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Journal articles on the topic 'Quaternion analysis'

Author: Grafiati

Published: 29 July 2024

Last updated: 30 July 2024

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1

Dai, Songsong. "Quaternionic Fuzzy Sets." Axioms 12, no. 5 (May 18, 2023): 490. http://dx.doi.org/10.3390/axioms12050490.

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A novel concept of quaternionic fuzzy sets (QFSs) is presented in this paper. QFSs are a generalization of traditional fuzzy sets and complex fuzzy sets based on quaternions. The novelty of QFSs is that the range of the membership function is the set of quaternions with modulus less than or equal to one, of which the real and quaternionic imaginary parts can be used for four different features. A discussion is made on the intuitive interpretation of quaternion-valued membership grades and the possible applications of QFSs. Several operations, including quaternionic fuzzy complement, union, intersection, and aggregation of QFSs, are presented. Quaternionic fuzzy relations and their composition are also investigated. QFS is designed to maintain the advantages of traditional FS and CFS, while benefiting from the properties of quaternions. Cuts of QFSs and rotational invariance of quaternionic fuzzy operations demonstrate the particularity of quaternion-valued grades of membership.

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2

Miškinis, P. "ON THE POSSIBLE EXISTENCE OF NEW FERMIONIC DEGREES OF FREEDOM IN D = 6." Mathematical Modelling and Analysis 8, no. 2 (June 30, 2003): 155–64. http://dx.doi.org/10.3846/13926292.2003.9637220.

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The bispinors formed by quaternion in D = 6 dimensional space‐time are proposed to be treated as new fermionic fields. The gauge nonabelian field is formulated by the quaternions. A new kind of physical object, an extended relativistic quaternionic membrane in D = 6, is discussed.

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3

Kim, Ji Eun, and Kwang Ho Shon. "The Regularity of Functions on Dual Split Quaternions in Clifford Analysis." Abstract and Applied Analysis 2014 (2014): 1–8. http://dx.doi.org/10.1155/2014/369430.

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This paper shows some properties of dual split quaternion numbers and expressions of power series in dual split quaternions and provides differential operators in dual split quaternions and a dual split regular function onΩ⊂ℂ2×ℂ2that has a dual split Cauchy-Riemann system in dual split quaternions.

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4

Chelnokov, Y. N. "Quaternion methods and models of regular celestial mechanics and astrodynamics." Applied Mathematics and Mechanics 43, no. 1 (January 2022): 21–80. http://dx.doi.org/10.1007/s10483-021-2797-9.

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AbstractThis paper is a review, which focuses on our work, while including an analysis of many works of other researchers in the field of quaternionic regularization. The regular quaternion models of celestial mechanics and astrodynamics in the Kustaanheimo-Stiefel (KS) variables and Euler (Rodrigues-Hamilton) parameters are analyzed. These models are derived by the quaternion methods of mechanics and are based on the differential equations of the perturbed spatial two-body problem and the perturbed spatial central motion of a point particle. This paper also covers some applications of these models. Stiefel and Scheifele are known to have doubted that quaternions and quaternion matrices can be used efficiently to regularize the equations of celestial mechanics. However, the author of this paper and other researchers refuted this point of view and showed that the quaternion approach actually leads to efficient solutions for regularizing the equations of celestial mechanics and astrodynamics.This paper presents convenient geometric and kinematic interpretations of the KS transformation and the KS bilinear relation proposed by the present author. More general (compared with the KS equations) quaternion regular equations of the perturbed spatial two-body problem in the KS variables are presented. These equations are derived with the assumption that the KS bilinear relation was not satisfied. The main stages of the quaternion theory of regularizing the vector differential equation of the perturbed central motion of a point particle are presented, together with regular equations in the KS variables and Euler parameters, derived by the aforementioned theory. We also present the derivation of regular quaternion equations of the perturbed spatial two-body problem in the Levi-Civita variables and the Euler parameters, developed by the ideal rectangular Hansen coordinates and the orientation quaternion of the ideal coordinate frame.This paper also gives new results using quaternionic methods in the perturbed spatial restricted three-body problem.

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Gadetska, Svitlana, Valeriy Dubnitskiy, Yuri Kushneruk, and Alexander Khodyrev. "Calculation of the results of basic arithmetic operations with quaternions defined in the interval form." Advanced Information Systems 7, no. 4 (December 3, 2023): 21–32. http://dx.doi.org/10.20998/2522-9052.2023.4.03.

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The goal of the work. Creation of an EXCEL-oriented calculator for determining the results of basic arithmetic operations with quaternions, which are given by hyperbolic interval numbers. The subject of research is the set of quaternions and arithmetic operations defined on this set. Research method: Algorithmic and numerical analysis of procedures for performing basic arithmetic operations with quaternions given by hyperbolic interval numbers. The obtained results. The definition of a quaternion, the coefficients of which are interval numbers, is given. It was obtained the ratios for determining in the interval form the following characteristics: quaternion norm, quaternion modulus, quaternion vector part modulus, quaternion argument, quaternion sign was obtained. It was obtained the ratios for determining in the interval form the following basic arithmetic operations: sum of quaternions, difference of quaternions, multiplication of a constant value by a quaternion, scalar product of quaternions, product of vector parts of quaternions, product of quaternions, vector product of quaternions, left and right division of quaternions. The ratio for calculating the determinants of the second order, the elements of which are defined in the interval form, is obtained. The ratio for the approximate calculation of determinants of arbitrary order, the elements of which are defined in the interval form, is obtained. It is shown that the operations of multiplication (division) of interval numbers and raising them to an integer power are expedient to perform when they have a hyperbolic form. It is advisable to perform the addition (subtraction) operation with interval numbers given in the classic form or in the CENTER-RADIUS system. It is better to use the last form in the case of determining the coefficients of the quaternions based on the results of technological calculations. Screenshots of the formulas for determining the vector product of quaternions are given, provided that the coefficients at their orths are represented by interval numbers.

