Search Results (507)

We present novel proofs of the Riemann hypothesis by extending the standard complex Riemann zeta function into a quaternionic algebraic framework. Utilizing λ-regularization, we construct a symmetrized form that ensures analytic continuation and restores critical-line reflection symmetry, a key structural property of the Riemann ξ(s) function. This formulation reveals that all nontrivial zeros of the zeta function must lie along the critical line Re(s) = 1/2, offering a constructive and algebraic resolution to this fundamental conjecture. Our method is built on convexity and symmetrical principles that generalize naturally to higher-dimensional hypercomplex spaces. We also explore the broader implications of this framework in quantum statistical physics. In particular, the λ-regularized quaternionic zeta function governs thermodynamic properties and phase transitions in Bose–Einstein condensates. This quaternionic extension of the zeta function encodes oscillatory behavior and introduces critical hypersurfaces that serve as higher-dimensional analogues of the classical critical line. By linking the spectral features of the zeta function to measurable physical phenomena, our work uncovers a profound connection between analytic number theory, hypercomplex geometry, and quantum field theory, suggesting a unified structure underlying prime distributions and quantum coherence. Full article

This study investigates the tracking control of quadrotor micro aerial vehicles using nonlinear model predictive control (NMPC), with primary emphasis on the implementation of a real-time embedded control system. Apart from the limited memory size, one of the critical challenges is the limited processor speed on resource-constrained microcontroller units (MCUs). This technical issue becomes critical particularly when the maximum allowed computation time for real-time control exceeds 0.01 s, which is the typical sampling time required to ensure reliable control performance. To reduce the computational burden for NMPC, we first derive a nonlinear quadrotor model based on the quaternion number system rather than formulating nonlinear equations using conventional Euler angles. In addition, an implicit continuation generalized minimum residual optimization algorithm is designed for the fast computation of the optimal receding horizon control input. The proposed NMPC is extensively validated through rigorous simulations and experimental trials using Crazyflie 2.1^®, an open-source flying development platform. Owing to the more precise prediction of the highly nonlinear quadrotor model, the proposed NMPC demonstrates that the tracking performance outperforms that of conventional linear MPCs. This study provides a basis and comprehensive guidelines for implementing the NMPC of nonlinear quadrotors on resource-constrained MCUs, with potential extensions to applications such as autonomous flight and obstacle avoidance. Full article

The health of road tunnel linings directly impacts traffic safety and requires regular inspection. Appearance defects on tunnel linings can be measured through images scanned by cameras mounted on a car to avoid disrupting traffic. Existing tunnel lining mobile scanning methods often fail in image stitching due to the lack of corresponding feature points in the lining images, or require complex, time-consuming algorithms to eliminate stitching seams caused by the same issue. This paper proposes a mobile scanning method aided by collimated lasers, which uses lasers as corresponding points to assist with image stitching to address the problems. Additionally, the lasers serve as structured light, enabling the measurement of image projection relationships. An inspection car was developed based on this method for the experiment. To ensure operational flexibility, a single checkerboard was used to calibrate the system, including estimating the poses of lasers and cameras, and a Laplace kernel-based algorithm was developed to guarantee the calibration accuracy. Experiments show that the performance of this algorithm exceeds that of other benchmark algorithms, and the proposed method produces nearly seamless, measurable tunnel lining images, demonstrating its feasibility. Full article

Duplex ball bearings are common components in space satellite mechanisms, and their behaviour impacts the overall performance and reliability of these systems. During rocket launches, these bearings suffer high vibrational loads, making their dynamic response essential for their survival. To predict the dynamic behaviour under vibration, simulations and experimental tests are performed. However, published models for space applications fail to capture the variations observed in test responses. This study presents a multi-degree-of-freedom nonlinear multibody model of a hard-preloaded duplex space ball bearing, particularized for this work to the case in which the outer ring is attached to a shaker and the inner ring to a test dummy mass. The model incorporates the Hunt and Crossley contact damping formulation and employs quaternions to accurately represent rotational dynamics. The simulated model response is validated against previously published axial test data, and its response under step, sine, and random excitations is analysed both in the case of radial and axial excitation. The results reveal key insights into frequency evolution, stress distribution, gapping phenomena, and response amplification, providing a deeper understanding of the dynamic performance of space-grade ball bearings. Full article

