Laser trackers are highly precise, efficient, and reliable measurement instruments. Due to constraints of object size, measurement range, and observation environment, laser trackers need to collect observation coordinates from different stations, which must then be registered into a unified geographical or instrumental coordinate system (Fig. 1). The accuracy and reliability of registration during station transfer are crucial for subsequent data processing. Considering random errors in both original and target coordinate systems, the symmetric similarity transformation model is suitable for data modeling. The total least-squares (TLS) estimator can provide optimal parameter estimates for such a model. Unfortunately, unfavorable observation conditions often lead to outliers, which can render registration parameter estimates unreliable and affect subsequent data processing.

To overcome the adverse effects of outliers, a robust version of the TLS estimator was studied in detail. In this work, the symmetric similarity transformation model was adopted to model observed coordinates in both original and target coordinate systems. During the implementation of the TLS estimation algorithm, constraint equations in the symmetric similarity transformation model were converted into pseudo-observation equations, which reduced algorithm design complexity and improved numerical stability. Note that “total residuals” are more suitable for predicting observed residuals. Therefore, we applied a weight selection iterative method for the TLS problem based on “total residuals”. Meanwhile, the IGGⅢ equivalent weight scheme was chosen for re-weighting operations. It ensures robustness by setting a down-weighting area and a rejection area, and inherits the high normal validity of the TLS estimator by setting a constant weight area. The proposed robust TLS (RTLS) estimator maintains robustness in both observation and structure spaces, and is expected to perform well against unpatterned outliers.

Four parameter estimation methods—least squares (LS), robust LS (RLS), TLS, and RTLS—were compared in both simulation and field experiments. In the simulation experiment, two contamination models were adopted: 1) random errors were added to coordinates in both original and target systems, followed by random introduction of three outliers into specific coordinates; 2) a Gaussian mixture model was used to simulate observation errors (Table 1). Results from 100 Monte Carlo simulations include sequences and means of registration parameter absolute deviations, as well as registration accuracy (Fig. 2, Fig. 3, and Table 2). Both LS and TLS lack robustness, with outliers distorting their estimates. Regrettably, RLS is also susceptible to outlier influence. Considering the coupling effects of random errors and outliers, RTLS exhibits the optimal statistical performance.

Field data, collected by two Leica LTD840 laser trackers during the Hefei Light Source renovation project, were used to verify the proposed RTLS algorithm (Table 3). Here, an outlier of 0.006° was added to the horizontal angle observation of P4 at Station A, and an outlier of 0.2 mm was added to the ranging observation of HCG08 at Station B. Since no true values of registration parameters exist, the posterior unit weight standard deviation was used for accuracy evaluation (Table 4). Absolute point errors from the four methods are shown in Fig. 4. With contaminated coordinates, the registration accuracy of LS and TLS degrades significantly. The RLS estimator assigned co-factor factors of 10¹⁰ and 175828.3 to the 8th and 13th observation equations, respectively; in contrast, RTLS assigned 10¹⁰, 10¹⁰, and 36.4 to the 7th, 8th, and 13th observation equations. RLS fails to adequately recognize outlier effects, whereas RTLS effectively identifies contaminated observation equations and mitigates or eliminates their adverse impacts on laser tracker station transfer registration.

The accuracy and reliability of laser tracker registration decisively influence subsequent data processing. However, unfavorable observation conditions make outliers unavoidable. The coupling of outliers and random errors distorts estimates from methods such as LS, RLS, and TLS. The proposed RTLS estimator is proven to reduce or even eliminate outlier adverse effects while retaining the high normal validity of TLS. Future research can focus on robust estimation for multi-station registration.