Application of Dual-Number Matrices to the Inverse Kinematics Problem of Robot Manipulators

Abstract

This paper presents the application of dual-number matrices to the formulation of displacement equations of robot manipulators with completely general geometry. Dual-number matrices make possible a concise representation of link proportions and joint parameters; together with the orthogonality properties of the matrices we are able to derive, in a systematic manner, closed-form solutions for the joint displacements of robot manipulators with special geometry as illustrated by three examples. It is hoped that the method presented here will provide a meaningful alternative to existing methods for formulating the inverse kinematics problem of robot manipulators.

Internal Displacements and Effects of Manufacturing Errors in the Clemens Coupling*

“…Such techniques have proven useful not only for displacement analysis but also velocity and dynamic analysis as reported by Bagci [12], Dimentberg [13], Pennock and Yang [14], Agrawal [15], Brodsky and Shoham [16], and Mladenova [17]. An overview of these methodologies for the dual-number modeling of mechanisms and some additional references are given by Fischer [18].…”

“…The differentialoperator matrix corresponding to the links with revolute joints is The expansion of Eq. (14) can be put into the form …”

Approaching Dual Quaternions From Matrix Algebra

“…At this point, it is important to realize that the 4D rotation matrix corresponding to a 3D homogeneous transformation matrix through F is not an approximation, it is an exact representation, and that the use of dual numbers to represent general 3D translations in terms of 4D rotation matrices is completely different from the standard approach based on 3D rotation matrices [41], [42], [43]. The use given here, while allowing a clear-cut distinction between translations and rotations, provides at the same time a neat connection with standard homogeneous transformations.…”

An historical review of the theoretical development of rigid body displacements from Rodrigues parameters to the finite twist

“…In 1985, Pennock and Yang [65] investigated the use of dual number matrices for transformation of coordinates of lines to solve the inverse kinematics problem of robot manipulators. In the following year, the property of the dual orthogonal matrix was revealed by McCarthy [66], leading to the development of a dual form of the Denavit-Hartenberg matrix [67] and a dual form of the Jacobian of a manipulator.…”