Rotation Matrix Research Articles

Abstract A comprehensive theoretical study of resonant and non-resonant charge transfer processes in the homonuclear Na + + Na(3 s ) collision system is provided. The electronic structure of the quasi-molecular system is calculated using the multireference single- and double-excitation configuration interaction (MRD-CI) method under both one-electron (OEM) and thirteen-electron (TEM) models. The radial and rotational coupling matrix elements are obtained with careful inclusion of electron translation factors (ETFs). Based on these data, fully quantum-mechanical scattering calculations were performed using the molecular orbital close-coupling (QMOCC) method to derive state-selective and total cross sections for electron capture over a broad energy range (0.2-5600 eV/amu). The calculated resonant charge-transfer cross sections show excellent agreement with available theoretical and experimental results. Non-resonant processes, which involve electron capture into excited states of Na + , are found to become increasingly important above several keV/amu. This work provides the first theoretical predictions for non-resonant state-selective cross sections in this system, offering a valuable reference for future experimental and theoretical studies in atomic collision physics.

BackgroundIndividuals with autism spectrum disorder (ASD) have a wide range of challenges related to sensory and cognitive awareness. In Ethiopia, the increasing prevalence of ASD underscores the need for a comprehensive understanding of the associated challenges and impairments, an area that has not been studied so far.ObjectiveThe objective of this study was to investigate the underlying patterns of sensory and cognitive awareness impairments in children diagnosed with ASD at autism centers in Addis Ababa, Ethiopia.MethodsAn institution-based cross-sectional study was conducted at the Nehemia Autism Center and the Nia Foundation in Addis Ababa, Ethiopia. The study included children aged 4 to 16 years who had a confirmed diagnosis of ASD. A total of 145 study participants involved in this study. Study subjects were identified in collaboration with staff and caregivers. Caregivers of the study subjects were approached by trained data collectors, and written informed consent was obtained. The sensory/cognitive awareness subscale of the Autism Treatment Evaluation Checklist (ATEC) was administered to caregivers. This questionnaire tool has been validated for the autism population in Ethiopia. A face‒to-face interview was conducted. Data analysis was conducted IBM SPSS Version 22 Statistical Software. Principal component analysis with varimax rotation was employed to examine the patterns of sensory and cognitive awareness impairments. The numbers of principal components and factors to be retained were determined by examining the Eigenvalues and scree plot. Eigenvalues greater than 1 were used. The variable composition of each factor was examined by analyzing the factor loadings in the rotated component matrix. High variable loadings above 0.3 were considered for each factor.ResultsThis study revealed five patterns of sensory and cognitive awareness impairments in children diagnosed with ASD. Pattern 1, limitation in social engagement and exploration (α = 0.822); Pattern 2 challenges in emotional awareness and cognitive responsiveness (α = 0.743); Pattern 3 challenges in story comprehension and creativity (α = 0.62); Pattern 4 difficulties in social reciprocity and reward (α = 0.34); and Pattern 5 trouble with focus and attention (α = 0.12). All of these patterns accounted for 60% of the total variance.ConclusionIn this study, five patterns of sensory and cognitive awareness impairments were identified. Clinicians and therapists may need to consider these patterns for more personalized and effective support of children with ASD.

ABSTRACTThe co‐rotational (CR) approach is a widely adopted and efficient strategy for geometrically nonlinear analysis. The Classic Formalism, which derives full variations from local element to global coordinates, has been refined over the years and is generally regarded as theoretically sound. This study presents an insight into CR planar solid elements. To isolate these issues from element‐specific effects, we focus on the simplest planar solid elements: four‐node quadrilaterals and three‐node triangles. A comprehensive investigation is conducted to provide a unified overview and systematic comparison of commonly used local element frame construction methods. The findings indicate that the Classic Formalism may exacerbate unrealistic asymmetric responses in symmetric problems, particularly when the local element frame lacks precision. To address this, we propose a direct force correction formalism that introduces a correction term to directly enforce spin equilibrium. On the one hand, the derivation process eliminates the need for a second variation in computing the tangent stiffness matrix; on the other hand, the corrected equal weights act on each node of elements and are directly related to the rotation matrix, rather than a variant of the rotation matrix. As a result, this formalism may reduce the asymmetry in symmetrical cases better than Classic Formalism. Importantly, the proposed formalism is broadly applicable across various element types and frame construction methods.

