The authors have proposed a robust linearisation method for determining the angles of sinusoidal signals generated by gratings. This scheme solves the problem of non-linear subdivision by compensating sinusoidal signal. The conventional coordinate rotation digital computer (CORDIC) algorithm is optimised by double-rotation iteration, and the calculation accuracy of arcsine and arccosine functions is improved. A pipeline preprocessing circuit based on the CORDIC is designed for the signal compensation and digital subdivision. The two processes are implemented on a field-programmable gate array chip, which exhibits a wide input range and good dynamic response. A theoretical analysis using stable signals from a function generator verifies that the subdivision system achieves the ideal subdivision effect. The subdivision system is further applied to an anglemeasuring device for a 16,384-line circular grating. The results of subdivision exhibit good linearity for the grating moiré signal. Compared to the angular measurement by a laser interferometer within the three grating lines, the angle-measurement accuracy of ±0.000138° (0.5″) and the relative error is ±0.63% over a travel angle of three grating lines. Notably, the measurement error of the subdivision system is only half of that in a similar commercial product (MicroE system). 2 Architecture of the grating subdivision system The grating sensor generates four sinusoidal signals with a phase difference of 90°, and the period is equal to the grating pitch. The

In this paper, the sinusoidal error compensation method based on particle swarm optimization (PSO) is researched to reduce the subdivision precision loss caused by sinusoidal error in a grating moiré signal. On the basis of the principle of arctangent subdivision of a grating moiré signal, the subdivision error caused by sinusoidal error is quantitatively analysed. In view of the sinusoidal error compensation method, the signal compensation method involves complex calculations and occupies many resources; therefore, a sinusoidal error angle compensation method is proposed. The principle of the sinusoidal error angle compensation method based on the PSO algorithm is explained in detail. Aiming at the large calculation amount of the PSO algorithm, the appropriate PSO parameters are selected experimentally, and the complexity of the PSO algorithm is minimized while ensuring the fitting accuracy. The parameters of the signal waveform equation are solved using the PSO algorithm on the field-programmable gate array (FPGA) platform. According to the parameters solved, a lookup table for sinusoidal error compensation is designed. The grating system platform is built to verify the effect of the compensation scheme on the FPGA platform. The results show that the compensation method can effectively reduce the sinusoidal error component in the signal, and the subdivision error is reduced from 0.95'' to 0.56'' in the time domain. Consequently, the zero-order, first-order, second-order and third-order components are considerably suppressed. The compensation scheme can realize the sinusoidal error compensation of the grating moiré signal and effectively improve the measurement accuracy of the grating encoder.

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