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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]

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]

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]

In this study, the screw motions are studied using dual quaternions with the help of different 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. [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]

• 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]

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]

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]

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]

• An algebraic synthesis method is proposed to construct novel single-loop 6R spatial mechanisms. • The mechanisms can achieve constant velocity transmission between designated input/output axes. • All the floating joints of the single-loop 6R spatial mechanisms rotate around fixed axes. • The transmission quality of the mechanisms is optimized within a specified range of motion. • Transmission characteristics of the single-loop mechanisms are verified by kinematics analysis. This paper addresses the algebraic synthesis of novel single-loop 6R spatial mechanisms, enabling constant -1:1 or 1:1 velocity ratio transmission between two adjacent axes, whether parallel, intersecting, or skew. Based on the motion polynomial over dual quaternions, an algebraic synthesis method including four steps is presented to construct and optimize single-loop 6R spatial mechanisms with a constant transmission ratio of -1:1 or 1:1 between arbitrarily designated input and output axes. Using this method, several novel single-loop 6R spatial mechanisms for constant velocity transmission are constructed by designating different poses and rotation directions of the input and output axes. Kinematics analysis of single-loop 6R spatial mechanisms is implemented to verify their transmission characteristics. The results reveal that the generated 1-DOF single-loop 6R novel mechanisms can indeed transmit motion with a constant transmission ratio of -1:1 or 1:1 between two adjacent parallel, intersecting, or skew axes. This work provides a framework for further investigation on single-loop mechanisms with special transmission characteristics. [ABSTRACT FROM AUTHOR]

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]

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]

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]

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]

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]

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]

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 this study, we obtain triplets from quaternions. First, we obtain triplets from real quaternions. Then, as an application of this, we obtain dual triplets from the dual quaternions. Quaternions, in many areas, it allows ease in calculations and geometric representation. Quaternions are four dimensions. The triplets are in three dimensions. When we express quaternions with triplets, our study is conducted even easier. Quaternions are very important in the display of rotational movements. Dual quaternions are important in the expression of screw movements. Reducing movements from four dimensions to three dimensions makes our study easier. This simplicity is achieved by obtaining triplets from quaternions. [ABSTRACT FROM AUTHOR]

It is well known that the automorphism group Aut(H) of the algebra of real quaternions H consists entirely of inner automorphisms iq : p → q ·p·q-1 for invertible q ∈ H and is isomorphic to the group of rotations SO(3). Hence, H has only inner derivations D = ad(x), x ∈ H. See [4] for derivations of various types of quaternions over the reals. Unlike real quaternions, the algebra Hd of dual quaternions has no nontrivial inner derivation. Inspired from almost inner derivations for Lie algebras, which were first introduced in [3] in their study of spectral geometry, we introduce coset invariant derivations for dual quaternion algebra being a derivation that simply keeps every dual quaternion in its coset space. We begin with finding conditions for a linear map on Hd become a derivation and show that the dual quaternion algebra Hd consists of only central derivations. We also show how a coset invariant central derivation of Hd is closely related with its spectrum. [ABSTRACT FROM AUTHOR]

This paper proposes an innovated approach to model an electromagnetic force which is an ideal control force without plume and light pollution for optical imaging system. The main scope is modeling the 3-axis coupled electromagnetic force using dual quaternion for its advantages in describing spiral motion with translation and rotation motion simultaneously. The derivation procedure relies on a far field model of the electromagnetic force, and also, the force is re-formulized. Final equations show the dual quaternion dynamics can express the relative motion caused by the electromagnetic force acting on the position and attitude integrally. A new mission concept of diffraction imaging system in GEO is discussed as application scenario of the new dynamics model. Two cases with different mass were simulated to verify the new model, and the results display the electromagnetic force is significant for a small mass spacecraft in GEO. • The electromagnetic force is expressed by dual quaternion. • A novel dynamics model is derived to describe GEO electromagnetic collocated satellite. • The motion of diffraction imaging system is modelled using dual quaternion. [ABSTRACT FROM AUTHOR]

• 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]

