Laser tracker error determination using a network ...

Laser tracker error determination using a network measurement

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Laser tracker error determination using a network measurement Ben Hughes, Alistair Forbes, Andrew Lewis, Wenjuan Sun, Dan Veal, and Karim Nasr National Physical Laboratory, Hampton Road, Teddington, Middlesex, TW11 0LW, UK

E-mail: [email protected]

Abstract. We report on a fast, easily implemented method to determine all the geometrical alignment errors of a laser tracker, to high precision. The technique requires no specialist equipment and can be performed in under an hour. The technique is based on the determination of parameters of a geometric model of the laser tracker, using measurements of a set of fixed target locations, from multiple locations of the tracker. After fitting of the model parameters to the observed data, the model can be used to perform error correction of the raw laser tracker data or to derive correction parameters in the format of the tracker manufacturer’s internal error map. In addition to determination of the model parameters, the method also determines the uncertainties and correlations associated with the parameters. We have tested the technique on a commercial laser tracker in the following way. We disabled the tracker’s internal error compensation, and used a five position, fifteen target network to estimate all the geometric errors of the instrument. Using the error map generated from this network test, the tracker was able to pass a full performance validation test, conducted according to a recognised specification standard (ASME B89.4.19-2006). We conclude that the error correction determined from the network test is as effective as the manufacturer’s own error correction methodologies. PACS: 07.60.Ly, 42.62.-b, 06.20.-f, 42.62.Eh, 07.05.Hd, 07.05.Tp Keywords: laser trackers, uncertainty, large volume metrology, geometric error, interferometer, absolute distance meter, network test, error mapping, mathematical modelling, coordinate metrology, performance verification. (Some figures in this article are in colour only in the electronic version).

1. Introduction Since the invention of the laser in the 1960s, and the re-definition of the metre in terms of the speed of light [1] in 1983, laser-based measuring systems have revolutionised dimensional metrology. Until the 1990s, most laser metrology tools operated along a fixed axis, making measurements of displacement, or a combination of displacement and rotation(s). The invention of the tracking laser interferometer [2], or laser tracker, combined the displacement measuring accuracy of a laser interferometer with the three-dimensional angular measuring capability of a theodolite or totalstation, to produce a measuring instrument capable of high accuracy, long range 3-D metrology.

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Functionally, a laser tracker is a portable coordinate measuring system that tracks a moving target reflector and measures the position of the target in spherical coordinates (d,). The radial distance, or range component, d, is typically measured by an interferometer (IFM) or an absolute distance meter (ADM). The IFM or ADM laser beam is steered to track the moving target, usually a spherically mounted retro-reflector (SMR), by a motorised gimbal mechanism. Angle encoders on the mechanism provide azimuth, , and elevation, , angles to the target. After correction of d for refractive index of the ambient air [3] using measured values of the air temperature, pressure and assumed humidity, the spherical coordinates of the target are reported to the user, either directly as (d, ), or after transformation to a Cartesian coordinate system (x, y, z). Additionally, control/analysis software will correct the target location for various offsets and alignment features of the target (e.g. radius of the SMR) to give the resulting point of contact with the item being measured. Some laser trackers are also fitted with a material temperature sensor that allows post hoc correction of measurement results, to a standard temperature, often 20 °C [4]. The beam steering mechanism and angular encoders within the laser tracker are subject to misalignments, offsets, non-linearities and eccentricities that lead to errors in the measured coordinates. For this reason, all laser tracker manufacturers provide online correction of these systematic effects, usually by software running in the laser tracker control system. The correction software relies on a model that describes the beam steering mechanism and its errors. The parameters of the model are usually derived from a combination of disparate calibrations performed at the factory during manufacture and simple procedures performed by the user on the shop floor, prior to using the instrument. The trade-off between time and complexity means that end-user procedures are not able to capture all the errors in the tracking mechanism simultaneously. Several of the error parameters, particularly those relating to angular scale errors, are considered fixed at the time of manufacture and cannot be updated by the end-user. We propose a simple measurement procedure that is able to determine all the alignment and angle encoder errors of a laser tracker and their corresponding uncertainties. The procedure requires no specialist equipment and can be performed by an end-user in less than an hour. The procedure is based on obtaining data from the laser tracker during a series of repeated measurements of a network of fixed target locations. The repeated measurements are made with the tracker in different positions. After data acquisition, a mathematical error model of the tracker is fitted to the observations, in order to determine the parameters of the model. The model is designed to represent the basic geometric errors of the tracker mechanism and of the angular scales. To obtain the most accurate data, the measurements are made in IFM mode.

2. Geometric errors of a laser tracker An idealised laser tracker is shown schematically in figure 1. The two rotation axes – azimuthal axis (standing axis, output and the elevation axis (transit axis, output ) are orthogonal and intersect at a point, taken to be the nominal origin of the spherical coordinate system. After passing through some form of launching or delivery mechanism, the laser beam emerges as if emitted from the point of intersection of the two axes and the beam direction is orthogonal to the transit axis. The

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two angular encoders are mounted coaxially with the mechanical rotation axes, one per axis, and each encoder’s scale is uniform in pitch and radially aligned with the rotation centre.

