Linear Pentapods with a Simple Singularity Variety

There exists a bijection between the configuration space of a linear pentapod and all points (u,v,w, px, py, pz)∈R located on the singular quadric Γ : u2+v2+w2 = 1, where (u,v,w) determines the orientation of the linear platform and (px, py, pz) its position. Then the set of all singular robot configurations is obtained by intersecting Γ with a cubic hypersurface Σ in R6, which is only quadratic in the orientation variables and position variables, respectively. This article investigates the restrictions to be imposed on the design of this mechanism in order to obtain a reduction in degree. In detail we study the cases where Σ is (1) linear in position variables, (2) linear in orientation variables and (3) quadratic in total. The resulting designs of linear pentapods have the advantage of considerably simplified computation of singularity-free spheres in the configuration space. Finally we propose three kinematically redundant designs of linear pentapods with a simple singularity surface.