The rigid motion involving both rotation and translation in the 3D space can be simultaneously described by a unit dual quaternion. Considering this excellent property, the paper constructs the Helmert transformation (seven-parameter similarity transformation) model based on a unit dual quaternion and then presents a rigid iterative algorithm of Helmert transformation using a unit dual quaternion. Because of the singularity of the coefficient matrix of the normal equation, the nine parameter (including one scale factor and eight parameters of a dual quaternion) Helmert transformation model is reduced into five parameter (including one scale factor and four parameters of a unit quaternion which can represent the rotation matrix) Helmert transformation one. Besides, a good start estimate of parameter is required for the iterative algorithm, hence another algorithm employed to compute the initial value of parameter is put forward. The numerical experiments involving a case of small rotation angles i.e. geodetic coordinate transformation and a case of big rotation angles i.e. the registration of LIDAR points are studied. The results show the presented algorithms in this paper are correct and valid for the two cases, disregarding the rotation angles are big or small. And the accuracy of computed parameter is comparable to the classic Procrustes algorithm from Grafarend and Awange (J Geod 77:66-76, 2003), the orthonormal matrix algorithm from Zeng (Earth Planets Space 67:105, 2015), and the algorithm from Wang et al. (J Photogramm Remote Sens 94:63-69, 2014). [ABSTRACT FROM AUTHOR]

Highlights • Space manipulators can act as actuators for spacecraft due to the dynamic coupling. • Pose control forces provides equality constraints for trajectory planning problem. • Convex optimization method found the globally optimal solution efficiently. • Balance manipulator compensates for coupling and environmental disturbance. Abstract This paper presents a trajectory planning algorithm for a space robot with dual-manipulators. Here one manipulator of the space robot captures a target, and another manipulator is free. In this case, this study uses one manipulator as the mission manipulator to capture the target, and another as the balance manipulator aiming at the compensation of the pose disturbance. For this method, a novel trajectory planning algorithm applied to the balance manipulator is presented. The trajectory planning problem is transformed into series of problems of the optimal state solution, and then the iterative algorithms for the trajectory planning are designed. In the iterative algorithms, the bias force on the spacecraft base caused by the balance manipulator is used as the compensation force. Then, to calculate the expected compensation force and torque, a pose control law for the spacecraft base is introduced. The expected compensation force and torque provide equality constraints for optimization problems, which implies that the trajectory planning algorithm compensates for not only the disturbance generated by the manipulator's motion, but also environmental disturbances. This is because the expected compensation force and torque depend on the pose change of the spacecraft base rather than the type of the disturbance. Numerical simulation was carried out to analyze the proposed trajectory planning method. It was observed that the method greatly reduces the disturbance of Manipulator A on the spacecraft base. These results validated the effectiveness of the proposed method for the trajectory planning to make the spacecraft base disturbance up to minimum. [ABSTRACT FROM AUTHOR]

In 2016, Yüuce and Torunbalcı Aydın [18] defined dual Fibonacci quaternions. In this paper, we defined the dual third-order Jacobsthal quaternions and dual third-order Jacobsthal-Lucas quaternions. Also, we investigated the relations between the dual third-order Jacobsthal quaternions and third-order Jacobsthal numbers. Furthermore, we gave some their quadratic properties, the summations, the Binet's formulas and Cassini-like identities for these quaternions. [ABSTRACT FROM AUTHOR]

The avoidance of singularities is critical to design and control of parallel robots. This paper aims at efficient determination of the maximal singularity-free joint space and workspace of a class of six-DOF parallel robots with six kinematic chains of same type. We represent the singularity-free joint space by a 6-cube and determine it firstly. The singularity-free workspace is generated by continuous motion of all active joints in the singularity-free joint space. As a result, the boundary of the workspace can be obtained with simultaneous consideration of position and orientation of the mobile platform. The size relation between the maximal singularity-free joint space and workspace is discussed. To efficiently determine the singularity-free joint space and workspace, we propose dual quaternion-based Jacobian matrices and construct an efficient algorithm. The algorithm detects singularities in a given joint space and simultaneously calculates its corresponding workspace. The computational costs of the proposed algorithm and the traditional one are compared using a 6-U P S parallel robot, leading to 9 seconds and 458 seconds respectively. Finally, both the maximal singularity-free joint space and workspace of a 6- P US parallel robot are determined to further demonstrate the effectiveness of the new approach. [ABSTRACT FROM AUTHOR]

