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Journal articles on the topic 'Derivatives of quaternions'

Author: Grafiati

Published: 29 July 2024

Last updated: 30 July 2024

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1

Kim, Ji-Eun. "Approximation of Directional Step Derivative of Complex-Valued Functions Using a Generalized Quaternion System." Axioms 10, no. 3 (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 th

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2

Montgomery-Smith, Stephen, and Cecil Shy. "Using Lie Derivatives with Dual Quaternions for Parallel Robots." Machines 11, no. 12 (2023): 1056. http://dx.doi.org/10.3390/machines11121056.

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We introduce the notion of the Lie derivative in the context of dual quaternions that represent rigid motions and twists. First we define the wrench in terms of dual quaternions. Then we show how the Lie derivative helps understand how actuators affect an end effector in parallel robots, and make it explicit in the two cases case of Stewart Platforms, and cable-driven parallel robots. We also show how to use Lie derivatives with the Newton-Raphson Method to solve the forward kinematic problem for over constrained parallel actuators. Finally, we derive the equations of motion of the end effecto

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3

Saima, Siddiqui, Bingzhao Li, and Samad Muhammad Adnan. "New Sampling Expansion Related to Derivatives in Quaternion Fourier Transform Domain." Mathematics 10, no. 8 (2022): 1217. http://dx.doi.org/10.3390/math10081217.

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The theory of quaternions has gained a firm ground in recent times and is being widely explored, with the field of signal and image processing being no exception. However, many important aspects of quaternionic signals are yet to be explored, particularly the formulation of Generalized Sampling Expansions (GSE). In the present article, our aim is to formulate the GSE in the realm of a one-dimensional quaternion Fourier transform. We have designed quaternion Fourier filters to reconstruct the signal, using the signal and its derivative. Since derivatives contain information about the edges and

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Kim, Ji Eun. "Calculation of Two Types of Quaternion Step Derivatives of Elementary Functions." Mathematics 9, no. 6 (2021): 668. http://dx.doi.org/10.3390/math9060668.

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We aim to get the step derivative of a complex function, as it derives the step derivative in the imaginary direction of a real function. Given that the step derivative of a complex function cannot be derived using i, which is used to derive the step derivative of a real function, we intend to derive the complex function using the base direction of the quaternion. Because many analytical studies on quaternions have been conducted, various examples can be presented using the expression of the elementary function of a quaternion. In a previous study, the base direction of the quaternion was rega

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5

Gogberashvili, Merab. "(2 + 1)-Maxwell Equations in Split Quaternions." Physics 4, no. 1 (2022): 329–63. http://dx.doi.org/10.3390/physics4010023.

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The properties of spinors and vectors in (2 + 2) space of split quaternions are studied. Quaternionic representation of rotations naturally separates two SO(2,1) subgroups of the full group of symmetry of the norms of split quaternions, SO(2,2). One of them represents symmetries of three-dimensional Minkowski space-time. Then, the second SO(2,1) subgroup, generated by the additional time-like coordinate from the basis of split quaternions, can be viewed as the internal symmetry of the model. It is shown that the analyticity condition, applying to the invariant construction of split quaternions

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6

LEO, STEFANO DE. "A ONE-COMPONENT DIRAC EQUATION." International Journal of Modern Physics A 11, no. 21 (1996): 3973–85. http://dx.doi.org/10.1142/s0217751x96001863.

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We develop a relativistic free wave equation on the complexified quaternions, linear in the derivatives. Even if the wave functions are only one-component, we show that four independent solutions, corresponding to those of the Dirac equation, exist. A partial set of translations between complex and complexified quaternionic quantum mechanics may be defined.

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7

ROTELLI, P. "THE DIRAC EQUATION ON THE QUATERNION FIELD." Modern Physics Letters A 04, no. 10 (1989): 933–40. http://dx.doi.org/10.1142/s0217732389001106.

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We develop a relativistic free wave equation on the quaternions, linear in the derivatives. Even if the wave function is only two-component, we show that there exists four complex-independent solutions corresponding to those of the Dirac Equation.

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8

Ghanbarpourasl, Habib. "Attitude reconstruction from strap-down rate gyros using power series." Journal of Navigation 74, no. 4 (2021): 763–81. http://dx.doi.org/10.1017/s0373463321000023.

