## 2 Cable-Driven Parallel Manipulators Kinematics  
Let's consider the general structure of CDPM illustrated in Figure 2 that contains a moving platform connected with a fixed platform through several flexible cables. The reference frame {\(W\)} is attached to the fixed platform and the moving frame {\(R\)} is attached to the center of mass of moving platform. For convenience in notation, the right-superscript denotes the frame where the mentioned object reference to, and any vector without a right-superscript is implied to reference to {\(W\)}. The general coordinate of {\(R\)} with respect to {\(W\)} is defined by the vector \(x\) = {\(P\)\(\theta\)}\(\mathbb{T}\) where \(P\) = {\(x_{p}\)\(y_{p}\)\(z_{p}\)}\(\mathbb{T}\) and \(\theta\) = {\(\alpha_{p}\)\(\beta_{p}\)\(\gamma_{p}\)}\(\mathbb{T}\) are the positioning vector of center of mass and rotational vector that represents Euler angle (\(z\)-\(y\)-\(x\)) of the moving platform, respectively. Let \(A_{i}\) and \(B_{i}\) be the positioning vectors that represent the end of the ith cable at the pole and at the moving platform, respectively. According to Figure 2, we have \(a_{i}\) = \(A_{i}B_{i}\) and \(b_{i}\) = \(PB_{i}\). If the general coordinate of the moving platform is given, then the direction of tension exerted on any cable can be obtained through Equation (1).  
\[\dot{a}_{i}=\frac{A_{i}-\left(P+\frac{W}{R}R^{R}b_{i}\right)}{\left\|A_{i}- \left(P+\frac{W}{R}R^{R}b_{i}\right)\right\|} \tag{1}\]  
where \({}_{R}^{W}\mathbf{R}\) is the rotational matrix that represents the orientational of {\(R\)} respect to {\(W\)} and it can be defined by (2).  
\[{}_{R}^{W}\mathbf{R}=\left[\begin{array}{cc}C_{\gamma p}C_{\delta P}&C_{ \gamma p}S_{\beta P}S_{aP}-S_{\gamma p}C_{aP}&C_{\gamma p}S_{\beta P}C_{aP}+S_{ \gamma p}S_{aP}\\ S_{\gamma p}C_{\delta P}&S_{\gamma p}S_{\delta P}S_{aP}+C_{\gamma p}C_{aP}&S_{ \gamma p}S_{\delta P}C_{aP}-C_{\gamma p}S_{aP}\\ -S_{\delta P}&C_{\delta P}S_{aP}&C_{\delta P}C_{aP}\end{array}\right] \tag{2}\]  
where \(C_{x}\) and \(S_{x}\) represent for \(\cos(x)\) and \(\sin(x)\), respectively.  
In order to balance the external wrench \(\psi_{e}\) that exerted on \(P\), the actuators mounted in the fixed frame are controlled to create cable tension forces. Moreover, the cable tension forces also affect cable lengths, while cable lengths are desired results in the kinematic problem of CDPM. Hence, the derivation of the relationship between \(\psi_{e}\) and cable tension forces is an interesting issue. Let \(m\) and \(n\) be the degree of freedom of moving platform and the number of cables used in system, respectively. Considering the static equilibrium system of the moving platform, the relationship between the cable tension force and external wrench is established in Equation (3).  
\[\mathbf{Wt}=\psi_{e} \tag{3}\]  
where \(\mathbf{W}\in R^{mxn}\) is the wrench matrix that has ith column vector is defined in (4), and \(t\) = {\(t_{1}\)\(\dots\)\(t_{2}\)} is the vector that represents the set of magnitude of cable tension forces with \(t_{i}\geq 0\), \(\forall i\) = {0, \(\dots\)\(,n\)}.  
Figure 1: Two configurations CDPM: (**a**) 1-redundantly actuated planar CDPM; (**b**) 2-redundantly actuated spatial CDPM.