Boeing Large Scale Optical Seminar, Seattle, 1998

Alignment and field check procedures for the Leica Laser Tracker LTD 500

Raimund Loser, Stephen Kyle Leica Geosystems AG, 5035 Unterentfelden, Switzerland

1. Abstract

Laser tracker systems are based on hi-tech instrumentation and have a wide range of industrial applications. In recent years a significant number of such systems have come into use and user’s expectations are growing. Along with technical improvements, system accuracy is one of the most important issues. Users can take control of a number of factors which influence system accuracy, in particular a software correction model which compensates for geometrical manufacturing tolerances through a set of alignment parameters.

This set of parameters can be determined by a few measurements which are part of an alignment procedure. The following paper explains optimal measurement procedures, the mathematics of the correction formulae and the influences of slightly different measurement setups. Especially where users must make alignment measurements in a confined or restricted area, knowledge about the effects and importance of these parameters is fundamental to the achievable accuracy.

In addition to basic system alignment, short field check measurements are described. These enable users to estimate system accuracy and improves their level of confidence in the results.

2. Introduction

The laser tracker combines different measurement technologies into one system. A prime result of the system accuracy requirement is that most of the individual components operate close to physical limitations. Unfortunately it is impossible to manufacture the tracker exactly according to design and the small deviations from design give rise to significant systematic errors. This is particularly relevant in the case of the mechanical arrangement of laser beams, mirrors, rotation axes, motors and encoders and without some form of compensation maximum accuracy cannot be attained.

A perfect design has some relatively simple alignment requirements. For example the laser beam should be parallel to the primary (standing) axis and the standing axis should intersect the transit axis at right angles. Deviations from such alignment requirements, caused by manufacturing tolerances, can be quantified by making suitable measurements and their negative effects on further measurements eliminated by software compensation. Corrected measurements then only show small random effects.

Leica has developed a correction model based on the physical alignment deviations which arise during tracker manufacture. The mathematical correction is described by 15 parameters which each correspond to some physical quantity. These are known as the alignment parameters of the tracker’s error model and the Axyz LTM software provides the functions to calculate them.

The Axyz LTM software also supports the necessary measurement functions to collect individual tracker data for the alignment process and subsequent parameter calculation. In addition to alignment computation, these functions can be used to monitor the adjustment of the laser tracker in the working environment through a field check. Users can therefore make field check measurements to assure themselves of the tracker's accuracy and improve confidence in the job results.

Finally, distances measured by the tracker's interferometer (IFM) or Absolute Distance Meter (ADM) must clearly be known to a high accuracy. For the IFM this requires the calculation of an accurately know distance to a fixed reference point on the tracker housing, known as the Home Point or Birdbath. For the ADM a datum position and scale factor are required to ensure compatibility with IFM distance values.

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3. Alignment and distance parameters and their mathematical correction

The 15 parameters of the main geometrical alignment model of the Leica laser tracker LTD500 can be grouped into three categories:

1. The first category handles 6 different offset parameters.

2. The second category includes 4 tilt deviations. 3. The third category has 4 encoder parameters of eccentricity and 1 vertical index offset.

For distance corrections the Birdbath distance represents the datum reference for the interferometer. A datum offset and scale factor are stored for the ADM.

The main alignment parameters represent deviations from an ideal position and so should ideally have zero values.

The Birdbath distance is the separation of reflector centre and tracking mirror centre and will depend on the physical size and manufacture of the particular reflector used. Typical values are around 150mm.

The ADM datum offset represents the separation of the ADM's internal zero position and the tracking mirror centre. The ADM is located about 700mm below the mirror and this is the approximate value of the offset.

Ideally there is no scale difference between ADM and IFM and so the multiplying scale factor should be approximately equal to 1.

The modelling described here mainly relates to the static case of locating a fixed reflector in 3D space. When the beam is stable and pointing at the centre of the reflector the readings from the angle encoders and interferometer or ADM should be "exact", i.e. in error only by very small random amounts. However the lack of alignment between the internal components will cause the measured angle and distance values to be in error. The objective of the modelling process is to apply corrections to the measured values which are functions of the main alignment parameters.

The main alignment model, Birdbath distance and ADM parameters are computed as three separate functions. Since the parameters of the main model can affect distances it may be helpful, if the current alignment model has changed significantly, to re-calculate distance parameters after the alignment procedure.