Accurate visual localization in complex indoor environments remains a significant challenge due to feature degradation and cumulative errors. In response, we propose PLPM-SLAM, a novel RGB-D SLAM framework that integrates orthogonal Manhattan plane constraints with point-line-plane joint optimization to enhance both robustness and accuracy. Unlike traditional approaches that decouple only the rotation matrix, PLPM-SLAM utilizes three mutually orthogonal planes to jointly decouple both rotation and translation, effectively mitigating global drift. To address scenarios lacking a complete Manhattan structure, we introduce a virtual plane construction strategy based on heterogeneous feature associations. Additionally, PLPM-SLAM incorporates both homogeneous (point-point, line-line, plane-plane) and heterogeneous (point-line, point-plane, line-plane) geometric constraints throughout the tracking and optimization processes. In unstructured environments, a vanishing-point–guided joint optimization model is employed to improve geometric consistency. Extensive evaluations on public datasets (TUM, ICL-NUIM) and real-world sequences demonstrate that PLPM-SLAM consistently outperforms ORB-SLAM3 in both structured and low-texture settings. Specifically, PLPM-SLAM achieves RMSE reductions of up to 82.77% and 92.16% on the public and real-world datasets, respectively.

Background and Objectives: The term “oncoplastic breast surgery” (OBS) incorporates plastic and oncologic concepts. Through the application of diverse mammoplasty approaches, the remaining breast tissue can be reconstructed, thereby enabling more extensive resections to be achieved with oncologically safe, margin-free outcomes. This study aims to assess the efficacy of the single incision lateral mammoplasty (SILM) technique as an oncoplastic approach for managing breast cancer located in the outer quadrant, in comparison with the matrix rotation flap (MRF) technique. Materials and Methods: This prospective randomized controlled study comprised 68 patients, who were randomized into two groups scheduled to undergo breast surgery: Group A constitutes the matrix rotation flap MRF group and Group B represents the single incision lateral sulcus mammoplasty (SLIM) group. A follow-up was planned for postoperative complications and esthetic outcomes. Results: The mean age of patients in Group A was 51.4 ± 9.4 years, compared with 52.6 ± 8.1 years in Group B. A total of 14.7% and 11.8% of patients in Group A reported a hematoma or seroma, respectively, which were higher than what was reported in Group B, where a hematoma and seroma were reported in 5.9% of patients. Additionally, 32.4% and 50% of patients in Groups A and B, respectively, reported excellent satisfaction. The evaluation with the Vancouver Scar Scale (VSS) revealed that esthetic outcomes were significantly better in Group B. Conclusions: Compared to the MRF procedure, the SLIM results in a much lower rate of postoperative hematoma, minor seroma, minimum blood loss, reduced areolar deviation, and improved breast symmetry. Both the MRF and SLIM techniques yield acceptable cosmetic outcomes. However, a longer-term follow-up is necessary to establish the definitive oncological equivalence between techniques.

This paper presents the results of modeling a combined process of radial-shear rolling and pressing in order to obtain a reinforcing profile at different rotation speeds of the screw matrix. The parameters of the stress-strain state in the zones of radial-shear rolling and the formation of a screw profile in the matrix on the surface and axial parts of the workpiece were analyzed. It has been established that the forced rotation of the screw matrix not only helps to increase the process performance, but also plays an important role in preventing the defects formation. The maximum value of the equivalent strain is achieved in a fixed matrix, the higher the rotation speed of the matrix, the lower the values of the equivalent strain. At the same time, in the case of a high rotation speed of the die, the minimum strain level is 10 approximately. Stress state analysis has shown that this deformation scheme creates a high level of compressive stresses, which are dominated by tensile stresses.

