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Electromagnetic formation flight technology utilizes electromagnetic force and torque between satellites to control satellites' relative position and attitude without consuming propellant, thus having broad application prospects. However, the dynamics of electromagnetic formation flying are nonlinear and strongly coupled, posing challenges to the 6-DOF control of electromagnetic formation and the magnetic dipole allocation of satellites. The frequency division multiplexing method can approximate decoupling formation dynamics by loading multi-frequency AC carriers onto satellite magnetic dipoles and is likely to achieve more control functions. A design method for AC carriers based on frequency division multiplexing is proposed to decouple formation dynamics while controlling electromagnetic force and torque. This paper further provides an analytical solution for the magnetic dipoles of the electromagnetic force/torque equations. In addition, in response to external disturbances, model uncertainty, and the inability to obtain relative velocity and angular velocity information of the formation system, this paper establishes a 6-DOF coupling model based on dual quaternion and implements tracking control using the active disturbance rejection control method. Based on the internal force characteristics of the electromagnetic formation, an expression for the equilibrium distribution of reaction wheel torque is derived to avoid some reaction wheels' rapid accumulation of angular momentum. Numerical simulation results show the effectiveness of the proposed control method. [ABSTRACT FROM AUTHOR]

Dual quaternions have wide applications in automatic differentiation, computer graphics, mechanics, and others. Due to its application in control theory, matrix equation A X B = C has been extensively studied. However, there is currently limited information on matrix equation A X B = C regarding the dual quaternion algebra. In this paper, we provide the necessary and sufficient conditions for the solvability of dual quaternion matrix equation A X B = C , and present the expression for the general solution when it is solvable. As an application, we derive the ϕ -Hermitian solutions for dual quaternion matrix equation A X A ϕ = C , where the ϕ -Hermitian extends the concepts of Hermiticity and η -Hermiticity. Lastly, we present a numerical example to verify the main research results of this paper. [ABSTRACT FROM AUTHOR]

This paper proposes a trajectory-optimization problem for spacecraft close proximity to a noncooperative target, aiming at the generation of a six-degree-of-freedom (DOF) trajectory with the fuel-optimal objective value and considering multiple constraints on the control magnitude, line-of-sight, and glide-slope. The line-of-sight and glide-slope constraints are coupled between translational and rotational motions. The dual quaternion is an effective method for establishing the translationally and rotationally coupled model, because it can represent the translation and rotation in an integrated manner. Therefore, in this study, the trajectory-optimization problem of spacecraft close proximity coupled with position and attitude is established using dual quaternions. Next, the close-proximity trajectory-optimization problem is converted into a nonlinear programming problem, which can be solved efficiently using well-developed algorithms such as convex optimization. However, the zero-order hold used in the discrete method of convex optimization is an equidistant dispersion, which cannot guarantee the satisfaction of constraints between discrete points. Therefore the pseudospectral convex method is proposed using nonequidistant collocation points to mitigate the problem of constraint violation between discrete points and improve the accuracy and computational efficiency of the algorithm. The proposed algorithm can be applied to tasks such as rendezvous and docking with noncooperative targets and close proximity. Finally, the effectiveness of the proposed method was validated via numerical simulation, and the results were compared with those of the existing approach, GPOPS. The results indicate that the proposed algorithm is superior to GPOPS in computational efficiency and objective values. [ABSTRACT FROM AUTHOR]

Abstract Nowadays a unit quaternion is widely employed to represent the three-dimensional (3D) rotation matrix and then applied to the 3D similarity coordinate transformation. A unit dual quaternion can describe not only the 3D rotation matrix but also the translation vector meanwhile. Thus it is of great potentiality to the 3D coordinate transformation. The paper constructs the 3D similarity coordinate transformation model based on the unit dual quaternion in the sense of errors-in-variables (EIV). By means of linearization by Taylor's formula, Lagrangian extremum principle with constraints, and iterative numerical technique, the Dual Quaternion Algorithm (DQA) of 3D coordinate transformation in weighted total least squares (WTLS) is proposed. The algorithm is capable to not only compute the transformation parameters but also estimate the full precision information of computed parameters. Two numerical experiments involving an actual geodetic datum transformation case and a simulated case from surface fitting are demonstrated. The results indicate that DQA is not sensitive to the initial values of parameters, and obtains the consistent values of transformation parameters with the quaternion algorithm (QA), regardless of the size of the rotation angles and no matter whether the relative errors of coordinates (pseudo-observations) are small or large. Moreover, the DQA is advantageous to the QA. The key advantage is the improvement of estimated precisions of transformation parameters, i.e. the average decrease percent of standard deviations is 18.28%, and biggest decrease percent is 99.36% for the scaled quaternion and translations in the geodetic datum transformation case. Another advantage is the DQA implements the computation and precision estimation of traditional seven transformation parameters (which still are frequent used yet) from dual quaternion, and even could perform the computation and precision estimation of the scaled quaternion. Graphical Abstract

