## 4 Control Strategy of CDPDR
\begin{aligned} &\dot{a}_{0}=\ell_{0}\|\mathbf{s}\|\\ &\dot{a}_{1}=\ell_{1}\|\mathbf{s}\|\|\mathbf{X}_{\text{PO}}\|\\ &\dot{\dot{a}}_{2}=\ell_{2}\|\mathbf{s}\|\Big{\|}\dot{\mathbf{X}}_{\text{PO }}\Big{\|}^{2}\end{aligned} \right. \tag{45}\]  
where \(\ell_{i}\) is the positive constant. The control flowchart of the proposed controller is shown in Figure 7. The actual trajectory of the motion platform is calculated by the encoder and compared with the desired trajectory to obtain the error value. The error value is the input of ANF-TSMC controller, and the output of controller is the motor torque. By adjusting the motor torque, the trajectory error of the motion platform is reduced. Then the Lyapunov method is applied to prove the stability of the proposed control strategy. **Lemma 1**. : _The tracking error can converge to zero in finite time by designing a fast terminal adaptive sliding mode, control law and adaptive law for the tracking problem of a nonlinear system._  
Proof of Lemma 1.: The error of the adaptive law is \(\widetilde{a}_{i}=a_{i}-\dot{a}_{i}\), and the Lyapunov function is selected:  
\[V=\frac{1}{2}s^{T}s+\sum_{i=0}^{2}\frac{1}{2\ell_{i}}(a_{i}-\dot{a}_{i})^{2} \tag{46}\]  
Further, the derivative of the above equation is  
Figure 7: Control flowchart of the proposed controller. \[\dot{V} = \mathbf{s}^{T}\dot{\mathbf{s}}+\sum\limits_{i=0}^{2}\frac{1}{t}(\dot{a}_{i}-a _{i})\dot{a}_{i}\] \[= \mathbf{s}^{T}\Big{(}\mathbf{M}_{1}^{-1}\Big{(}\mathbf{\tau}+\mathbf{G}_{1}+\mathbf{F} _{\mathbf{R}}-\mathbf{C}_{1}\dot{\mathbf{X}}_{\mathbf{PO}}\Big{)}-\ddot{\mathbf{X}}_{\mathbf{d}}+\mathbf{K}_ {1}\dot{\mathbf{e}}+\mathbf{K}_{2}\dot{\lambda}\big{(}\mathbf{e},\dot{\mathbf{e}}\big{)}\Big{)}+ \sum\limits_{i=0}^{2}\frac{1}{t}(\ddot{a}_{i}-a_{i})\dot{a}_{i}\]  
Apply Equation (42),  
\[\dot{V} = \mathbf{s}^{T}\Big{(}-\mathbf{\theta}\mathbf{s}-\Big{(}\ddot{a}_{0}+\dot{a}_{ 1}\parallel\mathbf{X}_{\mathbf{PO}}\parallel+\dot{a}_{2}\parallel\ddot{\mathbf{X}}_{\mathbf{ PO}}\parallel^{2}\Big{)}sign(\mathbf{s})\Big{)}+\mathbf{s}^{T}\mathbf{D}_{1}+\sum\limits_{i=0}^{2} \frac{1}{t}(\dot{a}_{i}-a_{i})\dot{a}_{i}\] \[\leq \mathbf{s}^{T}\Big{(}-\Big{(}\mathbf{a}_{0}+\mathbf{a}_{1}\parallel\mathbf{X}_{ \mathbf{PO}}\parallel+\dot{a}_{2}\parallel\dot{\mathbf{X}}_{\mathbf{PO}}\parallel^{2} \Big{)}sign(\mathbf{s})+\mathbf{D}_{1}\Big{)}+\sum\limits_{i=0}^{2}\frac{1}{t}(\dot{a }_{i}-a_{i})\dot{a}_{i}\] \[\leq \parallel\mathbf{D}_{1}\parallel\parallel\mathbf{s}\parallel-\Big{(}\bm {a}_{0}+\mathbf{a}_{1}\parallel\mathbf{X}_{\mathbf{PO}}\parallel+\dot{a}_{2}\parallel \dot{\mathbf{X}}_{\mathbf{PO}}\parallel^{2}\Big{)}\parallel\mathbf{s}\parallel\] \[+\frac{1}{t_{0}}(\dot{a}_{0}-a_{0})\dot{\theta}_{0}\parallel\mathbf{s }\parallel+\frac{1}{t_{1}}(\dot{a}_{1}-a_{1})\ell_{1}\parallel\mathbf{s}\parallel \parallel\mathbf{X}_{\mathbf{PO}}\parallel\] \[+\frac{1}{t_{2}}(\dot{a}_{2}-a_{2})\ell_{2}\parallel\mathbf{s} \parallel\dot{\mathbf{X}}_{\mathbf{PO}}\parallel^{2}\] \[V=\frac{1}{2}s^{T}s+\sum_{i=0}^{2}\frac{1}{2\ell_{i}}(a_{i}-\dot{a}_{i})^{2} \tag{46}\]0  
where \(\delta=a_{0}+a_{1}\|\mathbf{X}_{\mathbf{PO}}\|+a_{2}\Big{\|}\dot{\mathbf{X}}_{\mathbf{PO}} \Big{\|}^{2}-\|\mathbf{D}_{1}\|\). According to Equation (37), \(\delta>0\). According to the theory of finite time stability, the tracking error can reach the sliding mode surface in finite time, and then converge to zero in finite time [39]. Lemma 1 is proved. \(\square\)  
**Remark 1**: _In the actual situation, due to the external interference and sampling frequency, it will not be accurate to zero [40]. The adaptive parameters will increase infinitely, so the dead zone technique is used to adjust the adaptive law._  
\[\dot{a}_{0}= \begin{cases}\ell_{0}\|\mathbf{s}\|,&\text{if}\|\mathbf{s}\|\geq\varepsilon \\ 0,&\text{if}\|\mathbf{s}\|<\varepsilon\end{cases}  \tag{49}\]  
\[\dot{a}_{1}= \begin{cases}\ell_{1}\|\mathbf{s}\|\|\mathbf{X}_{\mathbf{PO}}\|,&\text{if}\| \mathbf{s}\|\geq\varepsilon\\ 0,&\text{if}\|\mathbf{s}\|<\varepsilon\end{cases}  \tag{50}\]  
\[\dot{a}_{2}=