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6

Kim, Ji-Eun. "Approximation of Directional Step Derivative of Complex-Valued Functions Using a Generalized Quaternion System." Axioms 10, no. 3 (August 30, 2021): 206. http://dx.doi.org/10.3390/axioms10030206.

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The step derivative of a complex function can be defined with various methods. The step direction defines a basis that is distinct from that of a complex number; the derivative can then be treated by using Taylor series expansion in this direction. In this study, we define step derivatives based on complex numbers and quaternions that are orthogonal to the complex basis while simultaneously being distinct from it. Considering previous studies, the step derivative defined using quaternions was insufficient for applying the properties of quaternions by setting a quaternion basis distinct from the complex basis or setting the step direction to which only a part of the quaternion basis was applied. Therefore, in this study, we examine the definition of quaternions and define the step derivative in the direction of a generalized quaternion basis including a complex basis. We find that the step derivative based on the definition of a quaternion has a relative error in some domains; however, it can be used as a substitute derivative in specific domains.

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7

Scheglov, Georgiy. "On the geometric interpretation of quaternions by cones." Geometry & Graphics 10, no. 3 (January 9, 2023): 23–34. http://dx.doi.org/10.12737/2308-4898-2023-10-3-23-34.

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The geometric interpretation of quaternions is considered. The visualization complexity of quaternions is due to the fact that these objects have four independent parameters. A literature analysis shows that the problem of geometric interpretation of quaternions has not been completely solved to date. The first section provides general provisions on quaternions and the necessary notations. The second section describes the classical geometric interpretation of quaternions by arcs on a unit sphere. The third section describes a new geometric interpretation and its application to the problem of a vector finite rotation. The geometric interpretation of the quaternion as the surface of a right circular cone is presented. This representation allow demonstrating it as a holistic object in which the scalar and vector parts are interconnected, taking into account their modules and signs. For the considered normalized quaternion, it is easy to understanding an important entity, the quaternion versor: in general, it is a cone, which in the limiting case of a pure scalar quaternion transform into a sphere, and in the limiting case of a pure vector quaternion transform into an ordinary vector. This distinctive feature of the proposed geometric interpretation makes it possible, even when projected onto a plane, to clearly distinguish visualization of the quaternions with a nonzero scalar part from pure vector quaternions, which is difficult to do in the other known interpretations. The representation of quaternions by cones clearly demonstrates the need for a double quaternion product, when the vector is rotated around an arbitrary axis. Images of quaternions as cones, spheres and vectors can be useful in the study of quaternion algebra, which is currently finding increasing use in engineering.

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Scheglov, Georgiy. "On the geometric interpretation of quaternions by cones." Geometry & Graphics 10, no. 3 (September 26, 2022): 23–34. http://dx.doi.org/10.12737/2308-4898-2022-10-3-23-34.

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The geometric interpretation of quaternions is considered. The visualization complexity of quaternions is due to the fact that these objects have four independent parameters. A literature analysis shows that the problem of geometric interpretation of quaternions has not been completely solved to date. The first section provides general provisions on quaternions and the necessary notations. The second section describes the classical geometric interpretation of quaternions by arcs on a unit sphere. The third section describes a new geometric interpretation and its application to the problem of a vector finite rotation. The geometric interpretation of the quaternion as the surface of a right circular cone is presented. This representation allow demonstrating it as a holistic object in which the scalar and vector parts are interconnected, taking into account their modules and signs. For the considered normalized quaternion, it is easy to understanding an important entity, the quaternion versor: in general, it is a cone, which in the limiting case of a pure scalar quaternion transform into a sphere, and in the limiting case of a pure vector quaternion transform into an ordinary vector. This distinctive feature of the proposed geometric interpretation makes it possible, even when projected onto a plane, to clearly distinguish visualization of the quaternions with a nonzero scalar part from pure vector quaternions, which is difficult to do in the other known interpretations. The representation of quaternions by cones clearly demonstrates the need for a double quaternion product, when the vector is rotated around an arbitrary axis. Images of quaternions as cones, spheres and vectors can be useful in the study of quaternion algebra, which is currently finding increasing use in engineering.

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9

Dargys, Adolfas, and Artūras Acus. "Exponential and logarithm of multivector in low-dimensional (n = p + q < 3) Clifford algebras." Nonlinear Analysis: Modelling and Control 27, no. 6 (October 19, 2022): 1129–49. http://dx.doi.org/10.15388/namc.2022.27.29528.

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The aim of the paper is to give a uniform picture of complex, hyperbolic, and quaternion algebras from a perspective of the applied Clifford geometric algebra. Closed form expressions for a multivector exponential and logarithm are presented in real geometric algebras Clp;q when n = p + q = 1 (complex and hyperbolic numbers) and n = 2 (Hamilton, split, and conectorine quaternions). Starting from Cl0;1 and Cl1;0 algebras wherein square of a basis vector is either –1 or +1, we have generalized exponential and logarithm formulas to 2D quaternionic algebras Cl0;2, Cl1;1, and Cl2;0. The sectors in the multivector coefficient space, where 2D logarithm exists are found. They are related with a square root of the multivector.

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10

Kou, Kit Ian, Jian-Yu Ou, and Joao Morais. "On Uncertainty Principle for Quaternionic Linear Canonical Transform." Abstract and Applied Analysis 2013 (2013): 1–14. http://dx.doi.org/10.1155/2013/725952.