To solve point cloud registration, a novel solution to planar feature-based registration of point clouds was proposed in this study according to dual quaternion description based on the constraint of planar feature. The normal vector of the homologous feature plane should be kept parallel according to the registration, and the points on the plane should satisfy the planar equation. Moreover, the modified Levenberg-Marquardt method was adopted to complete the registration model, so as to avoid inappropriate initial values from causing non-convergence of the iteration. Lastly, the robustness and accuracy exhibited by the method were verified using simulated and measured data. [ABSTRACT FROM AUTHOR]

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In this study, the screw motions are studied using dual quaternions with the help of di erent perspectives. Firstly, orthogonality definition of dual quaternions is given and geometric interpretation of orthogonality condition is made. Then, the definition of dual circle is given using orthogonal dual quaternions and it is proved that this dual circle can represent the set of all screw motions. Also, these given theorems are reinforced with some conclusions. In addition, it is seen that a dual quaternion represents a screw motions as a screw operator therefore, other dual quaternions derived from the same dual quaternion represent the same screw motions. Then, it is seen that a screw motions symbolized by a dual quaternion transforms one dual vector to another, and when the sign of the dual vectors changes, it provides the same screw motions. Consequently, the answer of the question “Which dual circles symbolizing screw motions are dual orthonormal to each other?” is given and an important conclusion is obtained regarding this.

The rigid body displacement mathematical model is a Lie group of the special Euclidean group SE (3). This article is about the Lie algebra se (3) group. The standard exponential map from se (3) onto SE (3) is a natural parameterization of these displacements. In technical applications, a crucial problem is the vector minimal parameterization of manifold SE (3). This paper presents a unitary variant of a general class of such vector parameterizations. In recent years, dual algebra has become a comprehensive framework for analyzing and computing the characteristics of rigid-body movements and displacements. Based on higher-order fractional Cayley transforms for dual quaternions, higher-order Rodrigues dual vectors and multiple vectorial parameters (extended by rotational cases) were computed. For the rigid body movement description, a dual tangent operator (for any vectorial minimal parameterization) was computed. This paper presents a unitary method for the initial value problem of the dual kinematic equation. [ABSTRACT FROM AUTHOR]

Objectives: The high-precision registration of point cloud data is the key to ensure the integrity of 3D data on the surface of spatial objects. To address the problem that there are differences in position, attitude and scale of cloud data from neighboring stations, a method is proposed to solve the registration model of point cloud described by the dual quaternion under the constraints of point-planar feature. Methods: First, the rotation matrix and translation vector of the spatially similar transformation are represented by the dual quaternion, based on which the scale factor is taken into account and the vertical and parallel spatial topological relationships exist between the vectors constructed by the points in the plane and the points out of the plane respectively and the normal vectors of the plane, and this is used as the constraint of the spatially similar transformation to construct the parity model based on the least squares criterion. Then the Levenberg-Marquardt method is introduced to solve the level-difference model to avoid the possible non-convergence of the iterations in the level-difference treatment due to the inappropriateness of the initial values or due to the fact that the real symmetric matrix constructed by the Jacobi matrix is close to singularity. Result: Two sets of experiments are compared and analyzed with the existing methods, and the experimental results show that the proposed method can effectively achieve point cloud registration. Conclusions: Therefore, the method that takes into account the scale factor under the point-planar feature constraint and uses the dual quaternion to realize the spatial similarity transformation has a strong practical value. [ABSTRACT FROM AUTHOR]

Dual quaternions have wide applications in automatic differentiation, computer graphics, mechanics, and others. Due to its application in control theory, matrix equation AXB=C has been extensively studied. However, there is currently limited information on matrix equation AXB=C regarding the dual quaternion algebra. In this paper, we provide the necessary and sufficient conditions for the solvability of dual quaternion matrix equation AXB=C, and present the expression for the general solution when it is solvable. As an application, we derive the ϕ-Hermitian solutions for dual quaternion matrix equation AXAϕ=C, where the ϕ-Hermitian extends the concepts of Hermiticity and η-Hermiticity. Lastly, we present a numerical example to verify the main research results of this paper.

Recently, with the development of temporal knowledge graph technology, more and more Temporal Knowledge Graph Embedded (TKGE) models have been developed. The effectiveness of TKGE largely depends on the ability to model intrinsic relation patterns and capture specific information about entities and relations. However, existing approaches can capture only some of them with insufficient modeling capacity, and none has a “deep” architecture for modeling the entries in a quadruple at the same dimension. In this article, we propose a more powerful KGE framework named DuCape, which combines a dual quaternion and capsule network in modeling for the first time to make up for the defects of existing TKGE models. In dual quaternion vector space, the head entity learns a k-dimensional rigid transformation parametrized by relation and time, falling near its corresponding tail entity. Further, we employ the embeddings of entities, relations, and time trained from dual quaternion vector space as the input to capsule networks. Experimental results on several basic datasets show that the DuCape model constructed in this article is superior to existing state-of-the-art models. [ABSTRACT FROM AUTHOR]

