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Optical encoders are sensors based on grating interference patterns. Tolerances inherent to the manufacturing process can induce errors in the position accuracy as the measurement signals stand apart from the ideal conditions. In case the encoder is working under vibrations, the oscillating movement of the scanning head is registered by the encoder system as a displacement, introducing an error into the counter to be added up to graduation, system and installation errors. Behavior improvement can be based on different techniques trying to compensate the error from measurement signals processing. In this work a new “

Optical encoders are sensors used to measure the relative displacement between two mechanical parts. They are widely used in a vast range of applications, such as for example in robotics [

One approach to improve encoder's accuracy consists of analyzing the mechanical behavior of the sensor's components when they are facing to each one of these different nature solicitations. Under this approach, Alejandre and Artes [

Another point of view to improve the accuracy of the sensor is that just based on error compensation from signal processing techniques. Sensor's error compensation techniques based on signal processing constitutes a widely studied research field. The main reason for this lies on the fact that significant improvements of accuracy can be obtained normally at a lower cost than that associated to the introduction of mechanical modifications in the encoder design. There are several methodologies to accomplish encoder's error compensation [

When the encoder is operating under vibrations there are certain particularities associated with the deterioration of the measurement signals that can be treated with “

A mathematical description of the encoder measurement signals is given by:
_{1} and _{2} are the amplitudes, _{1} and _{2} the signal's mean values, _{1} and _{2} the signal phases, _{1} and _{2} are functions that represent the shape of the signals,

If the above conditions are not satisfied metrological errors appear with a consequent detriment of encoder's accuracy. Sanchez-Brea and Morlanes [_{off}is the error due to non-zero mean values; _{amp} is the relative amplitude error and _{ph}, represents the error committed by the encoder when the phase between them is not exactly π/2. In case of non-zero mean values the Lissajous figure is not centered at the ideal (0,0) location and the error has the same frequency value of the measurement signals. In case differences between signals amplitudes exist, the Lissajous figure becomes an ellipse with the major axis at 0 or π/2 depending on if amplitude of signal _{A}_{B}

_{0} and _{180} generated in the photodetectors of a four field scanning encoder and the resultant measurement signal _{A}_{0} − _{180}) deteriorated because of the presence of a 700 Hz and 100 ms^{−2} sine vibration at the same time the encoder is working. For its generation, a 20 μm grating period encoder has been operated on an electrodyamic shaker. In this figure a harmonic superimposed to the measurement signals can be appreciated with a frequency equal to that of the excitation frequency. As it can be observed the amplitude of the harmonic is minimum when the photodetectors are entirely illuminated or minimally illuminated, _{exc} is the frequency of the vibration excitation and

The proposed methodology is based on the combined use of fitting techniques to the Lissajous figure and the use of LUT. Basically, the methodology consists of the estimation of each one the types of errors separately just to build a compensation signal that will be composed of the addition of the individual error contributions. The process is tackled in two steps. In the first step the metrological errors of the encoder due only to manufacturing tolerances are estimated. The second step deals with the compensation of the error due to vibrations.

For the generation of each one of the contributions to the compensation signal of the so-called metrological errors the use of the group of

From the ellipse parameters the term Δ_{1} can be estimated as
_{x}_{y}_{1} has been estimated as
_{x}_{y}_{1} just from the fitting process is a bit more difficult process. The difficulty lies on the fact that differences of relative amplitudes tend to rotate the major axis of the ellipse to 0 radians while the phase error tends to rotate it to π/4 radians. Because both amplitude and phase errors are present in this case, the resulting rotation of the ellipse would be just a combination of the two types of errors. Through the estimation of the maximum and minimum values that these error terms can have for this encoder, an interpolating surface is determined (_{1} from the interpolating surface. Once all the error terms have been estimated, the offset, the amplitude and the phase errors are generated according to the set of

It has to be taken into account that due to the fitting process itself and the use of an interpolation surface of limited resolution, the reconstruction of the error terms in this way will be approximate. However, as most of the error contribution is at the fundamental frequency and the first harmonic, typical error values associated with the reconstruction are around 5%, and less than 10% in any case.

A second step consists of the error estimation due to vibrations that produce the relative movement of the gratings. To accomplish this task it is necessary to previously elaborate a look up table (LUT) by means of the procedure described in the references [^{−2}.