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We generalize the linear canonical transform (LCT) to quaternion-valued signals, known as the quaternionic linear canonical transform (QLCT). Using the properties of the LCT we establish an uncertainty principle for the QLCT. This uncertainty principle prescribes a lower bound on the product of the effective widths of quaternion-valued signals in the spatial and frequency domains. It is shown that only a 2D Gaussian signal minimizes the uncertainty.

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Kartiwa, Alit, Asep K. Supriatna, Endang Rusyaman, and Jumat Sulaiman. "Review of Quaternion Differential Equations: Historical Development, Applications, and Future Direction." Axioms 12, no. 5 (May 16, 2023): 483. http://dx.doi.org/10.3390/axioms12050483.

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Quaternion is a four-dimensional and an extension of the complex number system. It is often viewed from various fields, such as analysis, algebra, and geometry. Several applications of quaternions are related to an object’s rotation and motion in three-dimensional space in the form of a differential equation. In this paper, we do a systematic literature review on the development of quaternion differential equations. We utilize PRISMA (preferred reporting items for systematic review and meta-analyses) framework in the review process as well as content analysis. The expected result is a state-of-the-art and the gap of concepts or problems that still need to develop or answer. It was concluded that there are still some opportunities to develop a quaternion differential equation using a quaternion function domain.

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12

Dzwonkowski, Mariusz, and Roman Rykaczewski. "Quaternion Feistel Cipher with an Infinite Key Space Based on Quaternion Julia Sets." Journal of Telecommunications and Information Technology, no. 4 (December 30, 2015): 15–21. http://dx.doi.org/10.26636/jtit.2015.4.979.

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In this paper Quaternion Feistel Cipher (QFC) with an infinite key space based on quaternion Julia sets is proposed. The basic structure of the algorithm is based on the scheme proposed in 2012 by Sastry and Kumar. The proposed algorithm uses special properties of quaternions to perform rotations of data sequences in 3D space for each of the cipher rounds. It also uses Julia sets to form an infinite key space. The plaintext is divided into two square matrices of equal size and written using Lipschitz quaternions. A modular arithmetic was implemented for operations with quaternions. A computer-based analysis has been carried out and obtained results are shown at the end of this paper.

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13

de Oliveira, Andre Schneider, Edson Roberto De Pieri, Ubirajara Franco Moreno, and Daniel Martins. "A new approach to singularity-free inverse kinematics using dual-quaternionic error chains in the Davies method." Robotica 34, no. 4 (July 24, 2014): 942–56. http://dx.doi.org/10.1017/s0263574714001982.

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SUMMARYThe manipulation in singular regions promotes an instantaneous reduction in mechanism mobility, which can result in some disturbances in the trajectory tracking. The application of the quaternionic elements for motion representation not only guarantees an orthonormal transformation but also results in the smallest variance and minimizes the acceleration peaks. The use of a unit quaternion avoids these phenomena, but there are dimensional limitations that make it impossible to translate the representation. This work presents a methodology for using dual quaternions in the analysis of robot kinematics using the Davies method, which avoids kinematic singularities and ensures the optimal torque profiles.

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Weng, Zi-Hua. "Forces in the complex octonion curved space." International Journal of Geometric Methods in Modern Physics 13, no. 06 (June 15, 2016): 1650076. http://dx.doi.org/10.1142/s0219887816500766.

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The paper aims to extend major equations in the electromagnetic and gravitational theories from the flat space into the complex octonion curved space. Maxwell applied simultaneously the quaternion analysis and vector terminology to describe the electromagnetic theory. It inspires subsequent scholars to study the electromagnetic and gravitational theories with the complex quaternions/octonions. Furthermore Einstein was the first to depict the gravitational theory by means of tensor analysis and curved four-space–time. Nowadays some scholars investigate the electromagnetic and gravitational properties making use of the complex quaternion/octonion curved space. From the orthogonality of two complex quaternions, it is possible to define the covariant derivative of the complex quaternion curved space, describing the gravitational properties in the complex quaternion curved space. Further it is possible to define the covariant derivative of the complex octonion curved space by means of the orthogonality of two complex octonions, depicting simultaneously the electromagnetic and gravitational properties in the complex octonion curved space. The result reveals that the connection coefficient and curvature of the complex octonion curved space will exert an influence on the field strength and field source of the electromagnetic and gravitational fields, impacting the linear momentum, angular momentum, torque, energy, and force and so forth.

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Li, Guozhi, Fuhai Zhang, Yili Fu, and Shuguo Wang. "Kinematic calibration of serial robot using dual quaternions." Industrial Robot: the international journal of robotics research and application 46, no. 2 (March 18, 2019): 247–58. http://dx.doi.org/10.1108/ir-10-2018-0221.

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Purpose The purpose of this paper is to propose an error model for serial robot kinematic calibration based on dual quaternions. Design/methodology/approach The dual quaternions are the combination of dual-number theory and quaternion algebra, which means that they can represent spatial transformation. The dual quaternions can represent the screw displacement in a compact and efficient way, so that they are used for the kinematic analysis of serial robot. The error model proposed in this paper is derived from the forward kinematic equations via using dual quaternion algebra. The full pose measurements are considered to apply the error model to the serial robot by using Leica Geosystems Absolute Tracker (AT960) and tracker machine control (T-MAC) probe. Findings Two kinematic-parameter identification algorithms are derived from the proposed error model based on dual quaternions, and they can be used for serial robot calibration. The error model uses Denavit–Hartenberg (DH) notation in the kinematic analysis, so that it gives the intuitive geometrical meaning of the kinematic parameters. The absolute tracker system can measure the position and orientation of the end-effector (EE) simultaneously via using T-MAC. Originality/value The error model formulated by dual quaternion algebra contains all the basic geometrical parameters of serial robot during the kinematic calibration process. The vector of dual quaternion error can be used as an indicator to represent the trend of error change of robot’s EE between the nominal value and the actual value. The accuracy of the EE is improved after nearly 20 measurements in the experiment conduct on robot SDA5F. The simulation and experiment verify the effectiveness of the error model and the calibration algorithms.