• The finite-time velocity-free problem of 6-DOF spacecraft control is considered. • A novel finite-time velocities observer is presented in the dual-quaternion form. • A new observer-based non-singular terminal sliding mode control law is proposed. • The finite-time stability of observer-alone and the system is strictly proved. • The control law can be applied to various tasks by designing the reference frame. This paper addresses the problem of spacecraft's finite-time six-degree-of-freedom(6-DOF) tracking control without velocity measurements. To estimate the unavailable incorporating angular and translational velocities (called the dual angular velocity), a novel observer with a simple structure and finite-time stability is presented in the dual quaternion frame. Then, combining with the estimated dual angular velocity, a continuous velocity-free 6-DOF control law based on the terminal sliding mode theory is presented to achieve the relative pose tracking objective within finite time, where the specially designed parameter ensures its non-singularity. In addition, by designing the desired reference frame, the proposed control law can be employed to various missions. With consideration of the observation errors, a rigorous proof is provided to demonstrate the finite-time stability of the observer-alone and the observer-based control law. Two typical numerical simulations, hovering and circumnavigation, are carried out to verify the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

This paper focuses on the development of three newly formed representations of dual attitude and their respective kinematics equations - Dual Classical Rodrigues Parameters, Dual Modified Rodrigues Parameters, Dual Principal Axis and Principal Angle (Dual Principal Rotation Vector). These three formulations are developed by building off of the theories and concepts used with Dual Quaternions and utilizing numerous attitude identities for quaternions to create new identities using dual quaternions and dual attitudes while detailing the process used throughout the derivation. The accompanying dual kinematic equations using these dual attitude parameters are also generated using their respective dual attitude coordinate systems. Furthermore, the new dual attitude representations with their respective, distinct kinematic equations and identities are validated against a simulated 6-DOF rigid body spacecraft control problem to verify equivalency across equations using the dual quaternion as a baseline.

We provide a complete classification of paradoxical closed-loop n-linkages, where n ≥ 6 , of mobility n - 4 or higher, containing revolute, prismatic or helical joints. We also explicitly write down strong necessary conditions for nR-linkages of mobility n - 5 . Our main new tool is a geometric relation between a linkage L and another linkage L ′ resulting from adding equations to the configuration space of L. We then lift known classification results for L ′ to L using this relation. [ABSTRACT FROM AUTHOR]

In order to meet the position and attitude requirements of spacecrafts and test masses for gravitational-wave detection missions, the attitude-orbit coordination control of multiple spacecrafts and test masses is studied. A distributed coordination control law for spacecraft formation based on dual quaternion is proposed. By describing the relationship between spacecrafts and test masses in the desired states, the coordination control problem is converted into a consistent-tracking control problem in which each spacecraft or test mass tracks its desired states. An accurate attitude-orbit relative dynamics model of the spacecraft and the test masses is proposed based on dual quaternions. A cooperative feedback control law based on a consistency algorithm is designed to achieve the consistent attitude tracking of multiple rigid bodies (spacecraft and test mass) and maintain the specific formation configuration. Moreover, the communication delays of the system are taken into account. The distributed coordination control law ensures almost global asymptotic convergence of the relative position and attitude error in the presence of communication delays. The simulation results demonstrate the effectiveness of the proposed control method, which meets the formation-configuration requirements for gravitational-wave detection missions. [ABSTRACT FROM AUTHOR]

Abstract Considering that a unit dual quaternion can describe elegantly the rigid transformation including rotation and translation, the point-wise weighted 3D coordinate transformation using a unit dual quaternion is formulated. The constructed transformation model by a unit dual quaternion does not need differential process to eliminate the three translation parameters, while traditional models do. Based on the Lagrangian extremum law, the analytical dual quaternion algorithm (ADQA) of the point-wise weighted 3D coordinate transformation is proved existed and derived in detail. Four numerical cases, including geodetic datum transformation, the registration of LIDAR point clouds, and two simulated cases, are studied. This study shows that ADQA is valid as well as the modified procrustes algorithm (MPA) and the orthonormal matrix algorithm (OMA). ADQA is suitable for the 3D coordinate transformation with point-wise weight and no matter rotation angles are small or big. In addition, the results also indicate that if the distribution of common points degrades from 3D or 2D space to 1D space, the solvable correct transformation parameters decrease. In other words, all common points should not be located on a line. From the perspective of improving the transformation accuracy, high accurate control points (with small errors in the coordinates) should be chosen, and it is preferred to decrease the rotation angles as much as possible. Graphical Abstract

The rigid body displacement mathematical model is a Lie group of the special Euclidean group SE (3). This article is about the Lie algebra se (3) group. The standard exponential map from se (3) onto SE (3) is a natural parameterization of these displacements. In technical applications, a crucial problem is the vector minimal parameterization of manifold SE (3). This paper presents a unitary variant of a general class of such vector parameterizations. In recent years, dual algebra has become a comprehensive framework for analyzing and computing the characteristics of rigid-body movements and displacements. Based on higher-order fractional Cayley transforms for dual quaternions, higher-order Rodrigues dual vectors and multiple vectorial parameters (extended by rotational cases) were computed. For the rigid body movement description, a dual tangent operator (for any vectorial minimal parameterization) was computed. This paper presents a unitary method for the initial value problem of the dual kinematic equation.

