Open Access Te Herenga Waka-Victoria University of Wellington
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Dual Numbers and Invariant Theory of the Euclidean Group with Applications to Robotics

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posted on 2021-11-13, 21:48 authored by Daher, Mohammed

In this thesis we study the special Euclidean group SE(3) from two points of view, algebraic and geometric. From the algebraic point of view we introduce a dualisation procedure for SO(3;ℝ) invariants and obtain vector invariants of the adjoint action of SE(3) acting on multiple screws. In the case of three screws there are 14 basic vector invariants related by two basic syzygies. Moreover, we prove that any invariant of the same group under the same action can be expressed as a rational function evaluated on those 14 vector invariants.  From the geometric point of view, we study the Denavit-Hartenberg parameters used in robotics, and calculate formulae for link lengths and offsets in terms of vector invariants of the adjoint action of SE(3). Moreover, we obtain a geometrical duality between the offsets and the link lengths, where the geometrical dual of an offset is a link length and vice versa.


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Te Herenga Waka—Victoria University of Wellington

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Te Herenga Waka—Victoria University of Wellington

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Doctor of Philosophy

ANZSRC Type Of Activity code

970101 Expanding Knowledge in the Mathematical Sciences

Victoria University of Wellington Item Type

Awarded Doctoral Thesis



Victoria University of Wellington School

School of Mathematics, Statistics and Operations Research


Donelan, Peter; McGuiness, Mark