Quaternions are used in various applications, especially in those where it is necessary to model and represent rotational movements, both in the plane and in space, such as in the modeling of the movements of robots and mechanisms. In this article, a methodology to model the rigid rotations of coupled bodies by means of unit quaternions is presented. Two parallel robots were modeled: a planar RRR robot and a spatial motion PRRS robot using the proposed methodology. Inverse kinematic problems were formulated for both models. The planar RRR robot model generated a system of 21 nonlinear equations and 18 unknowns and a system of 36 nonlinear equations and 33 unknowns for the case of space robot PRRS; both systems of equations were of the polynomial algebraic type. The systems of equations were solved using the Broyden–Fletcher–Goldfarb–Shanno nonlinear programming algorithm and Mathematica V12 symbolic computation software. The modeling methodology and the algebra of unitary quaternions allowed the systematic study of the movements of both robots and the generation of mathematical models clearly and functionally. Full article

Multi-tethered spacecraft formation refers to a group of spacecraft that are connected by tethers. These spacecraft work together to perform tasks, such as encircling and capturing small celestial bodies. When the multi-tethered spacecraft formation is in the process of encircling and capturing small celestial bodies, there is a significant risk of the tethers colliding with the soil (or surface material) of the small celestial body. Such a collision can affect the trajectory of the small celestial body detector. To address this issue, a coupled dynamic model has been proposed. This model takes the interaction between the tethers and the soil of the small celestial body into account. The discrete element method is used to establish the asteroid soil model, and the multi-body-tethered spacecraft system is simplified into a two-spacecraft system. The detector model is established by using the dual quaternion, and the tether model is established by using the chain rod model combined with the finite element method. Finally, a multi-condition simulation test is carried out. The results show that the influence of tether–soil coupling on the trajectory of the detector is mainly as follows: the influence of tether–soil interaction on the trajectory of the detector is mainly reflected in the displacement of the detector along the axial direction of the tether. Full article

With the development of robotics, robots are playing an increasingly critical role in complex tasks such as flexible manufacturing, physical human–robot interaction, and intelligent assembly. These tasks place higher demands on the force control performance of robots, particularly in scenarios where the environment is unknown, making constant force control challenging. This study first analyzes the robot and its interaction model with the environment, highlighting the limitations of traditional force control methods in addressing unknown environmental stiffness. Based on this analysis, a variable admittance control strategy is proposed using the deep deterministic policy gradient algorithm, enabling the online tuning of admittance parameters through reinforcement learning. Furthermore, this strategy is integrated with a quaternion-based nonlinear model predictive control scheme, ensuring coordination between pose tracking and constant-force control and enhancing overall control performances. The experimental results demonstrate that the proposed method improves constant force control accuracy and task execution stability, validating the feasibility of the proposed approach. Full article

An abstract set of one-dimensional (spinor-type) elements randomly oriented on a plane is introduced as a basic subgeometric object. Endowing the set with the binary operations of multiplication and invertible addition sequentially yields a specific semi-group (for which an original Cayley table is given) and a generic algebraic system which is shown to generate, apart from algebras of real and complex numbers, the associative hypercomplex algebras of dual numbers, split-complex numbers, and quaternions. The units of all these algebras turn out to be composed of basic 1D elements, thus ensuring the automatic fulfillment of multiplication rules (once postulated). From the standpoint of a three-dimensional space defined by a vector quaternion triad, the condition of a standard (unit) length of 1D basis elements is considered; it is shown that fulfillment of this condition provides an equation mathematically equivalent to the main equation of quantum mechanics. The similarities and differences of the proposed logical scheme with other approaches that involve abstract subgeometric objects are discussed. Full article

In this research work, we focus on the one-dimensional quaternion linear canonical transform (1-D QLCT). Under certain conditions, we first derive the symmetry property of the 1-D QLCT for real signals. The new form of the convolution theorem related to this transformation is proposed. We develop this convolution definition to derive the correlation theorem for the 1-D QLCT. We then show that the direct connection between the quaternion convolution and quaternion correlation definitions permits us to provide a different way for proving the correlation theorem concerning the 1-D QLCT. Finally, we present a simple application of the convolution theorem to the study of quaternion swept-frequency filter analysis. Full article

The stability of a class of quaternion-valued switching neural networks (QVSNNs) with time-varying delays is investigated in this paper. Limited prior research exists on the stability analysis of quaternion-valued neural networks (QVNNs). This paper addresses the stability analysis of quaternion-valued neural networks (QVNNs). With the help of some symmetric matrices with excellent properties, the stability analysis method in this paper is undecomposed. The QVSNN discussed herein evolves with average dwell time. Based on the Lyapunov theoretical framework and Wirtinger-based inequality, QVSNNs under any switching law have global asymptotic stability (GAS) and global exponential stability (GES). The state decay estimation of the system is also given and proved. Finally, the effective and practical applicability of the proposed method is demonstrated by two comprehensive numerical calculations. Full article