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de Farias, Cristiana Miranda. "A Novel Dual Quaternion Based Cost Effcient Recursive Newton-Euler Inverse Dynamics Algorithm." International Journal of Robotic Computing 1, no. 2 (December 1, 2019): 144–68. http://dx.doi.org/10.35708/rc1868-126255.

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In this paper, the well known recursive Newton-Euler inverse dynamics algorithm for serial manipulators is reformulated into the context of the algebra of Dual Quaternions. Here we structure the forward kinematic description with screws and line displacements rather than the well established Denavit-Hartemberg parameters, thus accounting better efficiency, compactness and simpler dynamical models. We also present here the closed solution for the dqRNEA, and to do so we formalize some of the algebra for dual quaternion-vectors and dual quaternion-matrices. With a closed formulation of the dqRNEA we also create a dual quaternion based formulation for the computed torque control, a feedback linearization method for controlling a serial manipulator's torques in the joint space. Finally, a cost analysis of the main Dual Quaternions operations and of the Newton-Euler inverse dynamics algorithm as a whole is made and compared with other results in the literature.

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Condurache, Daniel, Mihail Cojocari, and Ionuţ Popa. "Hypercomplex Quaternions and Higher-Order Analysis of Spatial Kinematic Chains." BULETINUL INSTITUTULUI POLITEHNIC DIN IAȘI. Secția Matematica. Mecanică Teoretică. Fizică 69, no. 1-4 (December 1, 2023): 21–34. http://dx.doi.org/10.2478/bipmf-2023-0002.

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Abstract This paper introduces a novel computational method for analyzing the higher-order acceleration field of spatial kinematics chains. The method is based on vector and quaternionic calculus, as well as dual and multidual algebra. A closed-form coordinate-free solution generated by the morphism between the Lie group of rigid body displacements and the unit multidual quaternions is presented. Presented solution is used for higher-order kinematics investigation of lower-pair serial chains. Additionally, a general method for studying the vector field of arbitrary higher-order accelerations is discribed. The method utilizes the “automatic differentiation” feature of multidual and hyper-multidual functions to obtain the higher-order derivative of a rigid body pose without need in further differentiation of the body pose regarding time. Also is proved that all information regarding the properties of the distribution of higher-order accelerations is contained in the specified unit hyper-multidual quaternion.

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Lv, Jiaojiao, Jinrong Wang, and Rui Liu. "Hyers-Ulam stability of linear quaternion-valued differential equations." Electronic Journal of Differential Equations 2023, no. 01-37 (February 27, 2023): 21. http://dx.doi.org/10.58997/ejde.2023.21.

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In this article, we study the Hyers-Ulam stability of the first-order linear quaternion-valued differential equations. We transfer a linear quaternion-valued differential equation into a real differential system. The Hyers-Ulam stability results for the linear quaternion-valued differential equations are obtained according to the equivalent relationship between the vector 2-norm and the quaternion module.

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Cao, Wensheng. "Quadratic Equation in Split Quaternions." Axioms 11, no. 5 (April 20, 2022): 188. http://dx.doi.org/10.3390/axioms11050188.

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Split quaternions are noncommutative and contain nontrivial zero divisors. Generally speaking, it is difficult to solve equations in such an algebra. In this paper, by using the roots of any split quaternions and two real nonlinear systems, we derive explicit formulas for computing the roots of x2+bx+c=0 in split quaternion algebra.

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NG, CHI-KEUNG. "On quaternionic functional analysis." Mathematical Proceedings of the Cambridge Philosophical Society 143, no. 2 (September 2007): 391–406. http://dx.doi.org/10.1017/s0305004107000187.

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AbstractIn this paper, we will show that the category of quaternion vector spaces, the category of (both one-sided and two sided) quaternion Hilbert spaces and the category of quaternion B*-algebras are equivalent to the category of real vector spaces, the category of real Hilbert spaces and the category of real C*-algebras respectively. We will also give a Riesz representation theorem for quaternion Hilbert spaces and will extend the main results in [12] (namely, we will give the full versions of the Gelfand–Naimark theorem and the Gelfand theorem for quaternion B*-algebras). On our way to these results, we compare, clarify and unify the term ‘quaternion Hilbert spaces’ in the literatures.

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Dinh, Doan Cong. "Fundamental solutions of the Stokes system in quaternion analysis." Ukrains’kyi Matematychnyi Zhurnal 74, no. 11 (December 26, 2022): 1492–505. http://dx.doi.org/10.37863/umzh.v74i11.6510.

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UDC 532.5 The method of quaternionic analysis in fluid mechanics was developed by several generations of mathematicians with numerous important results. We add a small result in this direction. Thus, we introduce a new reformulation of fundamental solutions of the Stokes system within the framework of quaternion analysis and construct integral representations for its solutions.

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Song, Guangjing, Weiyang Ding, and Michael K. Ng. "Low Rank Pure Quaternion Approximation for Pure Quaternion Matrices." SIAM Journal on Matrix Analysis and Applications 42, no. 1 (January 2021): 58–82. http://dx.doi.org/10.1137/19m1307329.

