Singularity-free computation of quaternions from rotation matrices in E4 and E3

A real orthogonal matrix representing a rotation in E4 can be decomposed into the commutative product of a left-isoclinic and a right-isoclinic rotation matrix. The double quaternion representation of rotations in E4 follows directly from this decomposition. In this paper, it is shown how this decomposition can be performed without divisions. This avoids the common numerical issues attributed to the computation of quaternions from rotation matrices. The map from the 4×4 rotation matrices to the set of double unit quaternions is a 2-to-1 covering map. Thus, this map cannot be smoothly inverted. As a consequence, it is erroneously assumed that all inversions should necessarily contain singularities that arise in the form of quotients where the divisor can be arbitrarily small. This misconception is herein clari¿ed. When particularized to three dimensions, it is shown how the resulting formulation outperforms, from the numerical point of view, the celebrated Shepperd’s method.Peer ReviewedPostprint (author's final draft

A fast branch-and-prune algorithm for the position analysis of spherical mechanisms

The final publication is available at link.springer.comDifferent branch-and-prune schemes can be found in the literature for numerically solving the position analysis of spherical mechanisms. For the prune operation, they all rely on the propagation of motion intervals. They differ in the way the problem is algebraically formulated. This paper exploits the fact that spherical kinematic loop equations can be formulated as sets of 3 multi-affine polynomials. Multi-affinity has an important impact on how the propagation of motion intervals can be performed because a multi-affine polynomial is uniquely determined by its values at the vertices of a closed hyperbox defined in its domain.Peer ReviewedPostprint (author's final draft

On Cayley's factorization with an application to the orthonormalization of noisy rotation matrices

The final publication is available at link.springer.comA real orthogonal matrix representing a rotation in four dimensions can be decomposed into the commutative product of a left- and a right-isoclinic rotation matrix. This operation, known as Cayley's factorization, directly provides the double quaternion representation of rotations in four dimensions. This factorization can be performed without divisions, thus avoiding the common numerical issues attributed to the computation of quaternions from rotation matrices. In this paper, it is shown how this result is particularly useful, when particularized to three dimensions, to re-orthonormalize a noisy rotation matrix by converting it to quaternion form and then obtaining back the corresponding proper rotation matrix. This re-orthonormalization method is commonly implemented using the Shepperd-Markley method, but the method derived here is shown to outperform it by returning results closer to those obtained using the Singular Value Decomposition which are known to be optimal in terms of the Frobenius norm.Peer ReviewedPostprint (author's final draft

Accurate computation of quaternions from rotation matrices

The final publication is available at link.springer.comThe main non-singular alternative to 3×3 proper orthogonal matrices, for representing rotations in R3, is quaternions. Thus, it is important to have reliable methods to pass from one representation to the other. While passing from a quaternion to the corresponding rotation matrix is given by Euler-Rodrigues formula, the other way round can be performed in many different ways. Although all of them are algebraically equivalent, their numerical behavior can be quite different. In 1978, Shepperd proposed a method for computing the quaternion corresponding to a rotation matrix which is considered the most reliable method to date. Shepperd’s method, thanks to a voting scheme between four possible solutions, always works far from formulation singularities. In this paper, we propose a new method which outperforms Shepperd’s method without increasing the computational cost.Peer ReviewedPostprint (author's final draft

A survey on the computation of quaternions from rotation matrices

The parameterization of rotations is a central topic in many theoretical and applied fields such as rigid body mechanics, multibody dynamics, robotics, spacecraft attitude dynamics, navigation, 3D image processing, computer graphics, etc. Nowadays, the main alternative to the use of rotation matrices, to represent rotations in R3, is the use of Euler parameters arranged in quaternion form. Whereas the passage from a set of Euler parameters to the corresponding rotation matrix is unique and straightforward, the passage from a rotation matrix to its corresponding Euler parameters has been revealed to be somewhat tricky if numerical aspects are considered. Since the map from quaternions to 3x3 rotation matrices is a 2-to-1 covering map, this map cannot be smoothly inverted. As a consequence, it is erroneously assumed that all inversions should necessarily contain singularities that arise in the form of quotients where the divisor can be arbitrarily small. This misconception is herein clarified. This paper reviews the most representative methods available in the literature, including a comparative analysis of their computational costs and error performances. The presented analysis leads to the conclusion that Cayley's factorization, a little-known method used to compute the double quaternion representation of rotations in four dimensions from 4x4 rotation matrices, is the most robust method when particularized to three dimensionsPreprin

A Survey on Dual-Quaternions

Over the past few years, the applications of dual-quaternions have not only developed in many different directions but has also evolved in exciting ways in several areas. As dual-quaternions offer an efficient and compact symbolic form with unique mathematical properties. While dual-quaternions are now common place in many aspects of research and implementation, such as, robotics and engineering through to computer graphics and animation, there are still a large number of avenues for exploration with huge potential benefits. This article is the first to provide a comprehensive review of the dual-quaternion landscape. In this survey, we present a review of dual-quaternion techniques and applications developed over the years while providing insights into current and future directions. The article starts with the definition of dual-quaternions, their mathematical formulation, while explaining key aspects of importance (e.g., compression and ambiguities). The literature review in this article is divided into categories to help manage and visualize the application of dual-quaternions for solving specific problems. A timeline illustrating key methods is presented, explaining how dual-quaternion approaches have progressed over the years. The most popular dual-quaternion methods are discussed with regard to their impact in the literature, performance, computational cost and their real-world results (compared to associated models). Finally, we indicate the limitations of dual-quaternion methodologies and propose future research directions.Comment: arXiv admin note: text overlap with arXiv:2303.1339

A bilinear formulation for the motion planning of non-holonomic parallel orienting platforms

This paper deals with the motion planning problem for parallel orienting platforms with one non-holonomic joint and two prismatic actuators which can maneuver to reach any three-degree-of-freedom pose of the moving platform. Since any system with two inputs and up to four generalized coordinates can always be transformed into chained form, this path planning problem can be solved using well-established procedures. Nevertheless, the use of these procedures requires a good understanding of Lie algebraic methods whose technicalities have proven a challenge to many practitioners who are not familiar with them. As an alternative, we show how by (a) properly locating the actuators, and (b) representing the platform orientation using Euler parameters, the studied path planning problem admits a closed-form solution whose derivation requires no other tools than ordinary linear algebra.Postprint (author’s final draft

Hand-eye calibration made easy through a closed-form two-stage method

© 2022 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting /republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other worksAn analysis of the existing hand-eye calibration methods reveals that most of them are far from trivial. And, what is worse, their intrinsic complexity makes it difficult to elucidate under which circumstances they fail to provide an accurate solution. Thus, although it might seem that hand-eye calibration problem is uninspiring because it is assumed to be well-solved, we show in this paper that there was still room for improvement, both in terms of simplicity and robustness. After reviewing the most representative methods, we analyze the situations in which they fail, and we introduce a simpler closed-form alternative that accurately solves the problem in all the identified critical circumstances. Its performance is evaluated using simulated and real experimental data.Peer ReviewedPostprint (author's final draft