Driven by the need for impact resistance and vibration reduction for mechanical devices in extreme environments, an all-metal vibration isolator with 6-degree-of-freedom (6-DOF) motion that is horizontally symmetrical was developed. Its stiffness and damping ability are provided by oblique springs in symmetrical arrangement and a metal–rubber elasto-porous damper. The spring is symmetrically distributed in the center axis of the support load surface. It is necessary to investigate the kinematics and the singularity before conducting multi-body dynamics analysis of the vibration isolator. Based on the theory of dual quaternions, the forward kinematics equations of the isolator were successively derived for theoretical kinematics modeling. In addition, an enhanced Broyden numerical iterative algorithm was developed and applied to the numerical solution of the forward kinematics equations of the vibration isolator. Compared with the traditional rotation-matrix method and Newton–Raphson method, the computational efficiency of the enhanced Broyden numerical iterative algorithm was increased by 680% and 290%, respectively. This was due to the enhanced algorithm without the calculations of any inverse matrix and forward kinematics equations. Finally, according to the forward kinematics Jacobian matrix, the position-singularity trajectory at a given orientation and the orientation-singularity space at a given position are calculated, which provides a basis for the algorithm of the 6-DOF vibration isolator to avoid singular positions and orientations. [ABSTRACT FROM AUTHOR]

This paper presents a modified model for hand–eye calibration based on dual quaternion algebra. By using dual quaternions to represent the rotations and translations of a rigid body simultaneously in the task space, the formulation is elegant for the analysis of the hand–eye equation. The hand–eye transformation derived in this study is represented in a compact manner, which uses a combination of the dual part and the real part of the dual quaternion. Although the hand–eye equation can be solved by using six elements of a dual quaternion without using its scalar parts, the scaler numbers in both the real and dual parts of a dual quaternion contain part of the pose information. The originality is based on the derivation of the construct of the identification algorithm of external parameters of the camera by using all eight elements of a dual quaternion. Then, the data transformation between the cameras of the dual-arm hand–eye robot system is presented. The corresponding results demonstrate that the proposed hand–eye calibration algorithm can process measurement data with noise and can also improve the identification accuracy to verify its efficiency. [ABSTRACT FROM AUTHOR]

On-orbit service spacecraft with redundant actuators need to overcome orbital and attitude coupling when performing proximity maneuvers. In addition, transient/steady-state performance is required to fulfill the user-defined requirements. To these ends, this paper introduces a fixed-time tracking regulation and actuation allocation scheme for redundantly actuated spacecraft. The coupling effect of translational and rotational motions is described by dual quaternion. Based on this, we propose a non-singular fast terminal sliding mode controller to guarantee fixed-time tracking performance in the presence of external disturbances and system uncertainties, where the settling time is only dependent on user-defined control parameters rather than initial values. The unwinding problem caused by the redundancy of dual quaternion is handled by a novel attitude error function. Moreover, optimal quadratic programming is incorporated into null space pseudo-inverse control allocation that ensures the actuation smoothness and never violates the maximum output capability of each actuator. Numerical simulations on a spacecraft platform with symmetric thruster configuration demonstrate the validity of the proposed approach.

Considering that a unit dual quaternion can describe elegantly the rigid transformation including rotation and translation, the point-wise weighted 3D coordinate transformation using a unit dual quaternion is formulated. The constructed transformation model by a unit dual quaternion does not need differential process to eliminate the three translation parameters, while traditional models do. Based on the Lagrangian extremum law, the analytical dual quaternion algorithm (ADQA) of the point-wise weighted 3D coordinate transformation is proved existed and derived in detail. Four numerical cases, including geodetic datum transformation, the registration of LIDAR point clouds, and two simulated cases, are studied. This study shows that ADQA is valid as well as the modified procrustes algorithm (MPA) and the orthonormal matrix algorithm (OMA). ADQA is suitable for the 3D coordinate transformation with point-wise weight and no matter rotation angles are small or big. In addition, the results also indicate that if the distribution of common points degrades from 3D or 2D space to 1D space, the solvable correct transformation parameters decrease. In other words, all common points should not be located on a line. From the perspective of improving the transformation accuracy, high accurate control points (with small errors in the coordinates) should be chosen, and it is preferred to decrease the rotation angles as much as possible. [ABSTRACT FROM AUTHOR]

Control of a perturbed satellite formation antenna array, with dual quaternion dynamics, is considered. The perturbations and their derivatives are assumed bounded with unknown bounds. The objective of this article is to employ continuous control algorithms which feature adaptive gain nonoverestimation. In this work, three novel dual quaternion-based adaptive continuous sliding mode control algorithms to control a perturbed satellite antenna array are assessed. These methods of adaptive sliding mode control are caspable of handling coupled perturbed systems, such as dual quaternion models. The efficacy of the proposed algorithms is verified via simulations and comparison to the available traditional PD Plus controller and classical sliding mode control methods. [ABSTRACT FROM AUTHOR]