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23

Kim, Ji, and Kwang Shon. "Expansion of implicit mapping theory to split-quaternionic maps in Clifford analysis." Filomat 35, no. 11 (2021): 3833–40. http://dx.doi.org/10.2298/fil2111833k.

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This paper presents the regularity of a split-quaternionic function and a corresponding split- Cauchy-Riemann system of a split quaternion. The properties of an inverse and an implicit mapping theory for a split-quaternionic map are investigated. In addition, the paper proposes a definition and expression for a split biregular mapping in an open set in C2. The obtained results are illustrated with some examples.

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Mansouri, Imed, and Mohammed Ouali. "QUATERNION REPRESENTATION OF THE POWER MANIPULABILITY." Transactions of the Canadian Society for Mechanical Engineering 35, no. 2 (June 2011): 309–36. http://dx.doi.org/10.1139/tcsme-2011-0018.

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In this paper, a quaternion formulation of the power manipulability is developed. Quantifying the mechanical transmissibility of a mechanism, this parameter was previously introduced as a new homogeneous performance index of robot manipulators; however, its evaluation requires complex manipulations, particularly those of quadri-vectors. Furthermore, the quadri-vector form fits exactly with the structure of quaternions. Hence, a quaternion based method may be useful for power performance indices evaluation. It will be shown that this method leads to a recursive algorithm that manipulates kinematics’ parameters as matrices defined over the real quaternion space. Therefore, a new mostly simplified procedure is elaborated, which is suitable to robots performance analysis related issues. The resultant algorithm is largely simpler and more efficient to evaluate the power based performance indices.

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Ahmed, Abubaker, Hehua Ju, Yang Yang, and Hao Xu. "An Improved Unit Quaternion for Attitude Alignment and Inverse Kinematic Solution of the Robot Arm Wrist." Machines 11, no. 7 (June 21, 2023): 669. http://dx.doi.org/10.3390/machines11070669.

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This paper introduces a new method for kinematic modeling of the robot arm by deriving a new elegant mathematical formula based on the axis vector with the tangent of the rotation angle. For this purpose, an innovative analytical quaternion is introduced through integration between Axis-Invariants and unit quaternion features named Ju-Gibbs quaternion, which expresses the body rotation with non-redundant parameters compared with the quaternions in literature. Two theorems based on the new form of the quaternion are developed and proved for the kinematic modeling of the robot arm. The first is attitude alignment, which is essential in multiaxial rotation systems. The second theorem for the wrist inverse kinematic (IK) solution is utilized to obtain the joint variables for the last joints of the end effector. In order to verify the effectiveness and accuracy of the proposed method, a numerical example and simulation of different structural configurations of robot and human arms are intensively studied. The novel quaternion provides a new tool for kinematic analysis and reduces the computational complexity of the kinematic solutions of the Robot-Arms wrist. Furthermore, the method laid a new foundation for the IKs of multi-axis systems based on Axis-Invariant and tangent quaternion.

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Merkulov, S. A. "Quaternion supermanifolds." Functional Analysis and Its Applications 27, no. 1 (January 1993): 66–68. http://dx.doi.org/10.1007/bf01768674.

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Li, Guozhi, Shuizhong Zou, Shuxue Din, and Bin Qi. "Modified Hand–Eye Calibration Using Dual Quaternions." Applied Sciences 12, no. 23 (December 6, 2022): 12480. http://dx.doi.org/10.3390/app122312480.

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This paper presents a modified model for hand–eye calibration based on dual quaternion algebra. By using dual quaternions to represent the rotations and translations of a rigid body simultaneously in the task space, the formulation is elegant for the analysis of the hand–eye equation. The hand–eye transformation derived in this study is represented in a compact manner, which uses a combination of the dual part and the real part of the dual quaternion. Although the hand–eye equation can be solved by using six elements of a dual quaternion without using its scalar parts, the scaler numbers in both the real and dual parts of a dual quaternion contain part of the pose information. The originality is based on the derivation of the construct of the identification algorithm of external parameters of the camera by using all eight elements of a dual quaternion. Then, the data transformation between the cameras of the dual-arm hand–eye robot system is presented. The corresponding results demonstrate that the proposed hand–eye calibration algorithm can process measurement data with noise and can also improve the identification accuracy to verify its efficiency.

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Zhang, Lina, Yu Sang, and Donghai Dai. "Accurate Quaternion Polar Harmonic Transform for Color Image Analysis." Scientific Programming 2021 (December 29, 2021): 1–9. http://dx.doi.org/10.1155/2021/7162779.

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Polar harmonic transforms (PHTs) have been applied in pattern recognition and image analysis. But the current computational framework of PHTs has two main demerits. First, some significant color information may be lost during color image processing in conventional methods because they are based on RGB decomposition or graying. Second, PHTs are influenced by geometric errors and numerical integration errors, which can be seen from image reconstruction errors. This paper presents a novel computational framework of quaternion polar harmonic transforms (QPHTs), namely, accurate QPHTs (AQPHTs). First, to holistically handle color images, quaternion-based PHTs are introduced by using the algebra of quaternions. Second, the Gaussian numerical integration is adopted for geometric and numerical error reduction. When compared with CNNs (convolutional neural networks)-based methods (i.e., VGG16) on the Oxford5K dataset, our AQPHT achieves better performance of scaling invariant representation. Moreover, when evaluated on standard image retrieval benchmarks, our AQPHT using smaller dimension of feature vector achieves comparable results with CNNs-based methods and outperforms the hand craft-based methods by 9.6% w.r.t mAP on the Holidays dataset.

