A Receding Horizon-type Solution of the Inverse Kinematic Task of Redundant Robots

Abstract

With the exception of specially designed kinematic structures, the inverse kinematic task of redundant robots cannot be solved in “closed form”. Redundancy is often needed to make the arm “dexterous”. In general numerical techniques have to be applied. A typical approach is finding the “differential solution” by formulating the Jacobian of the problem in the first step, and using some generalized inverse that selects a definite solution of the ambiguous set of the possible ones. Each of these solutions suffers from huge joint coordinate time-derivatives in the vicinity of the kinematic singularities that are evaded by “deforming” the original task. A quasi-differential approach was recently elaborated that, though calculates the Jacobian, instead “inverting” it, applies a fixed point iteration that automatically evades the singularities. However, its solution depends on the eigenvalues of the Jacobian, and it was found not “flexible enough” for all practical purposes. The Moore-Penrose solution can be generalized as an optimization task under constraints for quite complex cost functions. The hard constraint in this approach forces the exact solution of the inverse kinematic task when it is possible. Due to it the intent of “minimizing the costs” is “overridden” in the vicinity of the singularities. In the present paper, based on the formal structure of the Receding Horizon Controllers, an alternative solution is suggested that allows flexibility for relaxing the costs and the “hard constraints”, too. Its computational complexity to some extent is reduced via evading the technique of gradient reduction. Simulation results are presented for a 7 DoF robot arm.