Design and experimental validation of an ultra-precision Abbe-compliant linear encoder-based position measurement system

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Design and experimental validation of an ultra-precision Abbe-compliant linear encoder-based position measurement system Niels Bosmans ∗,1 , Jun Qian, Dominiek Reynaerts KU Leuven, Department of Mechanical Engineering, Division PMA, Celestijnenlaan 300B, 3001 Heverlee, Belgium

a r t i c l e

i n f o

Article history: Received 6 April 2016 Received in revised form 15 July 2016 Accepted 23 August 2016 Available online xxx Keywords: Abbe principle Linear encoder Ultra-precision Metrology Thermal error Uncertainty Mechatronic Machine tool Coordinate measuring machine (CMM)

a b s t r a c t The design and development of an Abbe-compliant linear encoder-based measurement system for position measurement with a targeted 20 nm uncertainty (k = 2) in machine tools and CMMs is presented. It consists of a linear scale and a capacitive sensor, mounted in line on an interface which is guided in the scale’s measurement direction and driven by a linear motor based on the output signal of the capacitive sensor. The capacitive sensor measures the displacement of a target surface on the workpiece table. The functional point, which is the center of a tool or touch probe, is always aligned with the scale and capacitive sensor such that this configuration is compliant with the Abbe principle. Thermal stability is achieved by the application of a thermal center between the scale and capacitive sensor at the tip of the latter, which prevents both components to drift apart. Based on this concept, a prototype of a one-DOF measurement system was developed for a measurement range of 120 mm, together with an experimental setup aimed at verifying the reproducibility of the system for changing ambient conditions of ±0.5 ◦ C and ±5%rh and the repeatability during tracking of a target surface over a short period of time. These experiments have shown that the measurement uncertainty of the one-DOF system is below 29 nm with a 95% confidence level. © 2016 Elsevier Inc. All rights reserved.

1. Introduction For centuries, the improvement of machine tool accuracy has been the key enabler for the development of new technologies. The accuracy of machine tools depends mostly on the positioning accuracy of the tool with respect to the workpiece. The largest influences on this positioning accuracy are thermo-mechanical errors and geometrical errors [1]. Thermo-mechanical errors arise due to expansion and deformation of the machine tool structure by temperature changes, while geometrical errors are mainly because of inaccuracies in the movement of the slides. Reduction of thermal errors is currently achieved by placing the machine in a temperature-controlled environment, by employing lowexpansion—yet expensive—materials for the machine structure and by modeling and compensating thermal errors [2]. Geometrical errors are generally reduced by applying air bearings or hydrostatic bearings [3], by Abbe offset reduction [4] or by error measurement and compensation [5]. However, the application of certain precision engineering design principles, such as functional separation of the force loop ∗ Corresponding author. E-mail address: [email protected] (N. Bosmans). 1 Member of Flanders Make (www.flandersmake.be).

and the metrology loop and the Abbe principle, could be more thoroughly exploited to increase the accuracy of machine tools even further [6–8]. These principles have been applied in several specialized ultra-precision machine tools and CMMs based on laser interferometer position measurement [9–14] and in a few CMMs utilizing linear encoders [15,16]. Kunzmann et al. [17] compared the uncertainty of laser interferometry in air to linear encoder position measurements. They concluded that the uncertainty of laser interferometers over a short time period increases with measurement stroke (Ls ) by 0.2 × 10−6 · Ls (coverage factor k = 2 [18]), while for linear encoders the stability was practically independent on the measurement stroke. For this reason, encoder grids are used in the latest generation of 193 nm wavelength lithography machines [19]. The application of laser interferometry in machine tools requires additional measures to maintain the stability of the refractive index [20], which renders these machines complex and less cost-effective. The application of linear encoders compliant with the Abbe principle in three DOFs in machine tools could enhance the accuracy considerably, but has not yet been investigated. A full three-DOF application of the Abbe principle with linear encoders has only been realized in a CMM [16], where it was limited by a small work volume. Therefore, the aim of this research is the development of a linear encoder-based position measurement system for machine

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tools that enables Abbe-compliant measurement in three DOFs. To enable sub-100-nm accurate machining for normal machine tools [21], at least five times lower measurement uncertainty is required, namely 20 nm (k = 2). This goal should be reached for displacements in the order of 100 mm in standard metrology room ambient conditions of 20 ± 0.5 ◦ C and ±5%rh. This paper presents the latest results in the development of this measurement system. The proposed concept has been selected from a number of embodiments which have been introduced in [22]. The chosen embodiment of the measurement system was discussed in [23], although no details on the design were given. Bosmans et al. [24] presented the performance of the system in tracking of a target surface and in [25] the origin of the dynamic errors of the measurement system was investigated. Bosmans et al. [26] elaborated on the calibration of the measurement system. The current work presents the final updated uncertainty budget and detailed design of the measurement system and elaborates on the experiments that were conducted to verify the budgeted uncertainty. First, Section 2 explains the measurement system concept. Consequently, Section 3 describes the mechanical design of the one-DOF prototype, called Moving Scale (MS) system. The sources of measurement uncertainty are identified and presented in an uncertainty budget in Section 4, where each uncertainty component is discussed in more detail. Section 5 discusses the design of an experimental setup for determination of the reproducibility under changing ambient conditions and the repeatability of the one-DOF MS system and encompasses the results of these experiments. Finally, an upper bound on the measurement uncertainty of the MS system is estimated.

Effective point (EP)

Metrology frame Tool

Scale on interface

Funtional point (FP) Workpiece Displacement sensor Guides

Effective point Reading head

Target surface

Fig. 1. Configuration of linear encoders such that the path of the functional point and the effective point of the scale are in line.

Tool spindle Metrology frame Moving Scale system Target surface Workpiece Base frame

x-slide

z

y-slide 2. Conceptual layout

x

y

z-slide

Fig. 2. Concept of Moving Scale system on a three-DOF machine tool.

The generalized Abbe principle [27], which was reformulated by Bryan [28], states that the path of the effective point (EP) of a displacement measuring system should be collinear with the path of the functional point (FP) whose displacement is to be measured. If this is not possible, either the slideways that transfer the displacement must be free of angular motion or angular motion data must be used to calculate the consequences of the offset. It has been derived in [13] that the FP of a system with a spherical end-effector, such as a touch probe or a spherical tool, equals the center of that end-effector. To achieve Abbe-compliance for a two- or three-DOF system with a minimum amount of linear encoders and moving elements, the FP should remain stationary with respect to the reading heads and coincident with the path of the EPs of the linear encoders. The EP of the linear encoder is the readout position on the scale. This kind of configuration can only be attained if the encoders measure the movement of target surfaces surrounding the workpiece while at the same time allowing these target surfaces to move relatively with respect to the encoders in a direction perpendicular to the encoders’ measurement direction. The proposed configuration is illustrated in Fig. 1 in two DOF. The scales are mounted on an interface that is guided in the measurement direction and the reading heads of the encoders are connected to a stationary metrology frame. To allow relative movement between the encoder and its respective target surface in a direction perpendicular to the encoder’s measurement direction, a non-contact displacement sensor is added in line with the scale in between the target surface and the scale. The scale interface is actively driven by a linear actuator based on the output of the displacement sensor. In this way, the scale interface follows the movement of the target surface in the measurement direction. The position measurement of the target surface xtarget is constituted by

the measurement of the linear encoder xMS and the displacement sensor xDS : xtarget = xMS + xDS .