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Rautenbach, H. M., and J. J. J. Roux. "Statistical analysis based on quaternion normal random variables." Suid-Afrikaanse Tydskrif vir Natuurwetenskap en Tegnologie 4, no. 3 (March 18, 1985): 120–27. http://dx.doi.org/10.4102/satnt.v4i3.1042.

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The quaternion normal distribution is derived and a number of characteristics are highlighted. The maximum likelihood estimation procedure in the quaternion case is examined and the conclusion is reached that the estimation procedure is simplified if the unknown parameters of the associated real probability density function are estimated. The quaternion estimator is then obtained by regarding these estimators as the components of the quaternion estimator. By means of a example attention is given to a test criterium which can be used in the quaternion model.

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Bahri, Mawardi, Resnawati, and Selvy Musdalifah. "A Version of Uncertainty Principle for Quaternion Linear Canonical Transform." Abstract and Applied Analysis 2018 (2018): 1–7. http://dx.doi.org/10.1155/2018/8732457.

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In recent years, the two-dimensional (2D) quaternion Fourier and quaternion linear canonical transforms have been the focus of many research papers. In the present paper, based on the relationship between the quaternion Fourier transform (QFT) and the quaternion linear canonical transform (QLCT), we derive a version of the uncertainty principle associated with the QLCT. We also discuss the generalization of the Hausdorff-Young inequality in the QLCT domain.

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Bahri, Mawardi, and Ryuichi Ashino. "A Convolution Theorem Related to Quaternion Linear Canonical Transform." Abstract and Applied Analysis 2019 (May 28, 2019): 1–9. http://dx.doi.org/10.1155/2019/3749387.

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We introduce the two-dimensional quaternion linear canonical transform (QLCT), which is a generalization of the classical linear canonical transform (LCT) in quaternion algebra setting. Based on the definition of quaternion convolution in the QLCT domain we derive the convolution theorem associated with the QLCT and obtain a few consequences.

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Mohanapriya, Arusamy. "Stability analysis of linear quaternion-valued differential equation using integral transform." Annals of Mathematics and Computer Science 22 (March 28, 2024): 88–99. http://dx.doi.org/10.56947/amcs.v22.274.

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In this article, we examine the stability of first-order linear quaternion-valued differential equations using the Mittag-Leffler-Hyers-Ulam approach. We achieve this by transforming a linear quaternion-valued differential equation into a real differential system. The stability outcomes for these linear quaternion-valued differential equations are determined through the use of quaternion module and Fourier transform techniques.

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Yin, Ming, Wei Liu, Jun Shui, and Jiangmin Wu. "Quaternion Wavelet Analysis and Application in Image Denoising." Mathematical Problems in Engineering 2012 (2012): 1–21. http://dx.doi.org/10.1155/2012/493976.

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The quaternion wavelet transform is a new multiscale analysis tool. Firstly, this paper studies the standard orthogonal basis of scale space and wavelet space of quaternion wavelet transform in spatialL2(R2), proves and presents quaternion wavelet’s scale basis function and wavelet basis function concepts in spatial scale spaceL2(R2;H), and studies quaternion wavelet transform structure. Finally, the quaternion wavelet transform is applied to image denoising, and generalized Gauss distribution is used to model QWT coefficients’ magnitude distribution, under the Bayesian theory framework, to recover the original coefficients from the noisy wavelet coefficients, and so as to achieve the aim of denoising. Experimental results show that our method is not only better than many of the current denoising methods in the peak signal to noise ratio (PSNR), but also obtained better visual effect.

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Plaksiy, Yuriy, and Yuriy Kuznyetsov. "Analytical-numerical modeling of the process of solid body orientation in quaternions through a sequence of Euler angles for accurate analysis of orientation algorithms in SINS." Bulletin of the National Technical University «KhPI» Series: Dynamics and Strength of Machines, no. 2 (December 21, 2023): 58–64. http://dx.doi.org/10.20998/2078-9130.2023.2.293474.

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Two concepts for constructing analytical angular test motions of a rigid body for testing orientation algorithms when designing strapdown orientation systems are considered. The first concept is based on representing the orientation quaternion in a sequence of three Euler angles. The second concept is based on a formalized representation of the quaternion as a superposition of trigonometric functions of linear arguments and does not have a clear visual interpretation through the angles of elementary rotations. Analytical expressions for the model angular velocity can be obtained from the inverted kinematic equation in quaternions. The general case of linear Krylov and Euler angles is considered, as well as the case when one of the angles does not change over time. Analytical-numerical modeling of the angular motion of a rigid body and an assessment of the accuracy of the algorithm for determining the quaternion based on fourth- and fifth-order expansions with preliminary application of the Miller algorithm were carried out. For this purpose, the test movement model is supplemented by modeling ideal information from the outputs of angular velocity sensors in the form of quasi-coordinates using analytical formulas for the apparent rotation vector. It is shown that fifth-order formulas provide an improved estimate of the accumulated computational drift compared to fourth-order formulas. Keywords: Euler angles, orientation vector, quaternion, reference model, test motion, quasi-coordinates, Miller orientation algorithm, numerical-analytical modeling, accumulated drift.

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Bahri, Mawardi, Ryuichi Ashino, and Rémi Vaillancourt. "Convolution Theorems for Quaternion Fourier Transform: Properties and Applications." Abstract and Applied Analysis 2013 (2013): 1–10. http://dx.doi.org/10.1155/2013/162769.

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General convolution theorems for two-dimensional quaternion Fourier transforms (QFTs) are presented. It is shown that these theorems are valid not only for real-valued functions but also for quaternion-valued functions. We describe some useful properties of generalized convolutions and compare them with the convolution theorems of the classical Fourier transform. We finally apply the obtained results to study hypoellipticity and to solve the heat equation in quaternion algebra framework.

