Dual Quaternion Research Articles

Dual complex structure-preserving algorithm of dual quaternion singular value decomposition and its applications

Dual complex structure-preserving algorithm of dual quaternion singular value decomposition and its applications

A Comparative Analysis Among Dynamics Modeling Approaches for Space Manipulator Systems

Abstract This paper presents a comparative analysis on the space manipulator systems dynamics modeling approaches, namely, the standard approach (SA) and the dual quaternion based dynamics modeling approach. A detailed analysis supported by the results from numerical simulations, comparing the two approaches in terms of operational count and execution time, has been presented to determine which approach is computationally and temporally efficient.

Reply to Bektaş, S. Comment on “Ioannidou, S.; Pantazis, G. Helmert Transformation Problem. From Euler Angles Method to Quaternion Algebra. ISPRS Int. J. Geo-Inf. 2020, 9, 494”

The comment disputes some of the inferences in the paper “Helmert Transformation Problem. From Euler Angles Method to Quaternion Algebra”, published in this journal. The key points in the dissent are the following: (1) The number of unknown parameters in the reverse transformation problem using dual quaternions. (2) The reliability of both data and the results. (3) There should be no differences between Euler angles and quaternion methods. Our response is summarized as follows: (1) The problem can be solved using either eight or nine unknown parameters. (2) All the data and results are real. (3) There should be differences between methods because of different calculations.

Pose-Constrained Control of Proximity Maneuvering for Tracking and Observing Noncooperative Targets with Unknown Acceleration

This paper proposes a pose control scheme of for proximity maneuvering for tracking and observing noncooperative targets with unknown acceleration, which is an important prerequisite for on-orbit operations in space. It mainly consists of a finite-time extended state observer and constraint processing procedures. Firstly, relative pose-coupled kinematics and dynamics models with unknown integrated disturbances are established based on dual quaternion representations. Then, a finite-time extended state observer is designed using the super-twisting algorithm to estimate the integrated disturbances. Both observation field of view and collision avoidance pose-constrained models are constructed to ensure that the service spacecraft continuously and safely observes the target during proximity maneuvering. And the constraint models are further incorporated into the design of artificial potential function with a unique minimum. After that, the proportional–derivative-like pose-constrained tracking control law is proposed based on the estimated disturbances and the gradient of the artificial potential function. Finally, the effectiveness of the control scheme is verified through numerical simulations.

Active disturbance rejection control for spacecraft relative motion in libration point orbits subject to actuator saturation and time-delays

This paper proposes a novel active disturbance rejection control (ADRC) scheme to achieve the spacecraft relative pose synchronization in the libration orbits under the constrations of the actuator saturation and time-delays. The relative kinematics and dynamics between two spacecraft are modeled using dual quaternions to avoid singularity. A set of dimensionless variables is introduced to improve computational accuracy. An extended state observer, consisting of a time-delay predictor and an anti-windup compensator, is designed to address actuator saturation and communication time delays. A composite controller is then proposed, based on the designed observer, to mitigate the impact of total disturbances. Finally, two illustrative numerical simulations, including a spacecraft close rendezvous and docking scenario and a formation flying scenario, are provided to verify the effectiveness of the proposed ADRC scheme.

Spectral Properties of Dual Unit Gain Graphs

In this paper, we study dual quaternion, dual complex unit gain graphs, and their spectral properties in a unified frame of dual unit gain graphs. Unit dual quaternions represent rigid movements in the 3D space, and have wide applications in robotics and computer graphics. Dual complex numbers have found application in brain science recently. We establish the interlacing theorem for dual unit gain graphs, and show that the spectral radius of a dual unit gain graph is always not greater than the spectral radius of the underlying graph, and these two radii are equal if, and only if, the dual gain graph is balanced. By using dual cosine functions, we establish the closed form of the eigenvalues of adjacency and Laplacian matrices of dual complex and quaternion unit gain cycles. We then show the coefficient theorem holds for dual unit gain graphs. Similar results hold for the spectral radius of the Laplacian matrix of the dual unit gain graph too.

