Adaptive solution of the inverse kinematic task by fixed point transformation

In a wide class of robots of open kinematic chain the inverse kinematic task cannot be solved by the use of closed-form analytical formulae. On this reason the traditional approaches apply differential approximation in which the Jacobian of the — normally redundant — robot arm is "inverted" by the use of some "generalized inverse". These pseudo-inverses behave well whenever the robot arm is far from a singular configuration, however, in the singularities and nearby the singular configurations they suffer from a singular or ill-conditioned pseudoinverse. For tackling the problem of singularities normally complementary "tricks" have to be used that so "deform" the original problem that the deformed version leads to the inversion of a well-conditioned matrix. Though the so obtained solution does not exactly solve the original problem, it is accepted as practical "substitute" of the not existing solution in the singularities, and an acceptable approximation of the exact solution outside the singular points. Recently, in [1], an alternative, quasi-differential approach was suggested that was absolutely free of any matrix inversion. It was shown that it converged to one of the — normally ambiguous — exact solutions at the nonsingular configurations, and showed stable convergence in the singular points when a "substitute" of the not existing solution was created. This convenient convergence was guaranteed by the use of the "exact Jacobian" of the robot arm. The interesting question, i.e. what happens if only an "approximate Jacobian" is available, and the motion of the robot arm is precisely measurable with respect to a Cartesian "workshop"-based system of reference, was left open. Now it is shown that the convergence properties of the method can be improved by the application of simple rotational matrices, and on this basis the iterative application of an "Adaptive Inverse Kinematics" becomes possible. This approach has the specialty that no complete information it needs on the Jacobian at a given point. It is content with the observable system behavior only along the realized motion, so it seems to be easily implementable. Its operation is demonstrated for an irregularly extended 6 Degree-of-Freedom (DoF) PUMA-type robot arm, that has 8 rotary axles.