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Academic literature on the topic 'Derivatives of quaternions'

Author: Grafiati

Published: 29 July 2024

Last updated: 30 July 2024

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Journal articles on the topic "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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8

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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Books on the topic "Derivatives of quaternions"

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Quaternionic Analysis: Functions of one quaternionic variable. Independent, 2023.

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Book chapters on the topic "Derivatives of quaternions"

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Fernandez, Arran, Cihan Güder, and Walaa Yasin. "On Fractional Quaternionic d-Bar Derivatives." In Trends in Mathematics, 23–31. Cham: Springer Nature Switzerland, 2024. http://dx.doi.org/10.1007/978-3-031-41665-1_3.

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Yuan, Xinyi, Shou-Wu Zhang, and Wei Zhang. "Introduction and Statement of Main Results." In The Gross-Zagier Formula on Shimura Curves. Princeton University Press, 2012. http://dx.doi.org/10.23943/princeton/9780691155913.003.0001.

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This chapter states the main result of this book regarding Shimura curves and abelian varieties as well as the main idea of the proof of a complete Gross–Zagier formula on quaternionic Shimura curves over totally real fields. It begins with a discussion of the original formula proved by Benedict Gross and Don Zagier, which relates the Néeron–Tate heights of Heegner points on X⁰(N) to the central derivatives of some Rankin–Selberg L-functions under the Heegner condition. In particular, it considers the Gross–Zagier formula on modular curves and abelian varieties parametrized by Shimura curves. It then decribes CM points and the Waldspurger formula before concluding with an outline of our proof, along with the notation and terminology.

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Conference papers on the topic "Derivatives of quaternions"

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Müller, Andreas, Zdravko Terze, and Viktor Pandza. "A Non-Redundant Formulation for the Dynamics Simulation of Multibody Systems in Terms of Unit Dual Quaternions." In ASME 2016 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2016. http://dx.doi.org/10.1115/detc2016-60191.

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Quaternions are favorable parameters to describe spatial rotations of rigid bodies since they give rise to simple equations governing the kinematics and attitude dynamics in terms of simple algebraic equations. Dual quaternions are the natural extension to rigid body motions. They provide a singularity-free purely algebraic parameterization of rigid body motions, and thus serve as global parameters within the so-called absolute coordinate formulation of MBS. This attractive feature is owed to the inherent redundancy of these parameters since they must satisfy two quadratic conditions (unit condition and Plcker condition). Formulating the MBS kinematics in terms of dual quaternions leads to a system of differential-algebraic equations (DAE) with index 3. This is commonly transformed to an index 1 DAE system by replacing the algebraic constraints with their time derivative. This leads to the well-known problem of constraint violation. A brute force method, enforcing the unit constraint of quaternions, is to normalize them after each integration step. Clearly this correction affects the overall solution and the dynamic consistency. Moreover, for unit dual quaternions the two conditions cannot simply be enforced in such a way. In this paper a non-redundant formulation of the motion equations in terms of dual quaternions is presented. The dual quaternion constraints are avoided by introducing a local canonical parameterization. The key to this formulation is to treat dual unit quaternions as Lie group. The formulation can be solved with any standard integration scheme. Examples are reported displaying the excellent performance of this formulation regarding the constraint satisfaction as well as the solution accuracy.

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Luna-Elizarrarás, M. E., M. A. Macías-Cedeño, M. Shapiro, Theodore E. Simos, George Psihoyios, and Ch Tsitouras. "On Relations between the Derivative and the Two-Dimensional Directional Derivatives of a Quaternionic Function." In Numerical Analysis and Applied Mathematics. AIP, 2007. http://dx.doi.org/10.1063/1.2790263.

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Xia, Yili, Cyrus Jahanchahi, Dongpo Xu, and Danilo P. Mandic. "The HC calculus, quaternion derivatives and caylay-hamilton form of quaternion adaptive filters and learning systems." In 2014 International Joint Conference on Neural Networks (IJCNN). IEEE, 2014. http://dx.doi.org/10.1109/ijcnn.2014.6889498.

