On the symbolic insimplification of the general 6R-manipulator kinematic equations

Abstract

When symbolically solving inverse kinematic problems for robot classes, we deal with computations on ideals representing these robot's geometry. Therefore, such ideals must be considered over a base field K, where the parameters of the class (and also the possible relations among them) are represented. In this framework we shall prove that the ideal corresponding to the general 6R manipulator is real and prime over K. The practical interest of our result is that it confirms that the usual inverse kinematic equations of this robot class do not add redundant solutions and that this ideal cannot be “factorized”, establishing therefore, Kova´cs [7] conjecture. We prove also that this root class has six degrees of freedom (i.e. the corresponding ideal is six-dimensional), even over the extended field K, which is the algebraic counterpart to the fact that the 6R manipulator is completely general. Our proof uses, as intermediate step, some dimensionality analysis of the Elbow manipulator, which is a specialization of the 6R.