(1)

Therefore, the linear actuator should only keep the displacement sensor within its measurement range such that tracking errors do not result in measurement errors. Fig. 2 shows how this concept can be applied on a machine tool in three DOF. Target surfaces are added to the workpiece table and the base frame is extended to integrate the MS systems. Metrology frames that hold the MS systems’ reading heads are connected to the base frame and ensure a stable, functionally separated closure of the metrology loop [6,7]. For low-end applications, the metrology frames could be omitted and the reading heads could be directly attached to the machine base frame. 3. Design of a one-DOF MS system prototype The geometrical arrangement of the major elements in a oneDOF prototype of the MS system with a measurement range Lm of 120 mm is shown in Fig. 3. The utilized measurement stroke Ls of the system equals 107 mm. This value originates from the intended use of the MS system for the z-axis of an ultra-precision five-axis grinding machine with a stroke of 107 mm [29]. Fig. 4 displays a picture of the prototype, excluding the metrology frame and the reading head. The linear encoder is a Heidenhain LIP 281 and a Lion Precision C5-D capacitive sensor is integrated as the displacement sensor.

Please cite this article in press as: Bosmans N, et al. Design and experimental validation of an ultra-precision Abbe-compliant linear encoder-based position measurement system. Precis Eng (2016), http://dx.doi.org/10.1016/j.precisioneng.2016.08.005

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Fig. 3. Layout of the one-DOF MS system prototype for a measurement range Lm of 120 mm.

Fig. 4. Prototype of the one-DOF MS system. The metrology frame and reading head are not shown.

To prevent that deformation of the motor and the guides due to thermal dilation or driving forces would distort the metrology loop components, the force loop and the metrology loop have been structurally separated [8]. The force loop, depicted in Fig. 5, enters the system from the base frame into the linear guide, goes through the guide and support frame to the linear motor and back to the base frame. The metrology loop, of which a section is portrayed in Fig. 6, enters the system through the gap between the target surface (not shown in the figure) and the capacitive sensor, passes the scale and the reading head and then exits the system via the reading head interface and metrology frame. The capacitive sensor probe is mounted on a stainless steel AISI type 303 interface, which corresponds to the material of the probe’s outer body such that there is no differential thermal expansion between these components. This interface is connected to an aluminum EN AC-5083 scale interface with three ball-in-V connections, of which one is a split V-groove to increase the static stability [30]. A schematic layout of both interfaces is shown in Fig. 7a. Fig. 7b depicts the prototype. Kinematic mounting of the scale, a Heidenhain LIP 281 with a measuring range of 120 mm, is achieved by constraining it using five free-rolling contacts. In the

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Fig. 5. Front view of the one-DOF MS system with indicated force loop.

Fig. 6. Top view of the one-DOF MS system with indicated section of the metrology loop.

z-direction, the scale is preloaded to three precision 4-mmdiameter steel balls, two in the front and one at the back of the scale. Preloading is accomplished by two magnets located close to the ball contact points, which are acting on two 1-mm-thick ferromagnetic sheets glued to the scale. Because the preload is not acting along the supporting points, the scale will deform, causing scale errors. These scale errors are calibrated on the machine as described in [26,31]. The scale is preloaded in its measurement direction to a Zerodur disk by using a ball and a leaf spring, located at the back of the scale. The contraction of the scale due to this preloading will be calibrated as well. The Zerodur disk, in turn, is constrained in the z-direction by three free-rolling contacts. In the x-direction, the disk is preloaded to the side-walls of a wedge-structure in the probe interface. These side-walls intersect at the probe tip, thus creating a thermal center at the tip between the scale-disk combination and the probe. Because the high tangential contact stiffness at the wedge surfaces would result in a considerable thermal drift (see Appendix A), flexures are added to decrease the tangential stiffness of the wedge structure. As a result of the thermal center, the scale and probe will not drift apart under uniform thermal

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Fig. 8. Induced non-linearity error of the scale due to expansion of the scale interface in the z-direction for a temperature change of 0.5 ◦ C.

is expanded with the same coverage factor k = 2. This gives the expanded measurement uncertainty Ui = k · ui = k · i .

(2)

One should note that throughout this paper, uncertainty is defined as the expanded measurement uncertainty with a coverage factor k = 2, unless specified otherwise. This coverage factor gives the halfwidth of the 95% confidence interval. The errors due to ambient changes are considered to have a uniform distribution. If the width of this√ distribution is equal to 2a, then the standard uncertainty equals a/ 3 [18]. The expanded uncertainty resulting from a uniform temperature distribution √ with a width of ±T for a sensitivity ci then equals 2 · ci · T/ 3. In this research, T is assumed to be 0.5 ◦ C. All uncertainty components originating from type B evaluated uncertainties [32], together with dynamic and calibration errors, are assumed to be normally distributed. If a component is calculated in an analytical way, the formula is given in the table. For other components, the corresponding section in this paper or a reference to the proof data is given. 4.1. Linear encoder uncertainty Fig. 7. Scale interface with a virtual zero expansion between the tip of the capacitive probe and the scale by application of a thermal center.

expansion of the probe interface and as such, a virtual zero expansion interface is created. The steel probe interface has a Biot number of approx. 0.002, indicating that a uniform thermal expansion for ambient temperature changes can be assumed. The scale and probe interface, in turn, are mounted kinematically using ball-in-V grooves to the motor-and-guide support frame to prevent that deformations of the latter would distort the scale interface. The reading head of the linear encoder is screwed onto an interface of the same stainless steel alloy as the reading head housing, which is in turn kinematically connected to an aluminum metrology frame. The design of this mount and the metrology frame is described in Section 5.2. The MS system is guided by linear roller bearings, while the drive system consists of a direct-drive linear motor. 4. Uncertainty budget of one-DOF MS system Based on the results of experiments on the critical components of the MS system, finite element analysis (FEA) and analytic calculation, an uncertainty budget [18] has been constructed for an MS system with a measurement range Lm of 120 mm and a utilized measurement stroke Ls of 107 mm. The uncertainty budget is summarized in Table 1. All uncertainty components are combined using the rootsum-square combinatorial rule. Before combining, the standard uncertainty ui , equal to the standard deviation  of the component’s probability density function, is calculated and each component

