## 4 Control Strategy of CDPDR  
From the foregoing, the dynamic model of CDPDR can be expressed as  
\[M_{e}\ddot{X}_{PO}+C_{e}\dot{X}_{PO}-G-F_{r}=\tau \tag{33}\]  
where \(\tau=J^{T}r^{-1}\tau_{e}\). In engineering, the real physical parameters of CDPDR cannot be obtained accurately, and only the nominal model of CDPDR can be obtained generally. The system inertia matrix \(M_{e}\), the Coriolis force and centrifugal force matrix \(C_{e}\) and gravity matrix \(G\) be further written as  
\[\left\{ \begin{array}{l}\mathbf{M_{e}}=\mathbf{M_{1}}+\mathbf{E_{M}}\\ \mathbf{C_{e}}=\mathbf{C_{1}}+\mathbf{E_{C}}\\ \mathbf{G}=\mathbf{G_{1}}+\mathbf{E_{G}}\end{array} \right. \tag{34}\]  
where \(E_{M},\mathbf{E_{C}}\) and \(\mathbf{E_{G}}\) are the modeling errors corresponding to \(M_{e}\), \(C_{e}\) and \(G\). \(M_{1}\), \(C_{1}\) and \(G_{1}\) are the modeled parts. Considering the modeling errors, the dynamic model can be further written as  
\[M_{1}\ddot{X}_{PO}+C_{1}\dot{X}_{PO}-\mathbf{G_{1}}-\mathbf{F_{R}}=\tau \tag{35}\]  
where \(\mathbf{F_{R}}=\mathbf{F_{r}}-\mathbf{E_{M}}\ddot{\mathbf{X}}-\mathbf{E_{C}}\dot{\mathbf{X}}+\mathbf{E_{G}}\) is the cumulative error total vector. Define the position error vector of the CDPDR motion platform as \(e=\mathbf{X}_{PO}-\mathbf{X_{d}}\), further the derivative of the error vector as \(\dot{e}=\dot{\mathbf{X}}_{PO}-\dot{\mathbf{X}}_{d}\) and \(\ddot{\mathbf{e}}=\ddot{\mathbf{X}}_{PO}-\ddot{\mathbf{X}}_{d}\). Combined with the dynamic model,  
\[\ddot{\mathbf{e}}=M_{1}^{-1}\Big{(}\tau+\mathbf{G_{1}}+\mathbf{F_{R}}-\mathbf{C_{1}}\dot{\mathbf{X} }_{PO}\Big{)}-\ddot{\mathbf{X}}_{d} \tag{36}\]  
The following assumptions are made in order to realize the adaptive fast terminal sliding mode control. **Assumption 1**. : _The cumulative error total vector is a bounded matrix [37], satisfying_  
\[\|\mathbf{D_{1}}\|=\Big{\|}M_{1}^{-1}\mathbf{F_{R}}\Big{\|}<a_{0}+a_{1}\|\mathbf{X_{PO}} \|+a_{2}\Big{\|}\dot{\mathbf{X}}_{PO}\Big{\|}^{2} \tag{37}\]  
_where \(a_{0}\),\(a_{1}\) and \(a_{2}\) are unknown positive constants and \(\|\cdot\|\) denotes the Euclidean norm of a vector._  
For the precise trajectory tracking control of the motion platform, a controller with strong robustness is designed based on the method of adaptive non-singular fast terminal sliding mode. Ref. [38] proposed an improved terminal sliding mode (SMC) manifold to solve the singularity problem caused by negative fractions. On this basis, the selected sliding surface is as follows:  
\[\mathbf{s}=\dot{\mathbf{e}}+\mathbf{K_{1}}\mathbf{e}+\mathbf{K_{2}}\lambda(\mathbf{e}) \tag{38}\]  
where \(\mathbf{K_{j}}=diag\big{(}k_{j1},\cdots,k_{jn}\big{)},j=1,2\) is a positive definite matrix, and \(\lambda(\mathbf{e})=\big{[}\lambda(e_{1}),\ \ \cdots,\lambda(e_{n})\big{]}^{T}\), which is specifically expressed as  
\[\lambda(e_{i})=\left\{  
where \(\dot{\lambda}\big{(}e,\dot{e}\big{)}=\Big{[}\dot{\lambda}\big{(}e_{1},\dot{e}_{1} \big{)}\quad\cdots\quad\dot{\lambda}\big{(}e_{n},\dot{e}_{n}\big{)}\Big{]}^{T}\),  
\[\dot{\lambda}\big{(}e_{i},\dot{e}_{i}\big{)}=\Big{\{} \begin{aligned} &(q/p)e_{i} ^{q/p-1}\dot{e}_{i},\quad\bar{s}_{i}=0\,or\,\bar{s}_{i}\
eq 0,|e_{i}| \geq\mu\\ &\zeta_{1}\dot{e}_{i}+2\zeta_{2}sign(e_{i})e_{i}\dot{e}_{i}, \quad\bar{s}_{i}\
eq 0,|e_{i}|\leq\mu\end{aligned}  \tag{41}\]  
Finally, the control law of CDPDR system can be designed as follows:  
\[\tau=\tau_{eq}+\tau_{sw} \tag{42}\]  
\[\tau_{eq}=\mathbf{M}_{1}\Big{(}\ddot{\mathbf{X}}_{d}-\mathbf{K}_{1}\dot{e}-\mathbf{K}_{2}\dot {\lambda}\big{(}e,\dot{e}\big{)}\Big{)}+\mathbf{C}_{1}\dot{\mathbf{X}}_{\text{PO}}- \mathbf{G}_{1} \tag{43}\]  
\[\tau_{sw}=-\mathbf{M}_{1}\Big{(}\mathbf{\beta}\mathsf{s}+\Big{(}\dot{a}_{0}+\dot{a}_{ 1}\|\mathbf{X}_{\text{PO}}\|+\dot{a}_{2}\Big{\|}\dot{\mathbf{X}}_{\text{PO}}\Big{\|}^{ 2}\Big{)}sign(\mathbf{s})\Big{)} \tag{44}\]  
where \(\mathbf{\beta}=diag(\beta_{11},\cdots,\beta_{nn})\) is a diagonal positive definite matrix. \(\dot{a}_{i}\) is the estimate of \(a_{i}\), and its adaptive law is given as  
\[\left\{