The Ninth Homotopy Class of Spatial 3R Serial Regional Manipulators

Singularities form surfaces in the jointspace of a serial manipulator. Paï and Leu (Paï and Leu, 1992, IEEE Trans. Rob. Autom., 8, pp. 545–559) introduced the important notion of generic manipulator, the singularity surfaces of which are smooth and do not intersect with each other. Burdick (Burdick, 1995, J. Mech. Mach. Theor., 30, pp. 71–89) proposed a homotopy-based classification method for generic 3R manipulators. Through this classification method, it was stated in Wenger, 1998, J. Mech. Des., 120, pp. 327–332 that there exist exactly eight classes of generic 3R manipulators. A counterexample to this classification is provided: a generic 3R manipulator belonging to none of the eight classes identified in (Wenger, 1998, J. Mech. Des., 120, pp. 327–332) is presented. The weak point of the proof given in (J. Mech. Des., 120, pp. 327–332) is highlighted. The counterexample proves the existence of at least nine homotopy classes of generic 3R manipulators. The paper points out two peculiar properties of the manipulator proposed as a counterexample, which are not featured by any manipulator belonging to the eight homotopy classes so far discovered. Eventually, it is proven in this paper that at most four branches of the singularity curve can coexist in the jointspace of a generic 3R manipulator and therefore at most eleven homotopy classes are possible.