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A Comparative Study on Testing of Wide-Range Angles
Precision 2-axis Digital Angular Measuring Instrument
J.H. Lau, B.Y. Kuu, Hui Hong J.K. Li
JSB Tech Pte Ltd, 20 Science Park II, Singapore 117674
August 2011 (revised February 2013)
Abstract – 2-axis precision digital leveling instrument has greatly simplified the task for installation, setting up andmaintenance of high precision CNC machines, angular measuring instruments and alignment-sensitive equipment toattain maximum positional stability. This paper presents a comparative study on using rotary table, sine bar and fixed
angle jigs to test precision wide-range angle 2-axis digital leveling instrument. Literature review is presented focusing on
identifying sources of measuring errors for these test equipment and their respective testing methodology. The findings
demonstrate that a set of high precision fixed-angle jigs is a relatively reliable test apparatus as it has least variableaffecting the uncertainty of the entire test system and well-understood material properties. In contrast, rotary table has
the merit of its ability to generate large number of precise incremental angles within the full-range of 360° for testing
wide-range precision digital angular measuring instruments.
Keywords –Sine Bar, Rotary Table, Fixed-angle Jig, Precision Digital Leveling Instrument1.
1.1

Rotary Table Mechanism and Angular Measuring System
Direct-drive Servo motor used in Rotary Table
A rotary table is the core component of an angular motion system that exhibits one degree of freedom(rotation about one axis). The basic construction of a rotary stage consists of a platform and a base, where
the platform is restricted to rotate within single axis with respect to its base.
Figure 1 illustrates the direct drive rotary motor used in a typical rotary table structure. Direct drive rotary
motors develop high torque and enable precision servo control over very small step of angles within 360°.
Their dynamic response is excellent as load is coupled directly to the drive, eliminating the need fortransmission components that introduce backlash, hysteresis, gear-tooth error or belt stretch [1].

Figure 1 Direct Drive Motor Application

1.2

[1]

Air Bearing Mechanism of a Rotary Table System
The limitation of plain bearing technology represents one of the age-old problems for mechanicalengineers. Plain and rolling element bearings developed in the last century have been subjected to radical
improvement and follow the emergence of air bearings in recent decades. Bearing technology is further
pushed to their technical limits by the stringent demands of applications in the semiconductor
manufacturing test instruments, high resolution medical equipment, and high-speed precision CNC
machinery. The key technical performance of air bearings such as near zero friction and wear, high speed© 2013 DIGIPAS USA LLC. All Rights Reserved

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Innovation & Technology Driven ™
and high precision capabilities, and no oil lubrication requirements are advantages for applications in the
precision equipment and instrumentation [2, 3].
Air bearings utilize a thin film of pressurized air to provide a „zero friction‟ load bearing interface between
surfaces to create the physical separation of the two surface areas as depicted in Figure 2. Being noncontact, air bearings avoid the traditional bearing-related issues of friction, wear, and lubricant handling,
and offer distinct advantages in precision angular positioning and high speed motion applications [2, 3].
The fluid film of the bearing is achieved by supplying a flow of air through the bearing face and into thebearing gap. This is typically accomplished through an orifice or a porous media that restricts the flow of
air into the gap as referred as R1 in Figure 2. The restriction is controlled such that, although the air isconstantly escaping from the bearing gap, the flow of pressurized air as specified is adequate to match theflow through the gap. It is the restriction through the gap, R2 that maintains the pressure under the bearing
and supports the working load. When air pressure was introduced to the gap without restriction R1, theflying height would change and thus the air consumption and the stiffness would vary accordingly [2]. This
restriction is referred to as air bearing compensation and is used to optimize the bearing performance withrespect to lift, load, and stiffness for a particular application as described subsequently.

