Quaternion Rotation Research Articles

Efficient conversion of Euler angles to quaternion for practical aerospace applications

Abstract Current methods for converting Euler angles to a rotation quaternion are computationally intensive and have limited practical applications in aerospace. This study introduces an efficient approach that directly converts Euler angles in any desired sequence to a rotation quaternion, eliminating the need for intermediate Euler angle transformation matrices, which leads to a significant improvement in computational efficiency. Moreover, it is general because it works with all 12 possible sequences of rotations. A case study is presented, demonstrating the application of the algorithm in converting between the launch frame and the missile frame of a vertical takeoff vertical landing (VTVL) rocket. The results show more than a ninefold improvement in computational efficiency compared to the classical Euler angle method while maintaining consistency of results.

Three-Dimensional Mapping of the Rotation of Interactive Virtual Objects with Eye-Tracking Data.

We present a method for real-time recording of human interaction with three-dimensional (3D) virtual objects. The approach consists of associating rotation data of the manipulated object with behavioral measures, such as eye tracking, to make better inferences about the underlying cognitive processes. The task consists of displaying two identical models of the same 3D object (a molecule), presented on a computer screen: a rotating, interactive object (iObj) and a static, target object (tObj). Participants must rotate iObj using the mouse until they consider its orientation to be identical to that of tObj. The computer tracks all interaction data in real time. The participant's gaze data are also recorded using an eye tracker. The measurement frequency is 10 Hz on the computer and 60 Hz on the eye tracker. The orientation data of iObj with respect to tObj are recorded in rotation quaternions. The gaze data are synchronized to the orientation of iObj and referenced using this same system. This method enables us to obtain the following visualizations of the human interaction process with iObj and tObj: (1) angular disparity synchronized with other time-dependent data; (2) 3D rotation trajectory inside what we decided to call a "ball of rotations"; (3) 3D fixation heatmap. All steps of the protocol have used free software, such as GNU Octave and Jmol, and all scripts are available as supplementary material. With this approach, we can conduct detailed quantitative studies of the task-solving process involving mental or physical rotations, rather than only the outcome reached. It is possible to measure precisely how important each part of the 3D models is for the participant in solving tasks, and thus relate the models to relevant variables such as the characteristics of the objects, cognitive abilities of the individuals, and the characteristics of human-machine interface.

EHPR: Learning evolutionary hierarchy perception representation based on quaternion for temporal knowledge graph completion

Research on temporal knowledge graphs garners attention due to the intricate connection between facts and dynamic temporal factors. However, existing research uses timestamp as auxiliary data for representation learning and directly integrate it into facts, resulting in the inability to capture the intrinsic connections between relations under time evolution. To handle these challenges, we propose the Evolutionary Hierarchy Perception Representation (EHPR), which first leverages the Hamilton product to perform rotational transformations on relation and entity over time, aiming to learn temporal relation and temporal entity with close interactions with time information. Later, EHPR is divided into two modules: (a) Rotating the head entity towards the tail entity using temporal relation through Hamilton product to model complex patterns with quaternion rotation capabilities. (b) Adopting an evolutionary hierarchical factor to capture the differences in modulus distribution between the temporal head entity and the temporal tail entity, aiming to manage the evolutionary hierarchical information between different temporal entities. In this way, EHPR not only utilizes the rich quaternion rotation capabilities to model various relation patterns but also further enables modeling of evolutionary hierarchical patterns through evolutionary hierarchy factors. Experiments show that EHPR achieves remarkable performance on six mature benchmarks compared to state-of-the-art models. Furthermore, we successfully transferred the core idea of EHPR into complex embeddings, showcasing the framework's adaptability. Compared to complex embedding models, EHPR also demonstrates stronger expressive abilities with the Hamilton operator, surpassing the performance of complex Hermitian operator.

