Defining degenerate robots symbolically

The solution of the inverse kinematics problem requires solving a non-linear system of equations, modulo the trigonometric identity s2+c2−1, where c=cos(θ),s=sin(θ) for a joint variable θ. A method based on the geometry of conics was presented by Smith and Lipkin [A summary of the theory and applications of the conics method in robot kinematics. Proc. 2nd. Int. Conf. on Advances in Robot Kinematics, Springer, Linz, 1990, pp. 81–88] establishing a sufficient condition for the inverse kinematics problem simplification of a 6R robot with the last three joint axes intersecting. Robots verifying such condition were named degenerate robots, because their determining degree two equation is a degenerate conic consisting of two parallel lines. On the other hand, functional decomposition is a mathematical concept that allows, in certain cases, the reduction of the solution of an equation to a sequence of lower-degree ones. In this paper we show that the Smith and Lipkin condition is equivalent to the functional decomposition of the determing equation, over the unit circle. We also present a necessary and sufficient condition for the decomposibility of a general third-degree sine–cosine equation.