Search Results (590)

Let $U,V\subset \mathbb{H}$ be bounded $C^1$ domains, and let f be quaternion-valued on $U\times V$. We study the mixed Cauchy–Fueter system $D_x^{L}f=0$ and $f{D}_y^R=0$ on product domains by iterating the classical one-variable Borel–Pompeiu formulas in an order consistent with quaternionic multiplication. Under closure regularity on $\overline{U}\times \overline{V}$, we prove an iterated representation formula and show that, in the biregular case, the boundary contribution reduces to the distinguished boundary $\partial U\times \partial V$. This leads to a distinguished boundary transform, ${\mathcal{T}}_{U,V}$, on continuous boundary data. We prove that ${\mathcal{T}}_{U,V}$ maps $C(\partial U\times \partial V;\mathbb{H})$ into $C^{\infty }(U\times V;\mathbb{H})$, establish compact subset estimates for mixed real derivatives, and derive a local approximation theorem within the transform range by finite sums of separated one-variable Cauchy transforms. The analysis is restricted to this representation framework. In particular, the paper does not address a general solvability theory for the mixed inhomogeneous system and does not characterize the full range of ${\mathcal{T}}_{U,V}$. Full article

An automotive welding unit is a modular production cell within a welding workshop that integrates industrial manipulators, welding equipment, fixtures, and control systems to perform specific welding and assembly tasks. A large number of industrial manipulators are utilized in the automotive welding unit. The capability to quickly plan a short and collision-free path in the workspace of the manipulator is of great importance for improving the manipulator’s intelligence level and production efficiency. The RRT* algorithm, based on random sampling, has been widely applied in path planning for high-dimensional manipulators due to its probabilistic completeness and powerful exploration capabilities. However, the RRT* algorithm performs poorly in spaces containing narrow passages. Research on the practical application of path planning for 6-DOF manipulators is still insufficient, particularly in planning posture. To solve these two problems, an improved RRT* algorithm is proposed in this paper. New sampling and node connection strategies are designed to improve the expansion and convergence speed of the random tree in spaces containing narrow passages. A distance-constrained posture quaternion interpolation method is presented to generate smooth and continuous paths for manipulators of the automotive welding unit. Simulations and experiments are carried out to validate the proposed method, which confirms that the method can plan collision-free paths for manipulators more quickly compared to other methods. Full article

The article reviews the methods of determining the angular position of a pilot’s helmet used on board modern aircraft, and analyzes the methods of determining the angular position of an object used in aviation spatial orientation and inertial navigation systems. A functional analysis of the NSC-1 Orion helmet-mounted targeting system developed at AFIT was performed. The main part of the work consists of the development of new, original mathematical models for determining the angular position of the pilot’s helmet using quaternions, simulation studies of these models, and experimental verification of their results. The stages necessary for the development of mathematical models and their proper testing for disturbances occurring in the measurement of gravitational acceleration (sensor errors and acceleration from maneuvers) and the magnetic field (sensor errors and the influence of the aircraft’s own magnetic field) are presented. Full article

Traditional pedestrian inertial navigation (PDR) algorithms usually assume that the carrying mode of a smartphone is fixed and remains horizontal, while ignoring the significant impact of dynamic changes in the carrying mode on heading estimation, which is the core element of PDR algorithms. In practical application scenarios, pedestrians often change their way of carrying smart terminals (e.g., calling) according to their needs, corresponding to the difference in the heading estimation method; especially when the mode is switched, it will cause a sudden change in heading, which will lead to a significant increase in the localization error if it cannot be corrected in time. Existing smart terminal carrying mode recognition methods that rely on traditional machine learning or set thresholds have poor robustness; lack of universality, especially weak diagnostic ability for mutation; and can not effectively reduce the heading error. Based on these practical problems, this paper innovatively proposes a PDR framework that tries to overcome these limitations. Based on this research purpose, firstly, this paper classifies four types of common carrying modes based on practical applications and designs a CNN-LSTM hybrid model, which can classify the four common carrying modes in near real-time, with a recognition accuracy as high as 99.68%. Secondly, based on the mode recognition results, a multi-layer heading correction strategy is introduced: (1) introducing a quaternion-based universal filter (VQF) algorithm to realize the accurate estimation of initial heading; (2) designing an algorithm to accurately detect the mode switching point and developing an adaptive offset correction algorithm to realize the dynamic compensation of heading in the process of mode switching to reduce the impact of sudden changes; and (3) considering the motion characteristics of pedestrians walking in a straight line segment where lateral displacement tends to be close to zero. This study designs a heading optimization method with lateral displacement constraints to further inhibit the drifting of the heading caused by the slight swaying of the smart terminal. In this study, two validation experiments are carried out in two different environment—an indoor corridor and a tree shelter—and the results show that based on the proposed multi-layer heading optimization strategy, the average heading error of the system is lower than 1.5°, the cumulative positioning error is lower than 1% of the walking distance, and the root mean square error of the checkpoints is lower than 2 m, which significantly reduces the positioning error and shows the effectiveness of the framework in complex environments. Full article

