Forward kinematics of planar parallel manipulators in the Clifford algebra of P2

This paper presents the forward kinematics of planar platform manipulators composed of three RPR chains. The Clifford Algebra of projective two space is used to represent planar displacements. Also known as planar quaternions, the representation allows us to formulate the kinematic constraints in an algebraic form with rich geometric content. The forward kinematics problem reduces to the intersection of three circular hyperboloids. An algorithm for parameterizing the forward kinematics solution is presented. The general kinematics function is shown to yield six solutions. Furthermore, it is shown that real solutions only exist if the roots of a certain polynomial are on the interval from −1 to 1. A series of platform architectures is then studied to show how the kinematics algorithm specializes in these cases. Degenerate architectures are also identified.