Knowledge graph aims to describe the concepts, the entities and the complex relations of them in the real world. Recently, a series of quaternion rotation models that usually considering the relation vector as the rotation between head and tail entities, have been extensively studied. The results showed these models had the advantages of simplicity and efficiency. However, they are quite weak in capturing the representation and the feature interaction between entities and relations, resulting in insufficient expressiveness of the underlying models, because these models only focus on the linear relations between entities and relations. In order to solve this problem, this paper proposes a novel knowledge graph embedding model called DualDE, which dynamically maps the dual quaternions to the knowledge graph. Specifically, DualDE uses a dynamic mapping mechanism to construct the entity transition vector and the relation transition vector, and continuously adjusts the embedding position of the entity vector in the dual quaternion space according to the dual quaternion multiplication rules. In addition, DualDE can dynamically construct a variety of complex relation types, such as one-to-many, many-to-one and many-to-many. The experimental results based on three standard data sets show that the DualDE model is superior to the existing knowledge graph embedding models on many metrics. [ABSTRACT FROM AUTHOR]

Robotic-assisted surgical system has introduced a powerful platform through dexterous instrument and hand-eye coordination intuitive control. The knowledge of laparoscopic vision is a crucial piece of information for robot-assisted minimally invasive surgery focusing on improved surgical outcomes. Obtaining the transformation with respect to the laparoscope and robot slave arm frames using hand-eye calibration is essential, which is a key component for developing intuitive control algorithm. We proposed a novel two-step modified dual quaternion for hand-eye calibration in this study. The dual quaternion was exploited to solve the hand-eye calibration simultaneously and powered by an iteratively separate solution. The obtained hand-eye calibration result was applied to the intuitive control by using the hand-eye coordination criterion. Promising simulations and experimental studies were conducted to evaluate the proposed method on our surgical robot system. We extensively compared the proposed method with state-of-the-art methods. Results demonstrate this method can improve the calibration accuracy. The effectiveness of the intuitive control algorithm was quantitatively evaluated, and an improved hand-eye calibration method was developed. The relationship between laparoscope and robot kinematics can be established for intuitive control. [ABSTRACT FROM AUTHOR]

The intersection of surfaces is a fundamental process in computational geometry and computer-aided design applications to build and interrogate complex shapes in the computer. This paper presents a novel and simple dual quaternion-based osculating circle DQOC algorithm to find the intersection curve of two regular surfaces based on the osculating circle concept and dual quaternions. Additionally, we expressed the natural equations of the intersection curve. We have also demonstrated the superiority of our method through numerical examples.

This article aims to investigate the coupled attitude and orbit tracking control problem of rigid spacecraft motion. A proportional-integral (PI)-type event-triggered active disturbance rejection control scheme is proposed for spacecraft coupled attitude and orbit control under the unknown internal uncertainties and external disturbances. First, the relative kinematics and dynamics are established using error dual quaternions to describe the attitude and orbit coupled motion of rigid spacecraft. Then, the PI-type event-triggering schemes are designed to reduce the communication rates between the pose sensors and the observers. Finally, the asymptotic stability of the closed-loop control system is proved. The simulation results show that the proposed closed-loop control system can achieve satisfactory tracking performance and reduce the occupancy of on-board communication resources to ensure the stable operation of the spacecraft. [ABSTRACT FROM AUTHOR]

3D coordinate transformation is frequently encountered in geodesy applications. In addition, it is frequently encountered in computer vision, robotics, engineering surveying, and GiS applications. Seven parameter Helmert transformation is one of the widely used 3D transformation algorithms. Nowadays, the quaternion algorithms are finding increasingly common usage due to some disadvantages of Helmert transformation based on Euler angles; in this article, a new 3D Cartesian coordinate transformation with the dual quaternion method is explained in detail, and its advantages over the classical transformation problem algorithm are emphasized. Quaternions are not preferred in 3D transformation applications due to the complexity of the subject. As a result, readers generally stay away from the subject of quaternion. In this article, a new dual quaternion algorithm with eight parameters (DQA) is presented. In addition, a MATLAB function DQA_3d_transformation will be introduced in the article so that readers can easily perform 3D transformations with dual quaternions. We hope that after reading this article, the reader’s perspective on dual quaternions may change slightly. [ABSTRACT FROM AUTHOR]

