## 7 Experiments and Results
### Simulation  
In reality, there are many means of 8-cable CDPM wiring, so we tried to offer the two most common configurations. Both configurations mentioned in this section have symmetric geometry. Figure 17a depicts the configuration where all eight wires are attached to top of the workspace and the other depicts the configuration where four wires are attached to top of the workspace and the remaining are attached to the bottom of the workspace. These configurations can be found in many prototypes. The first one is famous thanks to the introduction of CoGiRo, and also considers the optimal geometry of two redundant actuators suspended CDPM [22]. IPAnema1, presented by [PERSON], [PERSON] et al. in [23], is the well-known prototype of the second configuration illustrated in Figure 17b. In this experiment, the moving platform of simulated CDPM is expected to track on the desired trajectory illustrated in Figure 18 with the given constraint \(\left[ \begin{array}{cccc}z_{\mathbf{P}}&\alpha_{\mathbf{P}}&\beta_{\mathbf{P }}&\gamma_{\mathbf{P}}\end{array} \right]=\left[ \begin{array}{cccc}1.5&0&0&0 \end{array} \right]\). The 5 th order polynomial control equations are applied in order to generate this trajectory. Time-based velocities of the end-effector along the \(X\)-axis and \(Y\)-axis are illustrated in Figure 19.  
Because the mentioned CDPMs in this section are overconstrained systems, there is a presence of the non-unique pseudo inverse of (50). To illustrate the performance of our model during tracking on the desired trajectory, a simple numerical approach is employed to solve that issue. The goal of this procedure is to find out a smooth series of minimum tensions which also satisfies the desired trajectory  
Figure 19: The velocity of moving platform in desired trajectory. Figure 18: The trajectory of moving platform for simulation test. and constraints of our proposed model. Firstly, the particular solution \(\mathbf{t}_{x_{0}}=\left[ \begin{array}{cccc}T_{x_{1_{0}}}^{A}&T_{x_{2_{0}}}^{A}& \ldots&T_{x_{\delta_{0}}}^{A}\end{array} \right]^{T}\) of (49) is a minimum L\({}^{2}\) norm solution. The general solution of (49) can be expressed as (53)  
\[\mathbf{t}_{x}=\mathbf{t}_{x_{0}}+\mathbf{N}\boldsymbol{\lambda} \tag{53}\]  
where \(\mathbf{N}\) is the \(8\times 2\)-matrix that represents the orthonormal basis for the null space of \(\mathbf{H}\) and the particular solution \(\mathbf{t}_{x}\) of (49) depends on the value of \(\boldsymbol{\lambda}=\left[ \begin{array}{cccc}\lambda_{1}&\lambda_{2} \end{array} \right]^{T}\). The constraint of (53) is introduced, and can be written as (54)  
\[\mathbf{t}_{x}>\mathbf{t}_{x_{\min}},T_{x_{\min}}=\frac{-\mathbf{b}_{\pi_{i}}} {\mathbf{a}_{\pi_{i}}}+T_{x_{i}adj},T_{x_{i}adj}>0 \tag{54}\]  
where \(\mathbf{t}_{x_{\min}}=\left[ \begin{array}{cccc}T_{x_{1_{\min}}}^{A}&T_{x_{2 _{\min}}}^{A}&\ldots&T_{x_{\delta_{\min}}}^{A}\end{array} \right]\). In fact, the addition of \(T_{x_{i_{adj}}}^{A}\) to maintain the accuracy of our proposed model as we discussed in Section 3. Applying the constraint (54) to Equation (53), the feasible solution set \(\Theta_{\lambda}\) is obtained and it is surrounded by a planar polygon. The polygonal boundary of \(\Theta_{\lambda}\) is defined by the vertex set \(\Omega_{\lambda}=\{\lambda_{i}\}_{i=1}^{n}\).However, the obtained feasible solution set \(\Theta_{\lambda}\) is presented in the domain of \(\mathbf{N}\). Since we cannot directly determine suitable \(\mathbf{t}_{x}\) in that space, \(\Theta_{\lambda}\) is transformed to new set \(\Theta_{\mathbf{t}}\) presented in the null space of \(\mathbf{H}\) by executing the linear affine transformation (53) to set \(\Omega_{\lambda}\), resulting in a new vertex set \(\Omega_{\mathbf{t}}=\left\{\mathbf{t}_{x_{i}}\right\}_{i=1}^{n}\) that defines the \(\mathbb{R}^{8}\)-polygon. The kth particular solution \({}^{k}\mathbf{t}_{x}\) is selected based on \({}^{k-1}\mathbf{t}_{x}\).