Standing axis

Azimuth encoder



Transit axis

Reflector

Laser beam (0, 0, 0) Elevation encoder

140



Figure 1 - Schematic representation of an idealised laser tracker.

Standing axis

Azimuth encoder

Transit axis

aA,1, bA,1, aA,2, bA,2



Reflector

Laser beam ex, by,0, bz,0 , 

Elevation encoder

aE,1, bE,1, aE,2, bE,2

Figure 2 - Schematic representation of a laser tracker with geometric errors – see Table 1.

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In a real laser tracker, illustrated schematically in figure 2, imperfections such as misalignments, scale eccentricities and axis offsets are present (see Table 1), which can lead to errors in the measured angles and distances. If uncorrected, these errors will give rise to systematic errors in the measured coordinates of the reflector target. In practice, the observed sensor readings are adjusted by the tracker software/firmware controller, to correct for the known misalignment errors (the errors being determined during manufacture or shop floor calibration). The correction adjusts the measured parameters d, and  to give best estimates of the true parameters, d*, and .

3. Laser tracker performance testing and error modelling In order to give confidence to users that laser trackers are operating correctly, specification standards have been developed that are aimed at testing the performance of laser trackers with a view to identifying faults, the effect of miss-calibration or poor correction of systematic errors. The ASME B89.4.19 standard [5] prescribes a number of tests of laser tracker performance including an extensive set of length measurements and two-face1 repeatability tests. More recently, the draft VDI/VDE 2716 [6] and draft ISO 10360 part 10 [7] documents propose a similar set of tests. These standards are compared by Loser [8] who describes a method of realising reference lengths suitable for the prescribed tests. These specification standards test the length measurement capability of the tracker and compare it with the manufacturer’s maximum permissible error (MPE), giving a ‘pass/fail’ indication. The tests do not attempt to quantify instrument accuracy. Recognising this limitation, Sandwith and Lott [9] have proposed a test based on the Unified Spatial Metrology Network (USMN) feature within the third party control and analysis software package SpatialAnalyzer [10]. Their test uses measurements of a network of fixed points from a number of different instrument locations and bundles the data together to calculate the overall uncertainty of the individual sensor readings – range and two angles. This test gives a single uncertainty value for each sensor. Before one can correct sensor readings for mechanical offsets within the mechanism, it is necessary to understand how these offsets and alignments affect the individual sensor readings. Loser and Kyle [11] have described the misalignments for an older laser tracker design that uses a gimballed mirror for beam-steering. The fifteen parameters of their model are broadly categorized as offsets, tilt deviations and angle encoder parameters. Loser and Kyle derived formulae to correct the observed sensor readings for the influence of these error parameters and describe experimental setups – five in total – that can be deployed to measure the error parameters. More recently, Muralikrishnan et al [12] at the National Institute of Standards and Technology (NIST) have modified the Loser and Kyle error model to reflect a slightly different, more common, mechanical arrangement within current tracker mechanisms. They modelled a tracker in which the laser beam source is inside the rotating head - this design does not require use of a gimballed mirror. Muralikrishnan et al also derived the sensitivity of the length measurement and two-face tests,

1

The term ‘two-face’ refers to two measurements of the same target point – one with the tracker gimbals oriented conventionally (‘front face’), the second with the standing axis rotated 180 ° and the transit axis rotated to re-target the same point (‘back face’). During the movement between the two orientations, the tracker rotates and inverts the laser head or targeting mirror.

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prescribed in the ASME B89.4.19 standard, to the individual error parameters, and showed that some parameters are not detectable by these tests. Furthermore, although tracker manufacturers provide users with software-assisted methods for determining some of the parameters and updating of the internal error models, users are not typically provided with methods for determining the errors associated with the angle encoders, so there have been several different approaches proposed by users to overcome this problem. For example, Gassner and Ruland [13] developed a method for calibration of the azimuth angle encoder based on back-to-back calibration against a rotary table. The rotary table was calibrated using a polygon and an autocollimator. For practical reasons, this approach is not suited to elevation angle calibration. Martin et al [14] developed calibration instruments for both the azimuth and elevation angles of robotic total stations and laser trackers, based on precision rotary tables and a precision slideway. More recently Muralikrishnan et al have developed a method [15] of calibrating the low-order harmonic errors of the azimuth encoder using an un-calibrated, but stable length artefact. They demonstrate a standard uncertainty of 0.4 µrad using this technique, which is an extension of their earlier error model [12]. Unfortunately, these tests require access to specialist equipment and only provide information on a few error parameters at a one time.

