Fine Tuning of the Fixed Point Iteration-Based Matrix Inversion-Free Adaptive Inverse Kinematics Using Abstract Rotations

The Inverse Kinematic Task of Robots of redundant open kinematic chain normally does not have closed form analytical solution. The generally viable approach applies the Differential Inverse Kinematics in which the derivative of the nominal trajectory of the robot as well as the derivative of its internal generalized coordinates according to some scalar variable (that may be e.g. the time) are related to each other by the Jacobian of the arm due to the chain rule of differentiation. The traditional solutions compute some generalized inverse of this Jacobian that exists only in the non-singular positions, and does not behave well in the vicinity of the singularities where normally complementary tricks (practically the modification of the inverse kinematic task by replacing it with a solvable “deformed” version) are applied to obtain some “solution”. These modifications may degrade the precision of the solution in the nonsingular points. The idea of replacing the matrix inversion with Fixed Point Iteration (FPI) in solving the inverse kinematic task was suggested in 2016 on the basis of the assumption that the kinematic parameters of the robot are precisely known. It was shown that this approach automatically yielded well behaving solutions in, and in the vicinity of the singularities without the use of any “complementary deformation”. In 2017 it was realized that in the possession of an approximate parameter set of the kinematic model an adaptive inverse kinematic task solution can be developed on this basis if the pose and location of the last segment of the robot as well as the generalized coordinates can be measured. This approach used counterrotations to guarantee the convergence of the fixed point iteration. Later it cropped up that similar abstract rotations can be applied in the realization of the fixed point iterations, too. The so elaborated solution can be combined with the inclusion of free parameters that can be used a) for making a trade-off between the precision requirements for the tracked position and/or pose, and b) parameters that affect the “distribution” of the ambiguous solution between the rotations of the redundant generalized coordinates. The operation of the approach is exemplified by the use of a redundant 8 Degree of Freedom robot arm via simulations made in Julia ver. 1.0.3.&nbsp