An SI-traceable multilateration coordinate measurement system with half the uncertainty of a laser tracker

We have validated the performance of a prototype coordinate measurement system based on multilateration by comparing it to a laser tracker, i.e. a well-proven instrument widely used in the industry. After establishing the uncertainty budget of the different systems, we performed position measurements with both instruments on common targets. Using the estimated uncertainties associated with the measurements, we found that the multilateration system provided lower position uncertainties than the laser tracker: on average 18 µm versus 33 µm for distances up to 12 m. The uncertainties represented by confidence ellipsoids are compatible between the two systems: for confidence regions of 95% probability, they overlap as expected, i.e. in 94% of the cases. We also measured the length of a 0.8 m long reference scale bar with the multilateration system at an error of only 2 µm. This cross-comparison is a new and key step in the characterization of this SI-traceable multilateration system.


Introduction
Large volume metrology, i.e. the metrology for the dimensional measurements of large structures of several meters in size, covers a large number of industrial and research applications [1]. As examples, we cite assembly tolerances of aircraft wings that must be less than 0.3 mm [2] and the alignment requirements for particle accelerators, which are on the * Author to whom any correspondence should be addressed.
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order of a few tens of micrometers over distances of a hundred meters [3]. Thus, to produce, assemble, or characterize these large parts and systems, accurate coordinate measuring systems such as large coordinate measuring machines, laser trackers, or multilateration systems [1,4] are required.
In this context, a multilateration system with metrological traceability to the realization of the SI meter has been developed by Cnam [5][6][7]. This system has already been used for the reference point determination of a radio telescope at the geodetic observatory of Wettzell [8] as well as for the measurement of the movements of a six-axis industrial robot [9]. The purpose of the current study is to validate the performances of this multilateration system through comparisons with external references: a laser tracker and a scale bar. The comparison in this case means to measure the same points in space with two different systems, quantify the differences between these two measurements, and check whether the uncertainties of the two instruments are compatible. These cross-comparisons are a new and key step for the characterization of this prototype of multilateration system, a crucial step that was not realized until now.
This paper is organized as follows. The sections from 2 to 4 present, successively, the multilateration system, the laser tracker, and the common target used for the comparison. In section 5, the uncertainty budgets of the two coordinate measurement systems are established. After that, the experimental setup used for the comparison is described in section 6. The workshop where the measurements took place is thus presented and the position of the targets are defined. In section 7, the experimental protocol of the comparison is detailed. Then, the experimental results are presented in section 8: the performances of the two instruments are evaluated and the measured positions and their uncertainties are compared. Lastly, in section 9, the multilateration system performance is evaluated using a scale bar.

