  There exists a bijection between the configuration space of a linear pentapod
and all points $(u,v,w,p_x,p_y,p_z)\in\mathbb{R}^{6}$ located on the singular
quadric $\Gamma: u^2+v^2+w^2=1$, where $(u,v,w)$ determines the orientation of
the linear platform and $(p_x,p_y,p_z)$ its position. Then the set of all
singular robot configurations is obtained by intersecting $\Gamma$ with a cubic
hypersurface $\Sigma$ in $\mathbb{R}^{6}$, 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
$\Sigma$ 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.
