Closed-form Solution for the Direct Kinematics Problem of the Planar 3-RPR Parallel Mechanism

In general, it is not possible to determine the actual manipulator platform's pose of a parallel mechanism from its active joints' coordinates. This problem is usually solved by using additional numerical procedures or by additional system information from auxiliary sensors, providing several weaknesses including initial pose estimations, reference drives, or workspace limitations. In this paper, we therefore introduce a closed-form solution for the direct kinematics problem of the planar 3-RPR parallel mechanism by using only the orientations of two active joints and the manipulator platform, where P denotes active prismatic joints and R passive revolute joints.