement is a difficult metrology problem; a review on volumetric accuracy measurement schemes can be found in [9,10]. Among them, the multilateration measurement using a tracking interferometer has been accepted by the industry as a feasible way to directly measure a machine tool’s volumetric accuracy. The tracking interferometer (the term in [7]), or the laser tracker, is a laser interferometer with a steering mechanism to change the laser beam direction to automatically follow a retroreflector attached to the machine spindle (referred to as the ‘‘target’’ hereafter). Since the 1990s, many research works have been reported, e.g. [11,12]. Commercially available tracking interferometers include the one by Etalon AG and its application to machine tool error calibration has been extensively reported [13,14]. This multilateration measurement could be applied within a thermal test to periodically measure the machine’s volumetric accuracy. When four (or more) tracking interferometers are available, a single test can measure the TCP’s 3D position in real-time [15]. However, due to the instrument’s high cost, typical users only have a single tracking interferometer. In such a case, as

Fig. 1. Thermal influence on the volumetric accuracy over the entire workspace: an illustrative example.

S. Ibaraki et al. / CIRP Annals - Manufacturing Technology 65 (2016) 483–486

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suggested in [13], one must repeat at least four tests at different tracking interferometer positions. It typically takes more than an hour and thus it cannot be performed within a thermal test. This paper first proposes a scheme to identify 2D geometric errors of linear axes by single-setup tracking interferometer measurement (i.e. the interferometer’s position does not have to be changed). Since the measurement time is significantly reduced, it can be applied within a thermal test to periodically check the machine’s volumetric accuracy. It enables a user to visually observe how the machine’s trajectory is distorted by the thermal influence. 2. Proposal of single-setup tracking interferometer measurement to identify 2D geometric errors of linear axes

2.3. Proposed test procedure and algorithm 2.3.1. Test procedure Fig. 3 depicts the proposed test scheme. The tracking interferometer is placed on the same plane as the target’s nominal trajectory. The target stops at each command position, pi ¼ ½yk ; zl T (k = 1, . . ., Ny, l = 1, . . ., Nz) along a rectangular path. The static laser displacement, di 2 R, is measured at each stop. Then, the linear position erroring of each axis, EYY(yk) and Ezz(zl), is measured. This can be done by moving the target to the Y- (or Z-) direction at the same Z (or Y) position as the tracking interferometer, with the laser beam fixed to the Y- (or Z-) direction.

2.1. Conventional multilateration algorithm First, the conventional multilateration algorithm is briefly re viewed (see [12,13] for further details). For the ith target position, pi 2 R3 (i = 1, . . ., N) and the jth tracking interferometer position, Pj 2 R3 (j = 1, . . ., Nt), the laser displacement, dij 2 R, is measured (see Fig. 2). The objective is to estimate actual target positions, pi (i = 1, . . ., N). This problem can be formulated as: X 2 ðf ij ðxÞdij Þ (1) min x

i¼1;...;N;j¼1;...;Nt

Fig. 3. Proposed tracking interferometer test on YZ plane.

where f ij ðxÞ :¼ kpi P j kd0j . d0j represents the dead path length. x 2 R (3N+4Nt) is a set of the parameters to be identified: (2)

x ¼ b fpi gi¼1;...;N ; fP j gj¼1;...;Nt ; fd0i gj¼1;...;Nt c

Typically, the problem (1) is locally solved by an iterative approach. The difference from the traditional trilateration problem is: (1) tracking interferometer positions, Pj, are unknown. (2) Dead path lengths, d0j, are unknown. To solve the problem (1) under this condition, the tracking interferometer must be placed at four or more different locations (Nt  4). Machine spindle

Z

Retroreflector (cat’s eye) Y

X Tracking interferometer

Fig. 2. Multilateration measurement setup.

2.2. Objective and simplification of the problem To significantly reduce the measurement time, we propose a scheme to estimate target positions with a single tracking test (i.e. Nt = 1). This requires the simplification of the problem as follows: 1) The objective is to estimate the target’s 2D positions, i.e. pi 2 R2. 2) The linear positioning error of two linear axes is known. 3) The interferometer position, P 2 R2, is separately estimated. Considering the YZ plane as an example, the objective of the proposed scheme is to estimate geometric error parameters in Table 1, except for linear positioning error motions, EYY(yi) and EZZ(zi), which must be pre-calibrated. Table 1 Geometric error parameters of Y- and Z-axes [4]. EZY(yj) EYY(yj) EAY(yj) EYZ(zk) EZZ(zk) EA(0Y)Z

Straightness deviation of Y-axis at y = yj Linear positioning deviation of Y-axis at y = yj Pitch error motion of Y-axis at y = yj Straightness deviation of Z-axis at z = zk Linear positioning deviation of Z-axis at z = zk Squareness error of Z- to Y-axis

