Velocity and acceleration level inverse kinematic calculation alternatives for redundant manipulators

Abstract

For both non-redundant and redundant systems, the inverse kinematics (IK) calculation is a fundamental step in the control algorithm of fully actuated serial manipulators. The tool-center-point (TCP) position is given and the joint coordinates are determined by the IK. Depending on the task, robotic manipulators can be kinematically redundant. That is when the desired task possesses lower dimensions than the degrees-of-freedom of a redundant manipulator. The IK calculation can be implemented numerically in several alternative ways not only in case of the redundant but also in the non-redundant case. We study the stability properties and the feasibility of a tracking error feedback and a direct tracking error elimination approach of the numerical implementation of IK calculation both on velocity and acceleration levels. The feedback approach expresses the joint position increment stepwise based on the local velocity or acceleration of the desired TCP trajectory and linear feedback terms. In the direct error elimination concept, the increment of the joint position is directly given by the approximate error between the desired and the realized TCP position, by assuming constant TCP velocity or acceleration. We investigate the possibility of the implementation of the direct method on acceleration level. The investigated IK methods are unified in a framework that utilizes the idea of the auxiliary input. Our closed form results and numerical case study examples show the stability properties, benefits and disadvantages of the assessed IK implementations.

Solvability of a system of linear equations—an approach based on the generalized inverses determined by the Penrose equations ★

“…In order to update the list of references provided in [3] with the most recent articles taking advantage of the Moore-Penrose inverse, let us mention that: in [16] the inverse of a stiffness matrix was used to solve the characteristic finite-element equations in considerations dealing with modeling of elastic deformation and non-local brittle fracture of solids, in [17] the inverse of a correlation matrix was used in a study of correlated pulsar-pairs, in [18] the inverse of operator matrices in a Hilbert space framework was used to derive solutions of constrained linear equations using variational principles in considerations of a class of problems (called Z-problems) that arise in effective media theory, in [19] the inverse was used to derive numerical solutions to particular nonlinear ordinary differential equations and differential algebraic equations involved in neural networks analysis, in [20] the inverse of a key matrix was used to derive a novel image encryption algorithm, in [21] the inverse was used to construct an image fusion algorithm based on a matrix factorization, which aims at optimizing computational workload, in [22] the inverse was used to solve a set of linear equations resulting from a novel variant of the multiconfiguration perturbation theory proposed in order to ensure its pivot invariance, and in [23] the inverse was used to specify new mode time coefficient (equivalent to the best-fit solution in the least squares sense) for dynamic mode decomposition utilized to extract dominant modes of unsteady flow fields. Additionally, in [24] the Moore-Penrose inverse was used in comparison of alternative models of velocity and acceleration level inverse kinematic for redundant manipulators, with the inverse of a Jacobian matrix involved in each model; orthogonal projectors onto the fundamental subspaces associated with the Jacobian matrix were utilized in the analysis as well. Redundant manipulators were also referred to in [25], where two algorithms for a dynamic Moore-Penrose inversion were introduced and implemented.…”