## 2 Force Modeling for Machining Stability Analysis
The objective function utilized in the optimization process is expressed as:  
\[F_{m}=K_{m}aph^{\beta}\ \ \ \ \ m\in t,n,a \tag{1}\]0  
where \(\theta\) represents the model parameters being optimized, \(F_{ij}^{meas}\) is the measured cutting force at the sample index \(i\) along force component \(j\) (\(j\in\{X,\ Y,Z\}\)), \(F_{ij}^{pred}(\theta)\) is the predicted cutting force computed using the current model parameters and \(N\) is the total number of data samples. This function minimizes the sum of squared errors (SSE) between the measured and predicted forces, ensuring an optimal parameter fit for accurate cutting force modeling. The detailed GA settings are provided in \[F_{m}=K_{m}aph^{\beta}\ \ \ \ \ m\in t,n,a \tag{1}\]1 A population size of 100 was selected based on the recommendation by [PERSON] et al., who demonstrated that relatively small population sizes can maintain optimization performance while reducing computational cost in continuous domains [16]. In our application, increasing the population size beyond 100 did not yield any meaningful reduction in the sum of squared errors (SSE), yet significantly increased computation time. In this approach, surface vibrations are modeled using a two-degree-of-freedom spring-damper-mass system, as illustrated in Figure 4, which computes the displacement of the tooltip caused by the vibration modes of the robotic system. The fitted results are presented in Figures 5 and 6, where the identified parameters are \(K_{t}=5.7051\times 10^{6}\), \(K_{n}=4.5984\times 10^{6}\), \(K_{a}=-4.81485\times 10^{5}\), and \(\beta=0.4723\). Additionally, the dominant vibration modes of the robotic milling system are observed at approximately 250 Hz and 750 Hz, with a tooth passing frequency of 1000 Hz (i.e., \(\Omega=15\),000 rpm and \(N_{t}=4\)). The stiffness and damping ratios for the two vibration modes are summarized in \[F_{m}=K_{m}aph^{\beta}\ \ \ \ \ m\in t,n,a \tag{1}\]2 As shown in Figure 5, the mean force values are accurately predicted, demonstrating the model's reliability. Furthermore, the frequency responses in Figure 6 indicate that the proposed model effectively captures the system dynamics, making it suitable for machine learning-based data acquisition. It is important to note that the measured amplitude at the tooth-passing frequency is affected by the force/torque sensor's built-in post filter (cutoff frequency around 1200 Hz), which results in a lower measured amplitude compared to the predicted forces. \[F_{m}=K_{m}aph^{\beta}\ \ \ \ \ m\in t,n,a \tag{1}\]3 Unrolled geometry of the cutter with a helix angle. \[F_{m}=K_{m}aph^{\beta}\ \ \ \ \ m\in t,n,a \tag{1}\]4 Schematic plot of the two-degree-of-freedom spring-damper-mass system. \[F_{m}=K_{m}aph^{\beta}\ \ \ \ \ m\in t,n,a \tag{1}\]5 Comparison of measured and predicted milling forces using the proposed force model in (**a**) the x-direction, (**b**) the y-direction, and (**c**) the z-direction. (\(f=3\) mm/s) The numerical values shown in the legends represent mean cutting forces averaged over the time window displayed. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Representation (Genotype) & Continuous \\ Population Size & 100 \\ Selection Method & Roulette \\ Crossover Method & Scattered crossover \\ Crossover Rate & 0.8 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: General Setting of Genetic Algorithm. of 6 mm/min). By systematically adjusting these parameters, a dataset representing a wide range of machining conditions is constructed. This extensive dataset is essential for training a neural network model capable of predicting optimal feed rates under different machining conditions. For each combination of machining parameters, cutting force predictions are computed. However, not all generated data is directly used for neural network training. Only the data corresponding to stable machining conditions, identified through FFT-based magnitude minimization, is chosen for training the neural network.