Journal articles: 'Derivatives of quaternions'

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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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Montgomery-Smith, Stephen, and Cecil Shy. "Using Lie Derivatives with Dual Quaternions for Parallel Robots." Machines 11, no. 12 (November 28, 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 effector in dual quaternion form, which include the effect of inertia from the actuators.

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Saima, Siddiqui, Bingzhao Li, and Samad Muhammad Adnan. "New Sampling Expansion Related to Derivatives in Quaternion Fourier Transform Domain." Mathematics 10, no. 8 (April 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 curves appearing in images, therefore, such a sampling formula is of substantial importance for image processing, particularly in image super-resolution procedures. Moreover, the presented sampling expansion can be applied in the field of image enhancement, color image processing, image restoration and compression and filtering, etc. Finally, an illustrative example is presented to demonstrate the efficacy of the proposed technique with vivid simulations in MATLAB.

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Kim, Ji Eun. "Calculation of Two Types of Quaternion Step Derivatives of Elementary Functions." Mathematics 9, no. 6 (March 21, 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 regarded as the base separate from the basis of the complex number. However, considering the properties of the quaternion, we propose two types of step derivatives in this study. The step derivative is first defined in the j direction, which includes a quaternion. Furthermore, the step derivative in the j+k2 direction is determined using the rule between bases i, j, and k defined in the quaternion. We present examples in which the definition of the j-step derivative and (j,k)-step derivative are applied to elementary functions ez, sinz, and cosz.

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Gogberashvili, Merab. "(2 + 1)-Maxwell Equations in Split Quaternions." Physics 4, no. 1 (March 17, 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, is equivalent to some system of differential equations for quaternionic spinors and vectors. Assuming that the derivatives by extra time-like coordinate generate triality (supersymmetric) rotations, the analyticity equation is reduced to the exact Dirac–Maxwell system in three-dimensional Minkowski space-time.

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LEO, STEFANO DE. "A ONE-COMPONENT DIRAC EQUATION." International Journal of Modern Physics A 11, no. 21 (August 20, 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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ROTELLI, P. "THE DIRAC EQUATION ON THE QUATERNION FIELD." Modern Physics Letters A 04, no. 10 (May 20, 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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Ghanbarpourasl, Habib. "Attitude reconstruction from strap-down rate gyros using power series." Journal of Navigation 74, no. 4 (March 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 shown that the norm of quaternions’ derivatives in the Taylor series is bigger than the equivalent terms coefficients in the power series. In the proposed method, more terms can be used in the power series before the saturation of the coefficients and the error of the proposed method is less than that for other methods. The numerical results show that the application of the proposed method with quaternions performs better than other methods. The method is robust with respect to the noise of the sensors and has a low computational load compared with other methods.

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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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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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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 (August 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-commutativity of quaternion algebra. To address this issue, we introduce the generalized HR (GHR) derivatives which employ quaternion rotations in a general orthogonal system and provide the left- and right-hand versions of the quaternion derivative of general functions. The GHR calculus also solves the long-standing problems of product and chain rules, mean-value theorem and Taylor's theorem in the quaternion field. At the core of the proposed GHR calculus is quaternion rotation, which makes it possible to extend the principle to other functional calculi in non-commutative settings. Examples in statistical learning theory and adaptive signal processing support the analysis.

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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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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 (December 25, 2015): 559–67. http://dx.doi.org/10.5831/hmj.2015.37.4.559.

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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 (December 6, 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 the center line to reduce the full Cosserat beam model to a Kirchhoff beam model. We derive the equations of motion by Hamilton’s principle with an augmented Lagrangian. In order to fully discretize the beam model in space and time, we only consider piecewise interpolated configurations in the variational principle. This leads, after approximating the action integral with second order, to the discrete equations of motion. Here, it is notable that we allow the Lagrange multipliers to be discontinuous in time in order to respect the derivatives of the constraint equations, also known as hidden constraints. In the last part, we will test our numerical scheme on two benchmark problems that show that there is no shear locking observable in the discretized beam model and that the errors are observed to decrease with second order with the spatial step size and the time step size.

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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 (March 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 are thus parametrized in terms of quaternions and an original proportional derivative (PD) regulator is proposed in order to control the dynamical model. Numerical simulations will be performed on symmetrical flight configuration, proving the reliability of the proposed PD control technique.