Obviously one of the unique features of the tracker, which is responsible for its extensive application and the reason for its name, is that it can track a moving reflector. When the reflector is moving the beam is not pointing directly at its centre. Tracking works by detecting the return beam from the reflector on a Position Sensing Device (PSD). When the outward beam is on target the return beam is on the zero servo point of the PSD. If the reflector moves laterally the return beam is offset and moves to a different point on the PSD. This shift on the PSD is used to drive the tracking mirror and bring the beam back on target. During dynamic tracking there is therefore always some small offset from the zero point on the PSD. This must also be taken into account when correcting the readings.

The following sections describe these features in more detail.

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3.1 Alignment offset parameters

Transit axis offset, e Mirror offset, f Beam offset, Cover plate offset, O1x and O1y O2x and O2y

The alignment offset parameters are defined by the following conditions: • The transit axis should intersect the primary (standing) axis. The transit axis offset “e” describes

the mathematical correction value. • The plane of the mirror should be on the rotation centre where the axis intersect. The correction

parameter is called mirror offset “f”. • The laser beam should lie exactly on the standing axis. “O1x” and “O1y” are the components of

the residual offset error. • Refraction at the cover plate may move the beam off the standing axis. This offset is described by

the two components “O2x” and “O2y”.

Note Although beam offset parameters (O1x, O1y) can be accommodated by the model, in practice they can be set to zero and are not normally applied. This is due to use of the collar reflector during tracker initialization. If the outward beam is physically offset from the standing axis, the collar reflector finds the diametrically opposite position for the return beam on the PSD. When this point is used as the zero servo point the average position of the outward and return beams is identical with the standing axis and this is the actual condition required for zero beam offset.

3.2 Alignment tilt parameters

View from top of the laser tracker head

Mirror tilt, c Transit axis tilt, i Beam axis tilt, Ix and Iy

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The alignment tilt parameters are defined by the following conditions: • The plane of the mirror should be parallel to the transit axis. The angle “c” describes the residual

mirror tilt. • The transit axis should be perpendicular to the primary (standing) axis. The angle "i" describes the

residual transit axis tilt “i”. • The laser beam should also be parallel to the primary (standing) axis. The residual beam tilt has

two components “Ix” and “Iy”.

3.3 Encoder eccentricities and vertical index offset

Ey Ex 180

Ky Kx

Horizontal encoder Vertical encoder Vertical index offset, j eccentricity, Ex, Ey eccentricity, Kx, Ky

Errors due to angle encoders are covered by the following parameters: • The encoder centre and its rotation centre should be the same point. The horizontal circle encoder

eccentricity values “Ex” and “Ey” are the components of this deviation. • The vertical circle encoder also has two eccentricity components. The corresponding eccentricity

values “Kx” and “Ky” used in the model are actually twice the physical offsets due to reflection off the mirror.

• The reading on the vertical circle should be 90° when the beam points horizontally. The vertical index offset “j” defines the angle difference between this value and the actual reading. The parameter value is also twice the physical offset due to reflection off the mirror.

3.4 Any other alignment parameters required?

All the current parameters appear useful and offer a sufficiently complete description of the instrument.

Additional parameters have been evaluated and rejected. These include: • Standing and transit axis wobble which are equivalent to combinations of index error, mirror tilt and

transit axis tilt. • Zero servo point location on the PSD which is equivalent to beam offset

3.5 2-face positions

2-face positions are particularly useful for determining alignment parameters. In order to point at a target reflector, the tracker head can be positioned in two ways, face left or face right. The term comes from theodolite usage where the face of the vertical circle is either on the right or left when the telescope is pointed at the target. (The positions may also be called "1 and 2" or "forward and reverse".) Between the two positions there is a horizontal angle difference of 180°. If the zenith (vertical) angle is V in the first face then it is (360° - V) in the second face.

The importance of 2-face positions is that some alignment parameters change sign from one position to the other. The errors they cause then also change sign, making it easy to detect the source. Over half the alignment parameters have this property. These are:

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• Transit axis tilt i • Mirror tilt c • Cover plate offset O2x and O2y • Transit axis offset e • Horizontal circle eccentricity, Ex and Ey • Vertical circle eccentricity Ky • Vertical index offset j

Note Although these parameters have the property of error reversal this does not imply they can be individually detected. This is true in some cases but in others combinations will operate together.

3.6 Birdbath distance

The interferometer is the device which makes tracking possible but it can only measure a change of distance, not an absolute distance. What is required is the absolute distance to the centre of a reflector from the centre of the tracking mirror. By starting interferometric measurement from a point whose distance is already known, interferometric changes can be converted to absolute distances. The Birdbath offers a convenient fixed location on the tracker housing which provides this reference location.

Note The Birdbath distance depends on the physical dimensions of the reflector and its location in the Birdbath itself. The distance is therefore different for different reflector types.

