Fixed Point Iteration-based Problem Solution without the Calculation of the Jacobian

In technical applications typical problems arise that can be “differentially” treated by the calculation and subsequent inversion of certain Jacobi matrices. However, in many cases, the calculation of the inverse means no necessary computations because the solution is interesting only for a given, well defined input array, while by the use of the (generalized) inverse it can be obtained for an arbitrary input. In this view a matrix inversion-free quasi-differential solution of the inverse kinematic task of robots was recently suggested that showed nice and stable behavior in and in the vicinity of the kinematic singularities where the traditional, matrix-inversion-based approaches have difficulties and need complementary tricks to remain stable. In the case of a redundant robot this approach requires the calculation of the Jacobian. In the present paper it is shown that in the special case of quadratic Jacobians even the calculation of the Jacobian can be omitted. The present approach may have significance in the development of novel Receding Horizon Controllers in which replacement of Lagrange's original Reduced Gradient Method was suggested with a simple Fixed Point Iteration for the gradient of the auxiliary function of the optimization under constraints, because in it only quadratic Jacobians occur. As a simple example a 2 degree of freedom arm is considered via simulations to reveal the behavior of this approach. It is shown that by the use of a simple complementary norm reduction built in into the solution this approach is promising.