POLYNOMIAL SOLUTIONS TO THE DIRECT KINEMATIC PROBLEM OF PLANAR 3 DEGREE-OF-FREEDOM PARALLEL MANIPULATORS

This paper presents a general solution for the direct kinematics of planar three degree-of-freedom parallel manipulators. It has been shown elsewhere, using geometric considerations, that this problem can lead to a maximum of six real solutions. The formulation developed here leads to a polynomial of 6th order which is hence minimal. This is illustrated with an example, taken from the literature, for which six real solutions have been found. Moreover, for a special geometry in which the three joints on the platform and on the base are respectively aligned, it is shown here that the solution can be cascaded in two steps involving the solution of a cubic and quadratic, respectively. This particular class of planar parallel manipulators can therefore be solved in closed-form and no more than four real solutions have been found in this case. Examples of this class of manipulators are also solved.