## 6 Sigmoid Deformation Curve Interpolation  
Compared with the existing variable speed methods, such as trapezoidal variable speed, exponential variable speed, trigonometric function variable speed, etc. [18], the S-type variable speed methods such as exponential variable speed and trigonometric function variable speed are stable and reliable, and it is suitable for frequent acceleration and deceleration. Therefore, the S-type variable speed is adopted as the variable speed method in the variable region. As a common S-type mathematical squashing function, the sigmoid function can squash the input between 0 and 1, which is beneficial for the deformation of the sigmoid function in research. In the tool path planning of double-sided collaborative machining, the sigmoid deformation curve is obtained by deforming the sigmoid function. and this curve is used to complete the conversion between double-sided symmetrical machining and double-sided staggered machining in the S-type variable speed stage. As shown by the solid line in Figure 8, the sigmoid deformation curve includes the deceleration stage in the first half and the acceleration stage in the second half. A smooth and stable variable speed can be achieved by using the sigmoid deformation curve for interpolation in the variable region. Equation (22) is the mathematical expression of the sigmoid deformation curve. \[\left\{ \begin{array}{l}V(t)=\frac{K_{V}}{1+e^{\mathcal{A}_{0}(t-K_{T})}}+K, \quad t\in[0,2K_{T}]\\ V(t)=\frac{K_{V}}{1+e^{\mathcal{A}_{0}(3K_{T}-t)}}+K,\quad t\in[2K_{T},4K_{T}] \\ K=\frac{K_{V}e^{-K_{0}K_{T}}}{1+e^{-K_{0}K_{T}}}\end{array} \right., \tag{22}\]  
where \(K_{T}\) is time adjustment parameter that controls the range of the independent variable interval, which is equal to 0.25 times the variable speed time; \(K_{V}\) is the velocity adjustment parameter that controls the range of the dependent variable interval, which is equal to the initial velocity of the variable speed stage; \(K_{a}\) is acceleration adjustment parameter that controls the slope of the sigmoid deformation curve, which is solved by Equations (24) and (25); The constant term \(K\) represents the difference between the actual velocity and the initial and final velocity of the sigmoid deformation curve. Figure 7: (**a**) Double-sided symmetrical machining of the internal region. (**b**) Start of double-sided staggered machining. (**c**) Meeting of cutters on both sides. (**d**) End of double-sided staggered machining. The following displacement curve expression is obtained by integrating Equation (22) within the interval \([0,\,4K_{T}]\):  
\[\left\{ \begin{array}{ll}I(t)=-\frac{K_{V}}{K_{a}}\ln(1+e^{-K_{a}(t-K_{T})})+Kt,&t\in[0,2K_{T}]\\ I(t)=\frac{K_{V}}{K_{a}}\ln(1+e^{-K_{a}(3K_{T}-t)})+Kt,&t\in[2K_{T},4K_{T}]\\ K=\frac{K_{V}e^{-K_{a}K_{T}}}{1+e^{-K_{a}K_{T}}}\end{array} \right., \tag{23}\]  
Substitute the interval length \(L_{c}\) of the variable speed region into Equation (22) to simplify the monadic equation of acceleration adjustment parameter \(K_{a}\):  
\[2K_{V}\ln(1+e^{K_{a}K_{T}})-(0.5L_{c}+K_{V}K_{T})K_{a}+\frac{2K_{V}K_{T}}{1+e^ {K_{a}K_{T}}}=0, \tag{24}\]  
Use the numerical method to calculate Equation (24) and obtain the initial acceleration adjustment parameter \(K_{a}\). Due to the upper limit of the driving axis acceleration of the machine tool being \(a_{\max}\), it can be inferred from the derivative of Equation (22) that the slope of the sigmoid deformation curve is the highest at \(t=K_{T}\) and \(t=3K_{T}\), which the maximum acceleration value during the variable speed stage. Substitute \(t=K_{T}\) and \(t=3K_{T}\) into the derivative expression of Equation (22), and use the upper limit of acceleration \(a_{\max}\) to constrain \(K_{a}\):  
\[\frac{K_{V}K_{a}}{4}\leq a_{\max}, \tag{25}\]  
If the initial \(K_{a}\) satisfies Equation (25), substitute it as the final acceleration adjustment parameter into Equations (22) and (23) to obtain the expressions of the sigmoid deformation curve and displacement curve.