Forward kinematic problem of 5-RPUR parallel mechanisms (3T2R) with identical limb structures

This paper investigates the formulation of the forward kinematic problem of the 5-DOF parallel mechanisms performing three translation and two independent rotations with identical limb structures. The mathematical framework used in this paper is based on algebraic geometry where the forward kinematic problem is explored in a seven-dimensional kinematic space by the means of the so-called Study parameters. In this paper, the algorithms proposed for obtaining the forward kinematic expression is based on two approaches which differ by their concept of eliminating passive variables. The first one is based on the application of the so-called resultant method and the second one, as a novel approach referred to as linear implicitization algorithm, uses an elimination procedure based on solving several systems of linear equations. The forward kinematic problem is solved using the homotopy continuation and it is revealed that it has 1680 finite solutions. Moreover, the forward kinematic problem is studied in three-dimensional Euclidean space where a class of simplified designs are proposed which admit a simpler formulation for the forward kinematic problem. Finally, based on the simplified designs, for a nearly general design a univariate expression of degree 220 is obtained.