The dynamics of parallel Schonflies motion generators: the case of a two-limb system

The formulation of the mathematical model governing the dynamics of parallel Schonflies motion generators (SMGs) is the subject of this paper. These are robotic systems intended to produce motions that entail displacements of the Schonflies subgroup of rigid-body displacements. Such motions are representative of those produced by the serial robots termed SCARA, which involve three independent translations and one rotation about an axis of fixed direction. The main features of SMGs are illustrated with the aid of the McGill Schonflies motion generator. This robot is composed of two limbs, each being a four-joint R Pi Pi R - R representing a revolute, Pi a Pi joint, or a parallelogram linkage - kinematic chain, with only two joints actuated. One important feature of the McGill SMG is its mass distribution, as its moving parts account for about 10 per cent of the mass of its drive units, which are (a) fixed to the robot frame and (b) the only steel parts of the whole robot. Moreover, its joints account for roughly 10 per cent of the total mass of its moving parts, fabricated from aluminium, which justifies neglecting the joint inertia in the mathematical model. This feature calls for a formulation of the model in question in terms of the motor displacements, rather than the joint displacements, as the generalized coordinates of the model. Furthermore, in order to derive the model, the robot is decomposed into two subsystems, the drive and the linkage, the drive being decomposed, in turn, into two subsubsystems, the epicyclic gear train and the right-angled gearbox. The robot kinematics is first derived and then the dynamics model is formulated by means of the natural orthogonal complement. In the framework of this methodology, the inertia and what are called the Coriolis matrices of the mathematical model are additive, in the sense that they can be computed as the sum of the contributions of the different subsystems and subsubsystems; of a given mechanical system. The contributions of the subsystems and subsubsystems, in turn, can be computed as the sum of the contributions of the individual moving parts of these. Some general results applicable to all SMGs are derived, which leads to the simplification of the mathematical model of such systems.