The scale of the LIP 281 encoder is made of Zerodur, which has a coefficient of thermal expansion (CTE) ˛ZD = 0 ±0.1 × 10−6 /◦ C. This uncertainty on the CTE gives a measurement uncertainty of 3.5 nm and 2.4 nm for respectively the scale expansion and the zeropoint drift for ±0.5 ◦ C temperature changes. Air pressure variation causes a similar dilation of the scale, resulting in an uncertainty of 3.1 nm. A maximum air pressure variation of 8 kPa is assumed [33]. Zero-point drift resulting from air pressure variation is considered negligible because air pressure will not change considerably over the course of a machining task, which only lasts several hours. The scale is preloaded to the Zerodur disk and the capacitive probe interface using a ball and leaf spring at the back of the scale. Expansion of the aluminum scale carrier inflicts a small reduction in this preload, causing the scale to expand. The expansion is calculated by Eq. (16), in which Li is dependent on the size of the scale interface, which in turn is dictated by the measurement range Lm . For MS systems with a larger Lm , the interface will have a larger expansion, resulting in a larger expansion of the scale. The calculation is given in Table 1, with the Young’s modulus of the Zerodur scale EZD = 90.3GPa and Ascale = (15 × 2.9) mm2 , which results in a 0.8 nm uncertainty. Because the supporting points of the disk and scale are not in the same plane, expansion of the scale interface in the z-direction will result in relative movement of scale and disk in that direction. As a consequence, the scale will be pulled down at the contact point with the disk, causing the scale to bend. This bending deformation results in a scale non-linearity error of 7.4 nm for a 0.5 ◦ C temperature change (Fig. 8). For ±0.5 ◦ C uniform temperature changes, √ this leads to an expanded uncertainty (k = 2) equal to 2 · 7.4/ 3 = 8.5 nm [18].

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Table 1 Uncertainty budget of the one-DOF MS system prototype. U (k = 2) [nm] Uncertainty contribution (k = 2) [nm]

Random errors

Component

(Lm = 120 mm, Ls = 107 mm)

Linear encoder (LE) Scale temperature expansion Scale zero-point drift Air pressure variation Scale interface expansion

√ Lm · U˛ZD · UT /2 = [107 mm] · [0.1 × 10−6 /◦ C] · [2 · 0.5 ◦ C/ 3]/2 √ √ √ (Lm · U˛ZD · UT )/(2 2) = ([107 mm] · [0.1 × 10−6 /◦√C] · [2 · 0.5 ◦ C/ 3])/(2 2) (1 − 2ZD ) · Up · Ls /EZD = ((1 − 2 · [0.24]) · [8 kPa/ 3] · [107 mm])([91 GPa]) ◦ C] [0.011 N/␮m] √ [24.2 × 10−6 /◦ C] · ([62.5mm] + Lm ) · 2 · [0.5 · (Eq. (16)) E A

Scale deformation Reading head zero-point drift Zerodur stability (1 year) Periodic position error Length deviation calibration

Distr.

Reprod.

Unif. Unif. Unif. Unif.

3.5 2.4 3.1 0.8

Section 4.1 [31,34] √ √ −6 ZD · Ls / 3 = [0.03 × 10 ] · [107 mm]/ 3 (Bosse et al. [35]) [42] [148 nm/m] · Ls [26,31]

Unif. Unif. Unif. Gauss. Gauss.

8.5 1.9 2.1

[31,38] [43] [31]

Unif. Gauss. Gauss.

1.7

3

[0.011 N/␮m]+ ZD scale

Repeat.

Systematic errors

Ls

Capacitive displacement sensor (DS) Zero-point drift Noise Calibration Scale interface Zero-point drift

· 0.5 [(−8.1 + 44.8 · Lm ) nm/◦ C] · 2 √

Cosine error Other geometrical errors Combined uncertainty

0.8 1.0

Unif.

1.6

Unif. Unif. Gauss. Gauss. Gauss.

0.1 2.3

DS,FP · UMT,r (Rep.) | DS,FP · UMT,nr (Sys.) (Section 4.4) LE,DS · UMS,r (Rep.) | LE,DS · UMS,nr (Sys.) (Section 4.4) √ √ 2 2 Ls · ⊥ /(2 3) = [107 mm] · [200 ␮rad] /(2 3) [31]

Unif. Unif.



2

(Section 4.3.1)

16

Section 4.3.2 Section 4.3.3 [(14.2 + Lm · 0.032) nm/(m/s2 )] · [0.01 m/s2 ] [(14.2 + Lm · 0.032) nm/(m/s2 )] · [0.04 m/s2 ] Section 5.4.2

(3)

Motor heating Reading head heating Acceleration Vibration from guides Target surface tracking Geometrical errors Abbe offset DS w.r.t. FP Abbe offset LE w.r.t. DS

◦C

1.0

2

2

· · · + ([39 nm/m ] · Ls · Lm ) + (−4.7 + [26 nm/m] · Lm )

Temperature variation also has an effect on the internal optics of the linear encoder reading head. The temperature sensitivity of the LIP 281 reading head has been determined in [31,34], which has an average value of −12 nm/◦ C. However, it has also been found that temperature drift can easily be identified and compensated. This method is briefly described in Appendix B. The residual drift of a compensated reading head gives an uncertainty of 2.1 nm. Zerodur exhibits a long-term relaxation behavior which results in a continuous length reduction. For Zerodur material of more than 10 years old, this contraction is smaller than 0.03 × 10−6 /year [35]. If calibration intervals of 1 year are assumed, the contraction will lead to a measurement uncertainty of 1.9 nm. Due to the quality of the scanning and signal-processing electronics of the encoder, the position measurement will show a periodic error with a period equal to the length of one graduation interval. For the encoders used in this research, the Heidenhain LIP 281 [36], this error is specified within ±1 nm for a signal period of 0.512 ␮m. The periodic error will generally lead to a dynamic position error, of which the frequency depends on the traversing speed of the scale. Finally, the scale needs calibration to be traceable to the length standard and to reduce the non-linear scale errors. These errors are caused by deviation of the spacing of the grating lines from the true value due to manufacturing errors and mounting deformations. Bosmans [26,31] discusses the calibration of the scale. The calibration uncertainty amounts to 16 nm over a measuring stroke of 107 mm. 4.2. Capacitive sensor uncertainty The Lion Precision C5-D capacitive sensor and CPL-290 driver witness a drift due to temperature and humidity changes [37].

0.9 0.4

4.3 2.1

Unif. Gauss.

132 + ([154 nm/m] · Ls ) + ([21 nm/m] · Lm ) + · · · 2

0.2 0.7 7.6

2

1.4 0.3 11

7.9

17 21

Temperature and humidity sensitivity of this sensor have been investigated in [31,38]. According to the manufacturer, temperature sensitivity can be up to 4 nm/◦ C. Relative humidity sensitivity has been determined in [31] to be equal to 4.6 nm/%rh. It has been shown that both sensitivities can be compensated as in Appendix B, leading to a residual uncertainty of 1.7 nm. Noise of the capacitive sensor is specified by the manufacturer at 0.4 nm rms (root-mean-square), yielding an expanded uncertainty of 0.8 nm. As the scale and the capacitive sensor are mounted in line, the non-linearity of the capacitive sensor can be calibrated by using the scale as a reference measurement. It is assumed that the nonlinearity of the scale is negligible over the short stroke of the capacitive sensor. In [31] it is demonstrated that the calibration uncertainty of this procedure equals 1.0 nm.

