Upon first consideration of a Five-Axis machine, the fact that one rotational component may carry the other rotational component along with it during it's rotation, may cause one to conclude that when preforming the needed calculations, a distinction must first be made as to which axial component may be carrying the other [ as with dual head or dual table machines ], or if not at all [ as with head table machines ].

This paper will show that such a distinction is never necessary for the determination of the rotary coordinate values. Furthermore, it will show that all solutions [ for which there will be at least one pair, but more generally two pairs, if solutions exist ] for the two rotational axes on a Five-Axis machine can be intuitively calculated through the use of the following single one dimensional equation:

In this paper, the above equation will be explained and derived through an insightful problem description in which each of the two Five-Axis machine rotary axes will participate with equal status in a symmetrical manner.

In a previously published paper [1], the authors presented a methodology to simplify the calculations required to determine tool moves on a Five-Axis machine tool for the purpose of postprocessor development.  However, their method of finding solutions is not independent of the type of Five-Axis machine tool, and only achieves independence [2] through the use of a Six-Axis machine tool model, introducing an unnecessary additional complexity that distorts the model of the actual machine tool.

In this paper, we present a much simpler abstraction for the calculation of tool moves on a Five-Axis machine tool that does achieve independence of the type of Five-Axis machine tool.  This independence is achieved through the use of a paramaterized set of equations that span all Five-Axis machine types.  The parameters of these equations are for the specification of the relavant angular and relavant displacement values that are particular to the actual machine tool model.  Furthermore, this approach too, applies to 5-axis machine tools that have both non-orthogonal and positionally non-coplanar rotary axes, including the home tool axis.