Dual Quaternions Robotics: A) The 3R Planar Manipulator

Abstract

Kinematics analysis studies the relative motions, such as, first of all, the displacement in space of the end effector of a given robot, and thus its velocity and acceleration, associated with the links of the given robot that is usually designed so that it can position its end-effector with a three degree-of-freedom of translation and three degree-of-freedom of orientation within its workspace. This chapter presents mainly, on the light of both main concepts; the first being the screw motion or/ and dual quaternions kinematics while the second concerns the classical ‘Denavit and Hartenberg parameters method’ the direct kinematics of a planar manipulator. First of all, examples of basic solid movements such as rotations, translations, their combinations and general screw motions are studied using both (4x4) matrices rigid body transformations and dual quaternions so that the reader could compare and note the similarity of the results obtained using one or the other method. Both dual quaternions technique as well as its counterpart the classical ‘Denavit and Hartenberg parameters method’ are finally applied to a three degree of freedom (RRR) planar manipulator. Finally, we and the reader, can observe that the two methods confirm exactly one another by giving us the same results for each of the examples and applications considered, while noting that the fastest, simplest more straightforward and easiest to apply method, is undoubtedly the one using dual quaternions. As a result this work may as well act as a beginners guide to the practicality of using dual-quaternions to represent the rotations and translations ie: or any rigid motion in character-based hierarchies. We must emphasize the fact that the use of Matlab software and quaternions and / or dual quaternions in the processing of 3D rotations and/or screw movements is and will always be the most efficient, fast and accurate first choice. Dual quaternion direct kinematics method could be generalised, in the future, to more complicated spatial and/ or industrial robots as well as to articulated and multibody systems