4. The new model We now describe a variant of the NIST model and a simple test that can be used to determine all the error parameters described in our model, including the angle encoder errors. The method involves fitting a mathematical model describing the setup and the errors of the instrument to measurements of a network of fixed points. The key features of the technique are that no specialist hardware is required; the IFM (if fitted) can be used for improved accuracy; all parameters are determined simultaneously; random noise effects are averaged over many samples; and the uncertainties in the measured error parameters are calculated using statistically rigorous methods. The NPL error model [16] is similar to that proposed in the NIST paper [12], and to first order the models behave in the same way. However, there are some differences. The NPL model uses no linearisations so that the nonlinear effects are treated without significant approximation. In particular, the NPL model does not become singular for near vertical beam directions, whereas the NIST model involves terms such as 1/ sin(Vm ) that become singular for vertical beams corresponding to Vm = 0. The angle encoder errors are dealt with completely separately from the alignment errors and we use an explicit Fourier representation of the encoder error terms that is applicable to the latest designs of laser trackers (with laser source mounted on the gimbal axes). Our model caters equally well with data obtained from both front and back-face measurements and this feature is exploited to determine the geometric errors to a greater sensitivity. Before describing the observation equations, it is necessary to define our model coordinate system. Our geometric model relates the geometry of three axes: the azimuth axis of rotation A (the standing axis), the elevation angle rotation axis E (the transit axis) and the laser beam axis B. We assume that A is aligned with the Cartesian z-axis. This alignment sets four frame of reference constraints, leaving two degrees of freedom, namely rotation about the z-axis and height along the

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z-axis. We assume also that when =  = 0, the axis E lies in a plane parallel to the yz-plane, thereby fixing the rotation about the z-axis, and also that it intersects the x-axis, thus fixing the height along the z-axis. We further assume that when =  = 0, B lies in a plane parallel to the xyplane and that the beam source is at b0, nominally at the origin. With these assumptions, for =  = 0, the standing axis A is given by the location point (0,0,0)T and direction vector (0,0,0)T, and the transit axis E is given by location point e0 and direction vector nE,0 with

e x ,0  0   0  e 0 =  0 , n E ,0 = R x ( ) 1   1   0  0  

(1)

and B is given by location point b0 and direction vector nB,0 given by

bx ,0  1 1    b 0 = b y ,0 , n B ,0 = R z ( ) 0    bz ,0  0  0 

(2)

where we use short form notations Rx(), Ry() and Rz() for the Cartesian plane rotation matrices

0 1  R x    0 cos  0 sin 

  cos    sin   , R y     0  sin  cos   0

0 sin   cos   1 0  , R z     sin   0 0 cos  

 sin  cos  0

0 0 1

The parameters  and  represent geometric alignment offset errors of the laser tracker gimbals –  represents the angle of the transit axis in the yz-plane and  represents the beam axis angle in the xy-plane.

Laser tracker error determination using a network measurement

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7

z B

B





x

x z

y E

bz x

ex y

B

B x

z



E



by x

y

Figure 3 - Offset parameters of the error model – see Table 1. We assume that the observed values d, and  for each target point are related to their true values d*, and  according to the model equations (3) to (5).

d  1   d *    eD

(3)

   * a A,0   a A,q cos q *  bA,q sin q *   e A

(4)

   * a E ,0   a E ,q cos q *  bE ,q sin q *   eE

(5)

nQ

q 1 nQ

q 1

Here, represents a scale correction (e.g. for refractive index),  represents a displacement offset (e.g. laser dead-path), q represents the harmonic order of the Fourier series (maximum nQ), which is used to represent the angle encoder errors such as eccentricity and scale. The parameters eD, eA and eE are samples from a statistical distribution with expectation value zero and standard deviations D, A and E, respectively (i.e. they represent random noise). Parameter aA,0 is associated with fixing a frame of reference and is constrained to be zero. Noting that the x-component of b0, bx,0, can be compensated by changing the offset parameter , we constrain bx,0 instead to be zero. Examination of (1) through (5), shows that the remaining error model parameter terms which are non-zero are ex,0, by,0, bz,0, , , aE,0 and the other 4nQ Fourier terms. If we set nQ = 2, then the number of error parameters is 14, which is the same as for the NIST model. Equations (3–5) model how observations d, and are generated given the true values d*, and

. The model is completed by describing d*, and  in terms of the position of the tracker, the locations of the targets and the alignment and encoder errors. We first consider the rotations of the

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tracker axes required to point the laser at any particular target. At azimuth angle , the transit axis E() is moved to a position specified by the transformed locating point and direction vector

e = Rz ( )e0 , n E = Rz ( )n E ,0 *

*

0  = Rz ( ) Rx ( ) 1 0 *

(6)

The rotation matrix RE that corresponds to a rotation through angle  about the transit axis  E = ( ) is given by

RE = Rz ( * ) Rx ( ) Ry ( * ) Rx ( ) Rz ( * )

(7)

Therefore, if a point z is rotated by  about the axis E, it is moved to

zˆ = e  RE (z  e) Applying this transformation to Rz()b0 , the laser beam source b0 is rotated by  and  to



b( * ,  * ) = e  RE Rz ( * )b 0  e



b( * ,  * ) = Rz ( * )e0  Rz ( * ) Rx ( ) Ry ( * ) Rx ( )(b 0  e0 )