Rigid motion computation or estimation is a cornerstone in numerous fields. Attitude computation can be achieved by integrating the angular velocity measured by gyroscopes, the accuracy of which is crucially important for the dead-reckoning inertial navigation. The state-of-the-art attitude algorithms have unexceptionally relied on the simplified differential equation of the rotation vector to obtain the attitude. This paper proposes a functional iteration technique with the Rodrigues vector (named the RodFIter method) to analytically reconstruct the attitude from gyroscope measurements. The RodFIter method is provably exact in reconstructing the incremental attitude as long as the angular velocity is exact. Notably, the Rodrigues vector is analytically obtained and can be used to update the attitude over the considered time interval. The proposed method gives birth to an ultimate attitude algorithm scheme that can be naturally extended to the general rigid motion computation. It is extensively evaluated under the attitude coning motion and compares favorably in accuracy with the mainstream attitude algorithms. This paper is believed having eliminated the long-standing theoretical barrier in exact motion integration from inertial measurements. [ABSTRACT FROM AUTHOR]

We prove that every bounded rational space curve of degree d and circularity c can be drawn by a linkage with 92d-6c+1 revolute joints. Our proof is based on two ingredients. The first one is the factorization theory of motion polynomials. The second one is the construction of a motion polynomial of minimum degree with given orbit. Our proof also gives the explicit construction of the linkage. [ABSTRACT FROM AUTHOR]

A six-axis vibration isolation system is essential to high-precision space systems for attenuating vibrations on precise instruments. The kinematic optimal design is researched for the space six-axis vibration isolator via Stewart mechanism. Jacobian matrix is the basis of the kinematic performance index. However, the conventional Jacobian matrix is not usually dimensionally homogeneous due to the inhomogeneous physical units, caused by the different mathematical representations of the rotation and translation. In this paper, we propose a dual quaternion approach to derive the dimensionally homogeneous Jacobian matrix of a general six-axis parallel mechanism. Two quaternions are used to parameterize the rotation and translation of the platform. The dimensionally scaling factor for the generalized Jacobian matrix is defined as the ratio of the norms of the two quaternions. The dimensionally homogenous Jacobian matrix is then obtained and applied to the optimal design of the six-axis vibration isolator. The performance index of isotropy is considered to make the isolator minimum kinematic coupling in its working configuration. [ABSTRACT FROM AUTHOR]

Purpose The purpose of this paper is to propose an adaptive robust controller for coupled attitude and orbit control of rigid spacecraft based on dual quaternion in the presence of external disturbances and model uncertainties.Design/methodology/approach First, based on dual quaternion, a theoretical model of the relative motion for rigid spacecraft is introduced. Then, an adaptive robust controller which can realize coordinated control of attitude and orbit is designed in the existence of external disturbances and model uncertainties.Findings This paper takes advantage of the Lyapunov function which can guarantee the asymptotic stabilization of the whole system in the existence of parameters uncertainties. Simulation results show that the proposed controller is feasible and effective.Originality/value This paper proposes a coupled attitude and orbit adaptive robust controller based on dual quaternion. Simulation results demonstrate that the proposed controller can achieve higher control performance in the presence of parameters uncertainties. [ABSTRACT FROM AUTHOR]

This paper investigates the collision-free distributed coordination control of six-DOF spacecraft formation flying. Dual quaternions are used to develop the kinematic and dynamic models of relative translational and rotational motion between spacecraft. By assuming that the reference relative orbit and attitude information is only available to a subset of members in formation, an observer is introduced to estimate the reference information in finite time. With the estimates provided by the observer, a sliding mode controller is then designed for the coordination of six-DOF formation flying. Meanwhile, the artificial potential function method is employed to design evasive maneuvers in case of any collisions during orbital maneuvering. The overall stability of the closed-loop system is guaranteed by a Lyapunov-based method. The feasibility of the proposed control scheme is then demonstrated by a typical formation reconfiguration mission in a low Earth orbit environment. [ABSTRACT FROM AUTHOR]

The forward kinematics is the basis of the design and control of the parallel robots. This paper aims to provide an efficient solution to the forward kinematics of a class of six-degrees-of-freedom parallel robots for real-time applications. With a unit dual quaternion used as the generalized coordinates of the robot system, the forward kinematic equations are derived to be a set of quadratic ones. An efficient algorithm is proposed to get the actual solution to them. The convergence and singularity problems of the new algorithm have been discussed. We have provided a convergence strategy and revealed the internal relation of the singularity with that of the parallel robot, proving the feasibility of the algorithm and giving the working condition in the practical applications. The new algorithm have been compared to the Newton's method for an 8-U P S parallel robot, resulting in the time consumptions of 0.2187 milliseconds and 14.25 milliseconds respectively. And then we perform a simulation of the state-feedback control for an 8- P US parallel robot. The two examples present the applications of the new algorithm and demonstrate its validity and efficiency. [ABSTRACT FROM AUTHOR]