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AbstractThis paper introduces a power series based method for attitude reconstruction from triad orthogonal strap-down gyros. The method is implemented and validated using quaternions and direction cosine matrix in single and double precision implementation forms. It is supposed that data from gyros are sampled with high frequency and a fitted polynomial is used for an analytical description of the angular velocity vector. The method is compared with the well-known Taylor series approach, and the stability of the coefficients’ norm in higher-order terms for both methods is analysed. It is show

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9

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 (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 i

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10

Weng, Zi-Hua. "Forces in the complex octonion curved space." International Journal of Geometric Methods in Modern Physics 13, no. 06 (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 prop

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11

Xu, Dongpo, Cyrus Jahanchahi, Clive C. Took, and Danilo P. Mandic. "Enabling quaternion derivatives: the generalized HR calculus." Royal Society Open Science 2, no. 8 (2015): 150255. http://dx.doi.org/10.1098/rsos.150255.

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Quaternion derivatives exist only for a very restricted class of analytic (regular) functions; however, in many applications, functions of interest are real-valued and hence not analytic, a typical case being the standard real mean square error objective function. The recent HR calculus is a step forward and provides a way to calculate derivatives and gradients of both analytic and non-analytic functions of quaternion variables; however, the HR calculus can become cumbersome in complex optimization problems due to the lack of rigorous product and chain rules, a consequence of the non-commutati

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12

Roelfs, Martin, David Dudal, and Daan Huybrechs. "Quaternionic step derivative: Machine precision differentiation of holomorphic functions using complex quaternions." Journal of Computational and Applied Mathematics 398 (December 2021): 113699. http://dx.doi.org/10.1016/j.cam.2021.113699.

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13

KIM, JI EUN, and KWANG HO SHON. "THE DERIVATIVE OF A DUAL QUATERNIONIC FUNCTION WITH VALUES IN DUAL QUATERNIONS." Honam Mathematical Journal 37, no. 4 (2015): 559–67. http://dx.doi.org/10.5831/hmj.2015.37.4.559.

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14

Hante, Stefan, Denise Tumiotto, and Martin Arnold. "A Lie group variational integration approach to the full discretization of a constrained geometrically exact Cosserat beam model." Multibody System Dynamics 54, no. 1 (2021): 97–123. http://dx.doi.org/10.1007/s11044-021-09807-8.

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AbstractIn this paper, we will consider a geometrically exact Cosserat beam model taking into account the industrial challenges. The beam is represented by a framed curve, which we parametrize in the configuration space $\mathbb{S}^{3}\ltimes \mathbb{R}^{3}$ S 3 ⋉ R 3 with semi-direct product Lie group structure, where $\mathbb{S}^{3}$ S 3 is the set of unit quaternions. Velocities and angular velocities with respect to the body-fixed frame are given as the velocity vector of the configuration. We introduce internal constraints, where the rigid cross sections have to remain perpendicular to th

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15

Artale, Valeria, Cristina L. R. Milazzo, and Angela Ricciardello. "A quaternion-based simulation of multirotor dynamics." International Journal of Modeling, Simulation, and Scientific Computing 06, no. 01 (2015): 1550009. http://dx.doi.org/10.1142/s1793962315500099.

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The main problem addressed in this paper is the quaternion-based trajectory control of a microcopter consisting of six rotors with three pairs of counter-rotating fixed-pitch blades, known as hexacopter. If the hypothesis of rigid body condition is assumed, the Newton–Euler equations describe the translational and rotational motion of the drone. The standard Euler-angle parametrization of three-dimensional rotations contains singular points in the coordinate space that can cause failure of both dynamical model and control. In order to avoid singularities, all the rotations of the microcopter a

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16

Shi, Xiaoping, Xuan Peng, and Yupeng Gong. "Immersion and Invariance Adaptive Control for Spacecraft Pose Tracking via Dual Quaternions." Complexity 2021 (April 15, 2021): 1–18. http://dx.doi.org/10.1155/2021/6624222.