(8)

and the beam direction vector nB,0 is rotated to

n B ( * , * ) = RE Rz ( * )n B,0 1 n B ( * , * ) = Rz ( * ) Rx ( ) Ry ( * ) Rx ( ) Rz ( ) 0 0

(9)

Equations (8) and (9) give the physical origin and pointing direction of the laser beam, based on true values of the azimuth and elevation angles ( and ) and the beam and transit axis alignment offsets ( and ). Combining equations (3, 4, 5, 8, 9) and letting h be the vector of (6 + 4nQ) geometric error model parameters, then for any target location xi, we obtain







xi  b i , i , h  di n B i , i , h *

*

*

*

*



(10)

There are two solutions of equation 10: if (,) are solution parameters, then there is also a solution close to (+,  -). These two solutions correspond to the front-face and back-face measurement modes of the tracker. Equation (10) defines d*, , and  as functions of x and of the geometric error parameters in the vector h. Iterative techniques can therefore be used to solve (10) for d*, , and , given starting estimates derived from nominal tracker geometry, such as ex,0 =  =  = 0, b0 = (0,0,0)T. Table 1 summarises the individual parameters, hi, contained in h, as well as the additional parameters  and

 which determine d*.

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Table 1 - Error parameters of the model and their physical origin. i -

Parameter Description range offset 

-



1 2 3 4

ex by,0 bz,0

5

 

6 7 8 9 10 11 12 13 14

aA,1 bA,1 aA,2 bA,2 aE,0 aE,1 bE,1 aE,2 bE,2

Type distance

scale factor for range

value

transit (E) axis offset from standing (A) axis beam offset (y-direction) from origin beam offset (z-direction) from origin transit axis angle in yz-plane

distance distance distance angle

beam axis angle in xy-plane

angle

azimuth scale error, first order azimuth scale error, first order azimuth scale error, second order azimuth scale error, second order elevation angle offset elevation scale error, first order elevation scale error, first order elevation scale error, second order elevation scale error, second order

angle angle angle angle angle angle angle angle angle

5. Tracker calibration using the new model The error model parameters can be estimated from the measurement of a fixed set of targets xj, j = 1, …, nX from a number of tracker positions determined by locations pk and initial pointing directions k, with k = 1, …, nS. If di, i, and i are measurements associated with the jth target and the kth tracker position, then d*i, i, and i are determined by the multi-station form of equation (10)

R(α k ) R0 (x j  p k ) = b(i* , i* )  di*n B (i* , i* )

(11)

The observations di, i, and i obey the multi-station model equations

di = (1  r )di* (x j , p k , α k )  l  eD,i

(12)

nQ

i =  (x j , p k , α k )  a A,q cos qi  bA,q sin qi  eAi *

(13)

q =1

and nQ

i =  (x j , p k , α k )  aE ,0  aE ,q cos qi  bE ,q sin qi  eEi *

q =1

(14)

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where the random effects are associated with standard deviations  D,i ,  A,i and  E,i , respectively. We let c be the vector of all the configuration parameters pk, k, l r and h, and set

~ di (x j , c) = (1  r )di* (x j , p k , α k )  l

(15)

nQ

i (x j , c) =  (x j , p k , α k )  a A,q cos qi  bA,q sin qi  ~

*

(16)

q =1

and nQ

i (x j , c) =  (x j , p k , α k )  aE ,0  aE ,q cos qi  bE ,q sin qi . ~

*

(17)

q =1

Setting relative weights wD,i = 1/ D,i , etc, estimates of targets locations xj and the configuration parameters, including h, are found by solving a nonlinear least squares problem

min



~

~

~

2 (d  d (x , c))2  w2 (   (x , c))2  w2 (   (x , c))2  wD i j i j i j ,i i E ,i i A,i i

{x j },c i

 (Bc  c0 )T (Bc  c0 )

(18) subject to frame of reference constraints applied to c [17, 18, 19]. The term Bc – c0 accounts for any additional prior information about the configuration parameters, e.g., estimates of r, the scale corrections. In equations (15–17), the observations appear linearly in each equation so that the solution of equation (18) represents the maximum likelihood estimate of the parameters. Uncertainties associated with the observations can be propagated through to those associated with the fitted parameters {x j } and c [17]. The least squares solution determined in the network test produces estimates, h, of the error model parameters and a matrix Vh of the variances. These error parameter estimates along with  and  can be used to correct target estimates in the subsequent use of the tracker. As mentioned earlier, we bundle h with the other error parameters  and  into a configuration parameter vector, c = (,

h with its own associated variance matrix u 2 ( )    2 Vc =  u ( )    V h 

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If di, i, and i are associated with target xi , then they must also satisfy (10). We therefore regard ~ di as an estimate of d i with uncertainty D,i and treat the angle measurements similarly. We estimate d*i using

1 dˆi = (d i   ), 1  and i from nQ





ˆi = i   a A,q cos qi  bA,q sin qi , q =1

and i from nQ





ˆi = i  aE ,0   aE ,q cos qi  bE ,q sin qi . q =1

This allows us to estimate xi using xˆ i = b(ˆi ,ˆi , h)  dˆi n B (ˆi ,ˆi , h).