Figure 2 Flat Air Bearing [2]Because of their advantages over conventional bearings, air bearings are a natural choice for numerous
applications in precision linear and angular positional measurements, diagnostic and precision engineeringthat especially require clean room operating environments and high speed performance. The main
advantages of air bearings are listed below.
No.
1

MeritsZero Friction

2

Zero Wear

3

Straighter Motion

4

Silence andSmooth Operation

5

Higher Damping

6

Eliminates Oil

7

High Speed

Descriptions
Infinite resolution and very high repeatability are possible to beachieved due to zero static friction.
Non-contact means virtually zero wear between surfaces which resultsin consistent machine performance and low particle generation.
Conventional bearings subject to surface areas contact that are directly
influenced by surface finish and irregularities on the guide for linear or
angular motions. Air bearing being non-contact average these errors.
Roller and ball bearings generate noise and vibration during recirculating as hard elements become loaded/ unloaded and change
direction.
Being fluid film bearings, air bearings have a squeeze film damping
effect resulting in higher dynamic stiffness and improved
controllability.
Air bearings do not use oil lubrication and eliminating implications
associated with oil.
No balls or rollers to slip at high acceleration.

Table 1 The Advantages of Air Bearing over Conventional plain and Roller Bearings© 2013 DIGIPAS USA LLC. All Rights Reserved

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1.3

Angular-Positional Accuracy of Rotary Table for Angular measurement
In order to quantify the compounded uncertainty of the entire angular measuring system using rotary table,
the following describes some critical elements of errors that could affect the measurement accuracy of atypical single-axis digital leveling instrument.

1.3.1 Offset Error with reference to center of gravity (g)Figure 3.a and Figure 3.b show the initial setups for a typical rotary table before and after offset error
correction where the device under test (DUT) is loaded with a Digi-Pas® 2-axis precision master machinist
level model DWL-3500XY. The top surface of the L-plate is positioned in perpendicular to the center ofgravity. Hence, the offset error of the rotary table against the earth datum is eliminated (i.e. negligible withrespect to the resolution of DUT‟s of 0.001°) when the Granite table is leveled to 1 arc-second of accuracy
for both X and Y axis.

(a) Setup with no Offset Error(b) Setup with Offset Error

Figure 3 (a) Rotary Table Setups before and (b) after offset error correction[4]

1.3.2 Swash Error with reference to center of gravity (g)Figure 4 illustrates the Swash error, the surface (Y-axis) plane of the L-plate, where the DUT is placed on,
is not perpendicular to the center of gravity. Swash error occurs when the mounting plate contact surfaces
to the Rotary Table is not in absolute parallel to the earth datum. Swash error is separately dealt in the JSB
TECH Technical Paper “The Implication of Swash Error to Calibration System Uncertainty When TestingHigh Accuracy Inclinometer Utilizing MEMS Technology” [4].

Figure 4 Swash Error with reference to center of gravity (g)[4]

1.3.3 Encoder Scale Errors
Encoder error is composed of three types: (i) quantization error, (ii) instrument error, (iii) cycle
interpolation error [5]. Quantization error exists because the encoder cannot indicate motion occurring
within one resolution quantum at transition points. This is the highest frequency error component andrepeats every quantum of input motion. The quantization error is the deviation of the input shaft from the© 2013 DIGIPAS USA LLC. All Rights Reserved

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Innovation & Technology Driven ™
mid position for a given readout, with the maximum error of ±1/2 of the angular rotation between two
successive bits.
Instrument error is the sum of disc and reticle pattern errors and mechanical imperfections within theencoder. Factors such as bearing types, line to space ratio tolerances, substrate flatness, optical setup, andencoder alignment contribute to instrument error.
Cycle interpolation error is due to imperfections in the analog signals from the photo detector due to phaseshifts or dc offsets that cause position errors in the signals zero crossings, which affect the count produced
by a given amount of movement, as zero crossings are counted as a measure of movement.
1.3.4 Eccentricity Errors and Bearing Run Out Errors.
In order to achieve accurate angular measurement, a scale of uniform linear graduation must be positioned
at a consistent distance from the axis of rotation. Variations in radius, as caused by eccentricity of aperfectly round rotary scale, generate errors that vary once per revolution [1, 6]. These combine with othererrors within the system varying two or more times per revolution would result in scale distortion. Whenan angle encoder is removed and remounted on the same or a different mandrel, its accuracy would change
and the error recorded could vary accordingly. The difference would correspond to the error caused by thechange in eccentricity and higher order out-of-roundness of the encoder scale between its initial installation
for graduation and its reinstallation for use.
Precision instrument demand high rotating accuracy of the bearing as it affects the performance of the
instrument. Bearing run out error is often a key factor that affects rotation of encoder‟s axis in generating
accurate and stable angle for measurement. Irregular encoder code disk motion due to bearing run out
error could significantly degrade readout accuracy. The relationship between bearing run out, repeatability
and angular position depend on: (i) the relative dimensions of the inner and outer bearing races, (ii) the
number and diameter of its balls/rollers and (iii) the wear/adjustment of the bearing system [1, 6].
2.
2.1