The multilateration system
Multilateration is a technique for determining the position of a target T in space by measuring the distances between this target and known positions, corresponding in our case to measurement heads. From a geometric point of view, the position of T can be defined as the intersection of spheres centered at the measurement heads and of radii equal to the measured distances. In practice, the distances are determined by an absolute distance meter (ADM) and they are affected by zero-mean additive Gaussian noises. The coordinates of T are therefore rather determined by minimizing the following nonlinear residual function: (1) where ∥.∥ denotes the Euclidian norm of a vector and d i (x T ,y T ,z T ) are the distances measured between the m measurement heads H i and T. In practice, to solve the multilateration problem of formula (1), the number of heads should be at least equal to 4. The variables o i are the instrument offsets of the heads, i.e. additive constants corresponding to the distances between the electro-optic origin of the ADM and the location of the heads.
The multilateration system characterized in this study adopts a self-calibration method 3 . This means that the coordinates of the measurement heads H i and their instrument offset o i do not need to be known since they are determined automatically by the algorithm. This is possible as soon as more than nine targets are measured, which results in a system of equations with more observations than unknowns. In that case, formula (2) shows how the coordinates of the heads H i and targets T j are estimated by non-linear least squares: with m ⩾ 4 for the measurement heads and n ⩾ 10 for the targets.
In summary, a multilateration system is a coordinate measurement system using only distance information for determination of three-dimensional positions. However, such a system is complex to implement since it requires the installation of several measurement heads. In our case, the system, presented in figure 1, is composed of a common ADM and of four measurement heads.
The principle of our in-house developed ADM is based on the measurement of the phase shift φ of a light modulated in intensity by a radio-frequency (RF) carrier f RF at 5 GHz and propagated up to a retroreflective target. The measured distance d, proportional to the phase accumulated during this round trip, is therefore equal to: where c is the speed of light in vacuum, n air the air group refractive index through which the optical beam is propagated, and N an integer number corresponding to the number of synthetic wavelengths Λ = c/(n air × f RF ) within the distance to be measured. In practice, N is determined by a set of measurements at different RF carriers. One of the features of this ADM is its fiber-optic design based on optoelectronics components at 1550 nm, components from the telecommunication world, affordable and available in large quantities. The telemetric signal can thus be distributed through a fiber network before being emitted in free space in direction of a target by one of the measurement heads. For instance, the signal coming from the common ADM is shared between the four heads thanks to an optical switch. However, this approach also presents disadvantages since the four distance measurements to be performed for a given target are realized one after the other, in a time multiplex, which increases the measuring time and limits the measurements to static targets.
In this system, the measurement heads play both the role of a collimating system to propagate the optical beam over several tens of meters, and of an aiming system thanks to gimbal mechanisms for rotation of the laser beam in every direction of space around an invariant point as described in [5]. This multilateration system has been studied with a consistent metrological approach in [6]. For a distance measurement, the uncertainty contribution of the telemetric system is 2.1 µm (for a coverage factor k = 1, i.e. a confidence level of 68%) and that of a measurement head is 1.4 µm (k = 1). The latter is due to misalignments in its gimbal mechanism. This leads at the end to an uncertainty of 2.5 µm (k = 1). This value, validated by a comparison with an interferometric bench up to 35 m [7], does not depend on the measured distance.
This system can be used with two kinds of retroreflectors: hollow corner cube reflectors (CCR) [6] suitable for longdistance measurements (successfully tested up to 140 m), or glass spheres of refractive index 2 [7] (for distances up to 20 m). While the spheres induce high optical losses due to lower reflectivity and beam deflection at their output, they may be preferred since they offer a visibility from any angle 4 and are light weight.

The laser tracker
A laser tracker determines the position of a target in a spherical coordinate system by measuring the distance and the angles to the target. The distance to a target can be measured by an interferometer (IFM) or an ADM. The two orthogonal angles, the azimuth θ and the elevation ϕ, are measured by two angular encoders mounted along the mechanical axes of the gimbal mechanism of the laser tracker.
The laser trackers are a well-proven technology. In this study, a Leica AT960-LR from Hexagon and provided by RISE was used. Its performance as described in [10] on the basis of the ASME B89.4.19 standard, depends on two different contributions: the distance accuracy, ±0.4 µm + 0.3 µm m −1 for the IFM mode or ±10 µm for the ADM one, and the angular accuracy, ±15 µm + 6 µm m −1 . These specifications are reported as maximum permissible errors (MPE), but they can also be expressed as typical values by taking half the MPE as result. In this paper, it has been considered that these typical values are the uncertainties for a coverage factor k = 1.
Expressed as a single number, the uncertainty σ pos on a target position is calculated as follows 5 : For instance, at a distance of 5 m, this corresponds to a MPE of 64 µm, or 32 µm when expressed as an uncertainty at k = 1. The main limitation comes from the angle measurements, so at the end, there is not a big difference in terms of uncertainty between the IFM and ADM modes. However, the IFM mode is capable of high-speed measurements and generally preferred for continuous measurements of dynamic objects, while the ADM mode is more practical for static measurements since it allows the beam to be broken, and so interruptions in a series of measurements. Thus, in the case of this comparison between the multilateration system and the laser tracker, the ADM mode has been adopted.
Contrary to a multilateration system, a laser tracker does not need to measure a large number of targets before determining their positions. The coordinates of a target are known directly, without the application of an onerous algorithm. However, the laser trackers are sensitive to temperature gradients, a situation that can be seen outdoors or in harsh industrial environments such as a workshop without air conditioning. These gradients can induce beam deflections, and so errors in the angle measurements of the laser tracker, which can lead in some cases to position errors of several tens of micrometers as reported in [11,12]. Thus, for certain operations, the multilateration technique is more suitable as the length measurements are less susceptible to beam refraction than are the angular measurements: the knowledge of the angles is no longer necessary, only the distance data are required.
The Leica AT960-LR is compatible with two kinds of retroreflectors, hollow CCRs or cat eyes. The CCRs allow measurements up to 80 m for this laser tracker, while the cat eyes are suitable up to 18 m with an acceptance angle for the laser beam of ±75 degrees.

