# A New Method of Angle Measurement Error Analysis of Rotary Encoders
## 1 Introduction  
The precision measuring instruments that contain rotation joints and high-accuracy angle sensors, such as laser trackers, articulated arm coordinate measuring machines, total stations, and theodolites, have been widely used in the industrial fields of large-scale metrology [1; 2]. The measurement accuracy of these instruments is influenced significantly by the measurement accuracy of angle sensors. Rotary encoders are widely used as high-accuracy angle sensors. Much research has been carried out to improve angle measurement accuracy. [PERSON] et al. proposed an eccentricity error model, and they used a polygon and an autocollimator to detect the angle measurement error. The grating disk eccentricity error model parameters were estimated using the nonlinear least square method. The angle measurement error was decreased by correcting the eccentricity error [3]. [PERSON] et al. analyzed the angle measurement error data with the fast Fourier transform (FFT), and the constant was calculated based on particle swarm optimization (PSO) to overcome the non-convergence of the least square method [4]. [PERSON] et al. proposed a model based on the radial basis function (RBF) to avoid the work on complex analysis on source of error, but researchers have to get enough evenly distributed calibration data [5]. [PERSON] et al. proposed a method based on the adaptive differential evolution-Fourier neural network (ADE-FNN), based on the calibration device they devised [6]. However the calibration device cannot be used for calibrating a high-precision grating disk. [PERSON] et al. proposed a method based on the Fourier approach that includes transfer functions for the calibration of angle encoders. Different weighting schemes combining the measurement results to reduce the uncertainty of calibration was presented [7]. The cross-calibration methods mentioned above utilize external reference standards to acquire discrete calibration data to fit the correction function. [PERSON] et al. presented the error frequency of the eccentricity and runout of the rotating shaft; the reading heads in regularly distributed to compensate the installation eccentricity of the gratingdisk and runout error was proposed [8]. [PERSON] et al. developed a kind of self-calibration method based on a Fourier-based algorithm through a suitable geometric arrangement of multiple reading heads. The angle measurement error due to lateral shifts, graduation error, reading heads' positions, and non-uniform were analyzed [9]. [PERSON] et al. proposed an optimization-based arrangement of reading heads for self-calibration. The transfer function was used to obtain the relations between detectable Fourier components of angle measurement error and the angle interval of the reading heads. The optimal arrangements for two and three reading heads were analyzed, and the experiments were carried out to demonstrate this method. Then a four-heads separation technique was proposed to separate the encoder's graduation error from the influence of the radial error motion of the rotating shaft [10; 11]. The self-calibration methods mentioned above can improve the angle measurement accuracy by processing the difference values of multi reading heads without using a second auxiliary measuring device. The main error sources are not measured, and the angle measurement error model is not built in the black-box approaches mentioned above. [PERSON] et al. analyzed the error motion of the rotary stage and proposed that the angular positioning error of a spindle or a rotary stage is related to the instantaneous tilt and radial error motions of the rotating shaft based on the Abbe principle. Then the method was experimentally verified by the use of an optical polygon and an autocollimator [12]. [PERSON] et al. proposed a model of predicting the angular positioning error of a rotational mandrel caused by the coaxiality error of the mandrel and the radial error motion of the spindle.