## 2 Materials and Methods
### FTS Position Error Compensation  
Prior to turning tests, it was necessary to linearize the FTS positioning characteristics, which were unfavorably influenced by the piezoelectric inherent non-linearity and by errors from the DAC (digital/analog converter) of PLC. The compensation procedure involved, firstly, conducting a positioning accuracy test with the use of the Renishaw XL80 interferometer in accordance with ISO-230-2 standard for linear axes with a length not exceeding 2000 mm (test consisted of five bidirectional measurement cycles), with a travel range of \(0\div 1000\) um, stop intervals of 100 um. An additional overrun of 100 um for the first and last positions within the range was assumed. The measured deviations between real and set positions are shown in Figure 3a. A 6 th order polynomial relation was found to fit best (\(R^{2}=0.9897\)) when regressed mean values of measured positions for all 11 individual locations. The positioning compensation polynomial took the following form:  
\[\begin{split} y=3.788731^{-17}x^{6}+2.584442^{-13}x^{5}-4.755826^{- 10}x^{4}\\ +3.491071^{-7}x^{3}-9.961068^{-5}x^{2}+1.707137^{-2}x+1.342034\end{split} \tag{1}\]  
Ultimately, the signal controlling the FTS position was the difference between the preset value and the polynomial value for the preset value of \(x\). The positioning accuracy tests were repeated after implementing the FTS error compensation. The results of the positioning accuracy test prior to and after compensation acc. to ISO-230-2: mean reversal error \(\overline{B}\), range of mean bidirectional positioning error \(M\), systematic positioning error \(E\), positioning repeatability \(R\) and positioning error \(A\) are presented in Table 1.  
Compensation of the non-linear positioning characteristics of the piezoelectric drive and PLC allowed for significant reduction of errors and improving accuracy (Figure 3b). The polynomial model included in the compensation method facilitated the substantial change in the values of the mean bidirectional positioning error \(M\), systematic positioning error \(E\), and positioning error \(A\), which is presented in Table 1.  
The next preparatory step was to designate the drive frequency response. The position measurements were conducted using the XL80 interferometer for a sine curve with an amplitude of 10 V and frequency range from 1 to 100 Hz. Then, the amplitudes from 10 consecutive periods were averaged for each conducted measurement. The test results are presented in Figure 4.  
For a frequency lower than 8 Hz, the drive position amplitude is constant and equal to 1000 um. This limit is consistent with the manufacturer's specification. Above that frequency, the amplitude starts to decline with the dynamics of 4320 um per decade for a frequency below 20 Hz and 590 um per decade above this frequency. This phenomenon limits the maximum cutting depth of the tool for the machining of complex geometries at a high cutting speed.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Measure** & **Before Compensation** & **After Compensation** \\ \hline Mean reversal error \(\overline{B}\) [μm] & \(-0.345\) & \(-0.164\) \\ Range of mean bidirectional & & \\ positioning error \(M\) [μm] & 12.786 & 1.951 \\ Systematic positioning error \(E\) [μm] & 13.842 & 1.464 \\ Positioning repeatability \(R\) [μm] & 2.405 & 1.600 \\ Positioning error \(A\) [μm] & 14.570 & 2.533 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Results of the positioning accuracy tests according to ISO-320-2.