The common target
To perform the comparison between the multilateration system and the reference laser tracker, a common target has been used. This avoids adding new sources of errors related to the geometric corrections required to compare two distinct targets.
The common target, depicted in figure 1 and characterized in [5], is a hollow corner cube with aluminum coating. It is mounted on a motorized gimbal mechanism developed by Cnam in order to be able to orientate itself towards any coordinate measurement instrument. The mechanical imperfections related to the different angular orientations of this rotating reflector induces errors in the measured distances, which are characterized by a Gaussian distribution of standard deviation 3.9 µm (k = 1). This uncertainty contribution, σ reflector , should be included in the uncertainty budget of the distances measured by the ADM of the laser tracker and the ADM of the multilateration system.
For the comparison between the two coordinate measurement instruments, several target positions have been tested. In practice, a single target was used and moved from one position to another. Each position is related to a Leica tribrach, i.e. an adapter able to host a target on a Leica carrier.

Uncertainty budget of both systems
The table 1 shows a summary of the uncertainties of the two coordinate measurement systems, some of which have been presented in the previous sections.
In table 1, n air , which allows to deduce a mechanical distance from an optical path, is also taken into account. The latter, usually calculated using the semi-empirical Edlén's equation [13] or similar updated formulas like Bönsch and Potulski [14], depends on the air temperature, the atmospheric pressure, the partial pressure of water vapor, and the CO 2 content. The main contributions come from the air temperature with a factor of −0.95 µm/(m × • C) and from the atmospheric pressure with a factor of 0.27 µm/(m × hPa). In practice, these environmental parameters are measured using local sensors. It has to be noted that the air refractive index also depends on the vacuum optical wavelength, 795 nm for the laser tracker (ADM mode) and 1550 nm for the multilateration system, but this parameter does not have a large impact compared to the environmental contributions reported in table 2.
Assuming that the impact of the air refractive index can be neglected, and knowing that the uncertainties related to the ADM and the common target are independent Gaussiandistributed contributions, the uncertainty on a distance measurement is equal to 6.3 µm for the laser tracker and 4.7 µm for the multilateration system (k = 1).
As shown in table 1, it is easy to determine the uncertainty on a target position measured by the laser tracker: it is equal to the square root of the sum of the variances of three independent sources of error. On the contrary, this is more complex for a target position measured by the multilateration system since this partly depends on the algorithm that allows to move from distance data to position values. In this process, the geometry of the system, i.e. the positions of the targets relatively to the four measurement heads, plays an important role on the uncertainties. For instance, the final uncertainty on a measured position, when expressed as a single number (see below in formula (5)), is at least 1.5 times the uncertainty of the measured distances for an ideal arrangement [15].
The uncertainties on the positions determined by multilateration have been assessed as performed in [16]. In summary, once the coordinates H i and T j have been estimated by the multilateration algorithm with self-calibration, each head is processed independently to evaluate their position uncertainties. For this purpose, the partial derivatives of the distances ∥ H i -T j ∥ with respect to the coordinates of a studied head are calculated at the coordinates of that head. The resulting Jacobian matrices, one per head, describe how small changes in the coordinates of a head will change the distances derived from that position. It is then possible to find the covariance matrices of the head positions by a simple calculation as explained in [17]. With this first approximation, the multilateration problem becomes one where the head positions are known, but affected by zero-mean additive Gaussian noises of known covariance. Such a problem is then solved by applying the solution presented in [18]. In this way, the covariance matrix of each target position T j is obtained.
From these 3 × 3 covariance matrices, the uncertainty of each target is expressed by a single number as follows: where cov(T j ) is the covariance matrix of the target position j, and Trace is an operator that returns the sum of the diagonal elements of the covariance matrix. The mathematical solution for the uncertainty assessment of the positions has been validated in [16] thanks to comparisons with experimental measurements and Monte-Carlo simulations. However, the proposed approach considered measurement head and target positions determined by self-calibration with instrument offsets known. In our case the instrument offsets are determined by self-calibration, which means that no analytical solution is available. Therefore, the estimated uncertainties are probably optimistic, i.e. slightly lower than the reality. The comparison performed in this paper will therefore quantify this point.
After the establishment of the uncertainty budget, the next section focuses on the workshop where the comparison between the two instruments took place in order to define the positions of the targets relative to the instruments, but also to detail the environmental conditions of the experiment.
Depends on the target positions relative to the head positions

