Matrix inversion-free quasi-differential approach in solving the inverse kinematic task

The traditionally viable way for the solution of the inverse kinematic task for general open kinematic structure is the differential approach in which the Jacobian of the robot arm usually is inverted by the use of some generalized matrix inverse. These approaches suffer from the kinematic singularities nearby which these "inverses" are ill-conditioned and behave in a very inconvenient way. For dealing with the singularities complementary tricks used to be introduced that so deform the original task that the obtained "solution" behaves conveniently though it cannot solve the original task that does not have exact solution. In this paper an alternative approach is suggested that requires only the computation of the Jacobian but does not need the calculation of its "generalized inverse". Instead of that it applies an iterative sequence that has nice convergence properties. The method automatically handles the problem of the singularities, ambiguity, redundancies, and non-existing exact solutions without the application of any complementary trick or artificial parameter. Its operation is demonstrated for a simple 2 Degree of Freedom (DoF) arm, and for an 8 DoF arm, that is an irregular extension of a 6 DoF PUMA-type robot.