We can then propagate the uncertainties associated with the configuration parameters c, and the measurements di, i, and i, to obtain the variance matrix, Vi, associated with xˆ i [20, 21, 22].

6. Experimental realisation of the network In order to generate data for the least squares solution, we set up a network of target locations and tracker positions. To minimise the cost and complexity, we designed the network to use features and structures commonly available in user premises, namely the floor, two walls (intersecting at a corner) and a stable bench. Ideally the walls should be made of solid, (short-term) stable material (e.g. concrete, brick). Our laboratory had fibreboard walls, so we chose instead to fix the wallmounted targets to concrete pillars (which support the fibreboard walls) and to an artificially constructed frame, taking the place of one of the walls. To support the laser tracker, we used a commonly available wheeled tripod with rotating head and vertical lift capability, a magnetic base, and a trivet floor stand. We anticipate that most laser tracker users will have ready access to similar structures and mounting hardware. The relative locations of the tracker and targets were carefully planned in order to ensure that a) the whole angular range of the tracker was covered, in both azimuth and elevation, b) the target to tracker range varied from 0.27 m to 6.4 m, and c) that two ‘birdbath2 calibration’ lines of sight existed, one horizontal, and one on a vertical diagonal. This last requirement is designed to address one of the issues associated with laser tracker calibration, whereby the birdbath distance () and transit axis offset (ex) can be strongly correlated so that during parameter fitting their combined 2

‘Birdbath’ is a commonly used term which refers to the reference target location that is mounted on a small platform on the side of the laser tracker. The absolute distance from the laser beam origin to this point is a calibration parameter, , of the laser tracker error model.

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effect is well determined but the two individual values are poorly determined. A birdbath calibration [11] is usually performed using two target holders, arranged such that the tracker can measure the target-to-target separation from two locations - outside and inside, on the line joining the two targets (external and internal measurements). For the internal measurement, the tracker has to rotate in azimuth by 180°, whereas during the external measurement the azimuth angle remains fixed - the difference between the two results obtained is due to a combination of  and ex. By adding a second line-of sight measurement, but orientated at a different angle to the vertical (ideally around 45°), the correlation can be broken because  and ex have differing dependences on elevation angle. A simple way of achieving this layout in the target network is to fix one target holder to a wall, high up, and another to the edge of the bench. The tracker can then be placed on the trivet stand on the floor on the external line of sight between the two target holders, and then placed on the taller tripod, so that it is on the internal line of sight, as shown in figure 4. The remainder of the holders should be fixed to the walls, floor and bench, distributed throughout the available volume. The operator is free to choose the overall total number of nests, but a minimum of ten is recommended and the network will take too long to measure if there are too many. Laser tracker external position

Target 1

Laser tracker internal position

Target 2

Figure 4 -Measuring the birdbath error using a diagonal line of sight between two targets, set approximately 2 m apart. We found that an efficient experimental design consists of fifteen target holders with five tracker locations, according to the layout in figures 5 and 6. Tracker locations 0 and 1 are co-located, on top of the stand (position 1 is the same as position 0, but rotated), position 2 is on the stand, raised to full height, position 3 is on the bench and position 4 is on the trivet stand on the floor. Tracker location 0 is used to set the overall coordinate frame reference.

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Figure 5 - Network layout - elevation view. Blue diamonds represent target locations.

Figure 6 - Network layout - plan view. Tracker positions 0 and 1 are co-located. Our network used eight targets holders glued to the walls, one to the floor and six glued to the bench, at different heights. Two birdbath lines of sight were included: targets 6 & 8 and 9 & 10, viewed from tracker positions 0 & 3 and 0 & 4, respectively.

7. Demonstration using a commercial laser tracker In order to demonstrate the effectiveness of our technique, we used it to perform a full error determination of a commercial laser tracker (API model T3), using the supplied 0.5 inch SMR. We

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temporarily disabled the parameters of the tracker’s error map. Using the tracker’s interferometer, we performed the IFM volumetric tests prescribed in standard ASME B89.4.19 [5]. These tests took two hours to complete. The results are shown graphically in figure 7, which shows deviation from calibrated artefact length, for a series of measurements of the artefact, using several locations and orientations of both the tracker and the artefact. Length measurement errors of up to 1.9 mm can be observed which are far in excess of the 18 µm to 68 µm MPE values specified by the manufacturer for a properly operating tracker. These length measurement errors show the importance of calibrating the geometric errors of the tracker and compensating for these using a parametric model.