Experimental setup
The comparison between the multilateration system and the laser tracker was carried out on 1 February 2022 between 12:45 pm and 3:30 pm, i.e. for 2 h 45 min. The workshop where the experiment took place is depicted in figure 2: this was in a large volume of about 6.3 m (x) × 10.3 m (y) × 3.1 m (z), a room without air conditioning, but located in a basement where temperature variations remain small. For instance, and as shown in figure 3, the temperature, as measured by our most accurate sensor, changed by less than 0.5 • C during the comparison. However, from one sensor to another, i.e. from different locations in the room, we have observed concurrent temperature differences of up to 0.4 • C. The CO 2 content, not depicted in figure 3, was also recorded. It was equal to 530 ppm ± 80 ppm. The laser tracker has its own sensor for air refractive index compensation. It has recorded temperatures increasing from 18.7 • C to 18.9 • C, relative humidities increasing from 48.5% to 50.7%, and pressures decreasing from 1018.3 hPa to 1017.0 hPa. The differences between these environmental parameters and those used by the multilateration system are relatively small. The one that has the most impact is the pressure, with discrepancy up to 1.5 hPa. Such an error on the atmospheric pressure induces an error of 4.5 µm for a measured distance of 10 m.
In the end, it seems relevant to consider a length dependent uncertainty σ n of 0.6 µm m −1 for the distances measured with the laser tracker as well as those of the multilateration system. This corresponds to environmental uncertainties of 0.4 • C for the temperature, 1.5 hPa for the pressure, 5% for the humidity, and 80 ppm for the CO 2 content.
In this large volume, the measurement heads of the multilateration system have been placed as depicted in figures 2 and 4. This arrangement is quite far from an optimal configuration. Indeed, in a multilateration system, the uncertainties on the target coordinates depend on their positions relative to the locations of the measurement heads. For instance, to minimize the measurement errors, a solution consists in setting up the targets at the center of a regular tetrahedron as explained in [15]. Thus, in an ideal case, the measurement head H B would have been located at twice its height.
The ideal position of the laser tracker would have been in the middle of the working volume, but we have opted for a position outside this space to avoid any beam occlusion for the multilateration system. Thus, the laser tracker is located at the following coordinates, in meter: As shown in figures 2 and 4, 18 different target positions have been defined across the room. Most of them have been placed inside the volume formed by the four heads, but less favorable configurations for the uncertainties of the multilateration system have also been tested. For instance, the target position T 3 is located 4 m behind the measurement head H D . At the end, the distances measured the laser tracker are up 9.5 m, and those measured by the multilateration system are up to 11.6 m. The related uncertainties on the measured distances are thus between 6.4 µm and 8.5 µm for the laser tracker and between 4.7 µm and 8.4 µm the multilateration system.

