Due to the influence of object size and measurement environment, we usually use the laser tracker to measure at different positions and then fuse the observation data of different stations into the same coordinate system. The transformation error of the laser tracker directly affects the quality of the observation results, so it is significant to study the high-precision transformation adjustment method for the laser tracker. The traditional least squares (LS) method, as a widely used method, can only consider the error in the observation vector. This assumption does not satisfy the actual situation and the estimated parameters by LS are biased statistically. The existing weighted total least squares (WTLS) method can consider the random error in the coefficient matrix and observation vector at the same time. Although WTLS overcomes the shortcomings of the LS method, this method still has some problems. For example, the rotation matrix obtained by WTLS is not orthogonal and the error matrix by WTLS does not meet the structural characteristics.

To overcome the shortcomings of the existing methods, we propose a new improved adjustment method suitable for the transformation of laser tracker, which is called the structured constrained total least squares (SCTLS) method. This new method does not require any limitations for the statistical model and can take into account the correlation of coordinate observations. SCTLS uses a new structure matrix to describe the special structure corresponding to the error matrix so that the corrections of the same elements are consistent. At the same time, this method imposes restrictions on the parameters to be estimated to ensure the orthogonality of the rotation matrix. Because the transformation model is nonlinear, we use the Lagrange multiplier method to rigorously derive and give the iterative solutions of the algorithm in detail. Finally, the first-order accuracy of the parameters is evaluated. The proposed method is able to handle both similar and rigid coordinate transformations by controlling the number of constraints. We compare the differences between the proposed algorithm and the WTLS method to show that the SCTLS algorithm is more rigorous.

The simulations and measured experiments are designed to compare our algorithm with the LS method and WTLS method. In the simulations, we measure six observation points in two laser trackers stations. After adding random errors, the absolute biases corresponding to rotation and translation parameters for different methods are compared. The performance of the LS method is the worst, and the two absolute bias indicators of SCTLS are 64.4% and 62.2% of those of WTLS, respectively (Table 2), which means that the precision of our method has been significantly improved. The data from two stations in the storage ring of Hefei Light Source are selected to verify the effectiveness of our algorithm. The error of unit weight for SCTLS is 0.789582, which is smaller than that of the WTLS method. At the same time, it has been verified that the rotation matrix obtained by SCTLS is orthogonal and the corrections of repeated elements in the error matrix are consistent. However, the WTLS method does not obtain an orthogonal rotation matrix, which makes it difficult to calculate the rotation angle from the rotation matrix. At the same time, the error matrix of WTLS is not structured. The average point errors after the transformation of SCTLS and WTLS are 0.016 mm and 0.024 mm, respectively (Fig. 5). The above results in the simulations and measured experiments prove the effectiveness of our algorithm, which can improve the adjustment accuracy of the transformation parameters.

The proposed algorithm is more rigorous and makes some improvements based on the existing WTLS algorithm. By introducing constraint conditions and the structure matrix, the error accumulation in the transformation of the laser tracker can be effectively reduced. It should be noted that the proposed method is limited to the conversion of data between two measuring stations. In the actual process, multiple measuring stations need to be set up for measurement. In the next step, the proposed method will be extended to multi-station laser tracker data fusion, so that it will have higher application value in practical engineering problems.