Abstract Three-dimensional full-field scanning laser Doppler vibrometry (3D FFS LDV) represents one of the most significant non-contact vibration measurement technologies currently available. Among its core components, 3D alignment and laser vibrometry constitute the two most criti-cal technical elements. 3D alignment obtains the rotation matrices from the local probe coor-dinate system (PCS) to the global world coordinate system (WCS), thereby mapping non-orthogonal vibration components measured along three laser beam directions by three probes into the orthogonal WCS. However, existing alignment methods demonstrate significant de-pendence on calibration plates or additional measurement equipment, while exhibiting lim-ited adaptability to complex curved structures, thus constraining their engineering applica-tions. To address this issue, this paper proposes a 3D alignment algorithm based on the de-veloped mathematical model of 2D rotating mirror outgoing light rays that directly establish-es the global WCS on the measured object surface. This method utilizes the geometric con-straint relationships between multiple laser beam’s direction vectors and the WCS when mul-tiple probes illuminate the same point, thereby solving for the transformation matrix from the PCS to the WCS. This approach eliminates the dependence on calibration plates or feature points that is characteristic of conventional methods. Experimental results demonstrate that the method achieves maximum average relative errors of 2.85% and 6.53% in planar and curved surface alignment scenarios, respectively, while simultaneously enabling 3D full-field laser scanning. Compared to existing technologies, the proposed method exhibits significant advantages in terms of scenario adaptability and engineering practicality.

Given 2D point correspondences between an image pair, inferring the camera motion is a fundamental issue in the computer vision community. The existing works generally set out from the epipolar constraint and estimate the essential matrix, which is not optimal in the maximum likelihood (ML) sense. In this paper, we dive into the original measurement model with respect to the rotation matrix and normalized translation vector and formulate the ML problem. We then propose an optimal two-step algorithm to solve it: In the first step, we estimate the variance of measurement noises and devise a consistent estimator based on bias elimination; In the second step, we execute a one-step Gauss-Newton iteration on manifold to refine the consistent estimator. We prove that the proposed estimator achieves the same asymptotic statistical properties as the ML estimator: The first is consistency, i.e., the estimator converges to the ground truth as the point number increases; The second is asymptotic efficiency, i.e., the mean squared error of the estimator converges to the theoretical lower bound - Cramer-Rao bound. In addition, we show that our algorithm has linear time complexity. These appealing characteristics endow our estimator with a great advantage in the case of dense point correspondences. Experiments on both synthetic data and real images demonstrate that when the point number reaches the order of hundreds, our estimator outperforms the state-of-the-art ones in terms of estimation accuracy and CPU time.

This study investigates the dynamic performance of mountainous crawler tractors during small-radius slope steering, providing theoretical support for power machinery design in hilly and mountainous regions. Addressing the mechanization demands in complex terrains and existing research gaps, a steering dynamics model is established. The model incorporates an amplitude-varied multi-peak cosine ground pressure distribution, employs position vectors and rotation matrices to characterize 3D pose variations in the tractor’s center of mass, and integrates slope angle, soil parameters, vehicle geometry, center-of-mass shift, bulldozing resistance, and sinkage resistance via d’Alembert’s principle. Numerical simulations using Maple 2024 analyzed variations in longitudinal offset of the instantaneous steering center, bilateral track traction forces, and bulldozing resistance with slope, speed, and acceleration. Variable-gradient steering tests on the “Soil-Machine-Crop” Comprehensive Experimental Platform demonstrated model accuracy, with <8% mean error and <12% maximum relative error between predicted and measured track forces. This research establishes a theoretical foundation for predicting, evaluating, and controlling the steering performance/stability of crawler tractors in complex slope conditions.

A multi-stage combination trajectory design and pose switching control strategy is proposed to address the problem of vertical launch missiles flying around and attacking targets in the opposite direction. Firstly, a model for vertical launch tactical missile motion was established, by using quaternions to represent the attitude rotation matrix. By timely switching the rotation order, it was calculated as the missile attitude angle, adapting to the range of attitude angles for vertical launch diffraction attacks. Secondly, a segmented design was carried out for the diffraction attack trajectory, dividing the entire trajectory into four sections: vertical turning section, diffraction preparation section, diffraction turning section, and final strike section. The control strategies for pitch/yaw/roll three channels of each section of the trajectory were provided; and the transition models for different stages of trajectory were established, and the control methods under different control strategies were provided to achieve the stable flight diffraction and target attack of vertically launched missiles. Finally, taking a certain type of vertically launched tactical missile as an example, the effectiveness of the present method was verified through the comparative simulation. Monte Carlo simulation showed that the miss distance was less than 0.15 m, and the expected yaw angle error of the strike was less than 2°.