The classical Eneström–Kakeya Theorem restricts the location of the complex zeros of polynomials with real, positive, monotone increasing coefficients. That is, for ${\displaystyle p\left(z\right)=\sum _{v=0}^n{a}_v{z}^v}$, where $0\le a_0\le a_1\le \cdots \le a_n$, the zeros of p lie in the unit disk $|z|\le 1$ in the complex plane. Following the introduction of an analytic theory of functions of a quaternionic variable, this result was extended to polynomials of a quaternionic variable. Numerous generalizations of both the complex and quaternionic versions of the Eneström–Kakeya Theorem have appeared which modify the monotonicity condition and extend results to complex and quaternionic coefficients. We give a related theorem which generalizes several of the known results and includes them as corollaries. We impose a type-of-monotonicity condition on some of the real and imaginary parts of the coefficients of the polynomial. Full article

In this paper, we investigate a pair of Hermitian and anti-Hermitian solutions of a generalized Sylvester matrix equation $AXB+C\overline{X}D+EYF=G$ over commutative quaternion algebra by using a complex representation of commutative quaternion matrices, the Kronecker product, vec-operation, and Moore–Penrose-generalized inverse. We establish the necessary and sufficient conditions for the existence of solutions. Moreover, we derive explicit expressions when they are solvable. We also provide two numerical examples to illustrate the main results. Full article

To enhance the positioning accuracy of the inertial measurement unit (IMU)-based pedestrian localization, this study proposes an adaptive ensemble extended Kalman filter (EnEKF) that incorporates a distance constraint (DC). This study first introduces a dual foot-mounted IMU-based pedestrian localization system that employs two IMUs to measure the target human’s position. Second, an augmented data fusion model is developed by incorporating attitude quaternions from the inertial navigation system (INS) into the conventional INS error-state vector. Based on this new data fusion model, a DC-based EnEKF is designed. In this method, the EnEKF employs ensemble factors to address nonlinear and non-Gaussian characteristics inherent in the data fusion process. Then, the colored measurement noise (CMN) is considered, and the method is modified to form an EnEKF under CMN (cEnEKF). Moreover, the DC is employed to further restrict the INS-derived position estimates of the left and right feet obtained from the EnEKF algorithm. Finally, validation in two real-world scenarios confirms the effectiveness and superior performance of the proposed approach. Full article

This work presents the design of an attitude control experiment for onboard OPS-SAT-1 satellite execution, conceived with inherent extensibility to future mission architectures. OPS-SATs are ESA nanosatellite mission series designed as an in-orbit testbed for validating novel software and control techniques under real space conditions, OPS-SAT-1 being the first mission. Equipped with an advanced payload computer, OPS-SAT-1 enabled experimentation with innovative mission operations, including real-time attitude control strategies. Two attitude control algorithms, a modified Proportional–Integral–Derivative (mPID) and a fuzzy logic controller, were designed and implemented for the OPS-SAT-1. The design methodology applied to these controllers consisted of (i) modelling the space environment and satellite characteristics, (ii) assessing actuator feasibility, (iii) determining the operational ranges for attitude error and angular velocity, (iv) parametrizing controllers within these ranges, (v) fine-tuning controllers using multi-objective genetic optimization, and (vi) robustness analysis using the Monte Carlo method. Despite the technical issues related to communication with the OPS-SAT-1 hardware, which prevented the execution of the experiment in orbit, this work presents the simulation results that were obtained. These results indicate that fuzzy logic controllers may outperform PID controllers in terms of the accumulated error, settling time and steady-state error, whereas power efficiency appears to be less robust than in the PID. This suggest that a large uncertainty in the model could lead the PID to become more efficient. Near the nominal scenario, the fuzzy controller achieves superior error–cost trade-offs, enabling precise attitude stabilization with lower energy consumption. These findings suggest the potential advantages of modern control approaches compared to classical methods, which will be further assessed through future in-orbit experiments. Full article