Experimental protocol of the laser tracker comparison
The measurement protocol for the comparison of the two coordinate measurement systems comprised the following steps: 1. The target was placed in its initial position, on a leveled tribrach. 2. The four heads of the multilateration system started to aim at the target. 3. The retroreflector of the target was oriented in the direction of the head H A , then the ADM of the multilateration system measured the distance between this head and the target. 4.
Step 3 was repeated for the measurement heads H B , H C and H D . 5. The retroreflector was oriented in the direction of the laser tracker, and vice versa, then the laser tracker measured the distance and the angles that separated it from the target. 6. The target was moved into a new position. 7. Steps 2-6 were repeated for the 17 remaining target positions.
As explained above, the retroreflector was oriented in different directions at each position to aim at the different measurement heads of the multilateration system as well as the laser tracker. These changes of orientation of the retroreflector broke the optical beam of the laser tracker, which is why the ADM mode of the laser tracker was preferred for the distance measurements as explained in section 3.
Besides, the coordinates of a target measured by the laser tracker was the average of three successive position measurements, each of them being the average of two-face measurements. A two-face measurement consists in measuring a position from the front face of the laser tracker, then measuring it from its back face, which is achieved by changing the azimuth angle by 180 • , then adjusting the elevation angle. In the end, the difference between three successive position measurements for a given coordinate can be up to 24 µm with an RMS value of 8 µm for all the targets. Lastly, a warm-up time of 4 h was applied to the laser tracker before the first measurement.

The multilateration system
The coordinates of the 18 targets, as determined by the multilateration system, are obtained after applying a multilateration algorithm with self-calibration, which consists in processing the 4 × 18 measured distances to minimize the residual function in formula (2). The differences between the distances measured by the telemetric system and those calculated from the obtained coordinates were used to quantify the quality of the multilateration results in figure 5.
These errors are lower than 4.3 µm with a global standard deviation of only 1.5 µm. The multilateration algorithm has therefore perfectly converged. This also shows that the assessed uncertainties between 4.7 µm and 8.4 µm may be too pessimistic.

The laser tracker
The measurement heads were installed on ductile cast iron pipes fixed to the ground to guarantee a high stability, whereas the laser tracker was installed on a tripod (see figure 2). The pipes were installed several months before the measurement campaign, and it is assumed that their position does not to drift during the few hours of the measurement. On the contrary, the tripod of the laser tracker was installed on the day of the measurement and may move over time. In order to monitor the laser tracker stability during the experiment, two additional CCRs were mounted close to the ground and measured every hour. Their position, visible in figures 2 and 4, are assumed to be fixed. Thus, as shown by the curves in figure 6, which represents the distances that separate the laser tracker from these reference CCRs, the laser tracker moves over time, up to 30 µm for CCR 1 during the comparison of 2 h and 45 min. Such a drift is problematic for the comparison since it is of the same order of magnitude as the assessed uncertainties.
The coordinates of these two reference CCRs were studied in detail to explain these distance drifts. We have identified a continuous rotation of the laser tracker around its z-axis of 0.82 µrad per hour, and a translation of the laser tracker of few micrometers per hour. The latter is visible by correcting only the continuous rotation, as shown in figure 7. In these curves, after fitting by regression lines, the residuals have standard deviations along the x, y and z axes of, respectively, 2.9 µm, 1.7 µm and 3.5 µm.
In practice, since the positions of the reference CCRs are assumed to be fixed, the coordinates of all the targets measured by the laser tracker have been corrected. However, no additional uncertainty was considered for these corrections.

The position uncertainties
The position uncertainties σ pos of the 18 targets, noted σ LT and σ MLT to differentiate the laser tracker from the multilateration system, are presented in figure 8 in the form of single numbers. Those determined by the multilateration system are always lower than those measured by the laser tracker, with an average value of 18 µm versus 33 µm. For the positions measured by the laser tracker, their uncertainties increase with the measured distances by a factor of 4.2 µm m −1 . Thus, T 3 located at 9.5 m from the laser tracker, presents the highest uncertainty with a value of 52 µm. In the multilateration case, T 3 has again the highest uncertainty with a similar value of 51 µm, but this time due to a position largely outside the volume formed by the four measurement heads.

