This paper studies cable-driven parallel robots with less than six cables, in crane configuration. A geometrico-static model is provided, and the stability of static equilibrium is assessed within the framework of a constrained optimization problem. The method relies on ordinary linear-algebra routines, and it may be very simply applied to the most general architectures. Several examples are provided, concerning robots with a number of cables that range from 2 to 4.

Parallel mechanisms show desirable characteristics such as a large payload to robot weight ratio, considerable stiffness, low inertia and high dynamic performances. In particular, parallel manipulators with fewer than six degrees of freedom have recently attracted researchers' attention, as their employ may prove valuable in those applications in which a higher mobility is uncalled-for. The attention of this paper is focused on translational parallel mechanisms (TPMs), that is on parallel mechanisms whose output link (platform) is provided with a pure translational motion with respect to the frame. It deals with the general problem of the topological synthesis and classification of TPMs and it investigates both their constraint and direct singularities. In particular, it identifies for the first time special families of fully-isotropic mechanisms. Such manipulators exhibit outstanding properties, as they are free from singularities and show a constant orthogonal Jacobian matrix throughout their workspace. As a consequence, both the direct and the inverse position problems are linear and the kinematic analysis proves straightforward.

This paper presents a novel assessment of singularities of general parallel kinematic chains. Hierarchical levels in which different critical phenomena originate are recognized. At each level, the causes of singular events are identified and interpreted, and on their basis, a comprehensive taxonomy is proposed. First, the unactuated kinematic chain is studied. The concepts of leg and passive-constraint singularities are described, and stationary and increased instantaneous mobility configurations are identified. Then, a set of motorized joints is chosen. The effects of first-level singularities on the actuated chain are investigated, and further phenomena are identified, such as redundancy singularities and active-constraint singularities. The notions of reaction and action spaces are originally discussed. The consequences on the ability of the actuated chain to effectively govern its local and global freedoms are analyzed, and the complex interactions between the various singular events are studied. The instantaneous redundancy of actuators, which occur when these work either against each other or against joint constraints, is also evaluated. Finally, when the input-output mechanism is considered, the events described at the previous stages are interpreted within the perspective of the machine's desired use.