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Kim, Myoung-Jun, Myung-Soo Kim, and Sung Yong Shin. "A general construction scheme for unit quaternion curves with simple high order derivatives." In the 22nd annual conference. New York, New York, USA: ACM Press, 1995. http://dx.doi.org/10.1145/218380.218486.

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Turner, J. "Quaternion-based partial derivative and state transition matrix calculations for design optimization." In 40th AIAA Aerospace Sciences Meeting & Exhibit. Reston, Virigina: American Institute of Aeronautics and Astronautics, 2002. http://dx.doi.org/10.2514/6.2002-448.

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Zhang, Yi, and Kwun-Lon Ting. "Characterization of Instantaneous Point-Line Motions." In ASME 2004 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2004. http://dx.doi.org/10.1115/detc2004-57294.

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This article discusses systematically the characterization of instantaneous point-line motions, and the higher-order relationship between a point-line motion and the associated rigid body motions. The transformation of a point-line between two positions is depicted as a pure translation along the point-line followed by a screw displacement about their common normal and expressed with a unit dual quaternion referred to as the point-line displacement operator. The derivatives of the point-line displacement operator characterize the point-line motion to various orders with a set of characteristic numbers. Such a treatment leads to a consistent expression or unified treatment for the transformation of lines, point-lines, and rigid bodies. The relationships between point-line motions and rigid body motions are addressed in detail up to the third order.

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Li, Yafeng. "Color image restoration by quaternion diffusion based on a discrete version of the topological derivative." In 2014 7th International Congress on Image and Signal Processing (CISP). IEEE, 2014. http://dx.doi.org/10.1109/cisp.2014.7003777.

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Nurahmi, Latifah, and Dongming Gan. "Dynamic Analysis of the 3-RRPS Metamorphic Parallel Mechanism Based on Instantaneous Screw Axis." In ASME 2019 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2019. http://dx.doi.org/10.1115/detc2019-97851.

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Abstract The 3-rRPS metamorphic parallel mechanism can change its configurations thanks to the reconfigurable (rR) joint. The analysis in this paper will focus on one specific configuration where the moving-platform is able to perform 2-dof coupled rotational motions and 1-dof translational motion, which is well-known as 1T2R motion. In this configuration, the mechanism has two types of operation modes, i.e. x0 = 0 and x3 = 0, which have been extensively studied by many researchers. However, the dynamic behaviours of the mechanism in those two operation modes have not been studied. Accordingly, this paper presents the dynamic analysis of the 3-rRPS metamorphic parallel mechanism in both operation modes based on the Instantaneous Screw Axis (ISA). The types of operation mode are initially characterized by means of Euler-Quaternion parameters. The time derivative of transformation matrix is performed in each operation mode and the ISA can be determined. By using the ISA, velocities and accelerations of all points on the moving-platform can be evaluated, which become the foundation of the dynamic analysis in this paper. This approach can be applied to parallel mechanisms having multiple operation modes of different mobility.

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Kong, Xianwen. "Variable Degree-of-Freedom Spatial Mechanisms Composed of Four Circular Translation Joints." In ASME 2020 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2020. http://dx.doi.org/10.1115/detc2020-22332.

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Abstract This paper deals with the construction and reconfiguration analysis of a spatial mechanism composed of four circular translation (G) joints. Two links connected by a G joint, which can be in different forms such as a planar parallelogram, translate along a circular trajectory with respect to each other. A spatial 4G mechanism, which is composed of four G joints, usually has 1-DOF (degree-of-freedom). Firstly, a 2-DOF 4G mechanism is constructed. Then a novel variable-DOF spatial 4G mechanism is constructed starting from the 2-DOF 4G mechanism using the approach based on screw theory. Finally, the reconfiguration analysis is carried out in the configuration space using dual quaternions. The analysis shows that the variable-DOF spatial 4G mechanism has one 2-DOF motion mode and one to two 1-DOF motion modes and reveals how the 4G mechanism can switch among these motion modes. By removing one link from two adjacent G joints each and two links from each of the remaining two G joints, we can obtain a queer-rectangle and a queer-parallelogram, which are the generalization of the queer-square or derivative queer-square in the literature. The approach in this paper can be extended to the analysis of other types of coupled mechanisms using cables and gears and multi-mode spatial mechanisms involving G joints.

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