## 1 Introduction  
### Motivation  
The cable-driven parallel manipulators (CDPMs) have a similar structure to the traditional parallel kinematic machines that consist of a moving platform (end-effector), the robot's fixed platform, and many links used to connect the moving platform to the fixed platform. The main difference between them is that, in the CDPMs, the rigid links are replaced by the flexible light-weight cables, and their lengths are adjusted by winches that are usually mounted on the fixed platform. By changing the length of these cables, the position and orientation of the moving platform are controlled. In recent decades, CDPMs are of great interest to researchers, because they overcome many disadvantages of conventional parallel mechanisms, such as easy assembling and disassembling, simple configuration, high-load capacity, etc. By these dominant advantages, CDPMs are widely applicable in many fields: contour crafting system in construction [1; 2], air vehicle simulation [3], etc. The giant CDPM developed by [PERSON] and [PERSON] was applied in agriculture [4]. Aerial robotic cameras used for broadcasting are studied in [5]. Besides the advantages that it brings, the use of cables instead of rigid links also presents many issues for CDPM. In order to deal with a trajectory planning task for the moving platform while balancing an external wrench exerted at the moving platform, the cable must be in stretching, i.e., there must always be cable tension forces exerted in them. The relationship between the elongation of cable and cable tension forces was also coordinated in our previous work [6]. Due to the unilateral property of the cable tension force, the cable can only be pulled but not be pushed, resulting in the wrench matrix of CDPM perhaps being in full rank, but not providing any solution. This leads to the limitation of a workspace of this kind of mechanism. In practice, many redundant actuators are added, not only for increasing load capacity, but also for reducing the limitation of workspace. The cable failure in CDPM is also researched in many articles [7, 8], since this affects the limitation of workspace. In this study, we are interested in a set of poses of the moving platform in which CDPM can balance any external wrench by stretching the cables through winches. This set of poses is also called wrench-closure workspace (WCW). Details of other important workspaces of CDPM are clearly introduced in [9]. According to [PERSON] and [PERSON] (2006), WCW depends on the geometry of the mechanism only [10]. Hence, WCW is the prerequisite for an optimal design configuration and checking the validation for a solution of the trajectory planning task of CDPM. Therefore, WCW is the most important workspace and has been widely studied. Currently, the determination of WCW can be classified into two groups: the determination of the boundary of WCW and determination based on wrench closure condition (WCC). The first group directly sketches the WCW by using an analytical approach, while the second must be combined with some search algorithms to complete the same task. However, in this paper, the method that determines whether a pose of the moving platform belongs to WCW is of interest.