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-0.800

-1.000

Error / mm

-1.200

-1.400

-1.600

-1.800

-2.000 Any 0° 90° 180°270° 0° 90° 180°270° 0° 90° 180°270° 0° 90° 180°270° 0° 90° 180°270° 0° 90° 180°270° 0° 90° 180°270° 0° 90° 180°270° H 1m

H

V

RD

LD

H

3m

V

RD

LD

6m

Configuration

Figure 7 - B89 volumetric test results with tracker error map disabled. Configuration parameters are as prescribed in ASME B89.4.19 and relate to three tracker-to-test artefact distances (1 m, 3 m, 6 m) and four azimuth rotations of the tracker (0° to 270°). The orientation of the artefact (horizontal, vertical, and the left and right diagonals) is also indicated. Three measurements are made in each configuration (red, green blue). Next, we used the un-compensated tracker to measure our target network. Each target location was measured from all five tracker locations, excluding targets with obscured line of sight. Each target was measured in front sight and back sight mode using the IFM. The total measurement sequence took 55 minutes and 126 measurements were made in total (out of a maximum possible of 15 x 5 x 2 = 150). Figures 8 and 9 show the locations of the targets relative to the tracker coordinate system origin, with each target colour representing data obtained from one of the five different tracker locations.

Figure 8 - Elevation view of relative position of targets with respect to tracker origin.

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Figure 9 - Plan view of relative position of targets with respect to tracker origin. We used the data obtained from the measurements to solve equation (18) and obtained the model parameters and their uncertainties. These are reported in tables 2 and 3.

Table 2 - Determined error parameters – distances, scale factor, and their uncertainties. Symbol

  ex by,0 bz,0

Value -2.467 6 mm 1.000 4.988 µm 91.201 µm -13.318 µm

Uncertainty (k = 1) 0.403 µm 5  10-7 0.122 µm 0.654 µm 0.974 µm

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Table 3 - Determined error parameters - angles and their uncertainties. Symbol

 

Value / seconds of arc 16.751

aA,1 bA,1 aA,2 bA,2 aE,0 aE,1 bE,1 aE,2 bE,2

Uncertainty (k = 1) / seconds of arc 0.128

9.647

0.079

0.526 -0.562 0.177 1.530 0.154 -1.736 -0.677 0.375 0.437

0.064 0.080 0.073 0.090 0.223 0.152 0.183 0.214 0.179

The model solution also provides calculated values for the standard deviations ( D, A ,E) included in the model equations (3, 4, 5). We use a process of maximising a marginalised posterior distribution to perform a posteriori re-weighting of the displacement measurements relative to the angle measurements. This allows us to calculate a posteriori values for the noise parameters associated with the distance and angle sensors [22]. These are shown in table 4. Table 4 - A posteriori standard deviations of noise parameters associated with the sensors. Standard deviation D,A

Value 1.216 µm

D,R

0.312 seconds of arc

A,A

2.351 µm

A,R

0.485 seconds of arc

E,A

3.365 µm

E,R

0.694 seconds of arc

The model solution also gives uncertainties for the target locations (these ranged from 2.1µm to 4.9 µm magnitude); tracker location uncertainties (1.1 µm to 2.2 µm magnitude); and tracker pointing direction uncertainties (0.33 to 0.45 seconds of arc) – all quoted as standard uncertainties. We can also examine any correlations between error parameters by computing the Cholesky factor Lh = {ljk} of the variance matrix Vh = LhLhT. The off-diagonal elements of this matrix indicate the level of correlation between pairs of parameters. We found that the only significant correlations are between  and by, between aE,0 and bz and between aE,2 and aE,0. The first correlated pair relates to the location of the beam axis B in the plane parallel to the xy-plane. The second pair relates to the location of the beam axis in the xz-plane. The third correlation arises because the elevation angles,

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i, never exceed 57º from the horizontal, so that, for the total set of data, the mean of cos(2 i) is 0.69. The tracker is mechanically limited to operate over elevation angles from –60° to +77°, so only a small improvement on the achieved elevation angle range could be obtained by placing a target nest high up, close to one of the tracker locations. After calculation of the parameters, we then used our error model to perform post hoc correction of the measurements which were previously reported in figure 7. This is equivalent to operating the laser tracker using the NPL error model with values of the error parameters determined from the network test. The results are plotted in figure 10.

0.100

0.080

0.060

Error / mm

0.040

0.020

0.000

-0.020

-0.040

-0.060

-0.080

-0.100 Any H 1m

0°

90° 180° 270° 0° H

90° 180° 270° 0° V

90° 180° 270° 0° RD

90° 180° 270° 0°

90° 180° 270° 0°

LD

3m

H

90° 180° 270° 0° V

90° 180° 270° 0° RD

90° 180° 270° LD

6m

Configuration

Figure 10 - B89 volumetric test results from figure 7, post hoc corrected using NPL error model and parameters derived from the network measurement. Bold lines show the manufacturer supplied MPE values. The errors have been significantly reduced and the instrument now passes this performance verification test. For comparison, the manufacturer’s standard error map was then re-instated and the artefact re-measured using this error map. The results, plotted in figure 11 show a similar level of performance.

