The aim of this paper is to classify one family of 3R serial positioning manipulators. This categorization is based on the number of cusp points of quaternary, binary, generic and non-generic manipulators. It was found three subsets of manipulators with 0, 2 or 4 cusp points and one homotopy class for generic quaternary manipulators. This classification allows us to define the design parameters for which the manipulator is cuspidal or not, i.e., for which the manipulator can or cannot change posture without meeting a singularity, respectively.