Sine Bar Mechanism and Angular Measuring System
Sine Bar Mechanism
Figure 5 illustrates a typical sine bar consists of a hardened, precision ground body with two precision
ground cylindrical shafts fixed at each end. The distance between the centers of the shafts is precisely
controlled, and the top surface of the sine bar is parallel to a line through the centers of the two shafts. The
dimension between the two shafts is chosen to be a whole number and forms the hypotenuse of a triangle
when used in conjunction with a set of block gauge [7].

Figure 5 Sine Bar [7]When a sine bar is placed on a level surface (e.g. the surface of a granite table), its top face is required tobe parallel to that granite table‟s surface. When one side of the sine bar is raised by a known distance, by
utilizing gauge blocks, the top face of the bar is tilted by the same amount forming an angle that is
calculated by the application of the trigonometric sine rule [7].
As shown in Figure 6, two accurately lapped rollers are located at the extreme ends position [8]. The centerto center distance between the rollers is available for fixed and precisely controlled distance of, forexample, L=250 mm ±2µm.

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Figure 6 Position of two accurately lapped rollers with fixed center distance between the rollers [8]2.2Factors Affecting the Compounded Uncertainty of Sine Bar Angular Measuring System
The main contributing factors affecting measurement uncertainty of the entire sine bar angular test system
are (i) the sine bar geometric dimension accuracy, (ii) the block gauge and granite table dimensional
accuracy (iii) the test system initial positional set up and operating environment control.
i.

ii.
iii.

The sine bar geometric accuracy depends on (a) the inter-roller axis parallelism (b) the distance
between the rollers center L, (c) the roller‟s diameter and roundness (d) the flatness of the top
surface area of the sine bar, and (e) the parallelism of the sine bar top surface to the surface of the
granite table [7].
The peripheral equipment used in conjunction with sine bar; (a) the block gauge dimensional
accuracy (b) surface area flatness of the granite table.
The leveling of granite table surface and the alignment of its surface where the sine bar is placed
to prevent offset error and Swash error. The testing laboratory environments fulfill therequirements of the test system on e.g. dust particle count, temperature and humidity.

IS 5359-1969 specifies the basic requirements for dimensional size limit and tolerance of sine bar that
affect angular measurement error. The use of multiple block gauges in combination when measuring large
angles introduces compounding errors that affect the accuracy of using sine bar for measuring large angle
typically over 45°.
2.2.1 Example on the Effect of Block Gauge Height (H) to measure angle (θ)
Assuming that the length between roller to roller L = 200.0000mm, the combination block gauge required
height (H) of 68.4040mm in order to generate an angle of 20.000°, as shown in Figure 7. By using grade
zero block gauge with tolerance <0.5µm, the estimated maximum uncertainty value from the combination
block gauge is Huncertainy=0.00014°.

H = 68.4040±0.0005mm

Figure 7 Effect of Block Gauge Height (H) on uncertainty to measure angle (θ)

2.2.2 Example on the Effect of Roller Center Distance (L) to measure angle (θ)
It was given that the roller-to-roller distance, L is 200.000±0.003mm as shown in Figure 8. With largest
angle measurement at 20.000° and H=68.4040mm, the estimated largest uncertainty of sine bar/ plate roller
center distance is Luncertainty = 0.00032°.

