## 3 Investigation Methodology and Experiment Setup  
The magnitude of an SDE is usually determined by using a reference encoder or a laser interferometer. Independent position information from a reference device is compared with a tested linear encoder reading, and the difference between these measures is accepted as the error. In order to perform such a test, a high-accuracy reference is needed. Another method that was realized in this paper is based on a constant-speed test. The tested encoder reading head was driven at constant speed, and its electric-output signals were recorded using a digital oscilloscope. Because of high sampling frequency, analog encoder signals were represented as a group of discrete points. The relative position values inside one grating period could be calculated by putting these discrete points of A and B signals into Equation (1). Claiming that scanning speed is constant, and knowing the used sampling rate that must be selected by taking into account the Nyquist-Shannon sampling theorem and the acceptable size of the sample, it is easy to compose the theoretically true position values at these points. For example, if the scanning speed is 100 mm/s, the grating period of the tested encoder is 20 \(\upmu\)m, and the sampling frequency is 250 MHz, there are 50,000 sampling points per one period. The first point corresponds to a zero position value, and the last one (50,000) corresponds to 20 \(\upmu\)m. All other points increase with a step of 0.0004 \(\upmu\)m. Then, the calculated values according to the arctangent algorithm could be compared with theoretically determined \"true\" position values. The differences are accepted as metrological linear encoder errors inside one grating period. In order to obtain statistically reliable results, this procedure must be repeated for several periods or several separate measurements, and the average value must be used. The accuracy of this methodology highly depends on the instability of the scanning speed and the precision of the used equipment. However, this method is completely suitable to determine the magnitude level of the error and to notice its trend. A standard optical (4-field scanning) linear encoder was selected for this experimental investigation. The special technological setup was designed to determine its metrological errors under different scanning speeds. A motorized translation stage based on direct-drive technology was used to drive-test the encoder's reading head at various constant speeds with high accuracy and low friction. The designed setup is shown in Figure 3.  
The 3-phase ironless linear brushless servo motor of the translation stage was controlled by an ACS servo motion controller with a built-in driver. In the translation stage, an integrated noncontact  
Figure 3: Designed experiment setup. Tested linear encoder was connected to direct-drive linear translation stage through mounting bracket. linear optical encoder was used as a feedback system for high-resolution positioning and fluent motion control. Electric signals were sampled and recorded by using a digital oscilloscope. Schematic representation of the experiment setup is shown in Figure 4.  
Experiment setup consisted of:  
1. Motion controller: servo motion controller with a built-in driver ACS Motion Control SPiiPlusCMnt. 2. Linear-translation stage: motorized direct drive linear translation stage \"STANDA\" 8 MTL1401-300. 3. Tested linear encoder: optical (4-field scanning) linear encoder Precizika Metrology L18 (measuring length = 300 mm, grating period = 20 \(\upmu\)m). 4. Data acquisition and processing equipment. digital oscilloscope PicoScope 3000 and notebook with appropriate software. Collected data were processed with computer software MATLAB. Determined SDE dependency on different scanning speeds was statistically evaluated by using a linear-regression approach. In order to analyze the nature of these errors at each speed, a fast Fourier transformation (FFT) algorithm was used.