## 2 Ellipse-Based Trochoidal Toolpath Model  
A trochoid is a curve traced by a point fixed to a circle as it rolls along a straight line without slipping [2]. To generate the true trochoidal toolpath, a workpiece coordinate system (WCS) \(O\)-\(xyz\) is established, as shown in Figure 1. The coordinate system origin \(O\) is the rolling circle center at the initial revolution angle \(\theta=0\). The \(x\)-axis is parallel to the straight line \(y=s/(2\pi)\) for the case of down milling (where \(s/(2\pi)\) is the radius of the rolling circle, \(s\) is the trochoidal step). The \(z\)-axis is perpendicular to the upper surface of the workpiece and oriented outwards. The \(y\)-axis is determined by using the right-hand rule. Then, the equation of the cutter location \(C\) (\(x_{C}\), \(y_{C}\)) that corresponds to the fixed point can be expressed as follows:  
\[\overset{\rightarrow}{OC}=\left[x_{C}(\theta)\quad y_{C}(\theta)\right]= \left[\tfrac{s\theta}{2\pi}+r\sin\theta{-}r\cos\theta\right],\text{ for down- milling, and }r>\frac{s}{2\pi} \tag{1}\]  
where the parameter \(\theta\) represents the revolution angle, which rotates counterclockwise from the negative \(y\)-axis for down milling. The trochoidal radius \(r\) represents the distance from the cutter location to the synchronous rolling circle center.  
For the above model, the shape range of the toolpath is determined by the trochoidal radius \(r\) and the trochoidal step \(s\) (where \(r\) represents the revolution range and \(s\) represents the feed per revolution). It can be noticed that the revolution ranges in the \(x\)-axis and \(y\)-axis directions are not independent, they are bound as one control parameter, i.e., the trochoidal radius \(r\). For a fixed-width slot, the trochoidal radius \(r\) is uniquely determined once the cutter radius has been selected. However, for a practical slotting process, it is always desirable to have more toolpath control parameters to obtain better machining results, and furthermore, to enhance the advantages of trochoidal milling. Therefore, this paper proposes an ellipse-based trochoidal toolpath model by separating the revolution range from the \(x\)-axis and \(y\)-axis directions, as shown in Figure 2. The equation of the modified trochoidal toolpath is written as follows:  
Figure 1: True trochoidal toolpath model for down milling.  
\[\overset{\rightarrow}{OC}=\left[x_{C}(\theta)\quad y_{C}(\theta)\right]=\left[\frac{s \theta}{2\pi}+b\sin\theta-a\cos\theta\right]\text{, for down-milling, and }a\geq b\text{, }a>\frac{s}{2\pi} \tag{2}\]  
where \(b\) is the semi-minor axis of the fundamental ellipse and is oriented along the \(x\)-axis direction. \(a\) is the semi-major axis of the fundamental ellipse and is parallel to the \(y\)-axis direction.  
Upper boundary of the slot  
The semi-minor axis of the fundamental ellipse is arranged in the \(x\)-axis direction, which means that the toolpath is compressed in this direction. The degree of compression can be defined as a ratio \(\rho=b/a\). The advantage of this compression is that the toolpath is shortened, which will be discussed in detail later in Section 4.  
During the ellipse-based trochoidal milling process, the cutter rotates around the tool axis at a given spindle speed. The cutting-edge path can be regarded as a circle, whose equation is described as follows:  
\[[x-x_{C}(\theta)]^{2}+[y-y_{C}(\theta)]^{2}=R_{c}^{2} \tag{3}\]  
where \(R_{c}\) is the cutter radius.  
The upper and lower boundaries of the slot are given as follows:  
\[\left\{\begin{array}{l}y=\frac{w}{2}=a+R_{c}\text{, for the upper baundary}\\ y=-\frac{w}{2}=-(a+R_{c})\text{, for the lower baundary}\end{array}\right. \tag{4}\]  
where \(w\) is the slot width.  
Figure 2: Ellipse-based trochoidal toolpath model for down milling.