## 2 The FNA Method
### The Solution of the Nominal Acceleration
Therefore, the Hanning window function with the stopband attenuation \(-44\) dB in Appendix B can meet the requirements, and its expression is  
\[w(i)=\frac{1}{2}\left[1-\cos\left(\frac{2i\pi}{N-1}\right)\right],\ 0\leq i\leq N-1 \tag{40}\]  
\(N\) in the above formula can be obtained from the transition bandwidth formula of the Hanning window function, and its value is  
\[\frac{62\pi}{N}=\triangle\omega=\frac{2\pi\left(f_{\text{stop}}-f_{\text{pass }}\right)}{f_{\text{s}}}\Rightarrow N=\left\lceil\frac{3.1f_{\text{s}}}{f_{ \text{stop}}-f_{\text{pass}}}\right\rceil \tag{41}\]  
where \(f_{\text{s}},f_{\text{pass}}\), _and_\(f_{\text{stop}}\) are the sampling frequency, passband cutoff frequency, and stopband start frequency of the FIR low-pass filter, respectively. To ensure that Equation (34) is a position sampling sequence with \(O\) as the midpoint, \(N\) must be an odd number; therefore, the \(N\) obtained from Equation (41) needs to be corrected as  
\[N=2\left\lfloor\frac{N}{2}\right\rfloor+1 \tag{42}\]  
So far, the coefficients of the FIR low-pass filter have been obtained as  
\[h(i)=h_{\text{d}}(i)w(i)=\frac{1}{2}\frac{\sin\left[\omega_{\text{c}}(i-\tau) \right]}{\pi(i-\tau)}\left[1-\cos\left(\frac{2i\pi}{N-1}\right)\right],0\leq i \leq N-1 \tag{43}\]  
The unknown \(\tau\) in the formula can be obtained from the linear-phase characteristic of the FIR filter as  
\[\tau=\frac{N-1}{2} \tag{44}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Symbol** & **Value** \\ \hline Interpolation period & \(T_{\text{s}}\) & \(1\) ms \\ Command feedrate & \(F_{\text{cmd}}\) & \(3000\) mm/min \\ Spindle speed & \(F_{\text{ppm}}\) & \(12\),\(000\) r/min \\ Maximum tangential acceleration & \(a_{\text{T}}\) & \(417\) mm/s\({}^{2}\) \\ Maximum normal acceleration & \(a_{\text{n}}\) & \(222\) mm/s\({}^{2}\) \\ Transition error of the virtual arc & \(\sigma\) & \(10\) μm \\ Sampling frequency & \(f_{\text{s}}\) & \(1\) / \(T_{\text{s}}\) \\ Passband cutoff frequency & \(f_{\text{pass}}\) & \(20\) Hz \\ Stopband start frequency & \(f_{\text{stop}}\) & \(120\) Hz \\ Stopband attenuation & \(\delta_{\text{stop}}\) & \(-40\) dB \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Main parameters used in the simulation and experiment. At this point, in order to check whether the amplitude-frequency characteristics of the designed FIR low-pass filter meet the design requirements, the discrete Fourier transform of \(h(i)\) can be performed as  
\[H\Big{(}e^{i\omega}\Big{)}=\sum_{i=0}^{N-1}h(i)e^{-j\omega i} \tag{45}\]  
Then amplitude-frequency response at \(f_{\text{stop}}\) is obtained as  
\[\delta=20\text{lg}\left|H(e^{j2\pi\frac{f_{\text{stop}}}{f_{\text{s}}}})\right| \tag{46}\]  
If \(\delta\) is less than the \(\delta_{\text{stop}}\) in Table 1, the designed filter meets the requirement; otherwise, the value of \(N\) should be increased or another type of window function should be used. So far, the design of the FIR low-pass filter was completed and the filter coefficients \(h(i)\) shown in Equation (43) were obtained. By filtering the acceleration Sequence (37), the X-axis acceleration at point \(O\) can be obtained as  
\[\hat{A}_{x}=\sum_{i=0}^{N-1}h(i)\cdot a_{x}(i) \tag{47}\]  
Denoting the filter coefficients, \(h(i)\) are written in vector form:  
\[\mathbf{m}_{2}=\left[ \begin{array}{cccc}h(0)&h(1)&\cdots&h(N-2)&h(N-1)\end{array} \right] \tag{48}\]  
The value of \(\hat{A}_{x}\) can be obtained from Equation (37) and (47) as  
\[\hat{A}_{x}=\mathbf{m}_{2}\left( \begin{array}{cccc}M_{1}^{2}\\ \dfrac{M_{1}^{2}}{T_{\text{s}}^{2}}\\ \vdots\\ q_{x}(N-2)\\ q_{x}(N-1)\end{array} \right)=\overbrace{\left(\frac{\mathbf{m}_{2}M_{1}^{2}}{T_{ \text{s}}^{2}}\right)}^{\mathbf{m}_{3}}\left[ \begin{array}{cccc}q_{x}(0)\\ q_{x}(1)\\ \vdots\\ q_{x}(N-2)\\ q_{x}(N-1)\end{array} \right]=m_{3}\left[ \begin{array}{cccc}q_{x}(0)\\ q_{x}(1)\\ \vdots\\ q_{x}(N-2)\\ q_{x}(N-1)\end{array} \right] \tag{49}\]  
Similarly, the acceleration of the Y-axis and Z-axis at the point \(O\) can be achieved as  
\[\left\{