4.3. Scale interface uncertainty 4.3.1. Zero-point drift The scale interface design was converted into a finite element model. The Hertzian contacts of the kinematic connections were modeled as spring elements with corresponding stiffnesses. It is assumed that Zerodur has a CTE of 0 ppm/◦ C. The resulting expansion for a uniform heating of 1 ◦ C is shown in Fig. 9. The relative displacement between the probe tip and the front √ of the scale equals −2.7 nm, which results in an uncertainty of 2.7/ 3 = 1.6 nm [18] and proves the feasibility of the thermal center arrangement. The negative value arises from the fact that the flexure structures and probe interface also bend downwards due to the differential expansion between disk, probe interface and scale interface

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Fig. 9. Thermal expansion of the MS system for a uniform temperature change of 1 ◦ C. The scale and capacitive probe drift −2.7 nm relative to each other. Fig. 11. Probe interface with two thermistors for measuring the temperature gradient resulting from heating by the reading head.

Fig. 10. Temperature difference of the motor and the scale interface with respect to the environment for motor currents of 0, 0.2, 0.4, 0.6, 0.8 and 1 A.

in the z-direction. This causes a rotation of the flexure’s end face, while the connection between disk and probe interface lies 3.5 mm beneath the neutral line of the flexures, resulting in a negative displacement. An analytic model of the scale interface’s thermal expansion indicated that the largest contribution to the zero-point drift arises from the change in preload of the scale in x-direction due to expansion of the scale interface [31]. This contribution is given by Eq. (16) and equals ˛Al · ([62.5 mm] + Lm ) · UT ·

kpreload kpreload + ks,p

,

(3)

for which ˛Al = 24.2 × 10−6 /◦ C is the CTE of aluminum EN-AW 5083, kpreload = 10.8 N/mm is the stiffness of the preload spring at the back of the scale and ks,p = 5.83 N/␮m is the equivalent stiffness of the connection between the scale and the probe interface. The dependence of the zero-point drift of the scale interface on Lm then equals 44.8 nm/m. When Eq. (3) is combined with the results of the FEA, the sensitivity of the zero-point to temperature equals (−8.1 + 44.8 · Lm ) nm/◦ C. 4.3.2. Motor heating Heating of the linear motor results in an additional expansion of the scale interface and a temperature gradient in the probe interface. This behavior was experimentally verified. The MS system was blocked in one position and the linear motor current was varied from 0 A till 1 A, which corresponds to a power consumption between 0 and 4 W. The temperature difference of the motor forcer containing the coils and the scale interface with respect to the environment was measured using thermistors. Fig. 10 shows the temperature difference for motor current setpoints of 0, 0.2, 0.4, 0.6, 0.8 and 1 A, corresponding to a force of 0, 1.2, 2.4, 3.6, 4.7, and 5.9 N. From this figure, it can be derived that for a friction force of 5 N, the scale interface would heat up by 1.2 ◦ C, which results in a measurement error of −3.2 nm and an associated measurement uncertainty of 1.9 nm, as calculated in Section 4.3.1. For a roller bearing, friction would be below 1 N, which demands a five times lower current and

Fig. 12. Temperature gradient in the probe interface due to heating of the reading head. At the start of the measurements, the reading head was moved to the front of the scale, closest to the probe interface.

hence a 25 times lower heat dissipation. This results in a negligible measurement uncertainty of 1.9/52 = 0.07 nm. 4.3.3. Reading head heating The influence of the reading head heat transfer on the probe interface has also been investigated. The MS system was first positioned such that the reading head was located at the back of the scale, away from the probe interface. Next, the MS system was moved such that the reading head resided at the other end of the scale, next to the probe interface. Temperature of the probe interface was monitored by two thermistors that were fixed by aluminum tape as shown in Fig. 11. The resulting temperature gradient is depicted in Fig. 12. The gradient is approximately 15 mK, which results in a measurement √error of 4 nm and an associated expanded uncertainty (k = 2) of 4/ 3 = 2.3 nm. 4.3.4. Dynamic errors Accelerations cause the scale and capacitive sensor probe to displace relative to each other due to the limited stiffness of the kinematic connections between them. It was determined by a finite element analysis and experiments that this dynamic error equals 18 nm/(m/s2 ) and that the sensitivity depends on measurement range Lm as (14.2 + Lm · 0.032) nm/(m/s2 ) [25]. In the intended application for the MS system, which is an ultra-precision free-form grinding machine, the maximum acceleration is only 10 mm/s2 , so that the error only amounts to 0.2 nm. Nevertheless, experiments show that this sensitivity to acceleration leads to an increase of high-frequency errors in the position measurement due to highfrequency error motions and stick-slip of the guides. Although roller bearings provide a smooth motion, stick-slip and out-ofroundness of the roller elements still causes minute vibrations. Fig. 13 depicts the high-frequency accelerations of an MS system with roller bearings, which equal 0.02 m/s2 rms for a velocity of

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Fig. 13. Acceleration of the MS system for velocities ranging from 10 mm/s to 50 mm/s.

10 mm/s, resulting in dynamic errors of 0.36 nm rms and an associated uncertainty of 0.7 nm. Tracking of the target surface also gives rise to dynamic errors, as determined in Section 5.4.2, resulting in an uncertainty of 7.6 nm. 4.4. Geometrical errors Residual Abbe errors are caused by (1) misalignment DS,FP = 125 ␮m [31] of the capacitive displacement sensor (DS) with respect to the functional point (FP), i.e. the tool center, in combination with angular error motion of the machine tool slides, and (2) misalignment LE,DS = 100 ␮m [31] of the scale and reading head of the linear encoder (LE) with respect to the capacitive sensor in combination with angular error motion of the MS system guides. Repeatable angular error motions contribute to the systematic errors, while non-repeatable error motions have an effect on the repeatability. The repeatable angular error motion of the machine tool slides ( MT,r ) and the guides of the MS system ( MS,r ) have been estimated at ±30 ␮rad and ±18 ␮rad respectively [31]. The non-repeatable error motion  MT/MS,nr is assumed to be five times lower [7]. The error motions are considered to be uniformly √ distributed, with a corresponding uncertainty U = 2 · MT/MS / 3. The MS system measures displacement of the target surface in the direction normal to the target surface. Therefore, a cosine error is introduced by non-perpendicularity  ⊥ of the target surface with respect to the linear encoder [13]. The cosine error equals 2 /2 and is uniformly distributed over the measurement ±Ls /2 · ⊥ stroke. The maximum misalignment  ⊥ has been estimated at 200 ␮rad [31]. 4.5. Analysis of the uncertainty budget The combined measurement uncertainty of the MS system with a measurement range Lm of 120 mm and a utilized measurement stroke Ls of 107 mm equals 21 nm. Table 1 also gives the dependency of the uncertainty on Lm and Ls . The components that contribute most to the measurement uncertainty are (1) the calibration uncertainty, (2) the deformation of the scale due to temperature expansion of the scale interface and (3) the dynamic measurement error during tracking of the target surface. 5. Experimental determination of reproducibility and repeatability 5.1. Requirements of experimental setup The measurement uncertainty of the MS system that was estimated in the previous section needs to be verified by comparison to a reference system of which the displacement is known. A