This paper develops a structured analytical framework and a robust numerical methodology for the spectral stability of time-varying bilateral quaternion differential equations of the form $\dot{q}=A\left(t\right)q+qB\left(t\right)$. By systematically extending classical real matrix theory to non-commutative dynamical systems via exact isometric real representations, this study utilizes the Kronecker product of real adjoint matrices to rigorously elucidate the underlying tensor structure of the bilateral evolution operator. This tensor-based reformulation proves that the Floquet multipliers of the bilaterally coupled system can be strictly decoupled into the product of the spectra corresponding to the left and right unilateral subsystems. Second, a “Scalar-Vector Stability Separation Principle” based on logarithmic norms is proposed, demonstrating that the transient energy evolution of the system is governed exclusively by the Hermitian real parts of the coefficient matrices, remaining entirely independent of the anti-Hermitian imaginary parts (rotation terms). Furthermore, for constant-coefficient and slowly varying systems, the Riesz projection from holomorphic functional calculus is introduced to establish algebraic criteria for exponential dichotomies, thereby revealing a cubic scaling law that relates the robustness threshold to the spectral gap (${\epsilon }_0\propto {\beta }^3$). Numerically, a Quaternion Chebyshev Spectral Collocation Method (Q-CSCM) is embedded within this exact vectorization framework to ensure that the algebraic symmetries of the bilateral system are strictly preserved through the isomorphic mapping. By explicitly constructing the fully discrete Kronecker product matrix via the exact real vectorization isomorphism, discrete energy estimates are utilized to rigorously prove that the numerical scheme successfully inherits the intrinsic spectral accuracy of the Chebyshev approximation. Comprehensive numerical experiments demonstrate that, within the low-dimensional regime, this methodology exhibits substantial temporal approximation efficiency advantages and superior numerical robustness compared to an alternative Legendre spectral baseline, as well as traditional explicit and state-of-the-art implicit symplectic Runge–Kutta methods, particularly when solving stiff and critically stable problems such as nonlinear Riccati oscillators. Full article

In this paper, a non-decomposition approach is adopted to study the robust stability of quaternion-valued neural networks (QVNNs) with leakage, discrete, and neutral time delays, and the parameter uncertainty is also considered. The existence and uniqueness of the equilibrium of QVNNs are proved by the homogeneous mapping theorem. By constructing appropriate Lyapunov functions and employing quaternion modulus inequality techniques, sufficient conditions for the global robust stability of QVNNs are presented. Notably, the considered QVNN is treated as a whole rather than being decomposed into complex-valued neural networks (CVNNs) or real-valued neural networks (RVNNs), which faithfully reflects the internal connections between quaternion neurons. Two numerical examples are used to verify the validity of the obtained conclusions. Full article

This paper investigates the finite-time passivity and state estimation problem for quaternion-valued neural networks with two additive delays. By employing the Lyapunov method, several criteria are derived to ensure the finite-time passivity of the discussed system and the asymptotic stability of the error system. A novel controller is proposed to achieve finite-time passivity of the discussed system and a proportional–integral observer (PIO) strategy is adopted to tackle the state estimation problem. The direct approach is used to handle the quaternion-valued neural networks without decomposing them into real-valued or complex-valued systems, which substantially simplifies the analysis procedure. Moreover, various quaternion-valued inequalities are utilized in the analysis, contributing to reduced conservatism in the derived results. Finally, the theoretical results have been effectively demonstrated through two numerical simulation examples. Full article