Multiscale topology optimization for the design of spatially-varying three-dimensional lattice structure

This paper introduces three dimensional (3D) topology optimization specifically tailored for designing spatially-varying primitive-cubic (CP) type lattice structures. The developed design process consists of three steps: pre-processing, main processing, and post-processing. In the pre-processing step, a surrogate model between lattice geometry variables and effective elasticity tensor is constructed using an artificial neural network. This surrogate model is essential because it enables the quick calculation of the effective elasticity tensor of lattice microstructure. In the subsequent main processing step, multiscale topology optimization is performed. During this step, lattice microstructure size and rotation design variables are optimized together with the macrostructure density design variables. Macrostructures are designed using a well-established three-field density scheme with a SIMP formulation. Formulations for spatially-varying 3D lattice microstructure are newly developed based on a unit quaternion rotation representation scheme. In the final post-processing step, a spatially-varying microstructure is restored using a de-homogenization scheme, newly developed particularly for the primitive-cubic (CP) type 3D lattice structures. The effectiveness and robustness of the proposed formulations are validated through three design examples for the compliance minimization problem. Moreover, a designed lattice structure is fabricated using an additive manufacturing machine.

Research on Positioning Tracking Mode of Sports Rehabilitation Training Based on Self-Powered Sensor Based on Particle Swarm Optimization Algorithm

The theme of today’s world is peace and development. A stable external environment has made people’s average life expectancy gradually increased, and the world is rapidly aging. Aging has brought many problems, such as the increase in the number of patients with limb dysfunction due to various diseases, which has gradually increased the demand for rehabilitation training. With traditional rehabilitation training methods, the training scenes are single and boring, and patients are prone to resisting. This paper designs and implements a real-time rehabilitation training guidance system based on self-powered sensors for the rehabilitation training needs of stroke patients. The system uses self-powered sensors to collect human motion information in real time, and compares it with the key posture sequences in the standard motion library to obtain corresponding matching results and guide patients to perform correct rehabilitation training. Using the rotation quaternion of 25 bone points in the patient’s rehabilitation exercise to calculate and update the rotation quaternion of the corresponding bone point of the character model, the function of the character model to follow the patient’s mirror motion is realized. This allows patients to control the completion of their rehabilitation movements without the need for medical staff to accompany them. And the stability of the system is optimized based on the particle swarm optimization algorithm. After traversal optimization, the current sensitivity coefficient of the model is reduced by about 75% compared with that before the correction, indicating that the current stability of the model obtained at this time has been improved to a certain extent. However, in the regression model of the self-powered sensor established by the particle swarm optimization algorithm, its parameters are reduced by about 82% compared with those before the correction, which shows that the current stability of the model has been greatly improved at this time, and the operating current of the receiving loop has been greatly improved.

Revolutionary Strategy for Depicting Knowledge Graphs with Temporal Attributes

In practical applications, the temporal completeness of knowledge graphs is of great importance. However, previous studies have mostly focused on static knowledge graphs, generally neglecting the dynamic evolutionary properties of facts. Moreover, the unpredictable and limited availability of temporal knowledge graphs, together with the complex temporal dependency patterns, make current models inadequate for effectively describing facts that experience temporal transitions. To better represent the evolution of things over time, we provide a learning technique that uses quaternion rotation to describe temporal knowledge graphs. This technique describes the evolution of entities as a temporal rotation change in quaternion space. Compared to the Ermitian inner product in complex number space, the Hamiltonian product in quaternion space is better at showing how things might be connected. This leads to a learning process that is both more effective and more articulate. Experimental results demonstrate that our learning method significantly outperforms existing methods in capturing the dynamic evolution of temporal knowledge graphs, with improved accuracy and robustness across a range of benchmark datasets.

RiQ-KGC: Relation Instantiation Enhanced Quaternionic Attention for Complex-Relation Knowledge Graph Completion

A knowledge graph is a structured semantic network designed to describe physical entities and relations in the world. A comprehensive and accurate knowledge graph is essential for tasks such as knowledge inference and recommendation systems, making link prediction a popular problem for knowledge graph completion. However, existing approaches struggle to model complex relations among entities, which severely hampers their ability to complete knowledge graphs effectively. To address this challenge, we propose a novel hierarchical multi-head attention network embedding framework, called RiQ-KGC, which integrates different-grained contextual information of knowledge graph triples and models quaternion rotation relations between entities. Furthermore, we propose a relation instantiation method for alleviating the difficulty of expressing complex relations between entities. To enhance the expressiveness of relation representation, the relation is integrated by Transformer to obtain multi-hop neighbor information, so that one relation can be embedded into different embeddings according to different entities. Experimental results on four datasets demonstrate that RiQ-KGC exhibits strong competitiveness compared to state-of-the-art models in link prediction, while the ablation experiments reveal that the proposed relation instantiation method achieves great performance.