Abstract This paper proposes an improved null-space behavior control algorithm based on rotation matrix constraints for nonlinear systems. We take the kinematic model of unmanned aerial vehicles (UAVs) as an example. First, the UAV mathematical model is incorporated into the mission design of null-space behavior control (NSBC) to ensure that the mission instructions satisfy the system’s motion constraints, overcoming the limitations of the classical NSBC method, which relies on the single integrator model. Second, a formation controller was designed for each agent, which utilizes a self-updating estimator that tracks the expected speed after merging under disturbance conditions. Finally, numerical simulations indicate that the proposed algorithm can form a rigid formation.

Hierarchical RL-ESN control for autonomous underwater vehicle: 6-DOF large angle rotation maneuvering.

Accurate localization plays a critical role in enabling underwater vehicle autonomy. In this work, we develop a robust infrastructure-based localization framework that estimates the position and orientation of underwater vehicles using only range measurements from long baseline (LBL) acoustic beacons to multiple on-board receivers. The proposed framework integrates three key components, each formulated as a convex optimization problem. First, we introduce a robust calibration function that unifies multiple sources of measurement error—such as range-dependent degradation, variable sound speed, and latency—by modeling them through a monotonic function. This function bounds the true distance and defines a convex feasible set for each receiver location. Next, we estimate the receiver positions as the center of this feasible region, using two notions of centrality: the Chebyshev center and the maximum volume inscribed ellipsoid (MVE), both formulated as convex programs. Finally, we recover the vehicle’s full 6-DOF pose by enforcing rigid-body constraints on the estimated receiver positions. To do this, we leverage the known geometric configuration of the receivers in the vehicle and solve the Orthogonal Procrustes Problem to compute the rotation matrix that best aligns the estimated and known configurations, thereby correcting the position estimates and determining the vehicle orientation. We evaluate the proposed method through both numerical simulations and field experiments. To further enhance robustness under real-world conditions, we model beacon-location uncertainty—due to mooring slack and water currents—as bounded spherical regions around nominal beacon positions. We then mitigate the uncertainty by integrating the modified range constraints into the MVE position estimation formulation, ensuring reliable localization even under infrastructure drift.

The coherency matrix serves as a valuable tool for explaining the intricate details of various terrain targets. However, a significant challenge arises when analyzing ground targets with similar scattering characteristics in polarimetric synthetic aperture radar (PolSAR) target decomposition. Specifically, the overestimation of volume scattering (OVS) introduces ambiguity in characterizing the scattering mechanism and uncertainty in deciphering the scattering mechanism of large oriented built-up areas. To address these challenges, based on the generalized five-component decomposition (G5U), we propose a hierarchical extension of the G5U method, termed ExG5U, which incorporates orientation and phase angles into the matrix rotation process. The resulting transformed coherency matrices are then subjected to a five-component decomposition framework, enhanced with four refined volume scattering models. Additionally, we have reformulated the branch conditions to facilitate more precise interpretations of scattering mechanisms. To validate the efficacy of the proposed method, we have conducted comprehensive evaluations using diverse PolSAR datasets from Gaofen-3, Radarsat-2, and ESAR, covering varying data acquisition timelines, sites, and frequency bands. The findings indicate that the ExG5U method proficiently captures the scattering characteristics of ambiguous regions and shows promising potential in mitigating OVS, ultimately facilitating a more accurate portrayal of scattering mechanisms of various terrain types.

High-precision positioning is the foundation for the functionality of various intelligent agents. In complex environments, such as urban canyons, relative pose estimation using cameras is a crucial step in high-precision positioning. To take advantage of the ability of an inertial measurement unit (IMU) to provide relatively accurate gravity prior information over a short period, we propose a minimal solution method for the relative pose estimation of a stereo camera system assisted by the IMU. We rigidly connect the IMU to the camera system and use it to obtain the rotation matrices in the roll and pitch directions for the entire system, thereby reducing the minimum number of corresponding points required for relative pose estimation. In contrast to classic pose-estimation algorithms, our method can also calculate the camera focal length, which greatly expands its applicability. We constructed a simulated dataset and used it to compare and analyze the numerical stability of the proposed method and the impact of different levels of noise on algorithm performance. We also collected real-scene data using a drone and validated the proposed algorithm. The results on real data reveal that our method exhibits smaller errors in calculating the relative pose of the camera system compared with two classic reference algorithms. It achieves higher precision and stability and can provide a comparatively accurate camera focal length.