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Figure 8 Effect of Roller Center Distance (L) on uncertainty to measure angle (θ)

2.2.3 Example on the Effect of Sine Bar/ Plate Parallelism (P) to measure angle (θ)
Given that the largest sine bar/ plate surface parallelism is 2.5µm (H 0=0.0025mm) and at L=200.000mm,
as shown in Figure 9. It was estimated the initial offset angle from parallelism uncertainty of sine bar/ plateis θMax, P = 0.00072°.

Figure 9 Effect of Sine Bar/ Plate Parallelism (P) on uncertainty to measure angle (θ)

3.
3.1

Precision Fixed Triangle Jigs
Utilizing Trigonometry Principal in Determining Precision Fixed Triangle Jig
Trigonometry is a well-known branch of mathematics that studies triangles and the relationships between
the lengths of their sides and the angles between those sides. When one angle of a triangle is 90 degrees
and one of the other angles is known, the third is thereby fixed. The two acute angles therefore add up to
90 degrees: they are complementary angles. The shape of a triangle is completely determined, except
for similarity, by the angles. Once the angles are known, the ratios of the sides are determined, regardlessof the overall size of the triangle. When the length of one side is known, the other two are determined.
These ratios are given by the trigonometric functions of the known angle A, where a, b and c refer to thelengths of the sides in Figure 10.

c

b
a

A

(a)
(b)
Figure 10 Precision Fixed Angle Jigs (a) 4.993° (b) 15.001°

3.2.

Utilizing the Coordinate Measuring Machine (CMM) to measure Surface Flatness of the Fixed Triangle Jig
CMM is measuring equipment for determining the physical geometrical shape of a physical object.
Measurement points are defined by a probe attached to a moving z-axis in addition to the planar X & Y
axis. Probes may be mechanical, optical image or laser.
Figure 11 below shows the Mitutoyo LEGEX series CMM claimed to be able to achieve exceptionallyhigh dimensional accuracy among CMM family of 0.2µm with a resolution of 0.01µm [9].

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Figure 11 Coordinate Measuring Machine (CMM)
Mitutoyo LEGEX Series [9]Multiple points of measurement on each surface area of the fixed triangular jig are obtained to determine
the least square fit of surface plane and the geometric angle of each surface plane with respect to its baseplane. The precise angle between the base and hypotenuse planes of the fixed triangular jig is thendetermined by applying sine function of the trigonometric principle.
3.3. Surface Flatness in Precision Measuring Instrument
In manufacturing of precision parts and assemblies, surface flatness is a critical quality of machining
process to meet the required specification. Surface flatness is defined in terms of least squares fit to asurface plane and is also referred as statistical flatness. Granite, steel and ceramic surface plates arecommonly used in precision geometrical measurements and the determination of surface flatness is an
important basic task of dimensional metrology. The flatness of the surfaces plates is the angular variationsof a predefined surface area and these angular variations can be determined using autocollimator, laserinterferometer or CMM, depending on the required surface quality [10].
Another variable affecting surface flatness resulting from interaction between machining tool and workpiece is surface finishing quality. Several processes such as diamond turning, grinding, lapping, honing,
polishing, ion- and electron-beam machining and chemical treatment are effective methods to enhancesurface finishing quality [11]. Lapping technique is advantageous due to its ability to precise control ofremoving material from a targeted surface area and capable of producing high surface uniformity anddimensional accuracy. Lapping is a process that material is subtracted from a work piece by abrasion to
produce a desired dimension and surface finish. The lapping plate typically rotates at a constant speed andoperates in conjunction with abrasive material [12]. The process of lapping effectively removes subsurface
damages on work piece such as cutting tool imprints and surface grinding traces. Surface finishes in thenanometer range can be achieved by lapping, thus makes lapping an attractive method for materials
processing in precision machining for applications in high precision metrology.
3.4. Material Selection and Thermal Distortion in Precision Measuring Instrument
Appropriate choice of material with known thermal expansion coefficient for base structure of precision
measuring instrument is critical in determining the accuracy performance. Physical dimension changes inaccordance with its linear and nonlinear characteristics of material when subjected to machining processes
and operating temperature variation. Compensation methods are able to minimize some negative
implications of thermal distortion [13]; however stable material thermal properties of base material would
contribute to improved reliability of an instrument.
Metal alloys material is commonly used in the instrumentation base structure, such as aluminium alloys,
tungsten alloys, beryllium alloys, silver alloys and copper alloys, and is depending on the required
performance level of a particular instrument. Among ceramic material, silicon carbide, tungsten carbide
and in particularly graphite is of relatively low thermal distortion [13].