2.3.2. Algorithm to estimate geometric error parameters For the target’s command position, pi , the positioning error, Dpi, can be modelled by using the rigid-body kinematic model [9]:

Dp i ¼



EYY ðyk Þ þ EYZ ðzl ÞðEAð0YÞZ þ EAY ðyk ÞÞzj



EZY ðyk Þ þ EZZ ðzl Þ   EYY ðyk Þ  Akinematics e þ EZZ ðzl Þ

(3)

where e 2 R (2Ny+Nz+1) represents a vector containing all the geometric error parameters to be identified (i.e. the parameters in Table 1 except for EYY(yk) and EZZ(zl)). In this paper, the tool length is assumed constant and thus the influence of the angular error motion of Z-axis, EAZ(zi), is not included in Eq. (3). The influence of the position error in the X-direction (the direction normal to the plane concerned) is assumed negligibly small. e can be estimated by solving: min e

  2 X  EYY ðyk Þ kAkinematics e þ þ pi Pkdi EZZ ðzl Þ

(4)

i¼1;...;N

where P 2 R2 is the tracking interferometer position that must be pre-estimated. Table 2 compares the proposed scheme with the conventional multilateration algorithm. A potential uncertainty contributor for the proposed scheme comes from the assumption that the tracking interferometer position is known. This influence will be studied in Section 4. On the other hand, its clear advantage is shorter measurement time. Table 2 Comparison of the conventional multilateration and the proposed scheme.

Geometric errors to be identified Number of tracking tests required Tracking interferometer position Application

Conventional multilateration algorithm [13]

Proposed scheme

All errors in 3D

All errors in 2D

4 or more

One, with linear positioning errors pre-measured Has to be known prior to the test

Will be identified

Complete 3D geometric error calibration

Quick check ! thermal test

S. Ibaraki et al. / CIRP Annals - Manufacturing Technology 65 (2016) 483–486

A vertical machining centre of the configuration shown in Fig. 4 was measured. A commercial tracking interferometer, Laser TRACER by Etalon [13,14], was used. The command target positions are on a rectangular path in the XY plane within X500 mm  Y370 mm (shown by * marks in Fig. 5). By applying the present scheme, actual target positions are estimated as shown in Fig. 5 (red * marks). They are calculated by using the model (3) with the identified geometric parameters. The error from the command position is magnified 1000 times. The tracking interferometer was approximately at (X, Y) = (209, 553) mm. Then, for the comparison, the conventional multilateration test was performed. A 3D path, including the path in Fig. 5, was measured. Tracking tests were repeated with the tracking interferometer located at four different positions. Applying the algorithm in [13], target positions are estimated as shown also in Fig. 5 (blue * marks). Both estimated trajectories show a good match, except for slight difference (about 5 mm) in the X-axis linear positioning error. Recall that the proposed scheme directly measures linear positioning errors. The difference may be attributable to the machine’s thermal deformation during multiple tests.

Z

Table 3 Temperature measured near the front bearing of the spindle after 1 h spindle run at each speed. Spindle speed (min1) Temperature (8C)

Error sca 0 le: 50 µm

6000 34.8

Z

-100

18,000 47.1

Y Red: Cold

-200

Green: 6,000 min Blue: 12,000 min

-300

Y

Error scale: 50 um

-150

-250

-1

-1

Magenta: -1 18,000 min

-350 -500

X

12,000 37.0

Command position

-50

Y

Z

0 25.3

tested. The commanded target positions, total 40 points in Y450 mm  Z350 mm, are shown by marks in Fig. 6. In the experiment, Ezz(zl) was measured at each test, but EYY(yj) was measured only at the beginning of the entire thermal test, since our preliminary test showed that thermal influence on EYY(yj) is smaller than that on Ezz(zl). This is to reduce the measurement time. The tracking test (Fig. 3) and Ezz(zl) measurement took about 7 min.

Z [mm]

2.4. Experiment: comparison with the conventional multilateration

485

-400

-300

-200

-100

0

Y [mm]

X

Fig. 6. Estimated trajectories on YZ plane at the end of each spindle run for 1 h. The error from the command trajectory is magnified 1000 times. The schematic machine diagram illustrates the interpretation of structural deformation causing such a change in estimated trajectories.

3.2. Test result The target trajectories estimated by the proposed scheme are shown in Fig. 6. It shows that the TCP trajectory was distorted by about 50 mm at maximum, as the spindle temperature raised as shown in Table 3. It can be clearly observed that the spindle heat increased the straightness error motion of the Y-axis, and consequently, its pitch error motion changes the orientation of the Z-axis particularly at – Y side (see the schematic diagram in Fig. 6).

Fig. 4. Machine configuration. 50 0 -50

Error scale: 50 µm

Y mm

-100

* Command target positions

4. Uncertainty analysis

o Estimated by conventional multilateration algorithm (with 4 tracking tests)

-150

4.1. Contribution of estimation uncertainty of tracking interferometer position

o Estimated by proposed scheme (with single tracking test and direct measurement of Exx and Eyy)

-200 -250 -300 -350 -600

-500

-400

-300 -200 X mm

-100

0

100

Fig. 5. Comparison of estimated target trajectories on the XY plane by conventional and proposed multilateration algorithms.

