Robust, Fast and Accurate Solution of the Direct Position Analysis of Parallel Manipulators by Extra-Sensors

Parallel manipulators (PMs) are closed kinematic chains with one or more loops where only some pairs are actuated while the remaining are passive. In particular, they feature a fixed link (base) and an output moving link (platform) interconnected by at least two independent kinematic chains (legs) to form one loop. The most well known and commonly employed PMs (hereafter called UPS-PMs) feature n variable-length legs of type UPS (where U, P and S are for universal, spherical and prismatic pairs respectively). Equivalently, a revolute pair R could be used instead of the prismatic pair P in order to make the leg length variable (in this case the leg would be of type URS). These leg topologies provide the platform with six degrees of freedom with respect to the base. Although the definition of UPS-PMs requires n  2, in practice, neglecting overconstrained and redundantly-actuated manipulators, performance issues recommend 3  n  6. Indeed, UPS-PMs with only two UPS legs might exhibit a low stiffness against torques acting along the line joining the centers of the two spherical pairs, and their control would require the in-series placement of at least three actuators/sensors (one of them placed to control/measure at least one out of the three degrees of freedom of the spherical pairs) which reduces the overall manipulator dynamic and accuracy capabilities. On the other side, the use of more than six legs reduces the exploitable manipulator workspace for the increase of leg interference. Different sub-classes of manipulator architectures can be obtained according to the location of the centers of the U and S pairs in the base and in the platform respectively (Innocenti & Parenti-Castelli, 1994; Faugere & Lazard, 1995). General UPS-PM architectures feature distinct joint centers. Special architectures can be devised by setting some of the joint centers to be coincident. A schematic of a 6-DOF UPS-PM having six legs (n = 6) and general architecture is shown in Fig. 1. In the figure, the U pairs (connecting the legs to the base) and S pairs (connecting the legs to the platform) are depicted as grey and white dots respectively. Points Bi and Pi (i = 1, …, 6) represent the centers of the U and S pairs of the i-th leg on the base and on the platform respectively. The six legs of type UPS are represented by the telescopic rods BiPi (i = 1, …, 6). Accordingly, the length of the i-th leg is defined as the distance li = Pi - Bi. Manipulators with less than six DOF can be obtained from UPS-PMs by suitably eliminating or locking some of the leg kinematic pairs. For instance, considering a 6-DOF UPS-PM having six legs, elimination of four P pairs yields a 2-DOF PM having two legs of type UPS and four legs of type US. Well-known examples of UPS-PMs are as follows: 1) the 6-DOF UPS-PMs (Gough & Whitehall, 1962; Stewart, 1965; Cappel, 1967); 2) the 3-DOF spherical PMs (Innocenti & Parenti-Castelli, 1993); 3) the 2-DOF spherical PMs (Vertechy & Parenti-Castelli, 2006); and 4) the 1-DOF helicoidal PMs (Jacobsen, 1975). Because of their parallel architecture, UPS-PMs exhibit large payload-to-weight ratio, high accuracy, high structural rigidity and high dynamic capabilities, which make them excel as: a) fast and high precision robots in vehicle simulators (Gough & Whitehall, 1962; Stewart, 1965; Cappel, 1967), machine tools (Charles, 1995) and positioning systems (Schmidt-Kaler, 1992); b) passive Cartesian input devices in joysticks, master-slave teleoperation systems (Daniel et al., 1993) and other tracking devices (Geng & Haynes, 1994); c) force/torque sensors and generators in multi-axis sensors and motors (Gaillet & Reboulet, 1983; Nguyen et al., 1991; Lewis et al., 2002); d) mechanical transmissions in motion converters (Jacobsen, 1975); and e) orthopedic devices in fixations systems (Taylor & Taylor, 2000; Di Gregorio & Parenti-Castelli, 2002). Practical use of UPS-P...