## 2 Force Modeling for Machining Stability Analysis  
The experimental setup for the cutting force modeling process is illustrated in Figure 2. In this study, a voice-coil-motor-driven active end-effector is mounted on the flange of the UR10e manipulator, enabling precise control of the cutting spindle's position along the \(x\)-axis via the SERVOTRONIX CDHD2 servo drive. Three sets of experiments are conducted at feed rates of 120 mm/min, 180 mm/min, and 240 mm/min, with force data captured using an ATI-9105 force/torque sensor, which is installed beneath the fixture holding the workpiece. All experiments were carried out using a climb milling strategy, where the cutter rotates in the same direction as the feed motion. To model the instantaneous milling force of the measured system, [PERSON]'s exponential model is employed [15], which is expressed as  
\[F_{m}=K_{m}aph^{\beta}\ \ \ \ \ m\in t,n,a \tag{1}\]  
where \(F_{t}\), \(F_{n}\), and \(F_{a}\) are the instantaneous milling forces in the tangential, normal, and axial directions, respectively. Here, \(ap\) represents the axial depth of cut, \(h\) is the instantaneous chip thickness, and \(K_{t}\), \(K_{n}\), and \(K_{a}\) are the empirical parameters corresponding to each force direction. The exponent \(\beta\) is a common empirical parameter shared across all three force components. Figure 1: Proposed GA-XGBoost-based prediction workflow. The helix angle of the milling cutter can be approximated using a multiple straight-tooth model, as illustrated in Figure 3, in which the helical flutes are unrolled and represented as inclined straight lines to enable simplified analytical treatment. Across the axial direction, the cutter is discretized into multiple thin slices, each with a thickness of \(da\). Consequently, the milling force acting on each slice can be expressed as follows:  
\[dF_{x\ ij}=K_{t}daph_{ij}^{\beta}\sin\left(\phi_{ij}\right)-K_{n}daph_{ij}^{ \beta}\cos\left(\phi_{ij}\right) \tag{2}\]  
\[dF_{y\ ij}=K_{n}daph_{ij}^{\beta}\sin\left(\phi_{ij}\right)+K_{t}daph_{ij}^{ \beta}\cos\left(\phi_{ij}\right) \tag{3}\]  
\[dF_{z\ ij}=K_{a}daph_{ij}^{\beta} \tag{4}\]  
in which \(d\phi=(dap\tan\gamma)r^{-1}\) is the angular increment, \(\gamma\) denotes the helix angle. The instantaneous chip thickness is given by  
\[h_{ij}=f_{t}\sin\phi_{ij}+n(t-\tau)-n(t) \tag{5}\]  
where \(i=1,2,\cdots N_{t}\) represents the tooth index of the endmill, \(j=1,2,\cdots\text{round}(b/dap)\) is the slice index, \(f_{t}=60f/(\Omega N_{t})\) is the feed per tooth in m/tooth, \(f\) is the linear feed rate in m/s, \(\Omega\) is the spindle speed in rpm, \(N_{t}\) is the number of teeth per revolution. The terms \(n(t-\tau)\) and \(n(t)\) correspond to the previous and current surface vibrations due to regenerative chatter. Finally, the immersion angle for each slice is given by  
\[\phi_{ij}=\phi_{11}+\frac{2\pi}{N_{t}}i+d\phi \tag{6}\]  
The instantaneous milling force is then determined by summing the contributions from (2), (3), and (4), expressed as:  
\[F_{q}=\sum_{i=1}^{N_{t}}\sum_{j=1}^{\text{round}(\frac{b}{dt})}dF_{q\ ij}\ q\in x,y,z \tag{7}\]  
where \(dF_{m\ ij}=0\) if \(\phi_{ij}<\phi_{s}\) or \(\phi_{ij}>\phi_{e}\). Here, \(\phi_{s}=\pi-\cos^{-1}\left((r-b)r^{-1}\right)\) and \(\phi_{e}=\pi\) denote the entry and exit angles, respectively, under down-milling conditions, where \(b\) represents the radial depth of cut. Figure 2: Experimental Setup for cutting force modeling. To determine the empirical parameters of the exponential model, a genetic algorithm (GA) is employed to optimize the fit between the measured force data and the instantaneous milling force computed using (7).