3.6.1 Simple concept for calculating Birdbath distance

To calculate an absolute distance using the IFM, 2 points separated by a known length L are required. A reflector is tracked between the points. This provides a measurement of the subtended angle a and range difference d. The value d is the difference between the approximate absolute distances to each point, based on an assumed distance to the Birdbath. (Each distance measurement will be in error by the error in Birdbath distance but their difference correctly gives d.)

The following calculations correct the absolute distance D to the first point. This can then be compared with the provisional measured value and the difference used to correct the Birdbath distance.

( ) ( ) ( )Applying the cosine rule:

D2 + + − + =D d D D d a L2 22 cos

( )( ) ( )( ) ( )( )

Expanding and collecting terms:

2 1 2 12 1

022 2

D a Dd ad L

a− + − +

−−

=cos coscos

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( )( )( )

Solution to quadratic equation taking the positive root:

Dd d

d La

=

− + −−

−2

2 221

2cos

The interferometric distance d is inherently very accurate. It can also be shown that the effect of uncertainties in the measured angle, a, can be eliminated when a = 180°.

At optimal subtended angle:

Dd L

=− +

2

3.7 ADM offset and scale factor

Although the ADM can measure absolute distance to a fixed reflector at an unknown location, the datum position for this distance must be established. The measuring unit itself has an unknown position within the tracker housing and the components within the unit have an unknown datum position. What is required is to set a total datum offset so that absolute distance measurement is zero at the centre of the tracking mirror. A scale factor is also required to ensure complete compatibility with the IFM.

Because of its higher accuracy the IFM is used as a reference for the ADM. The technique compares several ADM distances with IFM distances to the same fixed points and calculates an instrument offset and scale factor to make the ADM values match the IFM values as closely as possible. The technique therefore requires accurate IFM values, i.e. the Birdbath distance must first be determined.

ADM measurements are affected by the material of the reflector, for example the glass in a tooling ball reflector. For this reason a hollow (air path) corner cube reflector must be used to determine the ADM parameters since this has no effect on the distance measurement. That in turn means that the Birdbath distance must first be accurately determined using the selected hollow corner cube.

Note In normal use, whenever an ADM measurement is made it is effectively used to reset the IFM distance and the actual distance is then recorded as an IFM measurement.

3.8 Additional internal parameters

In addition to alignment and distance parameters which users can calculate for themselves, depending on the geometrical state of the tracker or the environmental conditions, a number of other internal parameters are also required.

These internal parameters control the servo loop, motor speed and acceleration, position detection of the return beam on the PSD, as well as a number of ADM functions, in particular relating to correct frequency values. The optimal setting of these internal parameters is done at the factory or service centres or they are detected and set during the tracker's initialization procedure, prior to measurement. There is no need for direct modification by the user.

3.9 Alignment correction formulae The following formulae show corrections due to individual alignment parameters, together with the total correction for the measured value. Some re-arrangement of terms has been used in the total correction for programming convenience.

The vertical circle readings (Vm) indicated in the formulae need some clarification.

Readings of the vertical circle are often called vertical angles. In some industries vertical angles are measured positive up (angle of elevation) or negative down (angle of depression) from a horizontal pointing. In this case the zero value implies a horizontal pointing and the range of vertical angles is ±180°.

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It is common on Leica instruments for the zero pointing on the vertical circle to point up, i.e. towards the zenith. Angles in the vertical plane are then measured positive down and on through to 360°, i.e. they take only positive values in the range 0° - 360°. In this case a horizontal pointing is represented by the value 90° or 270°. These may also be called "vertical angles" but to avoid ambiguity the terms zenith angle or zenith distance are also used.

3.9.1 Formula to correct the measured horizontal angle

Correction due to transit axis tilt i:

−⎛⎝⎜

⎞⎠⎟i

Vmtan

2

Correction due to mirror tilt c:

−⎛⎝⎜

⎞⎠⎟

cVm

cos2

Correction due to beam tilt, Ix and Iy:

( ) ( )( )

Ix Hm Iy HmVm

cos sinsin

−

Correction due to beam offset, O1x and O1y:

( ) ( )( )

O x Hm O y HmDm Vm

1 1cos sinsin−

Correction due to cover plate offset O2x:

( )O x

Dm Vm2

sin

Correction due to horizontal encoder eccentricity, Ex and Ey:

( ) ( )Ey Hm Ex Hmsin cos−

Corrected horizontal angle:

( ) ( ) ( )( ) ( )

( ) ( )

Hcorr HmVm

Ix Hm Iy HmO x Hm O y Hm O x Hoff

Dm

iVm

c

Vm Ey Hm Ex Hm

= + − +− + +⎡

⎣⎢

⎤

⎦⎥

−

⎛⎝⎜

⎞⎠⎟ +

⎛⎝⎜

⎞⎠⎟

+ −

1 1 1

2

2

sincos sin

cos sin

sin

cossin cos

2

Note The additional term Hoff comes from the internally corrected PSD measurement and provides a further correction when measuring moving reflectors.