Nonlinear statics of three‐dimensional curved geometrically exact beams by a hierarchal quadrature element method

AbstractNonlinear static analyses of three‐dimensional (3D) curved geometrically exact beams are carried out using a hierarchal quadrature element method (HQEM) in this work. The initial value of the rotational quaternions is computed from an initial‐value problem with arc‐length as the “time” variable, so that the quaternions can be differentiated in subsequent computation. The 3D curved geometrically exact beams are first expressed by NURBS and then transformed to be expressed by the hierarchical quadrature bases. The geometrically exact beams are then analyzed by the same hierarchical quadrature bases, which conforms to the isogeometric concept. This procedure of analysis is new. The weak‐form quadrature element method (QEM) is a special case of the hierarchical quadrature element method (HQEM). It is found that the best accuracy may be obtained when the number of integration nodes is one degree less than the nodal points, instead of taking them the same as in the weak‐form QEM. Numerical tests through several examples show that the HQEM can present results with high accuracy using only a few degrees of freedom and is not sensitive to shear locking.

Long short-term memory prediction of user’s locomotion in virtual reality

Nowadays, there is still a challenge in virtual reality to obtain an accurate displacement prediction of the user. This could be a future key element to apply in the so-called redirected walking methods. Meanwhile, deep learning provides us with new tools to reach greater achievements in this type of prediction. Specifically, long short-term memory recurrent neural networks obtained promising results recently. This gives us clues to continue researching in this line to predict virtual reality user’s displacement. This manuscript focuses on the collection of positional data and a subsequent new way to train a deep learning model to obtain more accurate predictions. The data were collected with 44 participants and it has been analyzed with different existing prediction algorithms. The best results were obtained with a new idea, the use of rotation quaternions and the three dimensions to train the previously existing models. The authors strongly believe that there is still much room for improvement in this research area by means of the usage of new deep learning models.

Orientation Quaternion Norm Correction in SINS Algorithms: Calculation Normalization Schemes and Their Efficiency

The problem of correcting the norm of the computed orientation quaternion in the algorithms for the operation of strapdown inertial navigation systems is considered. Two existing approaches to the correction process are considered, the first approach is to normalize the rotation quaternion at the computation cycle, the second approach is to normalize the resulting quaternion. 5 well-known calculation schemes for norm correction are given. To simulate the test motion, we used an analytical quaternion kinematic rotation model based on a sequence of three rotations corresponding to the Krylov angles. The case of linear dependence of elementary rotation angles on time is considered. The model provides analytical representation of the projections of the angular velocity vector of the rigid body on the associated axes and the corresponding quasi-coordinates on the calculation cycle. The results of numerical simulation of the reference motion for a given set of frequencies are presented as dependences of the projections of the angular velocity vector of the rigid body on time and the constructed trajectories in the configuration space of the orientation parameters.
 To determine the rotation quaternion on a cycle, the Miller algorithm was used, which makes it possible to obtain an increase in the orientation vector based on ideal information from the angular velocity sensors in the form of quasi-coordinates. The transformation into a rotation quaternion occurs with the help of the corresponding expansions of the trigonometric functions of the true rotation angle (modulus of the orientation vector) in a series.
 Based on a numerical experiment, it is shown that the best result of correcting the norm of the calculated quaternion in the sense of the minimum error of the norm is given by one of the finite normalization schemes, for which there is no division operation and ensures the stability of the norm correction in time. The results of numerical simulation of the model rotational motion of a rigid body and the development of schemes for correcting the norm of the calculated orientation quaternion are presented.
 Keywords: orientation quaternion, Miller's orientation algorithm, SINS, norm error, computational drift, analytical reference model, quasi-coordinates, numerical simulation.