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Dong, Wenjun, Yujiao Huang, Tingan Chen, Xinggang Fan, and Haixia Long. "Local Lagrange Exponential Stability Analysis of Quaternion-Valued Neural Networks with Time Delays." Mathematics 10, no. 13 (June 21, 2022): 2157. http://dx.doi.org/10.3390/math10132157.

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This study on the local stability of quaternion-valued neural networks is of great significance to the application of associative memory and pattern recognition. In the research, we study local Lagrange exponential stability of quaternion-valued neural networks with time delays. By separating the quaternion-valued neural networks into a real part and three imaginary parts, separating the quaternion field into 34n subregions, and using the intermediate value theorem, sufficient conditions are proposed to ensure quaternion-valued neural networks have 34n equilibrium points. According to the Halanay inequality, the conditions for the existence of 24n local Lagrange exponentially stable equilibria of quaternion-valued neural networks are established. The obtained stability results improve and extend the existing ones. Under the same conditions, quaternion-valued neural networks have more stable equilibrium points than complex-valued neural networks and real-valued neural networks. The validity of the theoretical results were verified by an example.

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Chen, Shenglong, Hong-Li Li, Leimin Wang, Cheng Hu, Haijun Jiang, and Zhiming Li. "Finite-time adaptive synchronization of fractional-order delayed quaternion-valued fuzzy neural networks." Nonlinear Analysis: Modelling and Control 28 (June 19, 2023): 1–20. http://dx.doi.org/10.15388/namc.2023.28.32505.

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Based on direct quaternion method, this paper explores the finite-time adaptive synchronization (FAS) of fractional-order delayed quaternion-valued fuzzy neural networks (FODQVFNNs). Firstly, a useful fractional differential inequality is created, which offers an effective way to investigate FAS. Then two novel quaternion-valued adaptive control strategies are designed. By means of our newly proposed inequality, the basic knowledge about fractional calculus, reduction to absurdity as well as several inequality techniques of quaternion and fuzzy logic, several sufficient FAS criteria are derived for FODQVFNNs. Moreover, the settling time of FAS is estimated, which is in connection with the order and initial values of considered systems as well as the controller parameters. Ultimately, the validity of obtained FAS criteria is corroborated by numerical simulations.

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Agarwal, Ravi P., Hamed H. Alsulami, Erdal Karapınar, and Farshid Khojasteh. "Remarks on Some Recent Fixed Point Results on Quaternion-Valued Metric Spaces." Abstract and Applied Analysis 2014 (2014): 1–8. http://dx.doi.org/10.1155/2014/171624.

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Very recently, Ahmed et al. introduced the notion of quaternion-valued metric as a generalization of metric and proved a common fixed point theorem in the context of quaternion-valued metric space. In this paper, we will show that the quaternion-valued metric spaces are subspaces of cone metric spaces. Consequently, the fixed point results in such spaces can be derived as a consequence of the corresponding existing fixed point result in the setting cone metric spaces.

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HE, JIAN-XUN, and BO YU. "WAVELET ANALYSIS OF QUATERNION-VALUED TIME-SERIES." International Journal of Wavelets, Multiresolution and Information Processing 03, no. 02 (June 2005): 233–46. http://dx.doi.org/10.1142/s0219691305000804.

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In this paper we introduce quaternion-valued multi-resolution analysis. Applying the theory of matrix-valued wavelet analysis, we give the construction of scaling functions and wavelets by identifying the quaternion-valued function with the complex duplex matrix-valued function.

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Szczęsna, Agnieszka. "Quaternion Entropy for Analysis of Gait Data." Entropy 21, no. 1 (January 17, 2019): 79. http://dx.doi.org/10.3390/e21010079.

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Nonlinear dynamical analysis is a powerful approach to understanding biological systems. One of the most used metrics of system complexities is the Kolmogorov entropy. Long input signals without noise are required for the calculation, which are very hard to obtain in real situations. Techniques allowing the estimation of entropy directly from time signals are statistics like approximate and sample entropy. Based on that, the new measurement for quaternion signal is introduced. This work presents an example of application of a nonlinear time series analysis by using the new quaternion, approximate entropy to analyse human gait kinematic data. The quaternion entropy was applied to analyse the quaternion signal which represents the segments orientations in time during the human gait. The research was aimed at the assessment of the influence of both walking speed and ground slope on the gait control during treadmill walking. Gait data was obtained by the optical motion capture system.

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Alagöz, Yasemin, Kürşat Hakan Oral, and Salim Yüce. "Split quaternion matrices." Miskolc Mathematical Notes 13, no. 2 (2012): 223. http://dx.doi.org/10.18514/mmn.2012.364.

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Alagoz, Y., and G. Ozyurt. "Some properties of complex quaternion and complex split quaternion matrices." Miskolc Mathematical Notes 20, no. 1 (2019): 45. http://dx.doi.org/10.18514/mmn.2019.2550.

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Shang, Weiying, Weiwei Zhang, Hai Zhang, Hongmei Zhang, Jinde Cao, and Fawaz E. Alsaadi. "Finite-time lag projective synchronization of delayed fractional-order quaternion-valued neural networks with parameter uncertainties." Nonlinear Analysis: Modelling and Control 28 (February 22, 2023): 1–22. http://dx.doi.org/10.15388/namc.2023.28.30817.