Deformable patch-based garment design in immersive virtual reality

PurposeThis study aims to create and deform 3D garment apparel in an immersive virtual reality using head-mounted display and controllers. For this, adequate design methods for immersive virtual environment were explored and developed in order to confirm the suitability of the developed methods.Design/methodology/approachAn immersive virtual environment was prepared using Unreal Engine (UE) version 5.1 and Meta Human Lexi to create template garment that corresponds to the sizes of a human model. Dual quaternion skinning was adopted for pose deformation. For size deformation, patches were constructed with the measurement lines defined on Lexi. This patch-based approach was adopted not only for automatic generation but also for flat pattern projection of the template garment.FindingsThe research found that garment-making process can be brought into immersive virtual reality. Free use of one's hands and body made apparel deformation in an immersive environment conform with the real garment draping process.Originality/valueSimulating garment making in an immersive virtual reality has not previously been explored in detail. This research discovered, implemented and tested methods that best suit the environment where head-mounted display and controllers are essential in detail.

Solving the QLY Least Squares Problem of Dual Quaternion Matrix Equation Based on STP of Dual Quaternion Matrices

Dual algebra plays an important role in kinematic synthesis and dynamic analysis, but there are still few studies on dual quaternion matrix theory. This paper provides an efficient method for solving the QLY least squares problem of the dual quaternion matrix equation AXB+CYD≈E, where X, Y are unknown dual quaternion matrices with special structures. First, we define a semi-tensor product of dual quaternion matrices and study its properties, which can be used to achieve the equivalent form of the dual quaternion matrix equation. Then, by using the dual representation of dual quaternion and the GH-representation of special dual quaternion matrices, we study the expression of QLY least squares Hermitian solution of the dual quaternion matrix equation AXB+CYD≈E. The algorithm is given and the numerical examples are provided to illustrate the efficiency of the method.

An radicals construction technique based on dual quaternions and hierarchical transformers

At present, the exploration of Knowledge Graph Embedding (KGE) mainly focuses on static KGE models. Due to the diversity and complexity of vertical domain data, it is difficult for existing KGE models to achieve excellent performance. Specific KGE models must be constructed for research on vertical domain knowledge graphs. This paper explores the relationships between the composition of Chinese characters and their sounds, meanings, and forms by constructing the Chinese Radical Knowledge Graph (CRKG). CRKG was constructed from a unique perspective according to the characteristics of Chinese radicals and knowledge graphs. The minor components of Chinese characters are separated and constructed into the fourth dimension of the knowledge graph. In this paper, we propose a new Chinese Radical Knowledge Graph Embedding (CRKGE) model, RotAL, which introduces dual quaternions and hierarchical transformers into KGE. RotAL performs a translation-rotation operation on the head entity with the relation as a parameter, then does a translation operation on the head entity with the minor component as a parameter to be close to the tail entity. Finally, we feed the trained quaternions into hierarchical transformers to jointly learn entity-related composition and relational contextualization based on a head entity’s neighborhood. We demonstrate that our method can model and infer various relational patterns of CRKG, such as symmetric/antisymmetric, reflexive, inversion, and multiple. The experimental results on four CRKG datasets demonstrate that the RotAL model achieves the highest Hit@1 score, outperforming existing KGE models in accurately identifying the target entity on the first attempt. Furthermore, the RotAL model consistently achieves superior performance across other evaluation metrics, substantiating its significant advantage over existing models.

A Review on Modern Techniques in 3D Character Animation

3D character animation is a pivotal aspect of digital graphics, essential for various applications such as films, games, and simulations. This review paper explores advancements in character animation techniques, including linear blend skinning, dual quaternion skinning, and physics-based approaches. Emphasis is placed on recent innovations aimed at improving the realism and computational efficiency of character deformations, especially non-rigid transformations. By examining key methodologies and their applications, this paper provides a comprehensive overview of the current state and future directions in 3D character animation

TELS: Learning time-evolving information and latent semantics using dual quaternion for temporal knowledge graph completion

In temporal knowledge graphs (TKGs), the status of facts is intricately tied to the dynamic and precise nature of temporal factors. Existing research merely treats time as supplementary information, without considering the latent semantic changes caused by the positional changes of entities within specific relations in a temporal context. Furthermore, due to the coarse granularity of timestamps in existing TKGs, the number of multiple relations pattern among entities significantly increases, limiting model performance. This paper proposes a Time-Evolving Information and Latent Semantics model (TELS), which represents facts as dual quaternion embeddings to provide a compact and elegant representation. Specifically, we use timestamp dual quaternions, transforming the entity and relation into temporal entity and temporal relation through dual quaternion multiplication. Besides, we introduce semantic-aware dual quaternion to capture the latent semantics arising from the positional changes of entities within specific relations. Next, TELS consists of two parts: (a) We use semantic-aware dual quaternions to perform transformations on head entity and tail entity respectively through dual quaternion multiplication. Next, we utilize temporal relation to transform head entity to tail entity through dual quaternion multiplication. (b) We adopt an evolutionary hierarchical factor to encapsulate the differences in modulus distribution between the temporal head entity and temporal tail entity. In this way, TELS not only uses dual quaternions to handle key patterns and multiple relations pattern, but also handles evolutionary hierarchical patterns by capturing the modulus distribution differences between temporal entities. Meanwhile, TELS learns semantic-aware dual quaternion embeddings to capture the latent semantics endowed by relations to entities. Empirically, TELS can boost the performance over seven temporal knowledge graph benchmarks.

