Kinematics of a class of parallel manipulators which generates structures with three limbs

In this paper, the forward position analysis (FPA) of a class of parallel manipulators which can be modeled as structures with three limbs when the active joint rates, namely the generalized coordinates, are locked, is carried out using mixed and analytic procedures. The first option is a novel procedure which combines a numerical method and the closed-form solution of the geometry of a tetrahedron, that allows to find all the possible solutions for the FPA when specific requirements are satisfied. The second option is an analytic procedure capable of reducing a nonlinear system of three equations in three unknowns into a univariate 16th order polynomial equation, which evidently leads to a multiple solution. The introduction of linear equations, proposed in this contribution, obtained from the particular topology of these mechanisms simplifies considerably the FPA. In order to complement the kinematic analyses, the acceleration analysis of an exemplary parallel manipulator is approached by means of standard screw theory. Of course, as an intermediate step, this contribution also provides the velocity analysis of the chosen parallel manipulator. Finally, two numerical examples are provided.     

Kinematics of a five-degrees-of-freedom parallel manipulator using screw theory

In this work, the kinematic analysis of a five-degrees-of-freedom decoupled parallel manipulator is approached by means of the theory of screws. The architecture of the parallel manipulator under study is such that the translational motion of the moving platform, with respect to the fixed platform, is controlled by means of a central limb provided with two active prismatic joints while its rotational motion is controlled by means of a three-degrees-of-freedom spherical parallel manipulator. The forward position analysis is presented in semiclosed form solution applying recursively the Sylvester dialytic elimination method. On the other hand, the velocity and acceleration analyses are carried out using the theory of screws. Simple and compact expressions to compute the velocity state and the reduced acceleration state of the moving platform, with respect to the fixed platform, are easily derived in this contribution by taking advantage of the Klein form of the Lie algebra se(3). Finally, only few and slight modifications to the proposed method of kinematic analyses are required in order to approach the position, velocity, and acceleration analyses of parallel manipulators with similar topologies.