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This paper addresses the simultaneous attitude and position tracking of a target spacecraft in the presence of general unknown bounded disturbances in the framework of dual quaternions, which provides a concise and integrated description of the coupled rotational and translational motions. By virtue of the newly introduced dual direction cosine matrix, the dimension of the dual quaternion-based relative motion dynamics written in vector/matrix form can be lowered to six. Treating the disturbances as unknown parameters, a modular adaptive pose tracking control scheme composed of two separately

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17

Chelnokov, Yu N., Ya G. Sapunkov, M. Yu Loginov, and A. F. Schekutev. "Forecast and Correction of the Orbital Motion of the Space Vehicle Using Regular Quaternion Equations and Their Solutions in the Kustaanheimo–Stiefels Variables and Isochronic Derivatives." Прикладная математика и механика 87, no. 2 (2023): 124–56. http://dx.doi.org/10.31857/s0032823523020054.

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The regular quaternion equations of the orbital motion of a spacecraft (SC) proposed by us earlier in four-dimensional Kustaanheimo–Stiefel variables (KS-variables) are considered. These equations use as a new independent variable a variable related to real time by a differential relation (Sundman time transformation) containing the distance to the center of gravity. Various new regular quaternion equations in these variables and equations in regular quaternion osculating elements (slowly varying variables) are also constructed, in which the half generalized eccentric anomaly, widely used in c

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18

Wang, Chao, Zhien Li, and Ravi P. Agarwal. "Fundamental solution matrix and Cauchy properties of quaternion combined impulsive matrix dynamic equation on time scales." Analele Universitatii "Ovidius" Constanta - Seria Matematica 29, no. 2 (2021): 107–30. http://dx.doi.org/10.2478/auom-2021-0021.

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Abstract In this paper, we establish some basic results for quaternion combined impulsive matrix dynamic equation on time scales for the first time. Quaternion matrix combined-exponential function is introduced and some basic properties are obtained. Based on this, the fundamental solution matrix and corresponding Cauchy matrix for a class of quaternion matrix dynamic equation with combined derivatives and bi-directional impulses are derived.

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19

Takahashi, Kazuhiko, Eri Tano, and Masafumi Hashimoto. "Feedforward–Feedback Controller Based on a Trained Quaternion Neural Network Using a Generalised HR Calculus with Application to Trajectory Control of a Three-Link Robot Manipulator." Machines 10, no. 5 (2022): 333. http://dx.doi.org/10.3390/machines10050333.

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This study derives a learning algorithm for a quaternion neural network using the steepest descent method extended to quaternion numbers. This applies the generalised Hamiltonian–Real calculus to obtain derivatives of a real–valued cost function concerning quaternion variables and designs a feedback–feedforward controller as a control system application using such a network. The quaternion neural network is trained in real-time by introducing a feedback error learning framework to the controller. Thus, the quaternion neural network-based controller functions as an adaptive-type controller. The

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20

Xu, Dongpo, and Danilo P. Mandic. "The Theory of Quaternion Matrix Derivatives." IEEE Transactions on Signal Processing 63, no. 6 (2015): 1543–56. http://dx.doi.org/10.1109/tsp.2015.2399865.

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21

Eriksson, Sirkka-Liisa, and Heikki Orelma. "Quaternionic k-Hyperbolic Derivative." Complex Analysis and Operator Theory 11, no. 5 (2016): 1193–204. http://dx.doi.org/10.1007/s11785-016-0630-8.

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22

Rajchakit, Grienggrai, Pharunyou Chanthorn, Pramet Kaewmesri, Ramalingam Sriraman, and Chee Peng Lim. "Global Mittag–Leffler Stability and Stabilization Analysis of Fractional-Order Quaternion-Valued Memristive Neural Networks." Mathematics 8, no. 3 (2020): 422. http://dx.doi.org/10.3390/math8030422.

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This paper studies the global Mittag–Leffler stability and stabilization analysis of fractional-order quaternion-valued memristive neural networks (FOQVMNNs). The state feedback stabilizing control law is designed in order to stabilize the considered problem. Based on the non-commutativity of quaternion multiplication, the original fractional-order quaternion-valued systems is divided into four fractional-order real-valued systems. By using the method of Lyapunov fractional-order derivative, fractional-order differential inclusions, set-valued maps, several global Mittag–Leffler stability and

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23

Sobhani, Hadi, and Hasan Hassanabadi. "Scattering in quantum mechanics under quaternionic Dirac delta potential." Canadian Journal of Physics 94, no. 3 (2016): 262–66. http://dx.doi.org/10.1139/cjp-2015-0646.

