A dual quaternion algorithm of the Helmert transformation problem

Abstract

Rigid transformation including rotation and translation can be elegantly represented by a unit dual quaternion. Thus, a non-differential model of the Helmert transformation (3D seven-parameter similarity transformation) is established based on unit dual quaternion. This paper presents a rigid iterative algorithm of the Helmert transformation using dual quaternion. One small rotation angle Helmert transformation (actual case) and one big rotation angle Helmert transformation (simulative case) are studied. The investigation indicates the presented dual quaternion algorithm (QDA) has an excellent or fast convergence property. If an accurate initial value of scale is provided, e.g., by the solutions no. 2 and 3 of Závoti and Kalmár (Acta Geod Geophys 51: [245][246][247][248][249][250][251][252][253][254][255][256] 2016) in the case that the weights are identical, QDA needs one iteration to obtain the correct result of transformation parameters; in other words, it can be regarded as an analytical algorithm. For other situations, QDA requires two iterations to recover the transformation parameters no matter how big the rotation angles are and how biased the initial value of scale is. Additionally, QDA is capable to deal with point-wise weight transformation which is more rational than those algorithms which simply take identical weights into account or do not consider the weight difference among control points. From the perspective of transformation accuracy, QDA is comparable to the classic Procrustes algorithm (Grafarend and Awange in J Geod 77: [66][67][68][69][70][71][72][73][74][75][76] 2003) and orthonormal matrix algorithm from Zeng (Earth Planets Space 67:105, 2015. https://doi.

Helmert Transformation Problem. From Euler Angles Method to Quaternion Algebra

“…In this way, dual-quaternions are double numbers with their elements being quaternions. More specifically, they are defined as [14,15]: q = r + d•ε with r and d being quaternions and ε 2 = 0 with ε 0…”

Dual Quaternion Based Close Proximity Operation for In-Orbit Assembly via Model Predictive Control

“…As a preliminary of this work, we concisely present relevant physical concepts and mathematical rules of quaternion and dual quaternion. Readers can also refer to literature [31] for more details.…”

WTLS iterative algorithm of 3D similarity coordinate transformation based on Gibbs vectors

“…Apart from using rotations, there are other representation of R , such as unit quaternion (e.g., Horn 1987;Shen et al 2006; Zeng and Yi 2011; Závoti and Kalmár 2016), dual quaternion (e.g., Walker et al 1991;Wang et al 2014;Zeng et al 2018Zeng et al , 2019, direction cosine matrix (e.g., Chen et al 2004;Wang et al 2018), and the Rodrigues matrix or Gibbs vector (e.g., Zeng and Yi 2010;Závoti and Kalmár 2016;Zeng et al 2016;Kurt 2018). The first three approaches to the representation of R introduce functional constraints, for instance the norm of quaternion is unity as the unit quaternion representation is concerned.…”

“…This work is very popular in many fields, such as geodesy, engineering surveying, LIDAR, terrestrial laser scanning, photogrammetry, machine vision, etc. (Besl and McKay 1992;Crosilla and Beinat 2002;Horn 1987;Jaw and Chuang 2008;Kashani 2006;Krarup 1985;Marx 2017;Paffenholz and Bae 2012;Walker et al 1991;Wang et al 2014;Závoti and Kalmár 2016;Zeng 2014;Zeng et al 2018).…”