For patients with upper limb dysfunction after stroke, robot-assisted rehabilitation training plays an important role in functional recovery. The existing upper limb rehabilitation robots have some problems, such as complex mechanisms, insufficient compliance, and can only realize the rehabilitation training of shoulder and elbow joints in the horizontal plane. This research proposes a novel end-effector upper limb rehabilitation robot with three degrees of freedom. Two horizontal rotation freedoms are driven by motors and one vertical translation freedom is driven by a pneumatic cylinder. So it can realize the spatial rehabilitation training of shoulder and elbow joints. The rotation and translation transformation of the robot can be represented by a dual quaternion, which is concise in form and clear in the physical meaning. Therefore, this article adopts dual quaternions to complete the robot's kinematics modeling, inverse kinematics calculation, and terminal spiral motion trajectory planning. To improve the low-speed moving performance of the spiral motion, a sliding mode control strategy plus feedforward compensation is employed to control the displacement of the cylinder. Experiments show that the robot can realize proximal joints training and has good position tracking accuracy (tracking error is within 2 mm) with smoothness under the proposed control strategy, which can guarantee the accuracy and comfort of passive rehabilitation training, contributing to restoring the function of the impairment upper limbs. [ABSTRACT FROM AUTHOR]

This paper addresses the synthesis of one degree-of-freedom (1DOF) linkages that can exactly transmit angular motion between coplanar axes (i.e., parallel axes or intersectant axes) with arbitrarily prescribed constant velocity ratios. According to motion polynomials over dual quaternions and pure rolling models between two circles, an algebraic approach is presented to precisely synthesize new 1DOF linkages with arbitrarily prescribed constant velocity ratios. The approach includes four steps: (a) formulate a characteristic curve occurred by the pure rolling, (b) compute the motion polynomial of the minimal degree that can generate the curve, (c) deal with the factorization of the motion polynomial to construct an open chain, and (d) convert the open chain to a 1DOF linkage. Using this approach, several 1DOF planar, spherical, and spatial linkages for angular motion transmission between parallel axes or intersectant ones are constructed by designating various velocity ratios. Taking the planar and spherical linkages with a constant 1:2 velocity ratio as examples, kinematics analysis is implemented to prove their motion characteristics. The result shows that the generated linkages indeed can transmit angular motion between two coplanar axes with constant velocity ratios. Meanwhile, three-dimensional (3D)-printed prototypes of these linkages also demonstrate such a conclusion. This work provides a framework for synthesizing linkages that have great application potential to transmit motion in robotic systems that require low inertia to achieve reciprocating motion with high speed and accuracy. [ABSTRACT FROM AUTHOR]

In order to solve the problem of noise interference in camera calibration and robot forward kinematics solution, this study proposes a dual quaternion hand-eye calibration algorithm based on twice opposition-learning and random differential variation (ODHPO). The hand-eye calibration equation was innovatively rewritten in the form of dual quaternion, and the F-norm minimization model of the rotation and translation error function was constructed for optimization. At the same time, the penalty function method is introduced to effectively transform the constrained optimization problem in hand-eye calibration into an unconstrained problem, which is further solved by the ODHPO algorithm for global optimization. Compared with the singular value decomposition algorithm based on the traditional dual quaternion method, the ODHPO algorithm performs better in global optimization ability and convergence stability. Through numerical simulation and real robot hand-eye calibration experiments, it is proved that the proposed algorithm is superior to the traditional dual quaternion (CDQ) algorithm, the classical algorithms Tsai method and Navy method in terms of solution accuracy, sensitivity to the number of pose transformations and stability, demonstrating its potential for application and practical significance in robot vision system. ● An improved hunter-prey optimization algorithm with twice opposition-learning and random difference variation is proposed. ● A dual quaternion hand-eye calibration algorithm based on improved hunter-prey optimization is proposed. ● The effectiveness of the improved algorithm is verified by 12 benchmark test functions ● The proposed calibration algorithm is superior to other traditional calibration algorithms [ABSTRACT FROM AUTHOR]

The dual quaternion plays an advantageous role in the field of robotic kinematics and dynamics due to its compact expression. It is well known that quaternion can be easily implemented as algebraic operations on vectors through matrix operators. Similarly, the dual matrix operators can convert dual quaternion operations to matrix operations. However, it has not received adequate attention, and thus there are very few applications at present. With the dual matrix operators of dual quaternion, this paper re-verifies the equivalency between the conjugate formula of unit dual quaternion and dual Euler–Rodrigues formula. To combine this equivalence with the homomorphic mapping of Lie groups, a theoretical correlation of the current hand-eye calibration methods is established. Motivated by the attempt to establish a more complete scheme for hand-eye calibration, a new simultaneous method based on the conjugate formula of dual quaternion is proposed. The method verifies that the rotation problem can be extended to the rigid-body transformation problem via the dual algebra. Finally, the simulation and experimental validations for the hand–eye calibration method are given. • The conjugate formula of unit dual quaternion is re-verified via the dual operations. • A new simultaneous method is defined for hand-eye calibration. • The theoretical correlation with the existing methods is established. [ABSTRACT FROM AUTHOR]

In order to meet the position and attitude requirements of spacecrafts and test masses for gravitational-wave detection missions, the attitude-orbit coordination control of multiple spacecrafts and test masses is studied. A distributed coordination control law for spacecraft formation based on dual quaternion is proposed. By describing the relationship between spacecrafts and test masses in the desired states, the coordination control problem is converted into a consistent-tracking control problem in which each spacecraft or test mass tracks its desired states. An accurate attitude-orbit relative dynamics model of the spacecraft and the test masses is proposed based on dual quaternions. A cooperative feedback control law based on a consistency algorithm is designed to achieve the consistent attitude tracking of multiple rigid bodies (spacecraft and test mass) and maintain the specific formation configuration. Moreover, the communication delays of the system are taken into account. The distributed coordination control law ensures almost global asymptotic convergence of the relative position and attitude error in the presence of communication delays. The simulation results demonstrate the effectiveness of the proposed control method, which meets the formation-configuration requirements for gravitational-wave detection missions.

