## 1 Introduction  
Cable-Suspended Parallel Robots (CSPRs) are a class of parallel robots where the end-effector (EE) is supported by \(n\) extendable cables and in which the gravity force is used to keep cable tensions \(\tau_{1},\ldots,\tau_{n}\) positive. In this work, we focus in particular on robots that employ as many (taut) cables as the number of Degrees-of-Freedom (DoFs) of the EE. Such robots are often considered to move quasi-statically, that is, slowly enough that inertial effects can be disregarded. If a CSPR operates in static or quasi-static conditions, the EE must remain within the Static Equilibrium Workspace (SEW), namely, the set of poses where the EE can be in static equilibrium [1]. However, considering dynamic trajectories has become more and more important over the years. In [2], the authors defined the Dynamic Workspace (DW) as the set of EE poses that can be reached with positive cable tensions by taking advantage of inertia forces, with the EE moving in non-static conditions (nonzero velocity or acceleration): since the SEW is a subset of the DW, this expands the potential applications of CSPRs. For a dynamic trajectory to be feasible, the cable tensions must be positive at any instant, as the EE moves along the trajectory. This condition can be verified numerically, by solving the inverse dynamics problem at each time step; however, this approach can be computationally demanding, to the point that it is difficult to apply in real-time conditions. Another possibility is to use analytic methods: thus, several authors proposed trajectories defined by harmonic motions where the frequency must lie within a feasible range [3; 4; 5; 6; 7; 8; 9] that can be directly computed from the trajectory parameters. One of the first works in this field was [8], where the author considered several types of parametric dynamic trajectories; this work was extended and generalized in [7], where the authors considered elliptic paths in space, including circles and straight lines as special cases. Later works found similar results for cable robots having rotational DoFs [6] or overconstrained cable-driven parallel robots (CDPRs) [9]. Finally, it was proved in [10; 11] that the feasibility conditions found for the 3-cable, 3-DoF spatial robot with point-mass EE (considered in [3; 4; 5; 7; 8]) can be extended to a 6-cable, 3-DoF spatial robot with purely translational motion and a finite-size EE; this is a more realistic architecture for practical applications (for instance, a gripper can be attached on the EE [12] to carry an object). Operations that require dynamic manipulation of an object, such as throwing and catching, were attempted by both rigid-link robots and CSPRs. These techniques can be used not only to enlarge the workspace, but also to reduce operation times, which provides greater application possibilities. For rigid-link robots, some of the first works in this field were [13; 14], where the authors considered a one-joint manipulator performing throwing and batting operations on an object in the plane, and proved that pose and velocity could be completely controlled (within a subset of the state space) by underactuated manipulation. Other works on 1-DoF manipulators with similar dynamic motions were reported in [15], where the authors presented a 1-DoF robot prototype for tossing a ball towards a target, and [16], where a control strategy was proposed for a similar prototype that launched an object towards a target point. More recently, a concept similar to that presented in [13; 14] was applied to a 1-DoF robot capable of juggling a ball in three dimensions in an open-loop configuration [17]. For a general survey of dynamic manipulation using only unilateral constraints (which includes throwing movements), we refer the reader to [18]. Throwing can also be performed by more complex architectures, such as serial robotic arms: an early work is [19], where a 2-DoF robot moving in a vertical plane along an elliptical path was used to throw a ball and catch it back using visual feedback.