## 3 Quasi-Static and Inverse Kinematic Characteristics of Single Cable  
Since everything will be discussed in general frame \(\{A\}\) in this section, if any vector without a denotation of frame, then this vector respect to \(\{A\}\). For the convenience, \(\mathbf{T}=\left[ \begin{array}{cc}T_{x}&T_{z}\end{array} \right]^{T}\in\Upsilon\) denotes the tension acts on free extremity \(\mathbf{B}=\left[ \begin{array}{cc}x_{\mathbf{B}}&z_{\mathbf{B}}\end{array}  \right]^{T}\in\mathfrak{I}\), \(\mathfrak{c}=\left[ \begin{array}{cc}\rho&E&A\end{array} \right]^{T}\in \mathbf{N}\) denotes the properties of materials that are used in cable, \(\mathbf{S}=\left[ \begin{array}{cc}T_{x}&T_{z}&L\end{array} \right]^{T}\in\Delta\) denotes the solution of our study. According to (1), (2) and (4), the sagging cable model of Irvine in [17] can be used to archive the position of the free extremity \(\mathbf{B}\) of cable and can be expressed in (18). \[I:\Delta\times\mathbf{N}\rightarrow\mathfrak{I},\left[ \begin{array}{cc} \mathbf{S}&\varsigma\end{array} \right]^{T}\mapsto\mathbf{B} \tag{18}\]  
The main study of this section is try to figure out the feasible tensions and feasible unstrained lengths based on the desired position of the free extremity of cable \(\mathbf{B}^{d}=\left[ \begin{array}{cc}x_{\mathbf{B}^{d}}&z_{\mathbf{B}^{d}} \end{array} \right]^{T}\) with given \(\varsigma\) that can be obtained by deploying (19). \[f:\mathfrak{I}\times\mathbf{N}\rightarrow\Delta,\left[ \begin{array}{cc} \mathbf{B}^{d}&\varsigma\end{array} \right]^{T}\mapsto\mathbf{S} \tag{19}\]  
However, the exact (19) cannot be directly derived via (18) by its natural complexity. In previous research, based on given \(\mathbf{B}^{d}\) and the constant material of cable, numerical approaches have been deployed to find out feasible solution \(\mathbf{S}\). However, most of these methods cannot be used in real-time because of time-consumption. The approximated models were also introduced in research [13; 16; 20]. In this work, the approximated model of (19) will be constructed based on (18) by applying both the empirical statistical analysis approach and static analysis approach. Generally, the cost function of our approximated model can be express as Equation (20). \[\varepsilon=\left|x_{\mathbf{B}^{d}}-x_{\mathbf{B}}\right|+\left|z_{\mathbf{B}^{ d}}-z_{\mathbf{B}}\right| \tag{20}\]  
The optimum model is obtained if \(\varepsilon=0,\ \forall\mathbf{B}^{d}\in\mathfrak{T}\). However, it is impossible for any approximated model that converges to the actual model, i.e., we always have \(\varepsilon>0\). However, we restrict feasible solutions with the maximum error \(\varepsilon<\varepsilon_{\max}\). By using Irvine model with fixed \(\varsigma\) and \(L\), a set of feasible tension can be found to satisfy a given desired position \(\mathbf{B}^{d}\). The set of feasible tension of an instance is illustrated in Figure 3.  
**Statement 1**. : _Let \(\psi\) be the feasible tension set with given \(\varepsilon_{\max}\), \(L\), \(\mathbf{B}^{d}\) and \(\varsigma\) According to Irvine model, with given \(L\), \(\mathbf{T}\) and \(\varsigma\) there exists only one \(\mathbf{B}\), so (18) is the injective function. Similarly, (19) is also the injective function. Hence, if \(\varepsilon_{\max}\to 0\) then \(\left|\psi\right|\to 1\), i.e., the set has unique tension \(\mathbf{T}\in\psi\)._  
Collecting all of \(\psi\) with \(L\in\mathbb{R}^{+}\), we can get the feasible solution set \(\Psi\) of (19) with given \(\varsigma\) and considered \(\mathbf{B}^{d}\) as the input parameter. Figure 4 illustrates a specific case of \(\Psi\) with \(d=L-D\) and \(D\) is the \(L^{2}\)-norm of \(\mathbf{B}^{d}\). Figure 4: The feasible solutions \(\mathbf{S}\in\Psi\) and \(\psi\) is the blue segment line with \(\varepsilon_{\max}>0\). **Statement 2**. : _Let \(\Psi\subset\Pi\) be the feasible solution set with given \(\mathbf{B}^{d}\) and \(\varsigma\).