7

laser interferometer is a common solution as a reference system [39]. However, as can be derived from the results in [17], the short-term stability of a laser interferometer in air for a measurement stroke of 107 mm amounts to 26 nm, which is even higher than the envisioned measurement uncertainty of the MS system. Therefore, it is considered to be very difficult to determine the uncertainty of an MS system in motion with a laser interferometer in air. This is resolved by separately determining the uncertainty invoked by random measurement errors and systematic errors, as for random errors there is no need for traceability to the absolute length standard and hence no need for laser interferometry. The latter suggests the use of a more stable linear encoder for the reference measurement. Random errors can be subdivided into non-reproducible errors due to changing ambient conditions and non-repeatable errors over a short period of time. The latter include sensor noise and dynamic errors. The following sections will discuss the design of a setup and experiments conducted for determining these random errors. An estimation of the uncertainty from systematic errors can be found in [31]. 5.2. Layout of experimental setup Excepting laser interferometry, it is assumed that the MS system has the lowest measurement uncertainty compared to other state-of-the-art long-stroke displacement measurement systems. Therefore, it has been chosen to use an additional MS system, in which the capacitive sensor interface is replaced by a target surface, as the reference system. This will be called the Reference Scale (RS) system. The layout of the proposed experiment setup is illustrated in Fig. 14. The MS system is placed in line with the RS system. The layout has been configured such that most components could be produced in one machining setup, which drastically improves the alignment. The MS system follows the movement of the RS system either by control of the linear motor or by simply connecting both systems at the motor-and-guide support frame. The length of the scale of the RS system has been extended to 170 mm to allow for the placement of two reading heads on that scale for the same measurement stroke of 107 mm. The three reading heads are located on an aluminum metrology frame (MF) (Fig. 15). Reading heads 1 and 2 are L1 = 230 mm apart, while the distance between reading head 2 and 3 is L2 = 55 mm. Reading head 1 measures the position of the MS system, while reading head 2 and 3 simultaneously capture the position of the RS system. Because the scales are made of Zerodur, the difference between the positions indicated by reading head 2 and 3 can be used to compensate for the uniform thermal expansion of the MF. To ensure a uniform expansion, the metrology frame is made of aluminum EN AC-5083 because of its high thermal diffusivity. It is kinematically constrained to the base frame by three ball-in-V connections, of which one is a split V-groove to increase the static stability [30]. An estimation of the Biot number of the MF is in the order of 10−3 , which is far below the Biot limit of 0.1. It indicates that a uniform temperature distribution can be assumed for the MF as long as the ambient air does not have considerable thermal gradients. To decrease the influence of ambient temperature gradients and draughts on the MF, an aluminum shield is placed over the metrology frame, which is not shown in the figure. The dimensions of this shield and the spacing between the shield and MF were chosen based on values found in [13], in which the same principle is used to reduce thermal gradients. Additionally, the setup is placed inside an enclosure to prevent air flows coming from the metrology room air inlets from directly affecting the temperature of the setup. The reading heads are bolted onto a stainless steel AISI 430F interface. This interface is in turn kinematically constrained to the MF with ball-in-V connections, which are configured such that a

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Thermal center

Reading head

Interface

Cylinder V-Groove Fig. 16. Reading head interface.

Fig. 17. Layout of the experimental setup with a stationary target attached to the metrology frame.

Fig. 14. Layout of the experimental setup for determining the repeatability and reproducibility.

The measurement error, of which the variance determines the reproducibility and the repeatability of the setup, is defined as e = xMS − xRS

(4)

with xMS = x1 + xDS xRS = x2 +

Fig. 15. Metrology frame of the experimental setup.

thermal center is created that coincides with the x-position of the internal thermal center of the reading head’s scanning reticle. The connection is preloaded by two tension springs. Fig. 16 shows the design of the reading head interface.

L1 (x2 − x3 ) L2

(5) (6)

and in which x1 , x2 and x3 are the read-outs of the reading head of the MS system and the two reading heads of the RS system respectively, xDS is the measurement of the capacitive sensor and L1 and L2 are the distances between the reading heads as indicated in Fig. 14. The second term of (6) can be low-pass filtered to eliminate dynamic errors, because it should only compensate low-frequency thermal expansion errors. Since the RS system will introduce a reproducibility in the same order of magnitude as the MS system, the setup can be reconfigured such that the MS system measures a stationary target attached to the MF, which eliminates the major part of the uncertainty of the RS system from the equations. Fig. 17 illustrates this layout. For this configuration, xRS in Eq. (4) equals xRS =

L3 (x2 − x3 ) L2

(7)

and the RS system is only used for compensation of the thermal expansion of the MF.

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5.3. Uncertainty budget of the setup When determining the measurement uncertainty of the MS system UMS from the measurement error calculated by Eq. (4), the result also incorporates the uncertainty of determining the reference position Uref,M and Uref,S for the moving and stationary configuration respectively. Therefore, it is only possible to determine an upper boundary on UMS , given by UMS ≤ Umeas + Uref,M/S ,

(8)

in which Umeas is the expanded uncertainty (k = 2) derived from measurement data of e in Eq. (4). To determine the value of Uref,M and Uref,S , an uncertainty budget for the experimental setup was made, which is given in Table 2. This uncertainty budget consists of two parts. One part represents the reproducibility under a changing ambient temperature and humidity. In this part, dynamic errors are excluded, which can be achieved in practice by averaging over a sufficient amount of measurement samples. Reproducibility is considered for the moving configuration, corresponding to Eqs. (4)–(6) and the stationary target configuration, which corresponds to Eqs. (4), (5) and (7). The second part of the uncertainty budget represents the repeatability, in which ambient changes are assumed to be negligible, as measurements are taken over a short period of time. Repeatability is only considered in the moving configuration. A distinction is made between the relatively low-frequency components (<10 Hz) of the non-repeatable Abbe and cosine errors, and the high-frequency components (>10 Hz) of sensor noise and other dynamic errors. Since an envisioned application of the MS system lies in the generation of surfaces for EUV-lithography mirrors, the 10-Hz cut-off frequency was chosen based on the definition of the mid-spatial frequency range of finishing errors on EUV optics by Taylor et al. [40], which ranges from 1/mm to 1000/mm. If the velocity during machining is 10 mm/s, the transition frequency from figure errors to mid-spatial frequency errors is 10 Hz. The reproducibility and repeatability of the MS and RS system, UMS and URS respectively, can be calculated from Table 1, although a detailed calculation is omitted here. For the RS system, the measurement range Lm equals 120 mm and the measurement stroke Ls is 107 mm. The linear encoder reproducibility ULE originates from the zero-point drift of the reading head. The low-frequency repeatability is dictated by non-repeatable Abbe errors, while the high-frequency contribution corresponds to the periodic position error of ±1 nm (Section 4.1). The scale expansion (USC ) is caused by expansion of the Zerodur and the scale interface, and has also been derived in Section 4.1. ULE and USC determine the uncertainty of calculating the uniform expansion of the metrology frame over lengths L1 , L2 and L3 . The final uncertainty contributions arise from a residual temperature gradient in the metrology frame. A finite element analysis was conducted, of which the results are shown in Fig. 18. In the analysis, a temperature gradient of the surrounding air in the horizontal direction over the aluminum heat shield of 1 ◦ C was applied. This gradient corresponds to the assumed worst case situation in the metrology room. The resulting calculated temperature gradient in the metrology frame only differed 2.8 mK between the position of reading head 1 and reading head 3. A detailed calculation of the corresponding measurement error, which is 7.0 nm and 5.8 nm for the moving and stationary configuration respectively, can be found in [31]. The setup establishes a reference position for the MS system to measure. The uncertainty of this reference position is calculated for the moving (Uref,M ) and the stationary configuration (Uref,S ) respectively and given in Table 2. Finally, the combined measurement uncertainty of the MS system and the reference position is calculated as well.