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In this paper, the Schrödinger equation for quaternionic quantum mechanics with a Dirac delta potential has been investigated. The derivative discontinuity condition for the quaternionic wave function has been derived and the boundary conditions for the quaternionic wave function have been applied. Probability current densities for different regions of the problem have been determined along with reflection and transmission coefficients.

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24

Ekasasmita, Wahyuni, Mawardi Bahri, Nasrullah Bachtiar, Amran Rahim, and Muhammad Nur. "One-Dimensional Quaternion Fourier Transform with Application to Probability Theory." Symmetry 15, no. 4 (2023): 815. http://dx.doi.org/10.3390/sym15040815.

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The Fourier transform occupies a central place in applied mathematics, statistics, computer sciences, and engineering. In this work, we introduce the one-dimensional quaternion Fourier transform, which is a generalization of the Fourier transform. We derive the conjugate symmetry of the one-dimensional quaternion Fourier transform for a real signal. We also collect other properties, such as the derivative and Parseval’s formula. We finally study the application of this transformation in probability theory.

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25

Yu, Juan, Kailong Xiong, and Cheng Hu. "Synchronization Analysis for Quaternion-Valued Delayed Neural Networks with Impulse and Inertia via a Direct Technique." Mathematics 12, no. 7 (2024): 949. http://dx.doi.org/10.3390/math12070949.

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The asymptotic synchronization of quaternion-valued delayed neural networks with impulses and inertia is studied in this article. Firstly, a convergence result on piecewise differentiable functions is developed, which is a generalization of the Barbalat lemma and provides a powerful tool for the convergence analysis of discontinuous systems. To achieve synchronization, a constant gain-based control scheme and an adaptive gain-based control strategy are directly proposed for response quaternion-valued models. In the convergence analysis, a direct analysis method is developed to discuss the sync

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26

Lawrencenko, Serge, and Abdulkarim M. Magomedov. "Generating the Triangulations of the Torus with the Vertex-Labeled Complete 4-Partite Graph K2,2,2,2." Symmetry 13, no. 8 (2021): 1418. http://dx.doi.org/10.3390/sym13081418.

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Using the orbit decomposition, a new enumerative polynomial P(x) is introduced for abstract (simplicial) complexes of a given type, e.g., trees with a fixed number of vertices or triangulations of the torus with a fixed graph. The polynomial has the following three useful properties. (I) The value P(1) is equal to the total number of unlabeled complexes (of a given type). (II) The value of the derivative P′(1) is equal to the total number of nontrivial automorphisms when counted across all unlabeled complexes. (III) The integral of P(x) from 0 to 1 is equal to the total number of vertex-labele

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27

Xu, Dongpo, Hua Gao, and Danilo P. Mandic. "A new proof of the generalized Hamiltonian–Real calculus." Royal Society Open Science 3, no. 9 (2016): 160211. http://dx.doi.org/10.1098/rsos.160211.

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The recently introduced generalized Hamiltonian–Real (GHR) calculus comprises, for the first time, the product and chain rules that makes it a powerful tool for quaternion-based optimization and adaptive signal processing. In this paper, we introduce novel dual relationships between the GHR calculus and multivariate real calculus, in order to provide a new, simpler proof of the GHR derivative rules. This further reinforces the theoretical foundation of the GHR calculus and provides a convenient methodology for generic extensions of real- and complex-valued learning algorithms to the quaternion

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28

Chovancová, Anežka, Tomáš Fico, František Duchoň, Martin Dekan, Ľuboš Chovanec, and Martina Dekanová. "Control Methods Comparison for the Real Quadrotor on an Innovative Test Stand." Applied Sciences 10, no. 6 (2020): 2064. http://dx.doi.org/10.3390/app10062064.

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This article is a continuation of our previously published work that presented a comparison of nine attitude quaternion-based controllers of the quadrotor in simulation environment. In this article, the best three controllers were implemented into the real quadrotor. Namely proportional derivative (PD), linear quadratic regulator (LQR) and backstepping quaternion-based control techniques were evaluated. As a suitable test stand was not available on the basis of literature analysis, the article also outlines the requirements and the development of a new innovative test stand. In order to provid

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29

Pan, Jie, and Lianglin Xiong. "Novel Criteria of Stability for Delayed Memristive Quaternionic Neural Networks: Directly Quaternionic Method." Mathematics 9, no. 11 (2021): 1291. http://dx.doi.org/10.3390/math9111291.