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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 designed parts is then derived. One part is the adaptive disturbance estimator designed based on the immersion and invariance theory. Driven by the disturbance estimation errors, it can realize exponential convergence of the estimations and has the nice “parameter lock” property, which can hardly be expected in the conventional certainty equivalent adaptive controllers. The other part is a proportional-derivative-like pose tracking controller where the estimated disturbances are directly used. The closed-loop stability of the relative motion system under different kinds of disturbances is proven by Lyapunov stability analysis. Simulations and comparisons with two previous dual quaternion-based controllers demonstrate the novel features and performance improvements of the proposed control scheme.

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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 (March 1, 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 celestial mechanics and space flight mechanics, is used as a new independent variable. Keplerian energy and time are used as additional variables in these equations. These equations are used to construct quaternion equations and relations in variations of KS-variables and their first derivatives and in variations of Keplerian energy and real time; the isochronous derivatives of the KS-variables and of their first derivatives and the matrix of isochronous derivatives for the elliptical Keplerian motion of the spacecraft are found, which are necessary for solving the problems of predicting and correcting its orbital motion. The results of a comparative study of the accuracy of the numerical integration of the Newtonian equations of the spatial restricted three-body problem (Earth, Moon, and spacecraft) in Cartesian coordinates and the regular quaternion equations of this problem in KS-variables are presented, which show that the accuracy of the numerical integration of regular quaternion equations is much higher (by several orders) of the accuracy of numerical integration of equations in Cartesian coordinates. This substantiates the expediency of using regular quaternion equations of the spacecraft orbital motion and the quaternion equations and relations in variations constructed in the article on their basis for the prediction and correction of the orbital motion of a spacecraft.

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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 (June 1, 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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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 (May 2, 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 designed controller is applied to the control problem of a three-link robot manipulator, with the control task of making the robot manipulator’s end effector follow a desired trajectory in the Cartesian space. Computational experiments are conducted to investigate the learning capability and the characteristics of the quaternion neural network used in the controller. The experimental results confirm the feasibility of using the derived learning algorithm based on the generalised Hamiltonian–Real calculus to train the quaternion neural network and the availability of such a network for a control systems application.

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

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Eriksson, Sirkka-Liisa, and Heikki Orelma. "Quaternionic k-Hyperbolic Derivative." Complex Analysis and Operator Theory 11, no. 5 (December 30, 2016): 1193–204. http://dx.doi.org/10.1007/s11785-016-0630-8.

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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 (March 14, 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 stabilization conditions of considered FOQVMNNs are established. Two numerical examples are provided to illustrate the usefulness of our analytical results.

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Sobhani, Hadi, and Hasan Hassanabadi. "Scattering in quantum mechanics under quaternionic Dirac delta potential." Canadian Journal of Physics 94, no. 3 (March 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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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 (March 28, 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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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 (March 23, 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 synchronization without using the separation technique or reduced-order transformation. In particular, some Lyapunov functionals, composed of the state variables and their derivatives, are directly constructed and some synchronization criteria represented by matrix inequalities are obtained based on quaternion theory. Some numerical results are shown to further confirm the theoretical analysis.

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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 (August 3, 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-labeled complexes, divided by the order of the acting group. The enumerative polynomial P(x) is demonstrated for trees and then is applied to the triangulations of the torus with the vertex-labeled complete four-partite graph G=K2,2,2,2, in which specific case P(x)=x31. The graph G embeds in the torus as a triangulation, T(G). The automorphism group of G naturally acts on the set of triangulations of the torus with the vertex-labeled graph G. For the first time, by a combination of algebraic and symmetry techniques, all vertex-labeled triangulations of the torus (12 in number) with the graph G are classified intelligently without using computing technology, in a uniform and systematic way. It is helpful to notice that the graph G can be converted to the Cayley graph of the quaternion group Q8 with the three imaginary quaternions i, j, k as generators.

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Xu, Dongpo, Hua Gao, and Danilo P. Mandic. "A new proof of the generalized Hamiltonian–Real calculus." Royal Society Open Science 3, no. 9 (September 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 domain.

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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 (March 18, 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 provide a comprehensive overview, the hardware and software that was used is also presented in the article. The main contribution of this article is a performance comparison of the controllers, which was based on absolute quaternion (positioning) error and energy consumption.

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Pan, Jie, and Lianglin Xiong. "Novel Criteria of Stability for Delayed Memristive Quaternionic Neural Networks: Directly Quaternionic Method." Mathematics 9, no. 11 (June 4, 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 standards are gained to affirm the global exponential stability (GES) of the ES for the MQNN. Notably, compared to the existing work on QNN, our direct quaternionic method operates QNN as a whole and markedly reduces computing complexity and the gained results are more apt to be verified. The two numerical simulation instances are provided to evidence the merits of the theoretical results.

