Abstract: This paper analyzes and studies the errors of sine and cosine encoders using Lissajou plots. The sources of error in sine and cosine encoders are briefly introduced, mainly including assembly and adjustment errors and manufacturing errors, which significantly affect the measurement accuracy of the encoder. The analysis and study of sine and cosine encoder errors using Lissajou plots provides important guidance for reducing errors and improving encoder accuracy in practical applications.
Keywords: Sine/cosine encoder, Lissajou, error analysis, eccentricity error
1. Introduction
A photoelectric encoder is a precision instrument that converts the mechanical rotation angle on the output shaft into pulses or digital quantities through photoelectric conversion. Photoelectric encoders have advantages such as good stability, high precision, and digital interface, and are widely used in the fields of precision measurement and real-time control of civil, military, robotics and CNC machine tools [1-2]. Among them, the sine and cosine encoder is a photoelectric encoder that outputs orthogonal sine and cosine signals [3]. With the increase in application requirements, there are also higher requirements for the accuracy and reliability of encoders. Understanding the error sources of sine and cosine encoders and analyzing how to reduce errors is of great significance for improving encoder accuracy.
2. Sources of error
The sources of error mainly include component manufacturing errors and component assembly errors. Component manufacturing errors mainly include grating manufacturing errors, etc.; component assembly errors mainly include installation eccentricity and shaft wobble errors.
2.1 Component manufacturing errors
The grating is manufactured by a special photolithography equipment. Due to the error of the photolithography machine itself, the actual direction and position of the etched lines will deviate from the theoretical direction during the process of etching lines on the photolithography code disk. Factors such as the black-and-white contrast of the grating, the eccentricity of the etched lines and the spacing of the etched lines will affect the quality of the photoelectric signal to a certain extent, making the photoelectric signal not an ideal sine wave. When the installation eccentricity error of the encoder, the shaft wobble error, the interference and diffraction of light are sufficiently small, the accuracy of the encoder is significantly affected by the manufacturing error of the grating [5].
2.2 Component assembly and adjustment errors
2.2.1 Installation eccentricity error
During the assembly of the encoder, when the center of the installed grating disk does not coincide with the rotation center of the shaft system, an eccentricity phenomenon will occur, which will lead to eccentricity error during actual measurement. At present, structural schemes such as diameter-aligned reading heads or evenly distributed multiple reading heads are adopted [4], which can eliminate part of the installation eccentricity error to a certain extent, thereby improving the measurement accuracy of sine and cosine encoders.
2.2.2 Shaft wobble error
Shaft wobble error is one of the important factors affecting the accuracy of sine and cosine encoders. The shaft system is an important component of the encoder. It drives the grating to rotate precisely, creating relative motion between the grating and the slit, thus resulting in a regular change in the amount of light transmitted. If the main shaft experiences radial wobble and axial runout, eccentricity and changes in the spacing between the grating and the slit will occur, altering the contrast of the sine and cosine signals and affecting signal quality.
3. Error Analysis
3.1 Error Analysis Method
There are many factors that affect the accuracy of encoders. In addition to factors such as eccentricity error in grating manufacturing and installation and shaft wobble error[6], the quality of sine and cosine signals is the main factor affecting the accuracy of sine and cosine encoders. The factors affecting the quality of sine and cosine signals are mainly the amplitude of photoelectric signals, signal phase, DC component and waveform quality.
Lissajou graphs can intuitively and effectively observe the phase relationship between two sine and cosine signals, and are often used to analyze parameters such as DC level, phase, and amplitude of two quadrature signals.
Suppose that the waveforms of two sine and cosine signals A and B are as shown in formula (3-1). When they are applied to the X and Y axes of the oscilloscope respectively, the Lissajou diagram shown in Figure (1) can be observed.