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In this paper, we fixate on the stability of varying-time delayed memristive quaternionic neural networks (MQNNs). With the help of the closure of the convex hull of a set the theory of differential inclusion, MQNN are transformed into variable coefficient continuous quaternionic neural networks (QNNs). The existence and uniqueness of the equilibrium solution (ES) for MQNN are concluded by exploiting the fixed-point theorem. Then a derivative formula of the quaternionic function’s norm is received. By utilizing the formula, the M-matrix theory, and the inequality techniques, some algebraic sta

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30

Vepa, Ranjan. "Spacecraft Large Attitude Estimation Using a Navigation Sensor." Journal of Navigation 63, no. 1 (2009): 89–104. http://dx.doi.org/10.1017/s0373463309990075.

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In this paper we assume that we have measurements of a first difference of a typical satellite navigation carrier phase differential, which is a homogeneous quadratic function of the components of the attitude quaternion. We illustrate the determination of large or sustained attitudes using a dedicated unscented Kalman filter. The unscented Kalman filter structure was chosen for the dedicated filter because of its derivative free nature and other advantages. When a Gaussian distributed vector random variable is transformed to an equivalent quaternion it does not continue to be Gaussian distrib

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31

Giardino, Sergio. "Möbius Transformation for Left-Derivative Quaternion Holomorphic Functions." Advances in Applied Clifford Algebras 27, no. 2 (2016): 1161–73. http://dx.doi.org/10.1007/s00006-016-0673-y.

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32

Telegin, Aleksandr I. "Analytical solution of the first problem of dynamics of manipulators with rotational joints." Bulletin of the South Ural State University. Ser. Computer Technologies, Automatic Control & Radioelectronics 22, no. 2 (2022): 41–57. http://dx.doi.org/10.14529/ctcr220204.

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The problem of complexity of derivation and the cumbersome analytical form of equations of mathematical models of controlled body systems with explicit structural, kinematic, static and dynamic parameters is solved. First of all, this applies to the equations of dynamics on which the control systems are based. Our practical experience and theoretical results indicate that solutions to this problem should be sought in two directions. Firstly, in the direction of classifying body systems and using peculiarities of the representatives of the considered classes of body systems in terms of simplify

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33

Gori, Anna, and Fabio Vlacci. "On a Criterion of Local Invertibility and Conformality for Slice Regular Quaternionic Functions." Proceedings of the Edinburgh Mathematical Society 62, no. 1 (2018): 97–105. http://dx.doi.org/10.1017/s0013091518000226.

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AbstractA new criterion for local invertibility of slice regular quaternionic functions is obtained. This paper is motivated by the need to find a geometrical interpretation for analytic conditions on the real Jacobian associated with a slice regular function f. The criterion involves spherical and Cullen derivatives of f and gives rise to several geometric implications, including an application to related conformality properties.

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34

Kim, Ji Eun. "REPRESENTATION OF THE DERIVATIVE FOR SPLIT-QUATERNIONIC FUNCTIONS." Far East Journal of Mathematical Sciences (FJMS) 103, no. 12 (2018): 1921–29. http://dx.doi.org/10.17654/ms103121921.

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35

Kalauni, Pushpa, and J. C. A. Barata. "Reconstruction of symmetric Dirac–Maxwell equations using nonassociative algebra." International Journal of Geometric Methods in Modern Physics 12, no. 03 (2015): 1550029. http://dx.doi.org/10.1142/s0219887815500292.

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In the presence of sources, the usual Maxwell equations are neither symmetric nor invariant with respect to the duality transformation between electric and magnetic fields. Dirac proposed the existence of magnetic monopoles for symmetrizing the Maxwell equations. In the present work, we obtain the fully symmetric Dirac–Maxwell's equations (i.e. with electric and magnetic charges and currents) as a single equation by using 4 × 4 matrix presentation of fields and derivative operators. This matrix representation has been derived with the help of the algebraic properties of quaternions and octonio

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36

Wu, Qiong, Zhimin Yao, Zhouping Yin, and Hai Zhang. "Fin-TS and Fix-TS on fractional quaternion delayed neural networks with uncertainty via establishing a new Caputo derivative inequality approach." Mathematical Biosciences and Engineering 19, no. 9 (2022): 9220–43. http://dx.doi.org/10.3934/mbe.2022428.