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This paper discusses a class issue of finite-time lag projective synchronization (FTLPS) of delayed fractional-order quaternion-valued neural networks (FOQVNNs) with parameter uncertainties, which is solved by a non-decomposition method. Firstly, a new delayed FOQVNNs model with uncertain parameters is designed. Secondly, two types of feedback controller and adaptive controller without sign functions are designed in the quaternion domain. Based on the Lyapunov analysis method, the non-decomposition method is applied to replace the decomposition method that requires complex calculations, combined with some quaternion inequality techniques, to accurately estimate the settling time of FTLPS. Finally, the correctness of the obtained theoretical results is testified by a numerical simulation example.

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Bahri, Mawardi, and Samsul Ariffin Abdul Karim. "Some Essential Relations for the Quaternion Quadratic-Phase Fourier Transform." Mathematics 11, no. 5 (March 3, 2023): 1235. http://dx.doi.org/10.3390/math11051235.

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Motivated by the fact that the quaternion Fourier transform is a powerful tool in quaternion signal analysis, here, we study the quaternion quadratic-phase Fourier transform, which is a generalized version of the quaternion Fourier transform. We first give a definition of the quaternion quadratic-phase Fourier transform. We derive in detail some essential properties related to this generalized transformation. We explore how the quaternion quadratic-phase Fourier transform is related to the quaternion Fourier transform. It is shown that this relation allows us to obtain several versions of uncertainty principles concerning the quaternion quadratic-phase Fourier transform.

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45

El-Sayed Ahmed, Ahmed, Saleh Omran, and Abdalla J. Asad. "Fixed Point Theorems in Quaternion-Valued Metric Spaces." Abstract and Applied Analysis 2014 (2014): 1–9. http://dx.doi.org/10.1155/2014/258985.

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The aim of this paper is twofold. First, we introduce the concept of quaternion metric spaces which generalizes both real and complex metric spaces. Further, we establish some fixed point theorems in quaternion setting. Secondly, we prove a fixed point theorem in normal cone metric spaces for four self-maps satisfying a general contraction condition.

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Xiao, Xiaolin, Yongyong Chen, Yue-Jiao Gong, and Yicong Zhou. "2D Quaternion Sparse Discriminant Analysis." IEEE Transactions on Image Processing 29 (2020): 2271–86. http://dx.doi.org/10.1109/tip.2019.2947775.

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Carlone, Luca, Vito Macchia, Federico Tibaldi, and Basilio Bona. "Quaternion-based EKF-SLAM from relative pose measurements: observability analysis and applications." Robotica 33, no. 6 (April 1, 2014): 1250–80. http://dx.doi.org/10.1017/s0263574714000678.

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SUMMARYIn this work, we investigate a quaternion-based formulation of 3D Simultaneous Localization and Mapping with Extended Kalman Filter (EKF-SLAM) using relative pose measurements. We introduce a discrete-time derivation that avoids thenormalization problemthat often arises when using unit quaternions in Kalman filter and we study its observability properties. The consistency of the estimation errors with the corresponding covariance matrices is also evaluated. The approach is further tested on real data from theRawseeds datasetand it is applied within a delayed-state EKF architecture for estimating a dense 3D map of an unknown environment. The contribution is motivated by the possibility of abstracting multi-sensorial information in terms of relative pose measurements and for its straightforward extensions to the multi robot case.

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Chen, Wen-Juan, and Shao-Wen Yu. "RSVD for Three Quaternion Tensors with Applications in Color Video Watermark Processing." Axioms 12, no. 3 (February 22, 2023): 232. http://dx.doi.org/10.3390/axioms12030232.

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In this paper, we study the restricted singular-value decomposition (RSVD) for three quaternion tensors under the Einstein product, and give higher-order RSVD over the quaternion algebra, which can achieve simultaneous singular value decomposition of three quaternion tensors. Moreover, we give the algorithm for computing the RSVD of for quaternion tensors. What is more, we present a new blind color video watermarking scheme based on the forth-order RSVD over the quaternion algebra, and our numerical example demonstrates the effectiveness of the framework.

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Li, Tao, and Qing-Wen Wang. "Structure Preserving Quaternion Biconjugate Gradient Method." SIAM Journal on Matrix Analysis and Applications 45, no. 1 (January 22, 2024): 306–26. http://dx.doi.org/10.1137/23m1547299.

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EKATERINA A., GAVRILINA, and VELTISHEV VADIM V. "QUATERNION-BASED ATTITUDE CONTROL OF HIGHLY MANEUVERABLE UNMANNED UNDERWATER VEHICLE." CASPIAN JOURNAL: Control and High Technologies 54, no. 2 (2021): 106–12. http://dx.doi.org/10.21672/2074-1707.2021.53.1.106-112.

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Traditionally, unmanned underwater vehicles are operated at low angles of inclination (pitch and roll). However, there are tasks that require high maneuverability and controllability of the underwater vehicle in the entire range of orientation angles. At the same time, traditional control systems use the Euler angles and have limitations associated with the degeneration of the kinematic equations at a picth angle of ± 90 °, the problem of the non-uniqueness of the Euler angles, and the deterioration of the system performance at large inclinations of the vehicle. Thus, the question of developing a synthesis method for a control system for highly maneuverable unmanned underwater vehicles arises. The paper provides a review and comparative analysis of existing approaches to attitude control, on the basis of which an approach using quaternions is chosen. The law of attitude control based on quaternions was approved during field tests on a hybrid unmanned underwater vehicle "Iznos", developed at Bauman Moscow State Technical University. The results obtained during the tests are given in the article. The quaternion-based approach has better performance in comparison with the traditional control system. In addition, control system based on quaternion approach has a simple structure and can be used to increase the maneuverability of unmanned underwater vehicles with average requirements for positioning accuracy.

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