The comparison of the measured positions
In this part, the measurements performed by the multilateration system are compared with those performed by the commercial laser tracker to demonstrate that the in-house developed multilateration system works well and that its uncertainties have been properly assessed.
To compare the two sets of coordinates, the one obtained by the laser tracker with the one obtained by the multilateration system, a point-set registration using a Horn's quaternionbased algorithm [19] has been performed to express the different results in a unique system of coordinates. This consists in finding a translation vector t and a rotation matrix R that best match one collection of target coordinates to another one in a least squares sense:      To carry out properly this adjustment, the position uncertainties have been considered by introducing the weights w. Thus, the positions with high uncertainties are not as well aligned as those with low uncertainties: The uncertainties of the measured positions can be expressed by a single number, σ pos , as shown in table 1 with the example of the laser tracker. However, it is more rigorous to consider the geometry of the uncertainties in threedimensional space, which are defined by confidence ellipsoids. An example is shown in figure 9 for the case of T 15 . For the positions measured by the laser tracker, T LT , the axis of length 1.88 × σ distance corresponds to the direction of the optical beam of the laser tracker. The two other axes, radial to the direction of the optical beam, correspond to the angle measurements. Their lengths are equal to 1.88 × σ θ and 1.88 × σ ϕ , respectively. The factor 1.88 is due to the move from a one-dimensional to a trivariate error distribution [20]. For the positions measured by the multilateration system, T MLT , the determination of their confidence ellipsoid results from the covariance matrices obtained by applying the mathematical solution presented in [16].
Thus, in formula (7), the uncertainty σ LT is the uncertainty of T LT with respect to T MLT . This corresponds to the distance between the center of its confidence ellipsoid and the point of this ellipsoid in the direction of T MLT as depicted in figure 9. The same method is done for the determination of σ MLT .
Once the two sets of coordinates are expressed in a unique frame thanks to the weighted least squares detailed in formula (6), the results are compared. First, for each position, the distance between the point measured by the laser tracker and that point when it is measured by the multilateration system is depicted in figure 10, where the k = 1 uncertainty bars represent the values in figure 8. These distances are plotted relative At the end, the uncertainty bars at k = 1 of the two sets of coordinates overlap for nine positions. Positions 4,8,14 are excluded from these positions, but for less than 1.3 µm. When considering uncertainty bars at k = 2 (not depicted here), only the positions 15 do not overlap. Thus, 50% and 94% of data points lie within, respectively, one and two standard deviations. This demonstrated that the uncertainties of the two systems are compatible for k = 2 since we should have obtained a percentage of 95%.
It is also possible to examine the results in threedimensional space. In this case, the standard deviations of the differences in coordinates between the two systems are equal to 28.5 µm, 34.0 µm and 28.8 µm for, respectively, the x, y, and z axes. To better analyze this, figures 11 and 12 have been plotted. They represent, for each target, the coordinates determined by the multilateration system relatively to the position measured by the laser tracker, with confidence regions of 68% probability. For the presentation, we have chosen the laser tracker coordinates as the origin of the individual target reference frames.
When the results are observed in the xOz plane, the confidence ellipsoids of the two instruments overlap in most cases. More specifically, the z coordinates of the different target positions measured by the laser tracker and the multilateration system are very similar, only the positions 4, 15, 16 and 17 present differences higher than 44 µm when other are lower than 30 µm. In the xOy plane, the observed discrepancies are higher, especially for the y coordinates.
In three-dimensional space, ten positions over 18 have the confidence ellipsoids of the two coordinate measurement instruments that overlap, i.e. 56% of the positions have compatible confidence regions. This reveals that the threedimensional representation can change a little the results, but we still do not reach the expected 68%. These 10 positions are marked by green dots in figures 11 and 12. A majority of them are located at the center of the tetrahedron formed by the four measurement heads as shown in figure 4. The results seem therefore to partly depend on the position of the targets relatively to the location of the heads of the multilateration system. For confidence regions of 95% probability, only the position 15 has confidence ellipsoids that do not overlap. In other words, 94% of the positions have compatible confidence regions, as expected.
At the end, the results in three-dimensional space shows that the confidence regions of 68% probability seem slightly undervalued. Firstly, with the small number of measured targets, it is difficult to verify accurately whether the uncertainties  of the two instruments are compatible. Therefore, differences of a few percent can be explained in this way. Secondly, the position of the laser tracker drifted over time as pointed out in section 8.2. Corrections were made for the coordinates measured by the laser tracker, but no uncertainty was attributed to them. Thirdly, and as previously explained in section 5, the errors resulting from the determination of the instrument offsets by self-calibration was not taken into account. This point is probably the most important. Lastly, the point set registration algorithm probably also induces errors. So, there are additional uncertainty contributions that should have been considered, but these seem minor.