This paper presents a motion control approach with a focus on robotic manipulators based on screw theory and dual quaternions. The stability analysis of a general dual quaternion based controller has been capitalized to design an additional bounded twist controller. This controller is proposed to limit the maximum twist of the end-effector within a desired value, while preserving the accuracy achievable with high-gain feedback controllers. The proposed controllers could be useful for robotic tasks where curved motion is preferred over straight line motion. In that regard, the trajectories taken by the proposed controllers were analyzed for pose-to-pose control and some strategies have been provided for the proposed coupled controller to modify the natural trajectory. These behaviors were verified on a real robot as well as in simulation, comparing the experimental results with conventional decoupled controllers. The proposed controllers achieved smooth coupled motions which can be useful for tasks such as pick and place and assembly operations. Moreover, our coupled controllers based on screw theory need less actuator motion than the conventional decoupled approach for certain situations. • Screw motion controllers are introduced, based on dual quaternions and screw theory. • A generalized dual quaternion controller for meeting different behaviors is proposed. • A general stability proof is introduced for designing dual quaternion controllers. • A comparison is provided between coupled and decoupled controllers for curved motion. • Controllers based on screw theory tend to drive manipulators in screw paths. [ABSTRACT FROM AUTHOR]

The authors consider current problems of the modern theory of dynamic systems on manifolds, which are actively developing. A brief review of such trends in the theory of dynamic systems is given. The results of the algebra of dual numbers, quaternionic algebras, biquaternions (dual quaternions), and their application to the analysis of infinitesimal neighborhoods and infinitesimal deformations of manifolds (schemes) are presented. The theory of differential-algebraic equations over the field of real numbers and their dynamics, as well as elements of trajectory optimization of respective dynamic systems, are outlined. On the basis of connection in bundles, the theory of differential-algebraic equations is extended to algebraic manifolds and schemes over arbitrary fields and schemes, respectively. [ABSTRACT FROM AUTHOR]

This paper exploits screw theory expressed via unit dual quaternion representation and its algebra to formulate both the forward (position+velocity) kinematics and pose control of an n -dof robot arm in an efficient way. Efficiency is in less computer memory usage, in fast computation of the equations, in singularity-free representation of task space, in robustness to numerical errors, and in compactness of the representations. The formulation is simple, intuitive and straightforward to implement. We validated this formulation experimentally on a 7 dof robot arm. [ABSTRACT FROM AUTHOR]

We give a constructive proof for the existence of a unique rational motion of minimal degree in the dual quaternion model of Euclidean displacements with a given rational parametric curve as trajectory. The minimal motion degree equals the trajectory's degree minus its circularity. Hence, it is lower than the degree of a trivial curvilinear translation for circular curves. [ABSTRACT FROM AUTHOR]

This paper presents the implementation of the high performance adaptive control algorithm for 6-DOF systems described by dual quaternions. The fast adaptation is achieved from the introduced stable adaptation dynamics in the closed-loop system which cannot be expected in the conventional adaptive controller designed upon the certainty equivalence framework. The development of the proposed adaptive control algorithm relies on the two newly introduced algebraic operations. The one is a dual filter for a dual relative error signal. The dual filter essentially has a linear low-pass filter structure and enables us to recast the system dynamics into a linear parameter-affine system dynamics in dual quaternions combined with dual control inputs. The other is an isomorphism between a dual quaternion and a vector for the error measurement. Using this isomorphism, the stability proof of the closed-loop dual quaternion system based on the proposed Lyapunov-like function is rewritten in the form of familiar matrix/vector algebra. In order to demonstrate the performance improvement, numerical simulation results are given, which compare the proposed adaptive control method with the conventional one from the certainty-equivalence principle. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to introduce the application of dual quaternions in one interesting problem in structural biology, i.e., the description of protein structure. The secondary protein structure is a specific geometric shape and the description uses Chasles theorem which states that any rigid body displacement can be described by a screw motion. We will briefly introduce the theory of dual quaternions in connection with the screw motion. Consequently, it is shown that modeling based on dual quaternions is an elegant mathematical method and a compact formula for the description of secondary protein structure is derived using the dual quaternion calculus. [ABSTRACT FROM AUTHOR]

