The 3-RPS parallel manipulator from an algebraic viewpoint

• Study parametrization and algebraic constraint equations
• Global analysis of the kinematic behavior of the 3-RPS parallel manipulator
• Two operation modes are detected, analyzed and geometrically interpreted.
• Complete description of singularities and transitions between operation modes
• Analysis of singularity sets in kinematic image space and joint space is performed.

The 3-RPS parallel manipulator is a three degree of freedom parallel manipulator, which was introduced by K. Hunt in 1983 as one of the lower mobility parallel manipulators. Since then the 3-RPS gained a lot of attention in literature, but most of the articles on this manipulator use screw theory to explain its local kinematic behavior. An algebraic approach via Study's kinematic mapping reveals interesting global properties of this type of manipulator. The global kinematic behavior of the manipulator is described by algebraic equations, so called constraint equations. In the kinematic analysis these equations are manipulated using methods of algebraic geometry and are interpreted geometrically. It is shown, that the forward kinematics of the 3-RPS has in general 16 solutions in the field of complex numbers. Its workspace splits into two, essentially different operation modes. A geometric and kinematic interpretation of both modes is given. Furthermore it is shown that a transition between the operation modes is possible under certain circumstances. The operation modes are detected via a primary decomposition of the ideal corresponding to the constraint equations. Conditions for singular poses are derived from the constraint equations by discussing the Jacobian of the set of constraint equations. Finally, for singular poses of each operation mode as well as for singular poses belonging to both operation modes, a mapping into the space of joint parameters will be described. Images of singular poses of the manipulator under the mapping determine algebraic surfaces in the joint space, which are analyzed algebraically.