A study on dual quaternion based cooperative relative navigation of multiple UAVs with monocular vision-inertial integration

This paper addresses a cooperative relative navigation problem for multiple aerial agents, relying on visual tracking information between vehicles. The research aims to investigate a sensor fusion architecture and algorithm that leverages partially available absolute navigation knowledge while exploiting collaborative visual interaction between vehicles in mission flight areas, where satellite navigation-denied regions are irregularly located. To achieve this, the paper introduces a new approach to defining the relative poses of cameras and develops a corresponding process to secure the relative pose information. This contrasts with previous research, which simply linearized the relative pose information of aircraft cameras into navigation states defined in an absolute coordinate system. Specifically, the target pose in relative navigation is defined, and the pose of the camera and feature points are directly derived using dual quaternion representation, which compactly represents both translation and rotation. Furthermore, a mathematical model for the relative pose of the camera is derived through the dual quaternion framework, enabling an explicit pose formulation of relative navigation. The study investigates navigation performance in typical mission flight scenarios using an in-house high-fidelity simulator and quantitatively highlights the contributions of the proposed scheme by comparing the navigation error performance. Consequently, the proposed method demonstrates to have navigation accuracy in decimeter level even in GNSS-denied environments and an improved 3D Root Mean Square (RMS) error by 30% smaller than the conventional absolute navigation framework.

Research on datumn transformation model in particle accelerator component installation

When the important components of the particle accelerator are installed, it is required to locate the instrument in the global coordinate system by measuring the control points of the surrounding tunnels, which involves the datumn transformation between the global coordinate system and the station coordinate system, and the selection of the transformation model has gradually become a potential breakthrough in solving the problem. Four different models euler angle, Rodrigues matrix, dual quaternion and twelve parameters are used to analyze and compare the solution parameters, accuracy index, iteration efficiency, and the number of selected common points of the four models with the measured data of 2022 Hefei Light Source summer. The results show that the deviation of the four model parameters from the SpatialAnalyzer commercial software is no more than 3 μm for the translation parameters, and no more than 0.6” for the rotation angle. Among them, the twelve-parameters and dual quaternion two models are theoretically rigorous, with high and stable accuracy indexes, and the twelve parameters posteriori unit weight mean error shows obvious advantages. The twelve parameters and Rodrigues models have fewer iterations and higher efficiency. In summary, the dual quaternion model and twelve parameters method are preferred in the absence of better initial values. In addition, in the process of datumn transformation, along with the increase of the number of common points, the accuracies of all four model solutions show an increasing trend of first fast and then slow.

SE(3) Synchronization by eigenvectors of dual quaternion matrices

Abstract In synchronization problems, the goal is to estimate elements of a group from noisy measurements of their ratios. A popular estimation method for synchronization is the spectral method. It extracts the group elements from eigenvectors of a block matrix formed from the measurements. The eigenvectors must be projected, or ‘rounded’, onto the group. The rounding procedures are constructed ad hoc and increasingly so when applied to synchronization problems over non-compact groups. In this paper, we develop a spectral approach to synchronization over the non-compact group $\mathrm{SE}(3)$, the group of rigid motions of $\mathbb{R}^{3}$. We based our method on embedding $\mathrm{SE}(3)$ into the algebra of dual quaternions, which has deep algebraic connections with the group $\mathrm{SE}(3)$. These connections suggest a natural rounding procedure considerably more straightforward than the current state of the art for spectral $\mathrm{SE}(3)$ synchronization, which uses a matrix embedding of $\mathrm{SE}(3)$. We show by numerical experiments that our approach yields comparable results with the current state of the art in $\mathrm{SE}(3)$ synchronization via the spectral method. Thus, our approach reaps the benefits of the dual quaternion embedding of $\mathrm{SE}(3)$ while yielding estimators of similar quality.