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0.100

0.080

0.060

Error / mm

0.040

0.020

0.000

-0.020

-0.040

-0.060

-0.080

-0.100 Any

0°

H 1m

90° 180° 270° 0° H

90° 180° 270° 0° V

90° 180° 270° 0° RD

90° 180° 270° 0°

90° 180° 270° 0°

LD

3m

H

90° 180° 270° 0° V

90° 180° 270° 0° RD

90° 180° 270° LD

6m

Configuration

Figure 11 - B89 tests using manufacturer’s error map. Bold lines show the manufacturer specified MPE values. To understand how the uncertainties associated with the estimated error parameters h contribute to uncertainties in target locations, we performed a simulation of the measurement of 1000 points, between 1 m and 5 m from the tracker, at random azimuth and elevation angles, including front and back face measurements. After propagation of the parameter uncertainties, we calculated the RMS variance contribution from each influence factor and compared this with the overall target variance. Overall, we found that approximately 80 % of the variation is explained by the random effects associated with the angle measurements, and that the parameters, h, of our model contribute the remaining 20 % (between 0.1 % and 3.8 %, individually for each parameter). Thus, we are confident that our model correctly accounts for the major geometrical misalignments of the laser tracker.

8. Discussion The traceability of the measurements performed using our technique relies on either using a calibrated length standard as part of the network, or by performing prior calibration of the laser wavelength of the IFM, and verification of the IFM performance over the range of distances encountered to the targets in the network. We chose the latter alternative and calibrated the laser wavelength against a primary reference laser (used as a primary realisation of the metre) followed by back-to-back verification of the IFM against a calibrated laser interferometer. It should be noted that these two alternative traceability routes are already prescribed in the ASME B89.4.19 standard, so they should already be available to laser tracker users. Figures 10 and 11 show two effects which contribute to the overall performance of an errorcorrected laser tracker. Firstly, any trend in the data, from one configuration setting to the next

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represents residual errors which are not being corrected for by the error model. Secondly, scatter between the three measurements performed at each configuration indicates further random noise which is present. We tested the repeatability of placing the SMR into a fixed nest and observed around 4 µm variation in reported position. Observation of the target coordinates displayed by the tracker software whilst pointing at a stationary target showed that a typical lateral distribution of around 40 µm width could be seen at a 2.5 m range, whilst the width along the beam direction was around 4 µm. This is quite normal as the IFM range error is much smaller than the angular error, for all laser trackers. Shielding of the air path between the tracker and the target reduced the 40 µm width to around 23 µm. As the tracker to target distance is increased during the ASME B89 tests, the laser beam becomes more perpendicular to the artefact, and the lateral direction becomes more aligned with the measurand. This leads to the lateral variability in apparent target location playing a proportionally larger role in the error of the measured artefact length. This is evident in figures 10 and 11, for the tests performed at 6 m target-to-artefact separation. The measurement results reported on the extreme left of these figures involve the laser tracker almost in contact with the artefact and effect of the angular uncertainties is almost zero, as the beam is aligned with the measurand direction. The scatter in figures 10 and 11 is therefore explainable as a mixture of beam bending due to the atmosphere, angular sensor accuracy, and repeatability of the target nests. Our network test can be performed in under one hour and provides quantitative details of the laser tracker errors (gimbal offsets and mis-alignments and scale errors) and can be used to predict the performance of the tracker in a verification test, such as that preformed according to ASME B89.4.19. The IFM volumetric tests reported in figures 7 and 11 take around two hours to perform and they determine only the magnitude of length measurement errors (and two-face repeatability) of the laser tracker. Our technique can be used in several ways. Firstly, it could be used as an alternative to the performance verification techniques specified in standards such as ASME B89.4.19. Secondly, it could be used as a means to determine the residual errors of a laser tracker, after the manufacturer specified error compensation techniques have been applied. In this way, it could form part of a quality assurance programme, independent of the laser tracker manufacturer. Thirdly, with close cooperation with the tracker manufacturer, it should be possible to either implement the NPL error map directly in the tracker controller, or to provide a means of taking the NPL error map parameters, and converting the into the format used in the manufacturer’s error map, and then updating the error map in the controller. We note that the model allows realistic uncertainties associated with target coordinates that take into account the uncertainties associated with the alignment errors to be evaluated.

9. Conclusions We have described a new model of laser tracker geometric errors and shown how the parameters of the model can be determined, to high precision, using a simple series of repeated measurements of

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a set of fixed target locations, in under one hour. The model parameters are traceable via the laser wavelength and the uncertainties of the parameters are determined using rigorous methods. The model can either be used in place of the internal manufacturer-supplied error model, or used to correct for residual errors remaining after application of the manufacturer specified calibration routines. Our model determines not only the offsets and displacements of the gimbal axes, but also errors associated with the angular scales. It may be used as a simple technique for fast performance verification of a laser tracker, and integrated into a quality assurance programme aimed at monitoring laser tracker performance in between annual or biennial servicing. Our model shows that improvement to laser tracker performance is most easily achieved through improvement to the angular measurements of the target locations. The model described in this paper is applicable to most of the currently available models of laser tracker [23, 24, 25], and also to total stations, which have the laser mounted directly on the rotating head. The model does not apply to the older design of tracker [26], which differs mainly from newer designs by having the laser mounted within the tracker body and the laser beam reflected off a gimballed mirror. With some modifications, the current model could be adapted to cover the older tracker design. Acknowledgements The authors would like to thank Dr Quan Ma of Automated Precision Inc (API) for very useful discussion on the error model and practical aspects of the experimental processes described in this paper. We acknowledge funding received from the New Industrial Metrology Techniques (NIMTech) iMERA Plus project of the European Metrology research Programme (grant number 217257) and from the UK National Measurement System Programme for Engineering & Flow Metrology (2008-2011) References [1]