Structure‐preserving invariant interpolation schemes for invertible second‐order tensors

AbstractTensor interpolation is an essential step for tensor data analysis in various fields of application and scientific disciplines. In the present work, novel interpolation schemes for general, that is, symmetric or non‐symmetric, invertible square tensors are proposed. Critically, the proposed schemes rely on a combined polar and spectral decomposition of the tensor data , followed by an individual interpolation of the two rotation tensors and and the positive definite diagonal eigenvalue tensor resulting from this decomposition. Two different schemes are considered for a consistent rotation interpolation within the special orthogonal group , either based on relative rotation vectors or quaternions. For eigenvalue interpolation, three different schemes, either based on the logarithmic weighted average, moving least squares or logarithmic moving least squares, are considered. It is demonstrated that the proposed interpolation procedure preserves the structure of a tensor, that is, and remain orthogonal tensors and remains a positive definite diagonal tensor during interpolation, as well as scaling and rotational invariance (objectivity). Based on selected numerical examples considering the interpolation of either symmetric or non‐symmetric tensors, the proposed schemes are compared to existing approaches such as Euclidean, Log‐Euclidean, Cholesky and Log‐Cholesky interpolation. In contrast to these existing methods, the proposed interpolation schemes result in smooth and monotonic evolutions of tensor invariants such as determinant, trace, fractional anisotropy (FA), and Hilbert's anisotropy (HA). Moreover, a consistent spatial convergence behavior is confirmed for first‐ and second‐order realizations of the proposed schemes. The present work is mainly motivated by the frequently occurring necessity for remeshing or mesh adaptivity when applying the finite element method to complex problems of nonlinear continuum mechanics with inelastic constitutive behavior, which requires the consistent interpolation of tensor‐valued history data for the transfer between different meshes. However, the proposed schemes are very general in nature and suitable for the interpolation of general invertible second‐order square tensors independent of the specific application.

Effective Quaternion Rotation in An Ellipsoid through Sphere Transformation: A Linear Approach

The application of quaternion rotation in ellipsoids and spheres is an intriguing field with significant implications in computer graphics, robotics, and physics simulations. The necessity to decipher quaternion rotation in ellipsoid is crucial. This research aims to utilize the principles of quaternion rotation in spheres to compute the equivalent in the ellipsoid. This involves the application of a linear transformation to convert quaternion rotation in ellipsoids into spheres. By controlling the full rotation of quaternions, the spheres can be transformed back into ellipsoids, achieving the ultimate goal of controlling quaternion rotation within an ellipsoid. The results demonstrate that this approach effectively addresses the quaternion rotation on the ellipsoid and aligns with the fundamental properties of quaternions. Furthermore, it serves as a significant aid in implementing quaternion rotation on the ellipse.

Improving pedicle screw path planning by vertebral posture estimation

Objective. Robot-assisted pedicle screw placement in spinal surgery can reduce the complications associated with the screw placement and reduce the hospital return counts due to malfunctions. However, it requires accurate planning for a high-quality procedure. The state-of-the-art technologies reported in the literature either ignore the anatomical variations across vertebrae or require substantial human interactions. We present an improved approach that achieves pedicle screw path planning through multiple projections of a numerically re-oriented vertebra with the estimated posture. Approach. We proposed an improved YOLO-type neural network model (YOLOPOSE3D) to estimate the posture of a vertebra before pedicle path planning. In YOLOPOSE3D, the vertebral posture is given as a rotation quaternion and 3D location coordinates by optimizing the intersection over union of the vertebra with the predicted posture and the actual posture. Then, a new local coordinate system is established for the vertebra based on the estimated posture. Finally, the optimal pedicle screw path trajectory is determined from the multiple projections of the vertebra in the local coordinates. Main results. The experimental results in difficult cases of scoliosis showed that the new YOLOPOSE3D network could accurately detect the location and posture of the vertebra with average translation and orientation errors as small as 1.55 mm and 2.55°. The screw path planning achieved 83.1% success rate without breaking the pedicle cortex for the lumbar vertebral L1–L5, which is better than that of a doctor’s manual planning, 82.4%. With the clinical class A requirement to allow less than 2 mm out of the pedicle cortex, the success rate achieved nearly 100%. Significance. The proposed YOLOPOSED3D method can accurately determine the vertebral postures. With the improved posture prior, better clinical outcomes can be achieved for pedicle screw placement in spine internal fixation procedures.