Semiglobal Exponential Stability for Dual-Quaternion-Based Rigid-Body Tracking Control

Semi-Global Exponential Stability (SGES) is proven for the combined attitude and position rigid-body motion tracking problem, presenting a significant upgrade compared to previously proposed controllers which can only guarantee asymptotic stability. Dual quaternions are used to jointly represent the rotational and translational tracking error dynamics of the rigid body. A novel nonlinear feedback tracking controller is proposed, and a Lyapunov-based analysis is provided to prove SGES of the closed-loop dynamics. The analysis does not place any restrictions on the reference trajectory or the feedback gains. This stronger SGES result aids in further analyzing the robustness of the rigid-body system by establishing input-to-state stability (ISS) in the presence of time-varying additive and bounded external disturbances. Motivated by the fact that, in many aerospace applications, stringent adherence to safety constraints such as approach path and input constraints is critical for overall mission success, a framework for safe control of spacecraft is presented that combines the proposed feedback controller with control barrier functions. Numerical simulations are provided to verify the SGES and ISS results and also showcase the efficacy of the proposed nonlinear feedback controller in several nontrivial scenarios, including trajectory tracking, transposition and docking, flip maneuvers, collision avoidance, and rendezvous.

A High-Precision Hand-Eye Coordination Localization Method under Convex Relaxation Optimization.

Traditional switching operations require on-site work, and the high voltage generated by arc discharges can pose a risk of injury to the operator. Therefore, a combination of visual servo and robot control is used to localize the switching operation and construct hand-eye calibration equations. The solution to the hand-eye calibration equations is coupled with the rotation matrix and translation vectors, and it depends on the initial value determination. This article presents a convex relaxation global optimization hand-eye calibration algorithm based on dual quaternions. Firstly, the problem model is simplified using the mathematical tools of dual quaternions, and then the linear matrix inequality convex optimization method is used to obtain a rotation matrix with higher accuracy. Afterwards, the calibration equations of the translation vectors are rewritten, and a new objective function is established to solve the coupling influence between them, maintaining positioning precision at approximately 2.9 mm. Considering the impact of noise on the calibration process, Gaussian noise is added to the solutions of the rotation matrix and translation vector to make the data more closely resemble the real scene in order to evaluate the performance of different hand-eye calibration algorithms. Eventually, an experiment comparing different hand-eye calibration methods proves that the proposed algorithm is better than other hand-eye calibration algorithms in terms of calibration accuracy, robustness to noise, and stability, satisfying the accuracy requirements of switching operations.

A Review on the Applications of Dual Quaternions

This work explores dual quaternions and their applications. First, a theoretical construction begins at dual numbers, extends to dual vectors, and culminates in dual quaternions. The physical foundations behind the developed theory lie in two important fundamentals: Chasles’ Theorem and the Transference Principle. The former addresses how to represent rigid-body motion whereas the latter provides a method for operating on it. This combination presents dual quaternions as a framework for modeling rigid mechanical systems, both kinematically and kinetically, in a compact, elegant and performant way. Next, a review on the applications of dual quaternions is carried out, providing a general overview of all applications. Important subjects are further detailed, these being the kinematics and dynamics of rigid bodies and mechanisms (both serial and parallel), control and motion interpolation. Discussions regarding dual quaternions and their applications are undertaken, highlighting open questions and research gaps. The advantages and disadvantages of using dual quaternions are summarized. Lastly, conclusions and future directions of research are presented.

Kinematics characterizing with dual quaternion and parametric modeling of geometric error terms based on measuring path planning of CNC machine tools

Kinematics characterizing with dual quaternion and parametric modeling of geometric error terms based on measuring path planning of CNC machine tools

A Power Method for Computing the Dominant Eigenvalue of a Dual Quaternion Hermitian Matrix

In this paper, we first study the projections onto the set of unit dual quaternions, and the set of dual quaternion vectors with unit norms. Then we propose a power method for computing the dominant eigenvalue of a dual quaternion Hermitian matrix. For a strict dominant eigenvalue, we show the sequence generated by the power method converges to the dominant eigenvalue and its corresponding eigenvector linearly. For a general dominant eigenvalue, we establish linear convergence of the standard part of the dominant eigenvalue. Based upon these, we reformulate the simultaneous localization and mapping problem as a rank-one dual quaternion completion problem. A two-block coordinate descent method is proposed to solve this problem. One block has a closed-form solution and the other block is the best rank-one approximation problem of a dual quaternion Hermitian matrix, which can be computed by the power method. Numerical experiments are presented to show the efficiency of our proposed power method.

Jacobi method for dual quaternion Hermitian eigenvalue problems and applications

Jacobi method for dual quaternion Hermitian eigenvalue problems and applications