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4.
4.1

RESULTS AND DISCUSSION
Digi-Pas® DWL3500XY Test Using Nanotech® 350FG, Ultra-Precision 3-Axis Freeform® Generator
Figure 12 shows the Nanotech® 350FG ultra-precision rotary machine with ultra-precision rotary axes
utilized to test the DWL3500XY. The programmable linear resolution is 0.01 nanometer and rotary angle
resolution of 0.0000001° (0.00036 arc sec) with built-in thermally insensitive linear scale feedback systemof 34 Pico-meter resolutions. The Nanotech® 350FG allows swing capacity to 500mm diameter for offaxis and toric components, box-way hydrostatic oil bearing slides with 300mm of travel on Z, 350mm oftravel on X, 150mm vertical travel on Y, and an adaptive air bearing counterbalance assembly on thevertical axis for optimal servo performance [14].

Figure 12 Nanotech® 350FG, Ultra-Precision 3-Axis Freeform® Generator

The DWL 3500XY has a specification of 0.000° to 2.000° is ±0.001°, while the angles within 2.000° to
20.000° is ±0.003°. Table 2 illustrates an incremental angle test generated by rotary table. The set angles
and recorded readings from the DUT were taken for both positive and negative slops from 0.000° to
20.000°. The recorded readings observed from DUT and the rotary table generated angles indicate thatnumeric deviation fall within the upper and lower limit of DUT specification.

Table 2 DUT readings for Rotary Table

4.2

Digi-Pas® DWL3500XY Test Using Micrometer Sine Bar
The DWL 3500XY is further tested using a micrometer sine bar for small angles range (i.e. 0.000° to
3.056°) as shown in Figure 13. Another independent single-axis digital level of manufacturer „L‟ was also
simultaneously placed on the sine bar for measuring comparison. Readings displayed on both DUT and
device L were recorded against the micrometer sine bar set angles and tabulated as shown in Figure 14.

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Figure 13 Setup Configuration of Sine Bar for DUT and Device L

The resolution of leveling instrument from manufacturer L is 0.0001°, which is 10 times more precise than
DWL3500XY that has a resolution of 0.001°. Further, precise small angle incremental readings were also
taken from 0.0000° to 0.2800° as shown in Figure 15. The observed readings of DUT fell within Upperand Lower Limits of ±0.001. The readings from instrument L were also in consistent with the micrometer
generated readings.

Figure 14 Accuracy Test Comparisons for Angle Measurement Range from 0.0000° to 3.0560°

Figure 15 Accuracy Test Comparisons for Small Angle Measurement Range from 0.0000° to 0.2800°

4.3

Digi-Pas® DWL3500XY Test Using Precision Fixed-Angle Jig
A third party independent Singapore National Metrology Center calibrated precision fixed-angle jigs [15] of
4.993° and 15.001° were further utilized to test DWL 3500XY. The DUT was placed on top of the jig
surface and displayed readings were taken and recorded as shown in Figure 16. The result illustrates thatthe measured values displayed by the DUT is within the specification of ±0.003° for both jigs as shown in
Figure 17 and 18.

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Figure 17 Measurement reading of DUT on 4.933° fixed angle jig

Figure 16 DUT readings for 4.993°and 15.001°
fixed angle jigs, within tolerance of ±0.003°

5.

