The Forward Kinematics of 3-R $\underline {P}$ R Planar Robots: A Review and a Distance-Based Formulation

The standard forward-kinematics analysis of 3-RPR planar parallel robots boils down to computation of the roots of a sextic polynomial. There are many different ways to obtain this polynomial, but most of them include exceptions for which the formulation is not valid. Unfortunately, near these exceptions, the corresponding polynomial exhibits numerical instabilities. In this paper, we provide a way around this inconvenience by translating the forward-kinematics problem to be solved into an equivalent problem fully stated in terms of distances. Using constructive geometric arguments, an alternative sextic - which is not linked to a particular reference frame - is straightforwardly obtained with the need for neither variable eliminations nor tangent-half-angle substitutions. The presented formulation is valid, with no modification, for any planar 3-RPR parallel robot, including the special architectures and configurations - which ultimately lead to numerical instabilities - that cannot be directly handled by previous formulations.