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3.9.2 Formula to correct the measured vertical angle

Correction due to beam tilt, Ix and Iy:

( ) ( )[ ]− +Ix Hm Iy Hmsin cos

Correction due to beam offset, O1x and O1y:

( ) ( )−

+O x Hm O y HmDm

1 1sin cos

Correction due to cover plate offset O2y:

−O yDm

2

Correction due to transit axis offset e:

−⎛⎝⎜

⎞⎠⎟

⎡

⎣⎢

⎤

⎦⎥

22

2eDm

Vmcos

Correction due to mirror offset f:

−⎛⎝⎜

⎞⎠⎟

⎡

⎣⎢

⎤

⎦⎥

22

fDm

Vmcos

Correction due to vertical encoder eccentricity, Kx and Ky:

−⎛⎝⎜

⎞⎠⎟ +

⎛⎝⎜

⎞⎠⎟Kx

VmKy

Vmcos sin

2 2

Corrected vertical (zenith) angle:

( ) ( )[ ] ( ) ( )Vcorr Vm Ix Hm Iy Hm

O x Hm O y Hm O y VoffDm

VmDm

eVm

f Kx KyVm

= − + −+ +

−⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟ +

⎛⎝⎜

⎞⎠⎟+

⎡

⎣⎢

⎤

⎦⎥ +

⎛⎝⎜

⎞⎠⎟

sin cossin cos

cos cos sin

1 1 2

22

2 2

+

Notes The vertical index offset j is directly applied when reading the angle, i.e. the measured vertical angle Vm is already corrected for index error. The additional term Voff comes from the internally corrected PSD measurement and provides a further correction when measuring moving reflectors.

3.9.3 Formula to correct the measured distance

Correction due to transit axis offset e:

( )− e Vmsin

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Correction due to mirror offset f:

−⎛⎝⎜

⎞⎠⎟2

2f

Vmsin

Corrected distance:

Dcorr Dm Vm e Vm f= − ⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟+

⎛⎝⎜

⎞⎠⎟

22 2

sin cos

4. Measurement procedures to determine alignment and distance parameters

Axyz LTM application software allows users to make specific measurements to determine the above alignment and distance parameters in order to optimize measurement accuracy.

4.1 Alignment measurements

To calculate the 15 main instrument alignment parameters, a set of 2-face measurements to single points, together with a set of Ball Bar measurements, is required for processing by the optimizing routine.

The advantage of 2-face measurements has been mentioned earlier. They emphasize the error effects of over half the parameters.

The Ball Bar is a precisely manufactured bar which rotates a reflector around an accurate circle in an approximately vertical plane. It therefore provides an exact reference shape for controlling the alignment of the tracker's components.

4.1.1 2-face measurements

2-face measurements at near and far points:

The requirement here is to make 2-face measurements to a near point A and a far point B. Both points lie on line with the tracker, approximately at tracker height.

Recommendations for near and far points:

Near point A ideally < 0.4m

Far point B ideally > 5m

Zenith angles approximately 90°

Horizontal angle difference between A and B should be either zero degrees or 180°.

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2-face measurements on a vertical line:

2-face measurements on a vertical line should cover the whole range of vertical angle measurement and ensure that a range of distances is incorporated.

Recommendations for vertical line points:

At a range of 1.4m ideally establish 7 reflector positions in the range -40° to +40° with approximately 10° between each position.

The positions should be measured at 3 tracker ranges, approximately 1.4m, 2.0m and 2.5m. It is not necessary to measure all points at all ranges, as indicated in the diagram.

2-face measurements in the horizontal plane:

The requirement here is a number of evenly spaced 2-face measurements to roughly equidistant points at approximately tracker height.

Recommendations for horizontal plane points:

For good results, 16 evenly spaced measurements are recommended. They should be at approximately the same horizontal distance, somewhere between 1.5m and 2.5m.

If space is restricted:

Try rotating the tracker rather than moving the reflector.

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4.1.2 Ball Bar measurements

Recommendations:

Ideally the Ball Bar should be measured in 8 positions. 4 positions should be close to the tracker (1m or less) and 4 positions at between 3m and 5m from the tracker.

Positions should be roughly equally spaced. The Ball Bar centres then appear roughly at 45° intervals in the horizontal plane, as indicated in the diagram.

Ball Bar centres should be approximately at tracker height.

