I. Introduction
Cable-driven redundant parallel manipulators (CDRPM) has several attractive features and some advantage compared to conventional parallel manipulators. In spite of many advantages and promising potentials, there are many challenging issues in the design and development of cable robots. Workspace analysis is always a crucial issue in the design of any robotic manipulator. The uni-directional constraint imposed by cables causes the analysis of such robots much more difficult than that of conventional robotic manipulators. In general, workspace is defined as the set of all posture of manipulator that can reach by tense cables. In literature, four different types of workspaces have been introduced based on various definitions. A number of researchers addressed the set of postures that the end-effector can attain statically while only taking gravity into account [1]. In most of the analysis studied on this kind of workspace, numerical approaches are used to find out the corresponding workspace for a specific system [2]. Some researchers addressed a set of postures when cable robots are needed to exert specific force/moment combinations to interact with environment, besides maintaining its own static equilibrium. Ebert-Uphoff and Voglewede called this type of workspace as wrench feasible workspace [3]. Another workspace that has been introduced is the dynamic workspace along with a set of wrench called pseudo-pyramid, defined by Barrette and Gosselin as the set of all postures of the end-effector of the cable robot with specific acceleration requirement. The boundaries of this type of workspace are analytically formed for planar cable robots [4]. Finally, one of the most general workspace definitions is referred to the workspace in which any wrench can be generated at the moving platform while cables are in tension. Verhoeven and Hiller termed such workspace as controllable workspace [5]. This kind of workspace depends only on geometry of manipulator such as position of fixed and moving attachment point [6]. Therefore, it is interested in general design view point. Pham et al. proposed a “recursive dimension reduction algorithm” to check the force-closure condition of cable manipulators. The algorithm was derived based on convex analysis presented in [7]. Although the algorithm is systematic, no mathematical proof was provided to show that it is equivalent to the original force-closure theorem as defined in their cited reference [7]. Moreover, the algorithm is practically inefficient in computation. Ma and Diao described a systematic numerical method of verifying the existence of force-closure at a specific pose of a general 6-DOF cable manipulator with seven or more cables [8]. Gouttefarde and Gosselin addressed the same concept named Wrench-Closure Workspace (WCW), and determined the boundaries of the workspace for planar cable robots analytically [6]. They used geometric analysis of the null space of structure matrix to investigate the WCW. Null space analysis is the most common methods for determination of this kind of workspace, which is associated with challenges like extreme complexity of computations and increasing of the degrees of freedom and degrees of redundancy of the robot. In this paper among the alternative notions introduced for the workspace in which any wrench can be generated at the moving platform while cables are in tension, the term controllable workspace is used. All the proposed method including the analytical and numerical methods proposed to analyze this problem, suffers from a lack of physical interpretation of controllable workspace.