Inverse position analysis workspace determination and position synthesis of parallel manipulators with 3-RSR topology

Manipulators with 3-RSR topology are three-degree-of-freedom parallel manipulators that may be either spherical or mixed-motion manipulators. The inverse position analysis (IPA) and the workspace determination of 3-RSR manipulators are addressed by means of a new approach. The new approach is centered on a particular form of the closure equations called compatibility equations. The compatibility equations contain only the six coordinates (end-effector coordinates) which locates the end-effector pose (position and orientation) with respect to the frame, and the geometric constants of the manipulator. When the manipulator geometry is assigned, the common solutions of the compatibility equations are the end-effector coordinates which identify the end-effector poses belonging to the manipulator workspace. Moreover, they can be the starting point to easily solve the IPA. The presented compatibility equations can be also used to solve the position synthesis of the 3-RSR manipulator. This way of solving the position synthesis will demonstrate that only approximated solutions exist when more than eight end-effector poses are given.