Taking the Hexaglide-type parallel mechanism as the research object, the unit wrench screw of each limb is obtained by analyzing the geometric characteristics of the mechanism, and the first-order force influence coefficient matrix is derived based on the static equilibrium conditions. Using a zero-column construction method, the determinant of this matrix is made singular, thereby identifying the geometric conditions under which the mechanism becomes singular. Furthermore, by constructing linear dependencies among rows and columns, the conditions under which the Hexaglide mechanism does not exhibit configuration singularity are derived. On this basis, the three types of singularity conditions are unified into a parametric form, and a quantifiable geometric safety distance index is defined, which can be incorporated into an optimization model for singularity avoidance at the design stage. This work provides theoretical support and a methodological foundation for the design of such mechanisms.