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<abstract><p>This paper investigates the finite time synchronization (Fin-TS) and fixed time synchronization (Fix-TS) issues on Caputo quaternion delayed neural networks (QDNNs) with uncertainty. A new Caputo fractional differential inequality is constructed, then Fix-TS settling time of the positive definite function is estimated, which is very convenient to derive Fix-TS condition to Caputo QDNNs. By designing the appropriate self feedback and adaptive controllers, the algebraic discriminant conditions to achieve Fin-TS and Fix-TS on Caputo QDNNs are proposed based on quaternion

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37

Burnol, Jean-François. "Quaternionic gamma functions and their logarithmic derivatives as spectral functions." Mathematical Research Letters 8, no. 2 (2001): 209–23. http://dx.doi.org/10.4310/mrl.2001.v8.n2.a9.

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38

Luna-Elizarrarás, M. E., M. A. Macías-Cedeño, and M. Shapiro. "On the Derivatives of Quaternionic Functions Along Two-Dimensional Planes." Advances in Applied Clifford Algebras 19, no. 2 (2009): 375–90. http://dx.doi.org/10.1007/s00006-009-0165-4.

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39

Alipour, Mohammadreza, Farhad Fani Saberi, and Mansour Kabganian. "Inertia-free nonlinear attitude tracking with disturbance compensation using adaptive-sliding control based on quaternion algebra." SIMULATION 96, no. 1 (2019): 43–54. http://dx.doi.org/10.1177/0037549719854178.

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The main objective of this paper is to develop and simulate a nonlinear attitude tracking control algorithm. In this research, the designed controller is supposed to track the desired time varying attitude of a satellite in the presence of inertia uncertainties and external disturbances. Another restriction for this novel controller is that it should be implementable and more applicable for implementation in a real-time situation. In order to have an accurate and thorough model, the actuators are reaction wheels and the actuator dynamics are modeled in addition to spacecraft dynamics. By model

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Alipour, M. Reza, F. Fani Saberi, and M. Kabganian. "Modelling, design and experimental implementation of non-linear attitude tracking with disturbance compensation using adaptive-sliding control based on quaternion algebra." Aeronautical Journal 122, no. 1247 (2017): 148–71. http://dx.doi.org/10.1017/aer.2017.122.

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ABSTRACTIn this paper, a non-linear tracking control algorithm is extended. The control objective of this research is to track a desired time-varying attitude of a satellite in the presence of inertia uncertainties and external disturbances, in order to be more suitable for implementation in a real-world application. In this investigation, the actuators are reaction wheels and the actuator dynamics are modelled in addition to the spacecraft dynamics. Thus, the control signal is DC motor voltage which is the most fundamental control variable and can be generated easily by a motor driver in prac

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41

Alpay, Daniel, Fabrizio Colombo, and Irene Sabadini. "On a class of quaternionic positive definite functions and their derivatives." Journal of Mathematical Physics 58, no. 3 (2017): 033501. http://dx.doi.org/10.1063/1.4977082.

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42

Baksa, V. P., and A. I. Bandura. "On an attempt to introduce a notion of bounded index for the Fueter regular functions of the quaternionic variable." Matematychni Studii 60, no. 2 (2023): 191–200. http://dx.doi.org/10.30970/ms.60.2.191-200.

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There is introduced a concept of index for the Fueter regular function of the quaternionic variables. There are considered three approaches (Fueter, Sudbery and Mariconda) constructing the Fueter regular function from a holomorphic function of complex variable. Using Mariconda's approach there are constucted some analogs of such elementary functions as the exponent, the sine and the cosine. For the Mariconda analogs we proved that they have bounded index and their indices equal 1, 2, 2, respectively. Using recent results on sum of entire functions whose derivatives are of bounded index it is e

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Zhang, Yi, and Kwun-Lon Ting. "On Higher Order Point-Line and the Associated Rigid Body Motions." Journal of Mechanical Design 129, no. 2 (2006): 166–72. http://dx.doi.org/10.1115/1.2406086.