Fig. 18. Temperature distribution in the metrology frame caused by a temperature gradient of 1 ◦ C in the surrounding air.

From the uncertainty budget, it is clear that the stationary target configuration has a higher reproducibility because most of the influence of the RS system is excluded and the uncertainty of the compensation of the MF expansion is reduced thanks to a smaller L3 in comparison with L1 . Therefore, reproducibility of the zeropoint is preferably analyzed using this configuration. Nevertheless, it seems difficult to keep the measurement uncertainty of the reference position below the uncertainty of the MS system itself. This is mainly due to the uncertainty of the metrology frame expansion compensation. However, it should be noted that the uncertainty of the expansion of the metrology frame over L2 is only 5.6 nm for ±0.5 ◦ C temperature changes, which corresponds to a sensitivity of approximately 0.2 ppm/◦ C. This is just twice the maximum possible CTE of Zerodur (Class 2) and is very difficult to achieve by solely applying temperature compensation to correct for the thermal expansion of an aluminum frame. 5.4. Experimental results 5.4.1. Reproducibility In these experiments, the measurement error (Eq. (4)) was evaluated over a long period of time, together with measurements of the ambient temperature and humidity. The MS and RS systems were held stationary to exclude dynamic errors. The reproducibility was determined for two cases: 1. without numerical compensation for temperature and humidity changes, 2. with numerical compensation, where compensation is done by identifying a first-order double-input single-output transfer function from temperature and humidity to error and processing of the temperature and relative humidity measurements based on this transfer function. As the envisioned application of the MS system is an ultraprecision grinding process that could last for up to 5 h, the measurement uncertainty determined from the data is also considered over 5-h intervals. For shorter periods of time, the uncertainty will go down, while for longer time spans, it will rise. For each interval, the measurement error is calculated as the difference between the displacement measurement and the displacement at the beginning of the interval. If this is done for multiple intervals, a data set of possible errors within 5-h time periods is created. The rootsum-square (RSS) value of this data set is calculated to obtain the standard uncertainty. The sensitivity of the measurement error to temperature and humidity changes is determined by calculation of the steady-state value of the step response of a first order

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10 Table 2 Uncertainty budget of the experimental setup.

U (k = 2) [nm] Reprod. 1 Hz

Uncertainty contribution Components MS system RS system Linear encoder measurement Scale expansion over L2 Expansion L2 Expansion L1 Expansion L3 MF temp. gradient moving conf. MF temp. gradient stat. target conf.

UMS (Table 1) URS (Table 1) ULE (Section 4.1) USC (Section 4.1) UL2 =

Stat. target conf. (Fig. 17)

2 2 2 · ULE + USC

UL1 = L1 /L2 · UL2 UL3 = L3 /L2 · UL2 UMF,M [31] UMF,S [31]

Uncertainty of reference position Moving conf. (Fig. 14)



Uref,M =

 2 U + U2 + U2 RS 2 L1 2 MF,M

Moving conf.

UM =

Stat. target conf.

US =

>10 Hz

11 12 1.9 4.9

1.0 0.5 0.4 n/a

7.8 1.3 1.0 n/a

5.6

n/a

n/a

23 13 7.0 5.8

n/a n/a n/a n/a

n/a n/a n/a n/a

27

0.5

1.3

UL + UMF,S

14

n/a

n/a

 2 2 U + Uref,M  MS 2 2

29

1.1

7.9

18

1.0

7.8

Uref,S =

Combined uncertainty

Repeatability <10 Hz

3

UMS + Uref,S

Table 3 Calculation of the upper boundary of the reproducibility of the MS system according to Eq. (8). Component

U (k = 2) [nm]

Uncertainty of reference position: Uref,S Results of Case 2: Umeas,rpd,2

14 7.4

Ureprod ≤ Uref,S + Umeas,rpd,2 (Eq. (8))

21

origin of the temperature sensitivity lies mostly in the internal temperature drift of the reading heads. Humidity variation, on the other hand, has an effect on the capacitive sensor measurement [38]. In Table 3, the results of Case 2 for the stationary configuration are combined with the calculated uncertainty of the reference position to evaluate the upper boundary on the uncertainty. Uref,S is derived in Table 2 and equals the measurement uncertainty when determining the position of the stationary target surface. The upper boundary of the reproducibility of the MS system is determined using Eq. (8) and amounts to 21 nm. It can therefore be stated that the reproducibility of the MS system is below 21 nm with a 95% confidence level.

Fig. 19. Results of the experiments for determining the reproducibility in the stationary configuration.

transfer function from temperature and humidity to measurement error. This first order transfer function is determined by the Matlab function tfest on measurement data that has been filtered by a band-pass filter with cut-off frequencies of 1/(20 h) to decrease long-term drift and 1/(5 min) to decrease sensor noise. These procedures are explained in more detail in Appendix B. Because of its low reference position uncertainty Uref,S , reproducibility of the MS system was determined by the setup configured with the stationary target attached to the MF. In Fig. 19, the resulting measurements are shown for the two cases described above. In Case 1, the temperature sensitivity is calculated to be −43 nm/◦ C, while the humidity sensitivity is −7.7 nm/%rh, which would lead to an uncertainty Umeas,rpd,1 = 52 nm for ±0.5 ◦ C and ±5%rh ambient changes. These sensitivities are used for compensation in Case 2, which leaves a residual uncertainty Umeas,rpd,2 = 7.4 nm over intervals of 5 h. These uncertainties are calculated as in Appendix B. The

5.4.2. Repeatability Repeatability, which includes sensor noise, non-repeatable Abbe errors and dynamic errors, was verified by an experiment in which the MS system actively tracked the position of the RS system while moving the RS system by hand. The measurement range was traversed multiple times. Before the experiment, the non-linearity of the capacitive sensor was calibrated. The measurement error was then calculated by using Eq. (4): e = x1 + xDS − x2 −

L1 (x2 − x3 ) L2

(9)

with x1 = x1,r + x1,nr x2 = −˛Al L1 T (t) + x2,r + x2,nr x3 = −˛Al (L1 + L2 )T (t) + x3,r + x3,nr xDS = xDS,r + xDS,nr in which xi,r is the repeatable part of the measurement, caused by a repeatable length deviation, and xi,nr is the non-repeatable part of the measurement, caused by dynamic errors and measurement noise. The standard deviation of the non-repeatable parts of this measurement error gives the repeatability, which implies that the

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Fig. 20. Measurement results for determining the repeatability of the setup with the MS system tracking the movement of the RS system through control of the linear motor.

repeatable errors and temperature drift should be eliminated from the measurements. This is done in the following steps:

with max(x2 ) − min(x2 ) (j − 1) + min(x2 ) j ∈ [1, N + 1[ N x2,j ≡ x2 ∈ [Xj , Xj+1 [ x3,j ≡ x3 (x2,j ) xMF (t) ≡ a1 t + a2 t 2 + a3 t 3 + a4 t 4 tj ≡ t : x2 (t) ∈ [Xj , Xj+1 [. Xj =

(1) First, the metrology frame expansion xMF (t) = ˛Al L2 T(t) and the repeatable error eMF (x2 ) = x2,r (x2 ) − x3,r (x2 ) are determined by solving the following equations:

x2,j − x3,j =

eMF (Xj+1 ) − eMF (Xj ) Xj+1 − Xj

(x2,j − Xj ) + ...

