The quartic singularity surface of planar platforms in the Clifford algebra of the projective plane

In this paper, we study the workspace and singular configurations of a planar platform supported by three linearly actuated legs, the 3-RPR parallel manipulator. The constraint equations of the platform are formulated in the Clifford algebra of the projective plane, C+(P-2), which yields a manifold defining its set of reachable positions and orientations. We compute the jacobian of these equations and derive the algebraic equation of the surface of points in C+(P-2) for which this Jacobian is singular, called the singularity surface of the manipulator. For the general planar platform manipulator this surface is a quartic surface with a double line. For the special case of the "proportional" planar platform; the surface factors into two planes and a circular hyperboloid. For the special case of the "in-line" planar platform, this surface reduces to a quartic ruled surface. Further special cases of this surface are examined and found to consist of pairs of hyperbolic paraboloids. (C) 1998 Elsevier Science Ltd. All rights reserved.