An analytical method is presented to obtain all boundary surfaces to the accessible output set of four degree-of-freedom serial manipulators. The method is applicable to all manipulators that comprise combinations of prismatic and revolute joints. Position constraints of the end-effector of such mechanisms are formulated using the Denavit-Hartenberg representation. Examining the Jacobian of the underlying mechanism using a row-rank deficiency method yields sets of first-order singularities. These sets of singularities are substituted into the position constraint equations yielding parametric surfaces upon which the manipulator looses at least one degree of mobility. Singular curves are determined by intersecting singular surfaces. Due to the complexity of intersecting parametric singular surfaces, the resultant singular curves are numerically computed. Bifurcation points are identified and tangents are computed. Singular surfaces are partitioned into sub-surfaces that are studied for existence inside the accessible output set using a proposed perturbation technique. The result is a number of sub-surfaces that envelop the accessible output set. The theory presented is validated using a four-degree-of-freedom example.