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This paper presents a study on the higher-order motion of point-lines embedded on rigid bodies. The mathematic treatment of the paper is based on dual quaternion algebra and differential geometry of line trajectories, which facilitate a concise and unified description of the material in this paper. Due to the unified treatment, the results are directly applicable to line motion as well. The transformation of a point-line between positions is expressed as a unit dual quaternion referred to as the point-line displacement operator depicting a pure translation along the point-line followed by a sc

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44

Delgado, Briceyda B., and Jorge E. Macías-Díaz. "On the General Solutions of Some Non-Homogeneous Div-Curl Systems with Riemann–Liouville and Caputo Fractional Derivatives." Fractal and Fractional 5, no. 3 (2021): 117. http://dx.doi.org/10.3390/fractalfract5030117.

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In this work, we investigate analytically the solutions of a nonlinear div-curl system with fractional derivatives of the Riemann–Liouville or Caputo types. To this end, the fractional-order vector operators of divergence, curl and gradient are identified as components of the fractional Dirac operator in quaternionic form. As one of the most important results of this manuscript, we derive general solutions of some non-homogeneous div-curl systems that consider the presence of fractional-order derivatives of the Riemann–Liouville or Caputo types. A fractional analogous to the Teodorescu transfo

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Bandura, Andriy, and Oleh Skaskiv. "Slice Holomorphic Functions in Several Variables with Bounded L-Index in Direction." Axioms 8, no. 3 (2019): 88. http://dx.doi.org/10.3390/axioms8030088.

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In this paper, for a given direction b ∈ C n \ { 0 } we investigate slice entire functions of several complex variables, i.e., we consider functions which are entire on a complex line { z 0 + t b : t ∈ C } for any z 0 ∈ C n . Unlike to quaternionic analysis, we fix the direction b . The usage of the term slice entire function is wider than in quaternionic analysis. It does not imply joint holomorphy. For example, it allows consideration of functions which are holomorphic in variable z 1 and continuous in variable z 2 . For this class of functions there is introduced a concept of boundedness of

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46

Petrov, A. G. "On the Kinematic Description of the Motion of a Rigid Body." Прикладная математика и механика 87, no. 5 (2023): 711–19. http://dx.doi.org/10.31857/s0032823523050120.

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A system of ordinary differential equations is derived for a vector of finite rotation corresponding to Euler’s theorem: the vector of finite rotation is directed along the axis of finite rotation of a solid and its length is equal to the angle of plane rotation around this axis. The system of equations is explicitly resolved with respect to the time derivative of the components of the rotation vector. The right part of the system depends on the rotation vector and the angular velocity vector in the main axes. The equivalence of the obtained system of equations to the system of equations for q

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Gui, Haichao, and Anton H. J. de Ruiter. "Robustness Analysis and Performance Tuning for the Quaternion Proportional–Derivative Attitude Controller." Journal of Guidance, Control, and Dynamics 41, no. 10 (2018): 2308–17. http://dx.doi.org/10.2514/1.g003585.

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Benmansour, Jalal eddine, Elhassen Benfriha, and Abdelatif Bellar. "PD adaptive controller method for a three-axis stabilized rigid satellite attitude system." Algerian Journal of Signals and Systems 8, no. 1 (2023): 15–20. http://dx.doi.org/10.51485/ajss.v8i1.181.

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This paper deals with the attitude tracking control problem of rigid satellite with uncertainties of disturbances. An adaptive proportional derivative controller (PD) is proposed to deal with the influence of external disturbance in the attitude controller. The uncertain disturbances can be estimated through an adaptive algorithm and can be compensated in the proposed controller. The tracking error and the closed-loop system stability are ensured based on the Lyapunov analysis. using the representation of the spacecraft dynamics especially the quaternion properties.Simulation results can clear

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Luna-Elizarrarás, M. E., and M. Shapiro. "A Survey on the (Hyper-) Derivatives in Complex, Quaternionic and Clifford Analysis." Milan Journal of Mathematics 79, no. 2 (2011): 521–42. http://dx.doi.org/10.1007/s00032-011-0169-0.

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Aristizabal, Mauricio, Daniel Ramirez-Tamayo, Manuel Garcia, Andres Aguirre-Mesa, Arturo Montoya, and Harry Millwater. "Quaternion and octonion-based finite element analysis methods for computing multiple first order derivatives." Journal of Computational Physics 397 (November 2019): 108831. http://dx.doi.org/10.1016/j.jcp.2019.07.030.

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