The inverse kinematics solution of a dexterous robotic finger has a significant impact on the real-time control of the robotic hand. Therefore a rapid method for solving is needed. The classical homogeneous matrix transformation is the most popular method used in robot kinematics. However, for the multi degree-of-freedom (DOF) robotic finger, the matrix parameters cost much storage and the inverse matrix calculation requires a large amount of computational cost. So it is not conducive to the real-time control of the robotic hand. Therefore, a method based on dual quaternions is presented for analysing the kinematics of a multi-DOF (4-DOF) robotic thumb. Firstly, the kinematics equation is expressed by dual quaternions. Then the multivariate kinematic equations are converted to binary quadratic equations with methods of separating variables and variable substitution, which is relatively easy to obtain the closed-form solution of the inverse kinematics. Finally, it proves that the dual quaternions method has advantages over the homogeneous matrix transformation in storage and computational cost by the specific numbers for the robotic thumb, which is conducive to the real-time control of robotic hand. [ABSTRACT FROM AUTHOR]

After a review of some properties of dual quaternions, De Moivre's and Euler's formulas for the matrices associated with these quaternions are studied. In special case, De Moivre's formula implies that there are uncountably many matrices of unit dual quaternions satisfying An = I4 for n≥3. Also; we give the relation between the powers of matrices of dual quaternions. [ABSTRACT FROM AUTHOR]

A multi-robot collaboration system can complete more complex tasks than a single robot system. Ensuring the calibration accuracy between robots in the system is a prerequisite for the effective inter-robot cooperation. This paper presents a dual-robot system for orthopedic surgeries, where the relationships between hand-eye, flange-tool, and robot-robot need to be calibrated. This calibration problem can be summarized to the solution of the matrix equation of AXB=YCZ. A combined solution is proposed to solve the unknown parameters in the equation of AXB=YCZ, which consists of the dual quaternion closed-form method and the iterative method based on Levenberg–Marquardt (LM) algorithm. The closed-form method is used to quickly obtain the initial value for the iterative method so as to increase the convergence speed and calibration accuracy of the iterative method. Simulation and experimental analyses are carried out to verify the accuracy and effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

Abstract: Coordinated control problem of translation and rotation between two spacecraft is investigated in this paper. Using dual quaternion, we establish the relative coupled dynamic model in which the coupling effect between translational and rotational motion is indicated. Based on the logarithm of dual quaternion, a model-independent PD-like controller is proposed for spacecraft tracking control problem. The convergence of the closed loop system in the presence of external disturbances is proven theoretically. The validity of the proposed approach is demonstrated by numerical simulations. The advantages of the proposed controller are displayed by comparing with others. [Copyright &y& Elsevier]

We present a new formulation method to solve the kinematic problem of multiarm robot systems. Our major aims were to formulize the kinematic problem in a compact closed form and avoid singularity problems in the inverse kinematic solution. The new formulation method is based on screw theory and quaternion algebra. Screw theory is an effective way to establish a global description of a rigid body and avoids singularities due to the use of the local coordinates. The dual quaternion, the most compact and efficient dual operator to express screw displacement, was used as a screw motion operator to obtain the formulation in a compact closed form. Inverse kinematic solutions were obtained using Paden-Kahan subproblems. This new formulation method was implemented into the cooperative working of 2 Stäubli RX160 industrial robot-arm manipulators. Simulation and experimental results were derived. [ABSTRACT FROM AUTHOR]

In this paper, Euler's and De Moivre's formulas for complex numbers and quaternions are generalized for the dual quaternions. Also, the matrix representation of dual quaternions is expressed. [ABSTRACT FROM AUTHOR]

In this paper, it is investigated eigenvalues and eigenvectors of the dual Hamilton operators. Moreover, it is examined a special type dual quaternion equation using these eigenvalues and eigenvectors. Finally, it is given the nth power of a dual quaternion. [ABSTRACT FROM AUTHOR]

Abstract: A line symmetric motion is the motion obtained by reflecting a rigid body in the successive generator lines of a ruled surface. In this work we review the dual quaternion approach to rigid body displacements, in particular the representation of the group SE(3) by the Study quadric. Then some classical work on reflections in lines or half-turns is reviewed. Next two new characterisations of line symmetric motions are presented. These are used to study a number of examples one of which is a novel line symmetric motion given by a rational degree five curve in the Study quadric. The rest of the paper investigates the connection between sets of half-turns and linear subspaces of the Study quadric. Line symmetric motions produced by some degenerate ruled surfaces are shown to be restricted to certain 2-planes in the Study quadric. Reflections in the lines of a linear line complex lie in the intersection of the Study quadric with a 4-plane. [ABSTRACT FROM AUTHOR]