On the N-bar mechanism, or how to find global solutions to redundant single loop spatial kinematic chains

The author investigates the global sets of solutions for single loop inverse kinematic problems containing only independent rotational and translational degrees of freedom, providing a rational and compact method to obtain these sets. The proposed methodology relies on a deep understanding of spherical redundant mechanisms, shedding light on the basic algebraic structure underlying inverse kinematic problems associated with single loop kinematic chains. This methodology consists of finding the inverse kinematic solution to the n-bar mechanism. It is shown that one can model any kinematic loop equation as the loop equation derived from this mechanism after taking as many bars as needed and constraining some of the resulting degrees of freedom. The solution of any kinematic equation can be factored into a solution of both its rotational and translational components. While the solution of the former can be obtained quite easily, the later leads, in general, to inextricable formulae. Some important relationships between the two components are obtained, which make it simpler to obtain the solution to the translational component once the solution to the rotational one has been found