Figure 18 Measurement reading of DUT on 15.001° fixed angle jig

CONCLUSION
To ensure the mid-range (i.e. 0.01° to 0.001°) precision digital leveling instrument performs in accordance
within the specified accuracy, a Digi-Pas® 2-axis precision digital leveling instrument model DWL3500XY
was tested by using rotary table and sine bar. The accuracy performance of DWL3500XY was further
laterally compared against an independent manufacturer „L‟ leveling instrument and also tested with a third
party independent National Metrology Center calibrated fixed-angle jigs.
This paper presents the literature review of a comparative study on using rotary table, sine bar and fixed
angle jigs to assess precision wide measuring range angles of digital leveling instrument. The results fromcomparative tests utilizing the foregoing test methods showed that the DWL3500XY performed consistently
within its accuracy specification. The findings provided substantive evidence to suggest that:
1.

2.

3.

4.

5.

Conventional rotary table is single axis and suffers from compounding errors such as gear backlash,multiple gears, eccentric error and swash error. Direct-drive servo motor employing optical technology
improves the test system uncertainty significantly.
The construction of Sine bar is constrained by numerous physical dimensions such as top-bottom
surfaces parallelism, inter-round bar parallelism, surface flatness, inter-round bar distance tolerance,roundness/eccentric error compound to yield a considerable amount of uncertainty in the test system.
This method is not suitable for testing wide-range angles of high precision angular instruments having
stringent accuracy specification particularly when the angles under test becoming larger.
Fixed angle jig minimizes those limitations in Sine Bar and confines the sources for errors to (i) topbottom surfaces flatness and (ii) the distance between top surface plane and bottom plane. Multiple
measurement points of each surface and distance between the surfaces utilizing least square fit could
further improve the test system uncertainty.
Rotary table are capable of generating a considerable number of incremental precise angles of 0.001°
for a full 360° angles range. In contrast, fixed angle jig offered limited number of angles to test the 2axis digital angular measuring instrument. All test methods described require to test the instrument one
axis at a time.
The constraints of the foregoing angular test equipment are addressed and dealt in the accompanying
JSB TECH technical paper [16], which tackles ultra-precision 2-axis angular leveling measurement of 1
arc sec or lower utilizing nano-technology dimensional and angular interferometer instruments.

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6.

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[2] New Way® Air Bearings, “Air Bearing Application and Design Guide”, Rev. D, Dec. 2003.
[3] West Wind® Air Bearings, “Air Bearing Technology”, Second Edition, Jan. 2007.
[4] B. Y. Kuu, H.H. Li, “The Implication of Swash Error to Calibration System Uncertainty when Testing High Accuracy Inclinometer
Utilizing MEMS Technology”, JSB TECH PTE LTD, Oct. 2010.
[5] NASA Infrared

Telescope Facility (IRTF), “Encoders”, Institute of Astronomy, University of Hawai‟i, Pg. 3-4, Jun. 2003.

[6] G. G. Antony, “Gear Precision – How Positioning Accuracy and Repeatability is Effected by A Gear Head”, Neugart USA LP, Nov.
2005.
[7] A. K. Bewoor, V. A. Kulkarni, “Straightness, Flatness, Squareness, Parallelism, Roundness, and Cylindricity Measurements” inMetrology & Measurement, New Dehhi: Tata McGraw-Hill Education Private Limited, Pg. 84, 2009.
[8] K. V. S. Rajeswara Rao, “Comparators and Angular Measurements”, Dept. of IEM, R. V. College of Engineering, Aug. 2013.
[9] Mitutoyo‟s General Catalog, “Measuring Instruments Catalog”, No.E2012, Pg. N-3, May 2012.
[10] H. Haijema, J. Meijer, “Evaluation of Surface Plate Flatness Measurement”, European Journal Mech. Eng., Vol. 38, Pg. 165-172,
July 1993.
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[14] Moore Nanotechnology System, LLC, “Moore Nanotech® 350FG, Ultra-Precision Freeform® Generator Machine Features andSpecification Overview”, Rev. 0812.
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Centre, Mar. 2012.
[16] B. Y. Kuu, H.H. Li, “Characterizing and Testing of Ultra Precision 2-axis Digital Level”, JSB TECH PTE LTD, May 2012.

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