Clifford Algebra Space Singularities of Inline Planar Platforms

A Jacobian matrix of a general inline planar platform is studied. An inline planar platform is a manipulator with three legs, each with RPR joints, such that the revolute joints are free and align on each platform and the prismatic joints are powered. The configurations that cause the Jacobian matrix to become singular form a singularity surface that must be avoided for controllability. The Jacobian matrix is developed in the even Clifford algebra Cl+(P 2) of the projective space P 2 and its singularity surface is studied. A redundant planar platform manipulator is shown to have a block Jacobian matrix. A composite of singularity sets is developed for a redundant planar platform. A three-dimensional multi-platform manipulator is discussed.