Towards an efficient interval method for solving inverse kinematic problems

We present an algebraic analysis of the closure equation obtained for arbitrary single loop spatial kinematic chains, which allows us to design an efficient interval method for solving their inverse kinematics. The solution of a kinematic equation can be factored into a solution of both its rotational and its translational components. We have obtained general and simple expressions for these equations and their derivatives that are used to perform Newton cuts. A branch and prune strategy is used to get a set of boxes as small as desired containing the solutions. If the kinematic chain is redundant, this approach can also provide a discretized version of the solution set. The mathematics of the proposed approach are quite simple and much more intuitive than continuation or elimination methods. Yet it seems to open a promising field far further developments.