This thesis proposes to analytically express the positions of the legs (each leg has a length and two spin orientations) for a parallel leg mechanism in terms of a single variable. The only limitations imposed are the leg bases are all on the same reference plane and the tool orientation is normal to the mobile upper plate. This is done for a parallel leg mechanism having any number of legs. The variable chosen is the first leg's vertex position on the upper plate.

The state space is the space the configuration can occupy. By looking at a planar slice of the state space for a specified tool axis position and orientation on the upper plate, the different possible leg positions can be constructed. The three dimensional space can then be generated by overlaying a progressive set of planes.