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Relevant bibliographies by topics / Quaternion calculus

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Journal articles

Academic literature on the topic 'Quaternion calculus'

Author: Grafiati

Published: 29 July 2024

Last updated: 30 July 2024

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Journal articles on the topic "Quaternion calculus"

1

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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2

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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3

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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4

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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5

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

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Based on direct quaternion method, this paper explores the finite-time adaptive synchronization (FAS) of fractional-order delayed quaternion-valued fuzzy neural networks (FODQVFNNs). Firstly, a useful fractional differential inequality is created, which offers an effective way to investigate FAS. Then two novel quaternion-valued adaptive control strategies are designed. By means of our newly proposed inequality, the basic knowledge about fractional calculus, reduction to absurdity as well as several inequality techniques of quaternion and fuzzy logic, several sufficient FAS criteria are derive

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6

Li, Yongkun, and Xiaofang Meng. "Existence and Global Exponential Stability of Pseudo Almost Periodic Solutions for Neutral Type Quaternion-Valued Neural Networks with Delays in the Leakage Term on Time Scales." Complexity 2017 (2017): 1–15. http://dx.doi.org/10.1155/2017/9878369.

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We propose a class of neutral type quaternion-valued neural networks with delays in the leakage term on time scales that can unify the discrete-time and the continuous-time neural networks. In order to avoid the difficulty brought by the noncommutativity of quaternion multiplication, we first decompose the quaternion-valued system into four real-valued systems. Then, by applying the exponential dichotomic theory of linear dynamic equations on time scales, Banach’s fixed point theorem, the theory of calculus on time scales, and inequality techniques, we obtain some sufficient conditions on the

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7

Xu, Dongpo, Lina Zhang, and Huisheng Zhang. "LEARNING ALGORITHMS IN QUATERNION NEURAL NETWORKS USING GHR CALCULUS." Neural Network World 27, no. 3 (2017): 271–82. http://dx.doi.org/10.14311/nnw.2017.27.014.

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8

Pletinckx, Daniel. "Quaternion calculus as a basic tool in computer graphics." Visual Computer 5, no. 1-2 (1989): 2–13. http://dx.doi.org/10.1007/bf01901476.

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9

Danielewski, Marek, and Lucjan Sapa. "Foundations of the Quaternion Quantum Mechanics." Entropy 22, no. 12 (2020): 1424. http://dx.doi.org/10.3390/e22121424.

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We show that quaternion quantum mechanics has well-founded mathematical roots and can be derived from the model of the elastic continuum by French mathematician Augustin Cauchy, i.e., it can be regarded as representing the physical reality of elastic continuum. Starting from the Cauchy theory (classical balance equations for isotropic Cauchy-elastic material) and using the Hamilton quaternion algebra, we present a rigorous derivation of the quaternion form of the non- and relativistic wave equations. The family of the wave equations and the Poisson equation are a straightforward consequence of

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10

Shen, Shiping, Bing Li, and Yongkun Li. "Anti-Periodic Dynamics of Quaternion-Valued Fuzzy Cellular Neural Networks with Time-Varying Delays on Time Scales." Discrete Dynamics in Nature and Society 2018 (June 28, 2018): 1–14. http://dx.doi.org/10.1155/2018/5290786.

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A class of quaternion-valued fuzzy cellular neural networks with time-varying delays on time scales is proposed. Based on inequality analysis techniques on time scales, a fixed point theorem and the theory of calculus on time scales, the existence, and global exponential stability of anti-periodic solutions for this class of neural networks are established. The obtained results are completely new and supplement to the known results. Finally, a numerical example is given to illustrate the feasibility of our results.

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