Robotica (2007) volume 25, pp. 113–120. © 2006 Cambridge University Press doi:10.1017/S0263574706003055 Printed in the United Kingdom A new methodology for the determination of the workspace of six-DOF redundant parallel structures actuated by nine wires Carlo Ferraresi ∗ †, Marco Paoloni‡ and Francesco Pescarmona† †Dipartimento di Meccanica, Politecnico di Torino, Corso Duca degli Abruzzi, 24, Torino 10129, Italy. ‡Robotic Section of the Nuclear Fusion Unit, ENEA C.R. Casaccia, Via Anguillarese, 301, Roma 00060, Italy. (Received in Final Form: July 27, 2006, First published online: October 12, 2006) SUMMARY The WiRo-6.3 is a six-degrees of freedom (six-DOF) robotic parallel structure actuated by nine wires, whose character- istics have been thoroughly analyzed in previous papers in reference 2 . It is thought to be a master device for tele- operation; thus, it is moved by an operator through a handle and can convey a force reflection on the operator’s hand. A completely new method for studying the workspace of this device, and of virtually any nine-wire parallel structure actuated by wire is presented and discussed, and its results are given in a graphical form. KEYWORDS: Parallel robot; workspace; wire actuation. 1. Introduction Mechanical structures actuated by wires (wire robots) are characterized by the presence of a mobile platform (representing the end-effector) connected by several wires to a fixed frame; the wires are fixed to the platform, rolled over pulleys and stretched by motors fixed to the frame in order to exert forces and torques. At the same time, the position and orientation of the mobile platform can be determined by the measured wire lengths. Wire robots are parallel devices having wires as links, and belong to a set of fully parallel structures because every wire is an independent chain with one DOF. 1, 4–6 With respect to the traditional parallel structures, wire- actuated robots have several advantages: they allow great manoeuvrability, thanks to a reduced mass, and also promise lower costs with respect to traditional actuators. Furthermore, the stroke length of each linear joint does not follow the same restrictions as with conventional structures, because wires can be extended to much higher lengths, unwinding from a spool. This kind of a structure allows to comply with several needs in applications where conventional manipulation technology can be hardly used for technical or economical reasons. We could mention, for example, crane robots, 1, 4 high-speed manipulation robots 5 and force feedback devices to be used as masters in master–slave teleoperation systems. 6 ∗ Corresponding author. E-mail: carlo.ferraresi@polito.it Such devices offer many advantages, such as a simplified mechanical structure, very high speed, relatively large workspace and low inertia. However, it must be noted that wires can only pull objects and not push on them: this unilateral constraint compels the adoption of a redundant actuating mechanism. This can be seen as an analogy with the grasping problem for multi-finger systems with frictionless point contact; the forces exerted by the fingers on the grasped object are subject to the same unidirectional constraint. It has been stated 5, 6, 9, 13 that to obtain n degrees of freedom (DOFs) without external forces (or the gravity in the crane case) it is necessary and sufficient to use n + 1 wires; These devices are usually referred to as completely restrained position mechanism (CRPM), while devices with a higher number of wires are referred to as redundantly restrained position mechanism (RRPM). The study of the operative characteristics of wire-driven devices may present more difficulties than the traditional ones, in particular for the definition of their workspace and dexterity. The workspace is not simply the set of non-singular platform positions and orientations compatible with the joints limits, but it is also necessary that all forces and torques exerted in such platform poses should be obtainable only by means of a set of wire forces directed from the platform to the frame. Furthermore, the shape and the dimensions of the workspace, and the dexterity of these devices are greatly influenced by the number of wires and their geometrical disposition. It has been stated by several authors 5, 6, 12, 13 that in a certain pose of the end-effector of a six-DOF mechanism driven by m wires, it is possible to exert arbitrary force and moment, if and only if, the transpose of the inverse Jacobian (called the structure matrix with six rows and m columns) has a rank equal to six and if it is possible to find a vector belonging to its null space with all the components strictly positive. Practically speaking, after the evaluation of the null space base constituted by m − 6 vectors with m components, to decide if the considered pose belongs to the workspace at least one set of m − 6 coefficients must be found to form a linear combination of the m − 6 base vectors yielding a resultant with all the m components strictly positive. For the CRPM case (m = 7), the null space is a one- dimensional (1-D) space vector and then the examined pose belongs to the workspace if the vector chosen as base of the null space has all the seven components of the same sign. For