In this paper, the event-triggered integrated attitude and orbit control problem is studied for the distributed spacecraft formation system. First, the operation rules of dual numbers and dual quaternion are presented, and the attitude-orbit coupling error dynamics models for single spacecraft and spacecraft formation are established based on the dual quaternion. An event-triggered attitude-orbit coupled sliding mode tracking control law is proposed to stabilize the attitude-orbit control system, which can reduce the communication burden among spacecraft at the same time. Under the developed control scheme, the state trajectory of attitude-orbit control systems can arrive on the designed sliding surface in finite time, and the asymptotic stability of the overall formation control system can be ensured. Considering the fact that the inertia and mass parameters in practice may not be measured exactly, an adaptive sliding mode control law is further proposed. Finally, a simulation example is given to verify the validity and feasibility of the proposed spacecraft event-triggered control method. [ABSTRACT FROM AUTHOR]

This article is concerned with the development of two novel Kalman filters for satellite relative pose estimation. The relative pose, which is represented by a dual quaternion, is estimated from noisy lines-of-sight via a single camera, along with biases of linear velocity and angular rate measurements. The constraints on the dual quaternion are handled by brute-force and virtual measurement techniques. The partial reset analysis shows how to propagate the estimation error biases and covariances. Closed-form expressions for the Jacobians are provided. A quantitative observability analysis for the position states is provided. Compared with traditional representations, the dual quaternion model increases the position observability by a factor of four for each line-of-sight. The additive dual quaternion model enhances the observability of the rotation states thanks to a specific coupling term in the pose dynamics. Extensive Monte-Carlo simulations of a flight formation case verify that the proposed novel filters are asymptotically unbiased and statistically consistent for all practical purposes. They outperform the other candidate estimators in particular during the transient phase. The numerical simulations validate the observability analysis. [ABSTRACT FROM AUTHOR]

Point cloud registration refers to a critical step in point cloud preprocessing, which aims to uniformly represent the objects expressed by the two point sets. To solve spatial discrepancies in the collected artificial building point clouds, this study develops a point cloud registration method by employing dual quaternion description based on the point-linear feature constraint. First, the spatial transformation parameters are expressed by the dual quaternion, and the rotation matrix and translation vector are expressed simultaneously to avoid the error of separate calculations from accumulating. Subsequently, the registration model is built by complying with the constraints of coordinate equivalence after the registration of the same-name points, parallelism of direction vectors after registration of the same-name lines and the spatial geometric relationship between points and lines. On that basis, the scale factor is considered to register point clouds at different scales. Second, an optimized Levenberg–Marquardt method is adopted to solve the registration model for avoiding the iterative non-convergence attributed to inappropriate initial values in the solution. Lastly, the robustness and reliability exhibited by the proposed method are verified by performing two experiments with the simulated and measured data. As indicated from the experimentally achieved results, the joint constraint of point-linear features can achieve higher accuracy than the constraint of point or line features independently, and the combined point-linear features can register point clouds in high quality when point cloud data are scaled and partially missing. This study presents an effective registration method for manually registered auxiliary targets when they are difficult to deploy. [ABSTRACT FROM AUTHOR]

In this paper, Galilean orthogonal matrices in and are obtained with the help of unit quaternions. Moreover, Galilean orthogonal matrices in and are acquired. These matrices produce Galilelan motions in Galilean spaces. We investigate the invariance of the plane where shear motion is acting in Galilean and pseudo-Galilean spaces. Additionally, related examples of matrices that belong to both spaces are provided. With a similar method, dual Galilean orthogonal matrices are obtained by using unit dual quaternions. Finally, we strengthen our work with examples and draw their figures to explore visual representations. [ABSTRACT FROM AUTHOR]

Driven by the need for impact resistance and vibration reduction for mechanical devices in extreme environments, an all-metal vibration isolator with 6-degree-of-freedom (6-DOF) motion that is horizontally symmetrical was developed. Its stiffness and damping ability are provided by oblique springs in symmetrical arrangement and a metal–rubber elasto-porous damper. The spring is symmetrically distributed in the center axis of the support load surface. It is necessary to investigate the kinematics and the singularity before conducting multi-body dynamics analysis of the vibration isolator. Based on the theory of dual quaternions, the forward kinematics equations of the isolator were successively derived for theoretical kinematics modeling. In addition, an enhanced Broyden numerical iterative algorithm was developed and applied to the numerical solution of the forward kinematics equations of the vibration isolator. Compared with the traditional rotation-matrix method and Newton–Raphson method, the computational efficiency of the enhanced Broyden numerical iterative algorithm was increased by 680% and 290%, respectively. This was due to the enhanced algorithm without the calculations of any inverse matrix and forward kinematics equations. Finally, according to the forward kinematics Jacobian matrix, the position-singularity trajectory at a given orientation and the orientation-singularity space at a given position are calculated, which provides a basis for the algorithm of the 6-DOF vibration isolator to avoid singular positions and orientations.