Ball Bar plane should be vertical and facing the tracker to avoid oblique viewing

The Ball Bar locations should be alternately measured in face 1 and 2, as indicated.

Use a slow rotation speed.

Record points all around the circle.

If space is restricted:

Try to get at least 2 positions at different ranges with alternate face measurement.

It is also permissible to move the tracker rather than the Ball Bar. Rotating the tracker, for example, will place the Ball Bar in different measurement quadrants. This can also help to create a good distribution of positions in a confined test area.

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4.2 2-point method for Birdbath measurement

The 2-point method implements the simple concept of measuring two points with a known separation L by using 2 free-standing, unknown points.

The distance L between the points is first measured by placing the tracker close to the extended line of the points and using it almost as a pure interferometer to determine their separation L. The interferometric distance d is then almost equal to the separation L. Any existing errors in the Birdbath distance, or in the subtended angle (a), have only a small effect on the calculation of L which is therefore very accurate.

The tracker is then moved between the points such that the subtended angle (A) is almost 180°. This angle then accurately measured.

With this information the Bird Bath distance can be accurately calculated. The actual equations used in calculating absolute distance take full account of the geometry of the setup and are not a one-to-one implementation of the equations shown earlier for the simple concept.

Recommendations:

Alignment points at approximately tracker height.

In offset position 1 keep the subtended angle within 5° of zero.

In central position 2 keep the subtended angle within 5° of 180°.

The points can be close to the tracker.

4.3 Measurements for ADM parameters

These measurements must be made with a hollow corner cube reflector. The Birdbath distance for the selected reflector must first be calculated, if not already known to a good accuracy.

The idea is to measure several fixed reflector locations with both the interferometer (IFM) and ADM. Normally an ADM measurement simply re-sets the IFM distance and results in an IFM recording but a special recording mode preserves both sets of data.

Recommendations:

An absolute minimum of 2 test points is required for a solution and a minimum of 5 is recommended.

The points should be equally spaced out approximately on a line at tracker height, starting at a minimum distance of around 3.5m and ideally going up to the maximum range around 30m.

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4.4 Field checks (on-site or workshop checks)

When a measurement is made in face 2 it is converted into an equivalent face 1 measurement. If the tracker has good alignment parameters then pointings to a fixed reflector in both faces should be equal, apart from small random deviations.

When a reflector on a slowly rotating Ball Bar is tracked, the resulting set of points should fit very well to a circle.

Field checks (on-site or workshop checks) simply test the above conditions and should tell you if the system is producing good results. If 2-face checks start to show large angular differences or the Ball Bar test indicates a bad circle, then it may be time to re-calculate the parameters.

However it is still possible to get good measurement results even if there are, say, relatively large differences between 2-face measurements. This is particularly the case for differential measurements made within a localized area. For example the separation between two points will still be accurately measured if the errors are very similar at both points. An extreme example is the measurement of point separation along the laser beam. In this case the tracker functions almost as a pure interferometer and the result is very accurate. Even lateral point separations may be just as good when calculated with inaccurate alignment parameters.

Field checks can also be made for ADM parameters and the Birdbath distance.

Field check measurements are essentially the same as measurements made for alignment purposes. These are: • 2-face measurements • Ball Bar measurements • Measurements to calculate Birdbath distance • Measurements to calculate ADM parameters The differences with the same measurements made for alignment purposes are that field checks: • Do not need to be so comprehensive and well distributed • Do not enable you to re-define parameters

Although field checks do not permit users to re-define parameters they should indicate if a new alignment is necessary. Some recommendations follow.

4.4.1 Is the 2-face check OK?

Ideally the 2σ values in the instrument specification are:

Horizontal angle ±13cc

Vertical angle ±13cc

This means that typical values for differences should be not more than twice the above values. If several values are significantly higher a full alignment is recommended.

4.4.2 Is the Ball Bar check OK?

Ideally the 2σ values of the offsets from the best fitting circles, as stated in the instrument specification are:

At 1m: ±30μ (±0.0012 inch)

At 3m: ±90μ (±0.0035 inch)

The results show the maximum deviation from the best fitting circle which should be not more than twice the above values. If it is significantly higher a full alignment is recommended.

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4.4.3 Is the ADM check OK?

Ideally the 2σ value of the difference between an IFM and ADM values, as stated in the instrument specification is ±50μ (±0.002 inch).

If it is significantly higher the ADM parameters should be re-calculated.

4.4.4 Is the Birdbath distance OK?

Ideally the difference between the new check value and the existing value should be in the region of 10μ (±0.0004 inch) (2 σ value).

If it is significantly higher a new Birdbath calculation is required.