Some New Results on the Kinematics of 3R Serial Robots Using Nested Determinants

It has been recently shown that the singularity locus of a 3R robot, and in particular its nodes and cusps, can be algebraically characterized in terms of nested determinants. This neat and structured formulation contrasts with the huge and often meaningless formulas generated using computer algebra systems. In this paper we explore further this kind of formulation. We present two new results which we think are of interest by themselves. First, it is shown how Chrystal’s method, used to obtain the resultant of two quadratic polynomials, can be formulated as nested determinants. Second, it is also shown how the coefficients of the harmonic conic of two given conics, can also be expressed in the same form. These results lead to new formulations for the inverse kinematics of 3R robots, their singularity loci, their nodes, and some of their high-order singularities.