... + eMF (Xj ) + xMF (tj )

(10)

Because there are more data points and hence more equations than variables, this system of equations should be solved in the least-squares sense. By doing so, the measurement data are fit to a piecewise linear function dependent on x2 over N intervals

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and a fourth-order polynomial function dependent on time t. Eq. (10) can be rearranged and written in matrix-form as P



A1

=

y

A2





eMF

 =

a

⎡

m m1

⎤

⎢ ⎥ ⎢ .. ⎥ ⎣ . ⎦

(11)

mN with

⎡

x2,j − X2 0 ⎢ A1 = ⎢ .. ⎣ . 0

⎡

X1 − x2,j x2,j − X3 .. . 0

(X1 − X2 ) · t1 .. A2 = ⎣ . (XN − XN+1 ) · tN



eMF = eMF (X1 )

···

a = [ a1

a4 ]

a2

a3

··· .. . ···

··· ··· .. . x2,j − XN+1

⎤

0 0 ⎥ ⎥ .. ⎦ . XN − x2,j

⎤

(X1 − X2 ) · t1 4 .. ⎦ . 4 (XN − XN+1 ) · tN

eMF (XN )

T

T

This system is then solved for y in the least-squares sense as y = (PT P)

−1 T

P m.

(12)

(2) Calculation of the measurement error according to Eq. (9), but substituting x2 − x3 by xMF (t), results in e∗ = x1 + xDS − x2 −

L1 xMF = x1,r + x1,nr + xDS,r + xDS,nr − x2,r L2

− x2,nr = er∗ + x1,nr + xDS,nr − x2,r .

(13)

The repeatable part of this error, er∗ , consists of the scale length deviations and capacitive sensor errors that are dependent on position x1 . er∗ is determined by solving Eq. (13), which is accomplished by fitting a piecewise linear function in x1 over N = 100 intervals to e* . (3) The non-repeatable errors are then calculated by enr = e∗ − er∗ .

(14)

(4) In a next step, enr is split up into two frequency domains: one ranging between 0–10 Hz, called enr,l , the other one ranging from 10–500 Hz, which is enr,h . enr,l represents the errors that will result in figure errors while machining. enr,h , on the other hand, causes mid-spatial frequency errors on the workpiece. (5) From enr,l and enr,h , the repeatability is determined through calculation of the standard deviation. Fig. 20 provides an overview of the analysis steps for the conducted experiment. Fig. 20a shows the displacement of the MS system. Maximum velocity was 29.5 mm/s while maximum acceleration was 262 mm/s2 . Fig. 20b–f depicts steps (1)–(5) respectively. The standard deviation, calculated from the lowfrequency errors, was 2.2 nm. For the high-frequency errors it was 3.8 nm. Twice the standard deviation gives Umeas,rpt , which is used later to evaluate Eq. (8). It is assumed that the low-frequency errors are due to increased uncompensated non-linearities of the capacitive sensor because the variation of xDS is larger during tracking. Moreover, accelerations and velocities are also higher than expected, which causes additional errors. On the other hand, local figure errors of the target surface and non-perpendicularity of the surface with respect to the capacitive sensor could also give rise to errors in this order of magnitude. Additional experiments with a

Fig. 21. Power spectral density (PSD) and cumulative amplitude spectrum (CAS) of enr,h in Fig. 20f.

Table 4 Calculation of the upper boundary of the repeatability of the MS system according to Eq. (8). Repeatability (k = 2) [nm] Component

<10 Hz

>10 Hz

Uncertainty of reference position: Uref,M Umeas,rpt

0.5 4.4

1.3 7.6

∗ ≤ Uref,M + Umeas,rpt (Eq. (8)) Urpt

4.9

8.9



Combined: Urepeat =

2

∗ ∗ (Urpt,<10 ) + (Urpt,>10 ) Hz Hz

2

10

smoother target surface and a linear motor-controlled RS system have to be carried out to determine the origin of these errors. Calculation of the power spectral density (PSD) of enr,h offers more insight into the high-frequency errors, as depicted in Fig. 21a. For this figure, the data were not low-pass filtered at 500 Hz. Errors below 250 Hz are caused by motion errors of the guides. The PSD also shows peaks at 380 Hz and 555 Hz, which are the pitch and yaw eigenfrequencies of these guides. These high-frequency error motions of the guides result in accelerations that deflect the kinematic connection between scale and capacitive sensor. This PSD also allows to determine the rms error up to a certain bandwidth by calculating the cumulative amplitude spectrum (CAS) [41], which is shown in Fig. 21b. Table 4 summarizes the results of the repeatability experiments together with Uref,M , the uncertainty of determining the reference position (see Section 5.3 and Table 2). The measured repeatability amounts to 4.4 nm for the components below 10 Hz and 7.6 nm for those above 10 Hz. The upper boundary of the repeatability of the MS system is then derived based on the experimental results by using Eq. (8) and equals 4.9 and 8.9 nm respectively. The combined repeatability is below 10 nm with a confidence level of 95%. 5.5. Uncertainty of one-DOF MS system based on experiment data The reproducibility and repeatability that were calculated in Table 1 are the uncertainties calculated from the individual components’ contributions. The reproducibility and repeatability that

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Table 5 Upper boundary of the combined uncertainty of the MS system including the experimental results of Section 5.4. Component

U (k = 2) [nm]

Upper boundary of reprod. Ureprod (Table 3) Upper boundary of repeat. Urepeat (Table 4) Systematic errors (Table 1)

21 10 17

Combined uncertainty

29

13

Acknowledgments This work has been supported by a PhD grant from the Institute for the Promotion of Innovation through Science and Technology in Flanders IWT/101447, the EC FP7 FoF collaborative projects “MIDEMMA” (Grant agreement no. 285614) and “Hi-Micro” (Grant agreement no. 314055). The authors would also like to thank the company Heidenhain Gmbh for its support to this research. Appendix A. Thermal center stability

are discussed in this section are the ones derived from the experiments in Section 5.4. One could say that the combined uncertainty in Table 1 (21 nm) is a lower bound on the uncertainty of the MS system, while the combined uncertainty in that is derived in this section is an upper bound. The resulting reproducibility and repeatability from the experiments are given in Table 5, together with the systematic errors are caused by the uncertainty in calibrating the scale length deviation and by repeatable Abbe errors (Table 1). These contributions lead to a combined uncertainty of 29 nm for a 95% confidence level. The uncertainty of the MS system is therefore between 21 nm and 29 nm.

