Broadly applicable formulations for identifying control difficulties in positioning serial manipulators are developed based on the Implicit Function Theorem and acceleration analyses on singular surfaces. As a result, voids in the workspace are also identified. Rank deficiency conditions are applied to the positioning Jacobian with joint constraints included in the formulation. Geometric entities upon which the manipulator exhibits control difficulties are identified. These difficulties are associated with a admissible motion along a vector normal to the singular geometric entities. Definiteness properties of a quadratic form that is as a result of normal acceleration define control difficulties. Numerical examples are illustrated.