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Vepa, Ranjan. "Spacecraft Large Attitude Estimation Using a Navigation Sensor." Journal of Navigation 63, no. 1 (December 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 distributed. For this reason, a new predictor-corrector form of the unscented Kalman filter is proposed to maintain the normalization of the unscented mean quaternion estimate in the presence of additive disturbances. The results from our realistic simulations indicate that the large attitude of a spacecraft can be estimated to within 0·1% accuracy over large time frames. The filter is particularly useful for autonomous operations of spacecraft as well as in other applications where the process model is bilinear or second order in the states.

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Giardino, Sergio. "Möbius Transformation for Left-Derivative Quaternion Holomorphic Functions." Advances in Applied Clifford Algebras 27, no. 2 (April 29, 2016): 1161–73. http://dx.doi.org/10.1007/s00006-016-0673-y.

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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 (April 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 simplifying formalisms for deriving their equations of dynamics, as well as reducing the number of mathematical operations in the analytical representation of the equations of dynamics. Second, and in the direction of choosing the parameters of the state of bodies in which the analytical scalar-coordinate types of the equations of dynamics are written down. In this connection, it should be noted that the vast majority (more than 90 %) of industrial robots, as well as special-purpose robots, have a single open branch structure in which the bodies form rotational kinematic pairs of the fifth class (rotational articulations) with each other. If in such systems the poles of the bodies are chosen on the axes of their relative rotation, the interpole distances will be constant, which greatly simplifies the solution of the above problem. Regarding the choice of state parameters explicitly included in the equations of dynamics, it should be noted that quasi-velocities, i.e. projections of absolute angular velocities of bodies on the axes of their coupled coordinate systems, are the most suitable for these purposes. The point is that in the equations of kinematics, which close the equations of dynamics to a complete set of equations for solving a problem, one can always express quasi-velocities through any other parameters, for example, relative angles of rotation of bodies and their time derivatives, guiding cosines and their derivatives, quaternions, etc. If projections of absolute angular velocities of bodies on their axes are measured, for example, by gyroscopes on bodies, and the first problem of dynamics is solved, the solution formulas contain a minimum number of addition and multiplication operations. Thus, the goal of the study is to develop a simple method for deriving the analytical form of the equations of dynamics of manipulators with rotational joints in quasi-velocity, in which geometric, kinematic, static and inertial parameters of bo¬dies are explicitly expressed. The used research methods (vector and analytical mechanics of body systems, vector algebra, system analysis and methods of identity transformations) made it possible to reduce the derivation of the equations of dynamics of manipulators to formal actions of writing them out without performing complex mathematical operations of differentiation, magnification, calculation of vector operations, etc. The results of the study contain a proof of the general vector form of the equations of dynamics of manipulators in quasi-velocity with explicitly expressed interpole distances and parameters of mass distribution of bodies. Scalar-coordinate formulas and their simple special formulas for the case of parallel rotation axes of neighboring joints are derived for writing out the moments of driving forces in joints. As a special case, the formulas for writing out the equations of dynamics of manipulators on the plane were obtained. For them, the process of writing out the equations of dynamics is reduced to the specification of the number of bodies, their geometric and inertial parameters. Conclusion. The effectiveness of the outlined methods and the obtained formulas have been demonstrated by examples of deriving the equations of dynamics of a body with a single fixed point, a gyroscope in a gimbal, and an angular manipulator with three and six degrees of freedom in space. These results allow us to expect that the number of users of the proposed methods will grow. The positive experience of using these methods in the educational process in the disciplines “Fundamentals of Mechanics of Body Systems”, “Electromechanical Systems” and “Mechatronics” justifies our expectations.

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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 (August 28, 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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Kim, Ji Eun. "REPRESENTATION OF THE DERIVATIVE FOR SPLIT-QUATERNIONIC FUNCTIONS." Far East Journal of Mathematical Sciences (FJMS) 103, no. 12 (June 16, 2018): 1921–29. http://dx.doi.org/10.17654/ms103121921.

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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 (February 27, 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 octonions. Such description gives a compact representation of electric and magnetic counterparts of the field in a single equation.

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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 direct method, Lyapunov stability theory, extended Cauchy Schwartz inequality, Jensen inequality. Finally, the correctness and validity of the presented results under the different orders are verified by two numerical examples.</p></abstract>

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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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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 (March 19, 2009): 375–90. http://dx.doi.org/10.1007/s00006-009-0165-4.