Taking the look up table as a basis, it is proposed to build a signal to compensate the error due to vibrations as follows:

The above formula is based on the assumption that the error due to the relative movements between the gratings is an error that is superimposed to those so-called metrological errors. This way, _{me} is the mean of the metrological errors. The mean value of metrological errors is due to the imperfect phase conditions as it is the only one that produces a systematic component. The term _{gm} represents the amplitude of the error due to the relative movement of the gratings, which is obtained reading in the LUT of _{exc}, that can be determined from the harmonics of the FFT analysis of the measurement signals previously done. There is an uncertainty associated with this calculation due to the presence of the doublet as we have said in Section 2. Knowing that the doublet is nearly centered at the excitation frequency, it is possible to estimate _{exc} from the frequencies of both harmonics that form the doublet. Once these frequencies are determined it is also needed to calculate the phases associated to them, in order to establish the term Ø. As it has been stated in Section 2.2 the phases associated to the harmonics that form the doublet are almost equal or differ in practically half a cycle of the pure vibration harmonic sine at 700 Hz. In case that both phases differ, it is possible to compensate the error with both values, but if it is chosen the minimum phase the compensation signal (the sum of all the error contributions) has to be added to the position calculated by the arctangent algorithm, and subtracted otherwise. However, there is always some residual error after the compensation and to minimize the systematic component in this residual error certain considerations regarding the term Δ_{1} have to be made. They can be summarized as follows:
_{1}and Ø_{1} are the phases associated to the first and second peak that form the doublet at the excitation frequency; Ø is the phase associated to vibration error term _{vib} (_{1} is the increment or decrease in phase between the measurement signals with relation to the ideal 90° degrees (

In order to summarize and present the methodology in a clear manner for practical purposes, in this section the several steps needed to accomplish the whole procedure are listed as follows:

Operate the encoder for the whole measuring length registering the sensor's measurement signals with the source of vibration (for example, the machine in which the sensor is installed) turned off and calculate the ellipse that best fits the whole set of points that it is obtained when plotting both amplitudes of the measurement signals against each other. From this ellipse, compute Δ_{1}, Δ_{1} and Δ_{1} as explained in Section 3.1. The values obtained will be always the same values for the generation of the compensation signal.

Place the encoder in the shaker and set the acceleration level, frequency range of interest and sweep rate (frequency resolution wanted) to elaborate a look up table by means of the procedure described in references [

Turn on the machine or source of vibration that affects the performance of the encoder and put an accelerometer next on the machine and obtain the amplitude of acceleration and the frequencies at which the machine operates.

From the FFT of one of the measurement signals compute _{exc}, Ø and with _{exc} using the look up table of point 2 to calculate _{gm} by linear interpolation. Compute _{off}, _{amp}, _{ph} by the set of _{me} and _{vib} by _{comp}_{off} + _{amp} + _{ph} + _{vib}. Finally, add or subtract the compensation signal to the position determined by the arctangent algorithm applied to the deteriorated measurement signals following the criteria exposed by

Steps 1 and 2 and software to compute Step 4 could be developed by the encoder manufacturer. The Step 3 would be the only one that would be done in the industrial environment to obtain the acceleration and frequencies at which the machine operates. It has to be noted that in the methodology described no use of a sensor of higher accuracy or resolution has been necessary.

Finally, for the sake of clarity the methodology has been exposed considering that the vibration excitation is produced mainly in one direction, but the performance of the encoder can be different regarding the direction of vibration, as other modes of vibration of the encoder's components can be excited. This case is contemplated in the referenced work [

Results regarding the application of the above methodology are presented in ^{−2}). Concretely,

In

_{exc}, from the FFT of the measurement signals. This gives as a result that, if frequencies of the error and from the compensation signal do not match exactly, a certain phase is introduced between both, making the algorithm get worse as the encoder travels along the measuring length. Despite this, it is possible to reduce the error from a value of 2.3 μm to 1.3 μm for some peaks in case of 1,000 Hz of excitation frequency and from 1.8 μm to 0.9 μm in case of 1,400 Hz of excitation frequency. Instead of analyzing the performance of the method for single peaks it is more interesting to have an idea about how it performs for the whole period considered. From this point of view

In this work a new methodology for sine vibration error compensation of optical linear encoders has been presented. When the encoder operates subject to vibrations certain particularities arise that allow using an “

Results show that maximum and minimum errors at which the error oscillates are reduced considerably. When the encoder operates at a constant speed subjected to different excitations, it can be appreciated that the algorithm reduces the error committed by the encoder effectively, although the performance is not as good as in the case of increasing the velocity of the encoder instead the frequency of the excitation. We can conclude that the performance improves significantly for all the cases analyzed and it has to be noticed that even for the worst cases still there are significant improvements of around 50% for the parameter considered.

The proposed methodology allows an individualized treatment of the different types of errors that, when combined, permits one to obtain a compensation signal of similar shape and with similar values to the error that the encoder produces when it works under vibrations. The whole procedure is accomplished without the need to use a sensor of a higher accuracy or resolution. The simplicity of the different mathematical expressions used to calculate the different contributions to the final compensation signal together with the high computational efficiency of LUT procedures, makes the methodology suitable for use as an online procedure for compensating encoder error under sine vibration, once the error is characterized in a previous step off-line.

This work has been supported by the Ministry of Science and Education of Spain under Project DPI2007-64469.

(_{0} and _{180} generated in the photodetectors with resultant measurement signal _{A}_{A}

(_{1}; (

Error chart that can be used as a LUT to construct the vibration compensation signal.

Error compensation results. Constant excitation frequency (700 Hz) and increasing speed operation of the encoder (0.8 mm/s (

Error compensation results. Increasing excitation frequency (700 Hz (

Area enclosed by the error curves before and after the compensation.