## 1 Introduction  
Cable-driven robots, commonly called cable robots, are an important field in robotics. The parallel structure of cable robots leads to high payload capacity and high speeds of the end-effector throughout their workspace [1] due to low-moving masses. Moreover, cable robots stand out from other parallel robots since their links are constituted by cables wounded around actuated pulleys, allowing not only for a simple and cheap design but also for large workspace manipulation tasks. These advantages lead to a wide spreading of cable robots, which not only have been suggested for industrial applications [2] but also for home assistance and rehabilitation purposes [3] and for entertainment [4]. With respect to rigid-link parallel robots, the use of cables introduces an additional constraint due to their ability to resist tension but not compression, thus each wire can pull but not push the end-effector. This leads to safety concerns when cable robots are adopted in crowded areas or interact with humans, especially since an emergency stop procedure for these devices is not as simple to implement as in rigid-link robots. Although the damage occurring due to failures of cable-driven manipulators can be very severe, research on failure analysis and fault tolerance has not been sufficiently explored [5] in comparison to serial manipulators. Indeed, implementing an emergency procedure for these manipulators is not trivial, since stopping the actuators abruptly does not imply that the end-effector rests at a stable position, thus making failure analysis particularly challenging for cable robots. An analysis of failure modes was provided in [6] as an application of the proposed wrench-based analysis, which can determine if a failure is due to a negative tension, i.e., compression, or if the tension limits of a wire have been exceeded. In [7], the faults due to wire jamming, i.e., wire at a constant length, are also presented and the author proposed a certain degree of redundancy to provide fail-safe operation and fault tolerance. A very basic solution to improve safety in case of cable failure at the design stage can be admittedly achieved by increasing the number of cables far beyond the minimum needed: indeed, cable robots with many degrees of redundancy present a higher fault tolerance, particularly with a broken cable. However, this approach is far from being the optimal one, since, in addition to being more expensive, cables tend to obstruct the workspace and it can even be difficult to avoid interference between the cables. Therefore, it is important to safely control the robot in case of failure without increasing cable redundancy excessively. In [8], an emergency strategy for a two-dimensional cable robot is presented. This strategy aims to guide and stop the end-effector in a safe position on the basis of two different approaches: minimization of the kinetic energy and potential fields. In [9], a recovery strategy for a spatial 3-DOF four-wires cable robot is presented: the proposed approach aims at safely recovering the end-effector after a cable failure by controlling the robot with an elliptic trajectory even outside its residual Static Equilibrium Workspace (SEW) (i.e., a set of the end-effector poses, for a given motor mount configuration, where static equilibrium can be obtained while applying tension to all cables, assuming infinite maximum cable lengths and tensions [10]). Indeed, cable failure induces a variation in the SEW which should be taken into account by the recovery strategy algorithm. This is shown in [11], where a feasible motion leads the end-effector toward a Safe Pose (SP), where static equilibrium can be achieved. The end-effector is then stopped at the SP. The approach splits the recovery trajectory into two parts, each with different motion requirements, called the Connecting Path and Straight-Line Path, respectively.