  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.