(3-1 )
Figure (1) Lissajou diagram of signals A and B
By observing the Lissajou graphs of the two quadrature signals from the sine and cosine encoder, the error of the sine and cosine signals can be estimated. When there are factors such as amplitude, DC level, or orthogonality deviations in the two sine and cosine signals output by the encoder, the Lissajou graphs will be shifted or distorted. Figure (a) shows an ideal sine and cosine signal, and Figure (b) shows the Lissajou graph synthesized from the ideal sine and cosine signals.
(a) (b)
Figure (2) shows the ideal sine and cosine signals and the Lissajous graph.
3.2 Evaluation of Sine and Cosine Signals
The quality assessment of sine and cosine signals is mainly reflected in four aspects: DC level, equal amplitude, orthogonality, and sinusoidality.
3.2.1 DC Level Deviation
When there is a DC level deviation in the sine and cosine signals, the sine wave will have different zero-crossing values, which will cause a phase difference of subdivision [7]. When there is a DC level deviation in the sine and cosine signals, the Lissajou graph is two perfect circles with equal radii, but the center position is offset, as shown in Figure (3).
Figure (3) DC level offset
3.2.2 Unequal amplitude
The amplitude of sine and cosine signals is mainly affected by factors such as changes in the light intensity of the LED, fluctuations in the power supply level, and grating marking errors [8]. When the amplitudes of sine and cosine signals are not equal, the Lissajou graph is a circle with unequal radii, i.e. an ellipse, as shown in Figure (4).
Figure (4) shows unequal amplitudes.
3.2.3 Orthogonality
The phase difference between the two sine and cosine signals is not 90°, thus disrupting the orthogonality of the signals. Eccentricity and vibration of the grating disk can also disrupt the orthogonality of the signals, and the phase deviation also results in amplitude deviation. The orthogonality error of the two sine and cosine signals has a significant impact on the encoder's accuracy. When the orthogonality error is large, the encoder will exhibit errors such as code skipping. When the sine and cosine signals are not orthogonal, the Lissajous figures have inconsistent shapes in the four quadrants, as shown in Figure (5).
Figure (5) Signals are not orthogonal
3.2.4 Sinusoidality
Sine and cosine signals have the properties of sine waves[9], but there are various harmonics and noise in the actual output sine and cosine signals. These undesirable components destroy the sinusoidal nature of the sine and cosine signals. When there are harmonic components in the sine and cosine signals, the symmetry of the X and Y axes changes, and the shape of the Lissajou becomes irregular.
Currently, Changchun Yuheng Optics Co., Ltd. utilizes an integrated digital display platform to measure the amplitude error, offset error, and phase difference error of sine and cosine waveform signals, enabling error analysis and research on sine and cosine encoders. As shown in Figure (6):
Figure (6) shows the error values of the sine and cosine signal waveforms.
In this diagram, the green line represents the amplitude difference error, the yellow and white lines represent the DC offset errors of the Sine and Cosine signals, the blue line represents the phase difference angle error, and the red line represents the electrical angle error value of the final sine and cosine signals. The final angle error is calculated by averaging the sine and cosine signals over the entire sampling time and applying a function formula to the 360° signal.
(3-2)
In the formula, AS and AC are amplitude difference errors; OS and OC are offset errors; φ SERR and φ CERR are phase difference angle errors; and F SIN and F COS are waveform errors.
The ZND series sine and cosine encoders manufactured by Yuheng Optics Co., Ltd. employ a filter slit design and high-precision gratings and high-precision shaft system, which reduces the manufacturing errors of the shaft system and gratings, ensuring that the harmonic components in the sine and cosine signals are less than 1% and the electrical angle error is φ ±0.18°, resulting in high-quality sine and cosine signals.
5. Summary
This paper analyzes and studies the errors of sine and cosine encoders, introducing the impact of component manufacturing and assembly errors on their accuracy. Various imperfect components in the sine and cosine signals affect signal quality, leading to a decrease in encoder accuracy. The Lissajou graphical method is used to analyze the changes in the graphs when the sine and cosine signals contain DC components, phase deviations, amplitude inequalities, and harmonic content, laying the foundation for further improvements in encoder accuracy.
