## 3 Adaptive Sliding Mode Control (ASMC) Design
### ASMC Design  
In the actual processing of the FTS system, factors such as workpiece shape, material properties, and environmental changes often change, which brings challenges to the control system. Traditional control methods often struggle to cope with these changes because they often rely on predetermined models and parameters. Adaptive control has strong adaptability and can adjust control parameters in real time according to system dynamic characteristics and changes in the external environment, thereby maintaining the stability and efficiency of system performance. Therefore, we updated it based on the adaptive control proposed in the reference [29]. The adaptive law is as follows  
\[\dot{\hat{\eta}}=\varphi|\sigma| \tag{10}\]  
This article uses adaptive parameter estimation \(\hat{\eta}\) instead of the switching gain \(\eta\). This adaptive switching gain is designed to resist model parameter uncertainty and sudden external disturbances. \(\varphi\) is a positive constant that exponentially determines the convergence rate of the adaptive switching gain.  
Rewrite the control law (7) as  
\[u=\frac{1}{b}\big{[}a_{1}x_{1}+a_{2}x_{2}+\ddot{y}_{d}-\lambda_{1}(x_{2}-\dot{ y}_{d})-\lambda_{2}(x_{1}-y_{d})-\dot{\eta}\text{sgn}(\sigma)-k_{n}\sigma\big{]} \tag{11}\]  
By examining the adaptive rule (10), it can be seen that the adaptive effect depends on the control parameter \(\varphi\). Generally speaking, the larger the parameter \(\varphi\), the faster the adaptation process. In practical applications, due to the presence of sensor noise, an endless adaptive process may occur, resulting in the value of gain \(\hat{\eta}\) being too large. In order to avoid this problem, the following improvements are made to the adaptive rules in actual implementation.  
\[\dot{\hat{\eta}}=\left\{\begin{array}{ll}\varphi|\sigma|,&|e|>\varepsilon\\ 0,&|e|\leq\varepsilon\end{array}\right. \tag{12}\]  
where \(\varepsilon\) is the assigned threshold value of the tracking error bound. When the tracking error is suppressed within the bound of \(\pm\varepsilon\), the control parameters will maintain constant thereafter without adaption and the error will be kept by the bound \(\pm\varepsilon\). Otherwise, the control gains will be updated again. Thus, the position tracking error bound can be predefined, which is desirable for practical application.  
The defects of SMC itself and the delay and lag of control information in actual engineering applications cause chattering in the sliding mode of the system. For continuous systems, the quasi-sliding mode method is usually used to suppress the occurrence of this phenomenon, that is, by setting a certain \(\Delta\) neighborhood of the ideal sliding mode and limiting the motion trajectory of the system to this neighborhood. The following quasi-SMC method is generally used: the sign function in the ideal sliding mode is replaced by a saturation function.  
\[sat(\sigma/\Delta)\left\{\begin{array}{ll}\frac{\sigma}{\Delta}\;,&if|\sigma |\leq\Delta\\ sgn(\sigma)\;,&if|\sigma|>\Delta\end{array}\right. \tag{13}\]  
Among them, \(\Delta\) is called the boundary layer. Outside the boundary layer, the saturation function uses linearized feedback control, while inside the boundary layer, switching control is used.