GradPose: a very fast and memory-efficient gradient descent-based tool for superimposing millions of protein structures from computational simulations.

Computational simulations like molecular dynamics and docking are providing crucial insights into the dynamics and interaction conformations of proteins, complementing experimental methods for determining protein structures. These methods often generate millions of protein conformations, necessitating highly efficient structure comparison and clustering methods to analyze the results. In this article, we introduce GradPose, a fast and memory-efficient structural superimposition tool for models generated by these large-scale simulations. GradPose uses gradient descent to optimally superimpose structures by optimizing rotation quaternions and can handle insertions and deletions compared to the reference structure. It is capable of superimposing thousands to millions of protein structures on standard hardware and utilizes multiple CPU cores and, if available, CUDA acceleration to further decrease superimposition time. Our results indicate that GradPose generally outperforms traditional methods, with a speed improvement of 2-65 times and memory requirement reduction of 1.7-48 times, with larger protein structures benefiting the most. We observed that traditional methods outperformed GradPose only with very small proteins consisting of ∼20 residues. The prerequisite of GradPose is that residue-residue correspondence is predetermined. With GradPose, we aim to provide a computationally efficient solution to the challenge of efficiently handling the demand for structural alignment in the computational simulation field. Source code is freely available at https://github.com/X-lab-3D/GradPose; doi:10.5281/zenodo.7671922.

Flocks, Mobs, and Figure Eights: Swarming as a Lemniscatic Arch

Inspired by the natural mobbing behavior of birds, this work presents a novel, quasi-distributed swarming strategy called the Dynamic Lemniscatic Arch. It resolves the problem of producing globally-stable, evenly-spaced lemniscate (or, figure-eight) trajectories while relying on local interactions only. Such trajectories are advantageous in applications where energy consumption and mechanical strain must be minimized. Previous work in lemniscate curves has typically relied on predetermined trajectories, rather than on the emergent structure of the swarm. Furthermore, we enrich the traditional 2-dimensional lemniscate plane curve structure by forming an arch in the third dimension. This arch provides more consistent coverage in surveillance type tasks and, with minor variations in parameters, can be used to produce mobbing behavior. The technique relies on time-varying quaternion rotations linked to the positions of dynamically induced virtual agents. We provide a mathematical proof of stability, which demonstrates the swarm converges to the desired geometry. Simulations show that the strategy performs well with multiple agents and in numerous different configurations.

Codebook Design for PD-SCMA NOMA Systems

The hybrid power domain sparse code non-orthogonal multiple access (PD-SCMA) scheme multiplexes multiple users on a single resource element (RE) both in power and code domains, thus achieving a superior spectral efficiency. The multiplexing performance of such a scheme can be enhanced not only through deployment of sophisticated resource allocation schemes and low complexity robust multi-user detectors but also prudent multi-user multi-dimensional codebook design. By considering the key codebook design performance indicators (KPIs) in different channel use scenarios, this work assesses various known multi-dimensional constellation (MDC) design criterion on the performance of low-rate turbo-coded PD-SCMA NOMA system. In order to strike a balance with regard to good multi-dimensional modulation (MDM) and low complex detection, a low complex codebook design based on linear error correcting block codes is proposed. Unique user-specific rotation design based on quaternion rotation matrices is proposed with the aim of maintaining users symbol uniqueness for overloaded systems. Simulation results confirm that MDCs that satisfy design KPIs outperform others. Besides, the proposed user-specific rotation design remarkably enhances the BER performance of the block code based codebook design.