Comparison with the scale bar
Tocomplete the comparison, the length of a reference scale bar has also been measured by the multilateration system. This scale bar equipped with two nests is shown in figure 13. It has a nominal length of 800.169 mm ± 5 µm at 20 • C for a coverage factor k = 2. This value results from an interferometric calibration using the accompanied target, a spherically mounted retroreflector (SMR, which is a hollow CCR) of 1.5 inch diameter.
This new comparison has been realized 1 h after the measurements with the laser tracker, at 4:30 pm under a temperature of 19 • C, using a classical multilateration technique with the coordinates of the measurement heads and the instrument offsets determined previously. As depicted in figures 2 and 4, the scale bar has been positioned on the ground between the four heads. The used target was the 1.5 inch diameter SMR, oriented manually in the direction of the different measurement heads.
The interpoint distance between the two nests, when determined by the multilateration system, is equal to 800.166 mm ± 10 µm. The uncertainty on this interpoint distance has been calculated as the combined uncertainty of two independent positions. Let T 19 and T 20 be these two target positions. The uncertainty of T 19 is equal to the distance between the center of its confidence ellipsoid and the point of this ellipsoid in the direction of T 20 , and inversely for uncertainty of T 20 .
At the end, the error between the measured distance and the nominal value is equal to only 2 µm. A length correction of 1.2 × 10 −6 per kelvin was performed to take into account the thermal expansion coefficient of the scale bar made of Invar. error = 800.166 mm − 800.169mm (9) The position measurements of the multilateration system are therefore compatible with the length of the scale bar.

Conclusion
A multilateration system with a metrological traceability to the SI meter has been developed. This is an alternative to the spherical measurement systems, such as laser trackers, to reduce the errors linked to angle measurements in difficult environments with for example temperature gradients. Indeed, a multilateration system only requires distance information for determination of three-dimensional positions.
This multilateration system is based on an absolute distance meter and four measurement heads. When used with a hollow corner cube as target, it allows distance measurements up to 140 m with an uncertainty of 4.7 µm in a controlled environment, i.e. when the air refractive index is well known.
To validate the performance of this system, it was compared with two external references: a laser tracker and a scale bar. This was done in a large volume of about 6.3 m × 10.3 m × 3.1 m, a workshop without air conditioning, but with a temperature stable within 0.5 • C over several hours. In this environment, an additional uncertainty contribution of 0.6 µm m −1 was considered on the distance measurements for the air refractive index correction.
Eighteen positions of a common target were measured for comparison with the laser tracker, then two more for determination of the length of a scale bar. The distance differences between the 18 first target positions measured by the laser tracker and those determined by multilateration are equal to 50 µm on average for distances up to 11.6 m. Then, for the two last target positions corresponding to the scale bar, the interpoint distance differs from the nominal value by only 2 µm. This demonstrates the proper functioning of the developed multilateration system.
The position uncertainties assessed for the multilateration system, when expressed as a single number, were between 11 µm and 51 µm, values 1.8 times lower on average than those obtained by the laser tracker. The uncertainties of the two instruments were quite compatible, even if the confidence regions of the positions determined by multilateration have been slightly undervalued. Indeed, the uncertainty contribution of the determination of the instrument offsets by selfcalibration has not been taken into account.
Finally, in three-dimensional space, 56% of the measured positions have the confidence ellipsoids of the two systems that overlap, instead of the 68% probability represented by their confidence regions. However, as expected, this value reaches 94% for confidence regions of 95% probability. In this case, the impact of the errors resulting from the determination of the instrument offsets by self-calibration is likely to be negligible. Nevertheless, in the future, the instrument offsets will be determined in a different way, through additional calibration measurements, in order to take into account its contribution to the uncertainty.

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).