Multilateration with Self-Calibration: Uncertainty Assessment, Experimental Measurements and Monte-Carlo Simulations

Abstract

Large-volume metrology is essential to many high-value industries and contributes to the factories of the future. In this context, we have developed a tri-dimensional coordinate measurement system based on a multilateration technique with self-calibration. In practice, an absolute distance meter, traceable to the SI metre, is shared between four measurement heads by fibre-optic links. From these stations, multiple distance measurements of several target positions are then performed to, at the end, determine the coordinates of these targets. The uncertainty on these distance measurements has been determined with a consistent metrological approach and it is better than 5 µm. However, the propagation of this uncertainty into the measured positions is not a trivial task. In this paper, an analytical solution for the uncertainty assessment of the positions of both targets and heads under a multilateration scenario with self-calibration is provided. The proposed solution is then compared to Monte-Carlo simulations and to experimental measurements: it follows that all three approaches are well agreed, which suggests that the proposed analytical model is accurate. The confidence ellipsoids provided by the analytical solution described well the geometry of the errors.

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

“…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.…”

“…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.…”

“…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].…”

Methodology to Evaluate the Performance of Portable Photogrammetry for Large-Volume Metrology

“…Furthermore, the integration of LTs into metrological software allows the achievable uncertainty to be predicted in such a way that users can better plan measurements to ensure they are of high quality [2]. Several approaches focus on further optimising measurement uncertainty and process efficiency using LTs, such as multi-lateration [3][4][5] and multi-sensor architectures [6,7]. However, a measuring process based on LTs entails several challenges: the significant costs of the equipment, the high mechanical stability requirements over the observation period, and the need for highly experienced operators [8,9].…”