This paper presents a modified model for hand–eye calibration based on dual quaternion algebra. By using dual quaternions to represent the rotations and translations of a rigid body simultaneously in the task space, the formulation is elegant for the analysis of the hand–eye equation. The hand–eye transformation derived in this study is represented in a compact manner, which uses a combination of the dual part and the real part of the dual quaternion. Although the hand–eye equation can be solved by using six elements of a dual quaternion without using its scalar parts, the scaler numbers in both the real and dual parts of a dual quaternion contain part of the pose information. The originality is based on the derivation of the construct of the identification algorithm of external parameters of the camera by using all eight elements of a dual quaternion. Then, the data transformation between the cameras of the dual-arm hand–eye robot system is presented. The corresponding results demonstrate that the proposed hand–eye calibration algorithm can process measurement data with noise and can also improve the identification accuracy to verify its efficiency.

We propose a practical method for computing the singular value decomposition of dual quaternion matrices. The dual quaternion Householder matrix is first proposed, and by combining the properties of dual quaternions, we can implement the transformation of a dual quaternion matrix to a bidiagonalized dual number matrix. We have proven that the singular values of a dual quaternion matrix are same to the singular values of its bidiagonalized dual number matrix. Numerical experiment demonstrates the effectiveness of our proposed method for computing the singular value decomposition of dual quaternion matrices. [ABSTRACT FROM AUTHOR]

Abstract Topology and performance are the two main topics dealt in the development of robotic mechanisms. However, it is still a challenge to connect them by integrating the modeling and design process of both parts in a unified frame. As the properties associated with topology and performance, finite motion and instantaneous motion of the robot play key roles in the procedure. On the purpose of providing a fundamental preparation for integrated modeling and design, this paper carries out a review on the existing unified mathematic frameworks for motion description and computation, involving matrix Lie group and Lie algebra, dual quaternion and pure dual quaternion, finite screw and instantaneous screw. Besides the application in robotics, the review of the work from these mathematicians concentrates on the description, composition and intersection operations of the finite and instantaneous motions, especially on the exponential-differential maps which connect the two sides. Furthermore, an in-depth discussion is worked out by investigating the algebraical relationship among these methods and their further progress in integrated robotic development. The presented review offers insightful investigation to the motion description and computation, and therefore would help designers to choose appropriate mathematical tool in the integrated design and modeling and design of mechanisms and robots.

• An algebraic synthesis method is proposed to form novel multi-loop linkages for rotation scaling. • Multi-loop planar, spherical, and spatial linkages can be constructed using the proposed method. • The generated linkages can exactly transmit angular motion with prescribed transmission ratios. • Exact angular motion transmission between parallel, intersecting, or skew axes is achieved. • Transmission characteristics of the multi-loop linkages are proved by kinematics analysis. This paper deals with the algebraic synthesis and input-output analysis of one degree-of-freedom (1-DOF) multi-loop planar, spherical, and spatial linkages for increasing or decreasing angular motion with a desired constant transmission ratio between two adjacent parallel, intersecting, and skew axes respectively. Based on the rotation scaling model and motion polynomials over dual quaternions, an algebraic synthesis method including three procedures is proposed to exactly construct novel multi-loop linkages owning arbitrarily prescribed constant transmission ratios and input/output axes. To illustrate this method, several 1-DOF multi-loop planar, spherical, and spatial linkages for rotation scaling are synthesized by designating various input and output axes as well as transmission ratios. Taking some multi-loop planar, spherical, and spatial linkages as examples, input-output analysis is carried out to verify their transmission characteristics. The results demonstrate that the generated 1-DOF multi-loop linkages can indeed transmit motion with prescribed constant transmission ratios and input/output axes. This work provides a framework for further investigation on mechanisms performing specified tasks for motion transmission. [ABSTRACT FROM AUTHOR]

In this study, we are interested in the way quaternions to represent 3D and 4D rotations in Lorentzian space. We give a new method for obtaining a rotation matrix in Lorentzian space with the help of a unit quaternion. Furthermore, we prove that rotation matrices correspond to a quaternion leave invariant the same axis in Euclidean and Lorentzian space. Then, we introduce a semi‐orthogonal matrix representation of a quaternion curve in 4D space. Moreover, we provide applications and draw their figures to explore visual representations. Finally, due to the importance of the dual space in kinematics, robotics, and other areas related, we carry this work into their dual spaces by using a dual quaternion. [ABSTRACT FROM AUTHOR]