Rational Parametrization of Linear Pentapod's Singularity Variety and the Distance to it

A linear pentapod is a parallel manipulator with five collinear anchor points on the motion platform (end-effector), which are connected via SPS legs to the base. This manipulator has five controllable degrees-of-freedom and the remaining one is a free rotation around the motion platform axis (which in fact is an axial spindle). In this paper we present a rational parametrization of the singularity variety of the linear pentapod. Moreover we compute the shortest distance to this rational variety with respect to a suitable metric. Kinematically this distance can be interpreted as the radius of the maximal singularity free-sphere. Moreover we compare the result with the radius of the maximal singularity free-sphere in the position workspace and the orientation workspace, respectively.