3. Application: measurement of thermal influence on twodimensional motion trajectory 3.1. Test procedure and setup The test procedure was as follows: the proposed test presented in Section 2.3 was performed when the machine was cold. Then, the spindle was rotated in 6000 min1 for 1 h. After it, the spindle was stopped, and the same test was repeated. Repeat this with the spindle rotation speed 12,000 min1 and 18,000 min1 respectively for 1 h. Table 3 shows the temperatures measured near the spindle’s front bearing. The machine’s thermal compensation was turned off throughout the experiment. From the machine configuration in Fig. 4, it can be predicted that the YZ-plane has larger thermal influence and thus this plane was

Unlike the conventional multilateration algorithm, the tracking interferometer position, P, must be pre-calibrated in the present formulation. Its rough estimate can be calculated by solving the problem (4) for P with no geometric errors, i.e. min P

2 X   E ðy Þ  k YY k þ pi Pkdi EZZ ðzl Þ

(5)

i¼1;...;N

In reality, the machine does have geometric errors, and it causes the estimation error of P. It in turn gives the estimation error of target positions given by solving Eq. (4). The objective of this subsection is to study this uncertainty propagation. The uncertainty analysis procedure is summarised as follows: first, geometric errors of linear axes are modelled in the same manner as in [16]. Namely, each geometric error in Table 1 is modelled as a Fourier series (third-order in this analysis). The standard deviation of each Fourier coefficient is given based on e.g. tolerances in ISO 10791-3 [17]. When the machine’s geometric errors are randomly given, P is calculated by Eq. (5), and then geometric error parameters are estimated by solving Eq. (4) with this erroneous estimate of P. The expanded uncertainty in the estimation of geometric error parameters is evaluated by using the Monte Carlo simulation [18].

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S. Ibaraki et al. / CIRP Annals - Manufacturing Technology 65 (2016) 483–486

its influence on the difference between each estimated trajectory in the thermal test in Fig. 6 is negligibly small, since the tracking interferometer’s position is the same in all the tests. In other words, although the first estimated trajectory (‘‘Cold’’) may be subject to significant uncertainty, the thermal deviation in four estimated trajectories in Fig. 6 can be observed with much smaller uncertainty. Prior knowledge on the machine’s geometric accuracy would significantly reduce the uncertainty in the estimation of the tracking interferometer position. For example, if the squareness error of linear axes is pre-calibrated, the tracking interferometer position can be estimated with much smaller uncertainty. The feasibility of this approach will be studied in our future research. 5. Conclusion

Fig. 7. Uncertainty in estimated target position due to estimation error of the tracking interferometer position.

Fig. 7(a) shows the expanded uncertainty (k = 2) in the estimated target positions. The test setup is the same as in Section 2.4. Fig. 7(b) shows the uncertainty (k = 2) in target positions due to randomly given linear axis geometric errors, which is given as an input to the Monte Carlo simulation. When the given uncertainty in target positions is about 29 mm at maximum (Fig. 7(b)), the estimation uncertainty of target positions was about 23 mm at maximum (Fig. 7(a)). 4.2. Contribution of laser length measurement uncertainty The uncertainty in the length measurement by a laser interferometer is clearly among major uncertainty contributors. The length measurement uncertainty described in the Etalon Laser TRACER’s catalogue (0.2 mm + 0.3 mm/m) is used as an input to the Monte Carlo simulation. Its influence on the expanded uncertainty (k = 2) in the estimated target positions is shown in Fig. 8. In principle, this uncertainty also influences the conventional multilateration algorithm. On the other hand, the uncertainty in the previous subsection is only in the present scheme. The comparison between Figs. 7(a) and 8 shows that its contribution is similar in size as the length measurement’s uncertainty.

Fig. 8. Uncertainty in estimated target position caused by the uncertainty in laser length measurement.

4.3. Discussion The uncertainty caused by the estimation error of the tracking interferometer position can be significant. It is important to note that

This paper proposed a scheme to identify 2D geometric errors of linear axes by single-setup tracking interferometer measurement. This identification is possible because it requires (1) the linear positioning error of linear axes be directly measured in priori, and (2) the tracking interferometer position be estimated separately. Its advantage is its shorter measurement time. It was applied to a thermal test to periodically investigate the influence of the spindle heat to the machine’s trajectory on a 2D plane. No conventional thermal tests in [4] measure such thermal influence. The uncertainty analysis focused on the influence of the assumption (2) above. It showed that the uncertainty in estimated target positions caused by the estimation error of tracking interferometer position was about 29 mm at maximum, when the uncertainty in target positions was about 23 mm at maximum.

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