## 3 Dynamic Modeling of CDPDR
### Dynamic Model
Equation (20) is further combined into matrix form, which is expressed as  
\[\mathbf{M}\ddot{\mathbf{X}}_{\mathbf{PO}}+\mathbf{C}\dot{\mathbf{X}}_{\mathbf{PO}}-\mathbf{G}-\mathbf{F}_{r}=\bm {J}^{T}\mathbf{T} \tag{21}\]  
the dynamic Jacobian matrix in the inertial coordinate system, \(\mathbf{J}= \begin{bmatrix}(\mathbf{u}_{1}^{\mathbf{O}})^{\mathsf{T}}&\begin{pmatrix}\mathbf{P} _{1}^{\mathbf{O}}\times\mathbf{u}_{1}^{\mathbf{O}}\end{pmatrix}\end{bmatrix} \). The relationship between \(\mathbf{J}\) and the kinematic Jacobian matrix \(\tilde{\mathbf{J}}\) is  
\[\mathbf{J}=\tilde{\mathbf{J}}\mathbf{S}^{\mathsf{T}} \tag{22}\]  
In Equation (22),  
\[\mathbf{S}= \begin{bmatrix}\mathbf{R}_{\mathbf{S}}^{\mathbf{O}}&0_{3\times 3}\\ 0_{3\times 3}&\mathbf{R}_{\mathbf{S}}^{\mathbf{O}}\end{bmatrix}  \tag{23}\]  
On the basis of the dynamic model, the influence of the delay of the motor on the control accuracy is further considered. The motor model is established as follows:  
\[\mathbf{J}_{m}\tilde{\mathbf{\theta}}+\mathbf{V}_{m}\tilde{\mathbf{\phi}}+\mathbf{r}\mathbf{T}=\mathbf{ \tau}_{e} \tag{24}\]  
where \(\mathbf{J}_{m}\) is the moment of inertia matrix of the motor. \(\mathbf{V}_{m}\) is the matrix of the viscous damping coefficient of the motor. \(\mathbf{\theta}=\mathbf{r}^{-1} \begin{bmatrix}I_{1(t)}-I_{1(t-1)}&I_{2(t)}-I_{2(t-1)} &I_{3(t)}-I_{3(t-1)}&I_{4(t)}-I_{4(t-1)}\end{bmatrix} \) is the rotation angle of the winch, and \(I_{i(t)}-I_{i(t-1)}\) is the length difference of the \(i\)-th cable at time t and t -1. \(\mathbf{r}\) is the radius of the winch drum. \(\mathbf{\tau}_{e}\) is the motor torque matrix. In conjunction with Equations (21) and (24), the CDPDR system dynamic model can be expressed as  
\[\mathbf{M}(\mathbf{x})_{e}\ddot{\mathbf{X}}_{\mathbf{PO}}+\mathbf{C}(\mathbf{x},\dot{\mathbf{x}})_{e}\dot {\mathbf{X}}_{\mathbf{PO}}-\mathbf{G}-\mathbf{F}_{r}=\  
\(T_{\text{imax}}\) is determined by the strength of the \(i\)-th cable and the maximum power of the motor, which is used to ensure that the cable will not be broken due to excessive tension.  
Decompose the cable tension to  
\[\mathbf{T}=\mathbf{T}_{\text{S}}+\mathbf{T}_{\mathbf{H}} \tag{28}\]  
where \(\mathbf{T}_{\text{S}}\) is the cable tension special solution, and \(\mathbf{T}_{\mathbf{H}}\) is the cable tension general solution. The singular value decomposition of the structural matrix \(J^{T}\) is carried out; meanwhile, the generalized inverse \(J^{T+}=\mathbf{J}\left(J^{T}\mathbf{J}\right)^{-1}\) of the structural matrix is introduced. The special solution term and general solution term of tension are obtained as follows:  
\[\mathbf{T}_{\text{S}}=\mathbf{J}^{T+}\left(M\ddot{\mathbf{X}}_{\mathbf{PO}}+\mathbf{C}\dot{\mathbf{X}}_ {\mathbf{PO}}-\mathbf{G}-\mathbf{F}_{\mathbf{r}}\right) \tag{29}\]  
\[\mathbf{T}_{\mathbf{H}}=\mathbf{N}(\mathbf{J})\lambda \tag{30}\]  
where \(\mathbf{N}(\mathbf{J})\) is the one-dimensional null space basis of the structural matrix \(\mathbf{J}\). Substituting Equations (29) and (30) into Equation (27), then  
\[\underline{\lambda}\leq\lambda\leq\overline{\lambda} \tag{31}\]  
\[\text{where}\quad\underline{\lambda}=\min_{1\leq i\leq m}\left(\max_{1\leq i \leq m}\left(\max_{1\leq i\leq m}\frac{T_{i,\text{min}}-T_{S_{i}}}{N(l)_{i}}, \frac{T_{i,\text{max}}-T_{S_{i}}}{N(l)_{i}}\right)\right)\text{ is the lower limit of }\lambda.\]  
\(\overline{\lambda}=\max_{1\leq i\leq m}\left(\min_{1\leq i\leq m}\left(\max_{ 1\leq i\leq m}\frac{T_{i,\text{min}}-T_{S_{i}}}{N(l)_{i}},\frac{T_{i,\text{max }}-T_{S_{i}}}{N(l)_{i}}\right)\right)\) is the upper limit of \(\lambda\). Equation (31) constrains the value range of \(\lambda\), and the general solution of cable tension can be changed by adjusting the value of \(\lambda\). The cable tension optimization problem is transformed into a single objective optimization problem for \(\lambda\). The least-variance method is employed to solve this problem, and the following optimization model is established:  
\[