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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]

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]

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]

• 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]

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]

Geometric error is a crucial factor influencing the spatial accuracy of CNC machine tools. A novel methodology for modeling, measuring, and identifying geometric errors in multi-axis machine tools is proposed in this paper. Firstly, a synthetic volumetric error model is established by utilizing dual quaternions for multi-axis CNC machine tools. To characterize the relative position relationship between the tool and the workpiece, the kinematics model automatically incorporates the complex coupling between position and orientation motion in an implicit way, enabling a concise and compact representation of the kinematics. Secondly, measure path planning includes candidate measurement positions which are screened to obtain the optimal position group by using observation indices and the modified Detmax method. Thirdly, a parametric modeling method based on exponential cosine fitting is proposed for representing both angular and linear errors, and an improved sparrow search algorithm and nested parameter uncertainty optimization are established to process curve fitting–based optimization of the geometric error term. The fitting of the exponential cosine model is quantified with model uncertainty, and the nested uncertainty optimization method is employed to improve geometric error terms with a poor fitting effect. Finally, the effectiveness is demonstrated through experimental comparisons. The geometric error of a single axis has an average decreased of 69.7%, and the compensation rate of roundness driven by two-axis synchronization is an average of 68.7%. This method offers the advantage of quantifying the minimum optimal number of measurements and positions, improving the efficiency of parametric modeling. [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]

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]

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]

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]

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]

The goal pursued by this article, is to evaluate the potential of sliding-mode control and estimation techniques, to address fault tolerance against a large class of actuator faults, including loss of controllability of the faulty actuator, for autonomous rendezvous between a chaser spacecraft and a passive spacecraft on a circular orbit. The proposed solution is based on the dual quaternion formalism, to describe in a single equation, rotational and translational spacecraft dynamics, solar array flexible modes, propellant sloshing, the most dimensioning space disturbances, and their coupling. Such a modelling formalism enables to propose a six degree-of-freedom fault tolerant control architecture, which relies on the generalized super-twisting control algorithm nested with a nonlinear fault estimator. An anti-windup strategy based on polytope algebra is applied to the control algorithm, to prevent instability due to actuator saturation when faults occur. Asymptotic stability of the proposed fault-tolerant control scheme is formally proved with respect to a wide variety of faults, providing that the first derivatives of the fault estimation error versus time and the sliding surface, are bounded. Intensive simulations from a functional engineering simulator that accurately simulates the rendezvous mission, are presented in the paper, as well as capture-oriented criteria. The presented results demonstrate that the proposed fault-tolerant solution is able to cover any kind of thruster faults, including total loss of controllability of the faulty thruster, as well as solar array flexible modes, propellant sloshing, gravity gradient, the second zonal harmonic, atmospheric drag and magnetic disturbances. [ABSTRACT FROM AUTHOR]

Due to the high complexity of geo-spatial entities and the limited field of view of LiDAR equipment, pairwise registration is a necessary step for integrating point clouds from neighbouring LiDAR stations. Considering that accurate extraction of point features is often difficult without the use of man-made reflectors, and the initial approximate values for the unknown transformation parameters must be estimated in advance to ensure the correct operation of those iterative methods, a closed-form solution to linear feature-based registration of point clouds is proposed in this study. Plücker coordinates are used to represent the linear features in three-dimensional space, whereas dual quaternions are employed to represent the spatial transformation. Based on the theory of least squares, an error norm (objective function) is first constructed by assuming that each pair of corresponding linear features is equivalent after registration. Then, by applying the extreme value analysis to the objective function, detailed derivations of the closed-form solution to the proposed linear feature-based registration method are given step by step. Finally, experimental tests are conducted on a real dataset. The derived experimental result demonstrates the feasibility of the proposed solution: By using eigenvalue decomposition to replace the linearization of the objective function, the proposed solution does not require any initial estimates of the unknown transformation parameters, which assures the stability of the registration method. [ABSTRACT FROM AUTHOR]

Since pairwise registration is a necessary step for the seamless fusion of point clouds from neighboring stations, a closed-form solution to planar feature-based registration of LiDAR (Light Detection and Ranging) point clouds is proposed in this paper. Based on the Plücker coordinate-based representation of linear features in three-dimensional space, a quad tuple-based representation of planar features is introduced, which makes it possible to directly determine the difference between any two planar features. Dual quaternions are employed to represent spatial transformation and operations between dual quaternions and the quad tuple-based representation of planar features are given, with which an error norm is constructed. Based on L2-norm-minimization, detailed derivations of the proposed solution are explained step by step. Two experiments were designed in which simulated data and real data were both used to verify the correctness and the feasibility of the proposed solution. With the simulated data, the calculated registration results were consistent with the pre-established parameters, which verifies the correctness of the presented solution. With the real data, the calculated registration results were consistent with the results calculated by iterative methods. Conclusions can be drawn from the two experiments: (1) The proposed solution does not require any initial estimates of the unknown parameters in advance, which assures the stability and robustness of the solution; (2) Using dual quaternions to represent spatial transformation greatly reduces the additional constraints in the estimation process. [ABSTRACT FROM AUTHOR]