The developing modern economy and manufacturing industry have put forward a higher demand for related measuring instruments. However, the requirements for orthogonality of internal structures increase the difficulty of instrument manufacturing, and the production efficiency is reduced. To break through the limitations in the production and manufacturing of large-scale measuring instruments in China, the concept of non-orthogonal shafting measuring instruments is proposed. This kind of instrument does not require the shafting to be perpendicular or intersecting with each other, and the non-orthogonal shafting laser theodolite (N-theodolite) is a typical non-orthogonal shafting instrument. Much research has explored its measurement performance and related theories. However, due to the lack of the reference end on the laser axis for a single N-theodolite, the inverse kinematics model fails to be established accurately. The inverse kinematics model is necessary for precision theory research, which is convenient for data simulation of error spatial distribution. Besides, the guidance technology based on the inverse model can help improve the automatic measurement function of the N-theodolite. Therefore, to address the difficulty of calculating the rotation angle for the N-theodolite in inverse motion without a reference end, the linear model for the inverse motion to achieve fast and high-precision calculation of the rotation angle is proposed in this paper.

In this paper, the basic theory of Lie groups and Lie algebras is introduced to achieve fast and high-precision calculation of the rotation angle. First, based on the theory of Lie groups and Lie algebras, the kinematics model of the N-theodolite can be constructed. The coordinate transformation matrix is represented as the product of the exponentials formula (POE) with clear physical meanings. Second, the error model of N-theodolite can be obtained through the corresponding differential calculations, and the parts related to the rotation angle error component are preserved. Then, the constraint relationship between the spatial target point and the laser axis pose parameters is constructed, and the linear equations for solving the error correction value of the rotation angle can be established. Besides, the initial values can be quickly obtained through trigonometric functions. Finally, the high-precision rotation angle values are obtained by linear addition of the initial estimation values and the error correction values. The efficient and accurate linear inverse kinematics model of N-theodolite is established.

In this paper, the simulation and real experiments are carried out to verify the proposed linear inverse kinematics model of the N-theodolite. The simulation results show that the rotation angle error calculated by the proposed method approaches 0 (Table 2 and Fig. 7). The proposed method is completely proved to be feasible in principle, and the inverse rotation angles are calculated with extremely high accuracy. However, the interference from multiple error sources is reflected in the experimental results, and the mean rotation angle error in the actual experimental is less than 0.02 mrad (Table 3). The parameters in the proposed linear inverse kinematics model include shafting parameters and the coordinates of spatial points, which would inevitably affect the performance of the N-theodolite. The two-dimensional turntable is used to provide reference values for rotation angle, and the official data of the manufacturer shows that the angle error of the turntable is ±3″, approximately ±0.014 mrad. The calculated angle error by the proposed method is slightly higher than the angle error of the turntable. Therefore, the results of the proposed linear inverse kinematics model approximate the angle error output by the high-precision two-dimensional turntable under the existing experimental conditions, which can prove the feasibility and accuracy of the method proposed in this paper.

A linear inverse kinematics model is proposed to address the difficulty in the inverse motion angle calculation of N-theodolites. Based on the basic theory of Lie group and Lie algebra, the forward motion model and theoretical rotation angle error transmission model for the N-theodolite are established. By combining the constraint relationship between the measured target point and the spatial laser axis pose parameters, a linear equation for calculating the error correction value of the rotation angle is constructed, achieving high-precision calculation of the rotation angle. The feasibility of this linear inverse kinematics model has been verified through simulation experiments, and the average rotation angle error calculated from real experiments is 0.019 mrad and 0.013 mrad. Due to the influence factors such as spatial coordinate errors of laser points, internal shafting errors, and turntable angular errors, the accuracy of related calculations is limited, but the current problem of no reference end is solved by the proposed method, and the requirements of research related to error distribution are met. Further research will be carried out on the accuracy theory of N-theodolite measuring systems, and improving the accuracy of shafting parameters and rotation angles simultaneously will be a key focus.