## 1 Introduction
For similar 2-link, underactuated robot arms, other authors considered the problem of optimizing the pitching trajectory [20; 21], while in [22] a control strategy for the throwing motion was proposed and applied to the Pendubot prototype. Regarding higher-DoF serial systems, a serial-link robot launching a ball towards a target was explored in [23], where the authors reported a success rate of 40%; later, the authors of [24] reported up to 85% accuracy by using both a physical simulation and the predictions of a deep-learning network. A 6-DoF serial arm was used for dynamic release and catching of a ball (in a \"dirobling\" motion) by predicting its trajectory through a model of the free-flight phase using data from a visual tracking system [25]. A more general trajectory design approach for \(n\)-link manipulators was introduced in [26]; the algorithm considers both kinematic and dynamic constraints, and finds a solution (if any) with a known probability. Finally, the sensitivity of the launch trajectory was considered in [27], where the authors showed how to find the optimal launch trajectory in order to minimize the variance of the landing-point position. A number of works by [PERSON] et al. [28] considered the possibility of throwing an object from a gripper; the proposed application was rapid transportation of parts in a manufacturing system. In [29], this idea was developed specifically for throwing and catching cylinder-shaped objects. The position of the launched object in midair was observed by a visual tracking system [30]. A slightly different approach, compared to throwing robots, was used in so-called \"casting\" robots, proposed in [31]: here the authors design a planar, underactuated 2-DoF robot whose EE is a gripper attached through an extendable cable to a rigid-link mechanism. This design, further developed in [32], was used to prove the possibility of reaching a target point far from the robot base, by swinging the rigid mechanism and then launching the EE by releasing the string at a suitable time. Later, the same authors proposed in [33] the idea of controlling the EE motion in midair, by controlling the tension of the string to which the EE is attached. In [34], this approach was extended to a 3-DoF spatial parallel robot; later on, in [35], a casting robot was proposed using a serial robot arm mounted on a mobile robot. In [36], a robotic arm launched a probe to a target location while the probe orientation was kept fixed during the launch motion. A system that launched an anchor ball attached to a cable was considered in [37], together with a ropeway robot moving along the launched cable after its anchor had been fixed at the target position. Notably, in a casting robot, the cable may interact not only with the launched part, but also with the environment, for instance by wrapping itself around a pole [38]. A concept closely correlated to casting was explored in [39], where the authors considered a shooting robot that catapulted an object connected to an elastic cable with an impulsive force; an advantage of this approach is the ability to reach \"blind spots\" behind objects. A potential drawback of casting manipulators, where the gripper detaches from the robot, is that they require long cables that, especially in cluttered areas, can be difficult to safely reel back. In this work, we focus on a 3-DoF CSPR with a point-mass EE attached to three cables and carrying a gripper from which a mass, modeled as a frictionless point object, must be thrown towards a given target by opening the gripper at a suitable instant. The obtained results can be readily extended to a 3-DoF CSPR with a finite-size EE such as the one in [10], as long as some conditions on the robot architecture are respected. The CSPR with parallelogram-type actuation in [10] can perform purely-translational motion: this is useful, since keeping a constant orientation of the EE helps in controlling the launch motion of the object.