In this paper, we introduce and develop the concept of osculating curve pairs in the three-dimensional Minkowski space. By defining a vector lying in the intersection of osculating planes of two non-lightlike curves, we characterize osculating mates based on their Frenet frames. We then derive the transformation matrix between these frames and investigate the curvature and torsion relations under varying causal characterizations of the curves—timelike and spacelike. Furthermore, we determine the conditions under which these generalized osculating pairs reduce to well-known curve pairs such as Bertrand, Mannheim, and Bäcklund pairs. Our results extend existing theories by unifying several known curve pair classifications under a single geometric framework in Lorentzian space. Full article

For four-channel photoplethysmograms (PPGs), this paper employs quaternion-valued medians as features for performing non-invasive blood glucose estimation. However, as the PPGs are contaminated by noise, the quaternion-valued medians are also contaminated by noise. To address this issue, principal component analysis (PCA) is employed for performing the denoising. In particular, the covariance matrix of the four-channel PPGs is computed and the eigen vectors of the covariance matrix are found. Then, the quaternion-valued medians of the four-channel PPGs are found and these quaternion-valued medians are represented as the four-channel real-valued vectors. By applying the PCA to these four-channel real-valued vectors and reconstructing the denoised four-dimensional real-valued vectors, these four-dimensional real-valued vectors are denoised. Next, these denoised four-dimensional real-valued vectors are represented as the denoised quaternion-valued medians. Compared to the traditional denoising methods and the traditional feature extraction methods that are performed in the individual channels, the quaternion-valued medians and the PCA are computed via fusing all of these four-channel PPGs together. Hence, the hidden relationships among these four channels of the PPGs are exploited. Finally, the random forest is used to estimate the blood glucose levels (BGLs). Our proposed PCA-based quaternion-valued medians are compared to the median of each channel of the PPGs and other features such as the time-domain features and the frequency-domain features. Here, the effectiveness and robustness of our proposed method is demonstrated using two datasets. The computer numerical simulation results indicate that our proposed PCA-based quaternion-valued medians outperform the existing quaternion-valued medians and the other features for performing non-invasive blood glucose estimation. Full article

Quaternions extend the concept of complex numbers and have significant applications in image processing, as they provide an efficient way to represent RGB images. One interesting application is face recognition, which aims to identify a person in a given image. In this paper, we propose an algorithm for face recognition that models images using quaternion matrices. To manage the large size of these matrices, our method projects them onto a carefully chosen subspace, reducing their dimensionality while preserving relevant information. An essential part of our algorithm is the novel Jacobi method we developed to solve the quaternion Hermitian eigenproblem. The algorithm’s effectiveness is demonstrated through numerical tests on a widely used database for face recognition. The results demonstrate that our approach, utilizing only a few eigenfaces, achieves comparable recognition accuracy. This not only enhances execution speed but also enables the processing of larger images. All algorithms are implemented in the Julia programming language, which allows for low execution times and the capability to handle larger image dimensions. Full article

Gravity matching is a key technology in gravity-aided inertial navigation. The traditional Sandia inertial matching algorithm introduces linearization errors using the linear error model, which can diminish navigation accuracy. To address this issue, we propose a novel gravity-matching algorithm based on modified adaptive transformed cubature quaternion estimation (MA-TCQUE), designed for a nonlinear error model to enhance accuracy in gravity-aided navigation. Additionally, the proposed algorithm can estimate the measurement noise matrix demonstrating improved filtering stability in complex and dynamic environments. Finally, simulation and experimental results validate the advantages of the proposed matching algorithm compared to existing state-of-the-art methods. Full article

Since Hamilton proposed quaternions as a system of numbers that does not satisfy the ordinary commutative rule of multiplication, quaternion algebras have played an important role in many mathematical and physical studies. This paper introduces the generalized notion of Pauli Fibonacci polynomial quaternions, a definition that incorporates the advantages of the Fibonacci number system augmented by the Pauli matrix structure. With the concept presented in the study, it aims to provide material that can be used in a more in-depth understanding of the principles of coding theory and quantum physics, which contribute to the confidentiality needed by the digital world, with the help of quaternions. In this study, an approach has been developed by integrating the advantageous and consistent structure of quaternions used to solve the problem of system lock-up and unresponsiveness during rotational movements in robot programming, the mathematically compact and functional form of Pauli matrices, and a generalized version of the Fibonacci sequence, which is an application of aesthetic patterns in nature. Full article