6. Conclusion and future work This paper presented a novel concept and prototype development of a linear encoder-based position measurement system compliant with the Abbe principle. The general layout of a multiDOF system based on this concept was introduced. One of the linear encoder modules consists of a scale and a displacement sensor which are located in line on a carrier moving in the scale’s measurement direction. The position of the carrier is controlled by a linear drive based on the output of the displacement sensor, which measures the movement of a target surface attached to the workpiece table. This configuration allows the functional point, such as the tool center, to always be coincident with the path of the effective point on the scale, in this way fulfilling the Abbe principle for the entire measurement volume. The actively driven carrier eliminates the introduction of forces in the measurement loop. Consequently, the design of a one-DOF prototype for a 120 mm measurement range and a 107 mm utilized measurement stroke was discussed. A thermally stable design of the interface carrying the scale and the probe was conceived by applying the thermal center principle. An uncertainty budget of the one-DOF MS system estimated that the total combined measurement uncertainty was 21 nm, of which the calibration uncertainty of the scale, the deformation of the scale due to temperature variation and the dynamic errors during target-surface tracking were the largest contributors to the uncertainty. Subsequently, an experimental setup was built to verify the estimated measurement uncertainty. Experiments were carried out to check the reproducibility under varying ambient conditions and the repeatability of a moving system over a short time period. The results showed a reproducibility of the entire experimental setup of 7.4 nm, which gave an upper boundary of the reproducibility of the MS system of 21 nm. The repeatability was 4.4 nm for a measurement error below 10 Hz and 7.6 nm above 10 Hz, leading to an upper boundary of 10 nm. The combination of the experimental results with estimations of the systematic errors indicate that the measurement uncertainty of the MS system is below 29 nm with a 95% confidence level. Future research will be focused on the implementation of the MS system on the linear slides of a five-axis ultra-precision grinding machine to reach form accuracies below 0.1 ␮m and on the determination of the systematic errors by laser interferometry.

Since the thermal stability of the chosen interface concept relies on the stability of the thermal center, this design principle is revised here in more detail. In general, a kinematic connection creates a thermal center between two components at a defined point when all planes normal to the kinematic constraints intersect in that point. In practice, however, these connections will never have infinite stiffness in the constraint direction and also no zero stiffness in the other five DOF. When there is a thermal expansion between these connections, which act as spring elements, the connected components will displace such that the potential energy in these elements is minimized. This could cause a drift of the thermal center from its theoretic location. Fig. 22a illustrates this drift, which can be estimated for one direction in the following way. First, the combined stiffness of all constraints in that direction is calculated by k ,y =

(j · k i, )2 i

k i,

+

(j · k i,⊥ )2 i

k i,⊥

≈

(j · k i, )2 i

k i,

(15)

with j the unit vector in the considered direction and k i,j a vector in the spring direction with magnitude equal to the stiffness of spring

Fig. 22. Drift of the thermal center due to non-ideal constraints.

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with T the length of the interval. If there are data available over a time larger than T and multiple intervals can be considered, the standard uncertainty can be calculated by the root-mean-square of the RSS of the intervals:



i=1

u0 = Fig. 23. (a) Example of zero-point error e0∗ (t) and (b) corresponding machining error on the workpiece.

ki,j . The deflection of each constraint i due to thermal expansion now results in a force which acts on k ,y and causes a displacement of the thermal center equal to yTC,i = ˛TLy,i

ki,⊥,y ki,⊥,y + k ,y

= ˛TLi ri ,

(16)

in which ri is called the stiffness ratio of constraint i, Ly,i is the distance in y-direction from the theoretical thermal center to constraint i and ki,⊥,y =

2 (j · k i,⊥ ) .

k

(17)

i,⊥

The 1-D equivalent of Eq. (16) is depicted in Fig. 22b. The total thermal center drift is calculated by the combination of the drift for each individual non-ideal constraint: yTC =

yTC,i = ˛T

i

Ly,i

i

ki,⊥,y ki,⊥,y + k ,y

= ˛T

Li ri

(18)

i

N

(u∗0,i )2

N

,

(21)

in which N equals the number of considered intervals. B.2. Determination of temperature and humidity sensitivity To determine the temperature and humidity sensitivity, the measurement data were first filtered with a low-pass filter with cut-off frequency of 1/(5 min) to reduce noise and by a high-pass filter with a cut-off frequency of 1/(20 h) to cancel low-frequency drift. These frequencies were chosen arbitrarily, although the highpass filter frequency should be taken sufficiently high to obtain enough variation of the data. Subsequently, the Matlab function tfest was used to estimate first-order transfer functions with temperature and humidity as the input and the error as the output, resulting in transfer functions fT and fRH . The sensitivity values were calculated by S = lim f ∗ H(t), t→∞

(22)

with H(t) the unit step function. Next, the compensated error was calculated by ec = e − [(fT ∗ Tlp )(t) + (fRH ∗ RH lp )(t)]

(23)

To keep the thermal center drift as small as possible, the stiffness ratio should be minimized, which implies minimizing ki,⊥ for maximum ki, . For instance, a viable ki,⊥ for thermal drift below 10 nm/◦ C for the configuration of Fig. 22b with ˛ =24.5 ppm/◦ C, dimensions L1 = 30 mm, L3 = 100 mm, L4 = 100 mm and ki, = 5 N/␮m would be 3.7 N/mm. These considerations are taken into account in the detailed design of the scale and capacitive sensor interface, which is discussed in Section 3.

in which Tlp and RHlp are low-pass filtered temperature and humidity data with a cut-off frequency of 1/(5 min). The standard uncertainty caused by uncompensated errors uc is then calculated according to the procedure described in the previous section. Based on this value and the calculated sensitivity values, the combined uncertainty of both uncompensated errors and errors due to a maximum temperature drift of T = 0.5 ◦ C and a maximum humidity drift of RH = 5%rh equals

Appendix B. Measurement data processing

U=

The envisioned application of the MS system is an ultraprecision five-axis grinding machine, which is described in [29]. This machine has a maximum machining time of 5 h and therefore, in this paper, the measurement uncertainty due to temperature and humidity drift is calculated over this time period. A machining cycle would start with homing or referencing of the axes’ position measurement, after which machining is initiated. The measurement error of the reference or zero-point e0 (t) at the start of the machining interval will thus reset to zero and the resulting measurement error e0∗ over time equals (19)

of which an example is illustrated in Fig. 23a. This error will result in an error on a flat workpiece according to Fig. 23b. It is assumed that the zero-point was taken at the upper left corner of the workpiece. When the standard uncertainty is calculated from this error, it is not correct to take the variance of the error over this time period, since it will lead to an underestimation of the real uncertainty. Instead, the root-sum-square value should be calculated by



u∗0 =

1 T

 0

T

e0∗ (t)2 dt,

Uc2 +

4 2 2 RH 2 ), (S T 2 + SRH 3 T

(24)

in which uniform temperature and humidity distributions are assumed.

B.1. Uncertainty of the zero-point

e0∗ (t) = e0 (t) − e0 (0),



(20)

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Please cite this article in press as: Bosmans N, et al. Design and experimental validation of an ultra-precision Abbe-compliant linear encoder-based position measurement system. Precis Eng (2016), http://dx.doi.org/10.1016/j.precisioneng.2016.08.005