## 6 Case Studies and Simulation  
In this section, the proposed wrench closure condition is applied to search for the wrench-closure workspace of two configurations CDPM defined in Figure 1. For convenience, we define three types of WCW:  
* Total orientation wrench closure workspace (TOWCW): Other than the definition in [22], in this article, TOWCW presents the whole WCW when position and orientation of moving platform are considered as variables. Notice that, because of the limitation in visualization, we can plot the TOWCW of 3-dof CDPM. * Constant orientation wrench closure workspace (COWCW): For 6-dof CDPM, we can visualize the projection of its TOWCW on up to 3-dimensional workspace. If the projected space was only related to the position of moving platform, this projection is called COWCW. For 3-dof CDPM, COWCW is the section obtained at specific \(\gamma_{P}\). * Constant position wrench closure workspace (CPWCW): Similar to COWCW, CPWCW is the projection of TOWCW when the projected space was only related to the orientation of moving platform, i.e., position of moving platform is constant. As shown in Figure 7, WCW of the first configuration CDPM shown in Figure 1 is generated by point-wisely employing proposed WWC in the grid map, which has a dimension of 1 m \(\times\) 1 m \(\times\) 2\(\pi\) and resolution of 0.02 m \(\times\) 0.02 m \(\times\) 0.02 rad. Total orientation WCW of planar CDPM is a subset of 3-dimensional space (\(x\)-\(y\)-\(y\)-\(\gamma_{P}\)); it is illustrated in Figure 7a. Moreover, 2-dimensional sections of total orientation WCW of planar CDPM obtained at \(\gamma_{P}\) = 0\({}^{\circ}\), 0.052 rad and \(-\)0.052 rad are also plotted in Figure 7b-d. The orange, yellow and purple closed curves in Figure 7a represent the boundaries of slices that present in Figure 7b-d, respectively. However, this configuration is simple, since it has 3-dof and only one redundant actuator. Hence, it cannot verify the feature \"applicable for a completely or redundantly restrained CDPM\" of proposed WWC. The second configuration CDPM shown in Figure 1 is a 2-redundantly actuated spatial CDPM, and it can be used to reinforce the flexibility of proposed WWC. As shown in Figure 8, WCW of the second configuration CDPM shown in Figure 1 is generated by applying proposed WWC. Constant orientation WCW with \(\alpha_{P}\) = 0, \(\beta_{P}\) = 0 and \(\gamma_{P}\) = 0.052 rad, and constant position WCW with \(x_{P}\) = 0, \(y_{P}\) = 0 and \(z_{P}\) = 0 are shown in Figure 8a,b, respectively. As shown in Figures 7c and 8a, the cross-section perpendicular to the \(z\)-axis of COWCW of the spatial CDPM is trickly equivalent to the COWCW of the  
Figure 6: The overview of proposed algorithm to check the feasibility of certain pose. planar CDPM due to their similar dimensions. This comment is also verified by [PERSON] et al. in [16]. In Figure 8b, the CPWCW is not continuous and in practice, this type of CDPM just can work on a small set of orientational spaces. Figure 8: Wrench closure workspace of the second configuration shown in Figure 1. (**a**) constant orientation WCW with \(\kappa_{\text{P}}=0\), \(\beta_{\text{P}}=0\) and \(\gamma_{\text{P}}=0.052\) rad; (**b**) constant position WCW with \(x_{\text{P}}=0\), \(y_{\text{P}}=0\) and \(z_{\text{P}}=0\). Figure 7: Wrench closure workspace of the first configuration shown in Figure 1. (**a**) Total orientation WCW; (**b**) 0 rad-section; (**c**) 0.052 rad-section; (**d**) \(-0.052\) rad-section. In simulations, the time-consumption of proposed WCC used to determine whether a pose presents in WCW is also considered. Time-consumption is not the same in each performance due to two factors: size of wrench matrix and the interruption of algorithm. The first factor depends on configuration of CDPM, it is related to the number of used cables and DoF of moving platform. In the worst case, when a pose belongs to WCW and the algorithm for checking the feasibility of system of inequations must evaluate all I-minor of given wrench matrix.