The construction of large-scale, dynamic datasets for specialized domain models often suffers with problems of low efficiency and poor consistency. This paper proposes a method that integrates multi-role collaboration with automated annotation to address these issues. The framework introduces two new roles, data augmentation specialists and automatic annotation operators, to establish a closed-loop process that includes dynamic classification adjustment, data augmentation, and intelligent annotation. Two supporting tools were developed: an image classification modification tool that automatically adapts to changes in categories and an automatic annotation tool with rotation-angle perception based on the rotation matrix algorithm. Experimental results show that this method increases annotation efficiency by 40% compared to traditional approaches, while achieving 100% annotation consistency after classification modifications. The method’s effectiveness was validated using the WATER-DET dataset, a collection of 1500 annotated images from the water conservancy engineering field. A model trained on this dataset achieved an F1-score of 0.9 for identifying water environment problems in rivers and lakes. This research offers an efficient framework for dynamic dataset construction, and the developed methods and tools are expected to promote the application of artificial intelligence in specialized domains.

Accurate acetabular cup placement in total hip arthroplasty (THA) requires reliable registration to establish body-to-world coordinate transformation, yet traditional methods rely on bony landmarks or invasive pins. This paper presents an inertial measurement unit (IMU)-based registration system for THA without bony landmark reliance or invasive contact. The system hardware integrates a 3-axis accelerometer and a 3-axis gyroscope as a sensor module, with Extended Kalman Filter (EKF) applied for motion data fusion. The software employs a rotation matrix and Euler angle combined registration algorithm to compute coordinate transformations. Experiments using a 3-axis tilt table validate the IMUs angle estimation accuracy (RMSE 0.4000.423) and the robust acquisition of rotation axis vectors during registration. Although it is theoretically and experimentally feasible, the registration relies on the assumption of patientoperating table immobility. Pending further optimization and clinical validation, this portable, cost-effective, and radiation-free solution shows potential as a standard registration tool for THA navigation devices.

The quantum states of a spin 12 (a qubit) are parametrized by the space CP1∼S2, the Bloch sphere. A spin j for a generic j (a 2j+1-state system) is represented instead by a point in a larger space, CP2j. Here we study the state of a single angular momentum/spin in the limit j→∞. A special class of states, |j,n⟩∈CP2j, with spin oriented towards definite spatial directions, n∈S2, i.e., (J^·n)|j,n⟩=j|j,n⟩, are found to behave as classical angular momenta, jn, in this limit. Vice versa, general spin states in CP2j do not become classical, even at a large j. We study these questions by analyzing the Stern–Gerlach processes, the angular momentum composition rule, and the rotation matrix. Our observations help to better clarify how classical mechanics emerges from quantum mechanics in this context (e.g., with the unique trajectories of a particle carrying a large spin in an inhomogeneous magnetic field) and to make the widespread idea that large spins somehow become classical more precise.

Separating multiple incoherent sound sources presents a significant challenge in acoustic imaging. Existing methods, such as principal component analysis (PCA) applied to the cross-spectral matrix, yield "virtual" sources based on decorrelation. However, this approach often fails due to the existence of multiple solutions in the general case. State-of-the-art methods to address this issue involve computing an additional rotation matrix to enforce criteria such as least spatial entropy, or spatial orthogonality, a process that, while effective, significantly increases computational complexity. This work introduces a hybrid approach combining PCA and deep learning for separating source maps from virtual sources. By simulating sound sources in random quantities and locations, a neural network tailored to this task is trained. The permutation problem between PCA-derived virtual sources and pre-simulated labels, as well as the estimation of the number of sources, is addressed by framing source separation as a set prediction problem utilizing the Hungarian loss. This method is array geometry agnostic and frequency agnostic, allowing for robust performance across diverse array configurations and a wide frequency range. Trained solely on simulated data, the model demonstrates effective source separation on real-world datasets, highlighting the potential of integrating deep learning with existing methods.

Outcome of the Matrix Rotation Flap in Conservative Breast Surgeries.