This paper introduces a geometrico-static analysis of an intrinsically safe parallel manipulator called R-Min. This robot was designed to reduce the risk of injury during a collision with a human operator, thanks to an underactuated architecture which enables large internal displacements in case of a collision. Indeed, the R-Min architecture is based on a modification of the well-known planar five-bar mechanism, where additional passive joints are introduced on the distal links in order to create a planar seven-bar mechanism with two degrees of underactuation. These two additional degrees of freedom are passively driven through the use of a supplementary passive leg, in which a tension spring is mounted between the base and the end-effector. In this paper, the conditions satisfying the equilibrium and the stability of the mechanism are introduced, based on a geometrico-static analysis. The direct and inverse problems are then solved using a numerical approach. Solutions to both problems are presented and classified. One subset of solutions to the inverse problem is isolated and projected in the Cartesian space in order to obtain the payload-invariant workspace of the R-Min robot.