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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 (June 26, 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 modeling actuator dynamics, the control signal is direct current motor voltage, which is the most fundamental control variable, and can be generated easily by a motor driver in practical cases. To achieve a robust tracking of the desired time varying attitude, a sliding mode controller is designed and adaptive techniques are developed based on sliding mode control to overcome the inertia uncertainties and to estimate and compensate for external disturbances. Since the quaternion illustration of equations makes it more straightforward to deal with the mathematical operation in three dimensions, and there are advantages such as singularity rejection, the kinematic equations of the satellite are parameterized using quaternion parameters and a novel control law will be derived by using a new facilitating approach to controller design, which is based on quaternion algebra. Using this approach, it will be easier to deal with tedious mathematical operations and, in contrast with most of the previous studies, the terms corresponding to derivatives of the desired attitude are not neglected and the tracking capability is retained. The global stability of both methods is investigated and proved using the Lyapunov stability theorem.

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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 (November 20, 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 practical cases. To achieve robust tracking of the desired time-varying attitude, a sliding mode controller is designed, and adaptive techniques are developed based on sliding mode control to overcome the inertia uncertainties and to estimate and compensate external disturbances. The kinematic equations of the satellite are expressed using quaternion parameters, and a novel control law will be derived by using a new facilitating approach in controller design, which is based on quaternion algebra, because of quaternion advantages, such as singularity rejection. Using this approach it will be more comfortable to deal with tedious mathematical operations, and on contrary with most of the previous studies, the terms corresponding to derivatives of the desired attitude are not neglected, and tracking capability is retained. The global stability of both methods (Sliding Mode Control (SMC) and adaptive sliding) is investigated using Lyapunov’s stability theorem. In order to validate the control methods, first, Simulink-ADAMS co-simulation of a 3-DOF attitude control is used to verify the algorithm performance and integrity, and finally, the control strategy is implemented on the Amirkabir University of Technology (AUT) 3-DOF attitude simulator for different types of non-linear attitudes. Both co-simulation and implementation results clearly illustrate the designed attitude control algorithm’s excellent performance in the various manoeuvres.

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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 (March 2017): 033501. http://dx.doi.org/10.1063/1.4977082.

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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 (December 18, 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 established that the Fueter regular function constructed by Mariconda's approach is of bounded index, if the derivatives of its addends have bounded index. Also there was examined a function of the form $H(q)=f_1(x_0+ix_1)+jf_2(x_2+ix_3)$, where $f_1$ and $f_2$ are entire functions of complex variable. For the function $H$ it is proved its Fueter regularity and index boundedness if the first order derivatives of $f_1$ and $f_2$ have bounded index. Moreover, the index of the function $H$ does not exceed the maximum of indices of the functions $f'_1$ and $f'_2$ increased by $1$.

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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 (January 17, 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 screw displacement about their common normal. The derivatives of the point-line displacement operator characterize the point-line motion to various orders with a set of characteristic numbers. A set of associated rigid body motions is obtained by applying an instantaneous rotation about the point-line. It shows that the ISA trihedrons of the associated rigid motions can be simply depicted with a set of ∞2 cylindroids. It also presents for a rigid body motion, the locus of lines and point-lines with common rotation or translation characteristics about the line axes. Lines embedded in a rigid body with uniform screw motion are presented. For a general rigid body motion, one may find lines generating up to the third order uniform screw motion about these lines.

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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 (September 10, 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 transform is presented in this work, and we employ some properties of its component operators, developed in this work to establish a generalization of the Helmholtz decomposition theorem in fractional space. Additionally, right inverses of the fractional-order curl, divergence and gradient vector operators are obtained using Riemann–Liouville and Caputo fractional operators. Finally, some consequences of these results are provided as applications at the end of this work.

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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 (July 26, 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 L-index in the direction b where L : C n → R + is a positive continuous function. We present necessary and sufficient conditions of boundedness of L-index in the direction. In this paper, there are considered local behavior of directional derivatives and maximum modulus on a circle for functions from this class. Also, we show that every slice holomorphic and joint continuous function has bounded L-index in direction in any bounded domain and for any continuous function L : C n → R + .

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Petrov, A. G. "On the Kinematic Description of the Motion of a Rigid Body." Прикладная математика и механика 87, no. 5 (September 1, 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 quaternions is shown. The coordinates of the orts of the main axes of a rigid body in fixed axes are expressed in terms of the angles of final rotation and the components of angular velocity according to simple analytical formulas.

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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 (October 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 (June 30, 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 clearly illustrate the feasibility, the effectiveness and the performance of the planned control strategies, which have been validated by the Monte Carlo method, the results can be extended to other adaptive attitude control laws.

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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 (November 5, 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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Journal articles on the topic 'Derivatives of quaternions'

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