Quaternion Methods and Regular Models of Celestial and Space Flight Mechanic: Using Euler (Rodrigues-Hamilton) Parameters to Describe Orbital (Trajectory) Motion. II: Perturbed Spatial Restricted Three-Body Problem

The article considers the problem of regularizing the features of the classical equations of celestial mechanics and space flight mechanics (astrodynamics), which use variables that characterize the shape and size of the instantaneous orbit (trajectory) of the moving body under study, and Euler angles that describe the orientation of the used rotating (intermediate) coordinate system or the orientation of the instantaneous orbit, or the plane of the orbit of a moving body in an inertial coordinate system. Singularity-type features (division by zero) of these classical equations are generated by Euler angles and complicate the analytical and numerical study of orbital motion problems. These singularities are effectively eliminated by using the four-dimensional Euler (Rodrigues-Hamilton) parameters and Hamiltonian rotation quaternions. In this (second) part of the work, new regular quaternion models of celestial mechanics and astrodynamics are obtained that do not have the above features and are built within the framework of a perturbed spatial limited three-body problem (for example, the Earth, the Moon (or the Sun) and a spacecraft (or an asteroid)): equations of trajectory motion written in non-holonomic or orbital or ideal coordinate systems, for the description of the rotational motion of which the Euler (Rodrigues-Hamilton) parameters and quaternions of Hamilton rotations are used. New regular quaternion equations of the perturbed spatial restricted three-body problem are also obtained, constructed using two-dimensional ideal rectangular Hansen coordinates, Euler parameters and quaternion variables, as well as using complex compositions of Hansen coordinates and Euler parameters (Cayley-Klein parameters). The advantage of the proposed orbital motion equations constructed using the Euler parameters over the equations constructed using the Euler angles is due to the well-known advantages of the quaternion kinematic equations in the Euler parameters included in the proposed equations over the kinematic equations in the Euler angles included in the classical equations.

Vector methods for determination of rigid body orientation

Vector methods for determining the orientation of a rigid body using information about vectors in the reference and body-related coordinate systems are considered. The analysis is based on the least squares method in vector-matrix form. This approach allows us to consider the application of different methods of solving the problem from a single standpoint: matrix (using the directional cosine matrix), geometric (using the Gibbs vector), quaternion (using the quaternion of rotation). Numerical evaluation of the accuracy of eight orientation determination algorithms obtained on the basis of these methods was performed.

Quaternion to Euler angles conversion: A direct, general and computationally efficient method.

Current methods of the conversion between a rotation quaternion and Euler angles are either a complicated set of multiple sequence-specific implementations, or a complicated method relying on multiple matrix multiplications. In this paper a general formula is presented for extracting the Euler angles in any desired sequence from a unit quaternion. This is a direct method, in that no intermediate conversion step is required (no quaternion-to-rotation matrix conversion, for example) and it is general because it works with all 12 possible sequences of rotations. A closed formula was first developed for extracting angles in any of the 12 possible sequences, both "Proper Euler angles" and "Tait-Bryan angles". The resulting algorithm was compared with a popular implementation of the matrix-to-Euler angle algorithm, which involves a quaternion-to-matrix conversion in the first computational step. Lastly, a single-page pseudo-code implementation of this algorithm is presented, illustrating its conciseness and straightforward implementation. With an execution speed 30 times faster than the classical method, our algorithm can be of great interest in every aspect.

Graph Convolutional Network for 3D Object Pose Estimation in a Point Cloud.

Graph Neural Networks (GNNs) are neural networks that learn the representation of nodes and associated edges that connect it to every other node while maintaining graph representation. Graph Convolutional Neural Networks (GCNs), as a representative method in GNNs, in the context of computer vision, utilize conventional Convolutional Neural Networks (CNNs) to process data supported by graphs. This paper proposes a one-stage GCN approach for 3D object detection and poses estimation by structuring non-linearly distributed points of a graph. Our network provides the required details to analyze, generate and estimate bounding boxes by spatially structuring the input data into graphs. Our method proposes a keypoint attention mechanism that aggregates the relative features between each point to estimate the category and pose of the object to which the vertices of the graph belong, and also designs nine degrees of freedom of multi-object pose estimation. In addition, to avoid gimbal lock in 3D space, we use quaternion rotation, instead of Euler angle. Experimental results showed that memory usage and efficiency could be improved by aggregating point features from the point cloud and their neighbors in a graph structure. Overall, the system achieved comparable performance against state-of-the-art systems.