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[2]

Lau K C, Hocken R J 1987 Three and five axis laser tracking systems United States Patent 4714339

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Birch K P, Downs M J 1994 Correction to the updated Edlén equation for the refractive index of air Metrologia 31 315-316

[4]

ISO 1 2002 Geometrical Product Specifications (GPS) – Standard reference temperature for geometrical product specification and verification. (Geneva, Switzerland: International Organization for Standardization)

[5]

ASME B89.4.19 2006 Performance Evaluation of Laser-Based Spherical Coordinate Measurement Systems (New York, USA: The American Society of Mechanical Engineers)

[6]

VDI/VDE 2617 part 10 2009 (draft) Accuracy of coordinate measuring machines: Characteristic parameters and their checking: Acceptance and reverification tests of lasertrackers (Düsseldorf, Germany: Verein Deutscher Ingenieure)

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[7]

ISO/NP 10360-10 2010 Geometrical Product Specifications (GPS) – Acceptance and reverification tests for coordinate measuring systems (CMS) – Part 10: Laser Trackers for measuring point-to-point distances (Geneva, Switzerland: International Organization for Standardization - new work item proposal)

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Loser R 2009 Laser Tracker Accuracy Certification for Large-Volume Measurements CMSC Journal Autumn 2009, 14-19

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Sandwith S, Lott R, Large Volume Metrology Conference, LVMC 2007, Chester, UK (http://www.lvmc.org.uk) (visited 15 February 2011)

[10] SpatialAnalyzer is a traceable metrology, 3D graphical software platform from New River Kinematics, that is in common use in large volume metrology applications (http://www.kinematics.com) (visited 15 February 2011) [11] Loser R, Kyle S 1999 Alignment and field check procedures for the Leica Laser Tracker LTD 500 Boeing Large Scale Optical Metrology Seminar [12] Muralikrishnan B, Sawyer D, Blackburn C, Philips S, Borchardt B, Estler W T 2009 AMSE B89.4.19 Performance Evaluation Tests and Geometric Misalignments in Laser Trackers J. Res. Natl. Inst. Stand. Technol. 114, 21-35 [13] Gassner G, Ruland R 2009 Laser Tracker Calibration – Testing the angle Measurement System 15th International Geodatische Woche, Obergurgl, Austria. SLAC-PUB-13476 [14] Martin D, Chetwynd D G 2009 Angle calibration of robotic total stations and laser trackers Proc. XIX IMEKO World Conference (Lisbon, Portugal) 1893 – 1898 [15] Muralikrishnan B, Blackburn C, Sawyer D, Phillips S, Bridges R 2010 Measuring Scale Errors in a Laser Tracker’s Horizontal Angle Encoder Through Simple Length Measurement and Two-Face System Tests J. Res. Natl. Inst. Stand. Technol. 115, 291-301 [16] Hughes B, Sun W, Forbes A, Lewis A 2010 Determining laser tracker alignment errors using a network measurement CMSC Journal Autumn 2010, 26-32 [17] Forbes A B 2009 Parameter estimation based on least squares methods Data Modelling for Metrology and Testing in Measurement Science, Pavese F, Forbes A B eds., (BirkhauserBoston, New York) 147–176 [18] Forbes A B, Harris P M 2005 Uncertainty associated with co-ordinate measurements Laser Metrology and Machine Performance VII Shore P, ed., (Euspen, Bedford) 30–39 [19] Peggs G N, Maropoulos P G, Hughes E B, Forbes A B, Robson S, Ziebart M, Muralikrishnan B 2009 Recent developments in large-scale dimensional metrology J. Eng. Manuf., B, 223 571–595 [20] Forbes A B 2002 Efficient algorithms for structured self-calibration problems Algorithms for Approximation IV Levesley J, Anderson I, Mason J C, eds., (University of Huddersfield) 146–153 [21] Barker R M, Cox M G, Forbes A B, Harris P M, 2007 Modelling discrete data and experimental data analysis National Physical Laboratory Report DEM ES 018, (Teddington)

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[22] Forbes A B, Hughes E B, Sun W 2010 Weighting observations from multi-sensor co-ordinate measuring systems MathMet 2010 (PTB, Berlin) [23] http://www.apisensor.com/tracker3-usa (visited 15 February 2011) [24] http://www.faro.com/lasertracker/home (visited 15 February 2011) [25] http://www.hexagonmetrology.co.uk/leica-absolute-tracker_283.htm (visited 15 February 2011) [26] http://www.hexagonmetrology.co.uk/leica-absolute-tracker-at401_955.htm February 2011)

(visited

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