The ability to reliably press physical buttons is a common requirement in robotics. Conventional vision-based methods often suffer from positional errors during execution if the end-effector’s approach is not perpendicular to the target surface. This paper proposes a novel dual-quaternion-based visual servoing method that enables robots to reach desired poses and enhances accuracy in robotic button-pressing. Our method acquires target pose information (position, depth and surface normal direction) from the RGB-D camera and converts it into dual quaternion representation to construct the visual servoing control system. The image Jacobian matrix for the dual quaternion pose is then computed. The dual-quaternion-based visual servoing ensures that the pressing direction and the optical axis of the coaxially mounted camera remain perpendicular throughout the pressing motion, thereby eliminating misalignment between the actual contact point and the visually identified target. By representing spatial displacements in SE(3) with dual quaternions, our method enables more compact, concise, and efficient pose representation and computation throughout the visual servoing process. Experimental results demonstrate that, compared to conventional methods, our technique achieves more efficient visual servoing control, significantly improving both positioning accuracy and computational efficiency. Full article

Accurate and continuous outdoor pedestrian positioning using smartphones remains challenging in complex environments like urban canyons, where Global Navigation Satellite System (GNSS) signals are frequently degraded or blocked, and Pedestrian Dead Reckoning (PDR) suffers from cumulative errors. To address this, this paper proposes a novel fusion method based on a Robust Adaptive Cubature Kalman Filter (RACKF). The core of our approach is a two-stage filtering architecture: the first stage employs a quaternion-based RACKF to optimally fuse gyroscope and magnetometer data for robust heading estimation; the second stage performs the core fusion of GNSS observations with an enhanced 3D PDR solution. Key innovations include an adaptive noise estimation strategy combining fading and limited memory weighting, a robust M-estimator-based mechanism to suppress outliers, and the integration of differential barometric height measurements. Experimental results demonstrate that the proposed method achieves a horizontal positioning accuracy of 3.28 m (RMSE), outperforming standalone GNSS and improving 3D PDR by 25.97% and 10.39%, respectively. This work provides a practical, infrastructure-free solution for robust smartphone-based outdoor navigation. Full article

High-agility spacecraft require time-efficient attitude maneuvers under strict actuator- and system-driven saturation limits on angular rate and angular acceleration. Analytical methods for attitude profile generation are attractive for on-board use because of their deterministic structure and low computational burden; however, depending on boundary conditions and sequential constraint-enforcement logic, they may yield either infeasible commands that violate constraints or overly conservative commands that underutilize available authority and unnecessarily prolong maneuver time. In contrast, numerical optimization-based methods can produce (near-)minimum-time solutions but are often too iterative and tuning-sensitive for real-time deployment. The proposed method produces an iteratively refined closed-form solution. The inner loop yields a closed-form solution for a given set of parameters, while the outer loop updates the parameter set via an iterative rescale step. The resulting finite-jerk (jerk-limited) profiles are intended for use in a feedforward–feedback architecture to mitigate terminal mismatch induced by quaternion-kinematics linearization and acceleration-related variable mappings. Numerical studies evaluate the proposed method using representative single-case examples and Monte Carlo simulations with comparisons against a baseline analytical method and a numerical optimization-based method. These results indicate that the proposed approach substantially improves feasibility and optimality such that it achieves maneuver times close to those of numerically optimized solutions, while maintaining a semi-closed-form structure. Full article

Quaternion Fractional Moment (QFM) descriptors are widely used in geometric pattern recognition due to their ability to encode multi-channel image information and exhibit invariance properties. However, their robustness under real-world acquisition variability, particularly photometric noise, remains insufficiently understood. Based on the Lipschitz stability theorem, which defines a strong, linear form of stability for dynamical systems, applied to one of our previous works, this article improves upon it by introducing a robustness-driven analysis framework that models feature extraction as a stochastic process, where bounded spatio-temporal perturbations generate multiple descriptor realizations for each pattern. Descriptor robustness is directly quantified in feature space using a novel normalized dispersion stability metric. Furthermore, a Lipschitz stability theorem is formally established and proved, providing theoretical guarantees of descriptor robustness under bounded perturbations. Experiments conducted on Moroccan–Andalusian geometric patterns with p4m and p6m symmetry groups demonstrate that the proposed framework achieves high intrinsic stability (${\sigma }_{norm}$ = 0.042 ± 0.010), while preserving state-of-the-art classification performance (Macro-F1 = 0.589 vs. 0.570 under σ = 0.05 noise). These results confirm that robustness is an intrinsic and measurable property of the descriptor, independent of classifier performance. The proposed framework provides both theoretical and methodological support for reliable geometric pattern recognition in cultural heritage imaging under real-world conditions. Full article