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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]

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]

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]

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]

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]

This paper develops a relative state estimation method for two spacecraft based on monocular vision measurement, where the leader spacecraft of three-dimensional shape is observed by a calibrated camera fixed on the follower spacecraft. The two-dimensional image of the leader spacecraft with multiple geometric features is obtained by the camera. Multiple geometric features including points, lines and circles are described under dual number algebra, and the observation models are proposed based on the geometric projection relationships of these features. By the proposed method, the geometric features appeared on the leader spacecraft can be employed to estimate the relative state between the two spacecraft. Another contribution of this paper is to develop a six-degree-of-freedom relative motion dynamics using dual number. The dynamics model describes the relative motion between arbitrary points on the spacecraft, and both the kinematically and dynamically coupling effects are considered. Based on the derived process model and observation model, an extended Kalman filter is presented to estimate the relative state between the two spacecraft. In addition, an unscented Kalman filter is also developed to avoid complicated calculation of derivations. Finally, numerical simulations are performed to evaluate the proposed EKF and UKF approaches based on multiple geometric feature measurement. Simulation results verify the effectiveness of the proposed method, and show that by using EKF or UKF technique, the relative translation and rotation estimation errors converge rapidly with the desired high estimation accuracy. Comparison simulations are made with the situations using less feature measurement, which demonstrate that employing multiple feature measurement provides superior estimation performance than methods with less feature measurement. [ABSTRACT FROM AUTHOR]

Based on the dual-quaternion description, a smooth six-degree-of-freedom observer is proposed to estimate the incorporating linear (translational) and angular velocity, called the dual-angular velocity, for rigid bodies. To establish the observer, some important properties of dual vectors and dual quaternions are presented and proved, additionally, the kinematics of dual-transformation matrices is deduced, and the transition relationship between dual quaternions and dual transformation matrices is subsequently analyzed. An important feature of the observer is that all estimation states are ensured to be $\mathcal {C}^{\infty }$ continuous, and estimation errors are shown to exhibit asymptotic convergence if the signals to be estimated are bounded. Furthermore, to achieve tracking control objectives, the proposed observer is combined with an independently designed proportional-derivative-like feedback control law (using full-state feedback), and a special Lyapunov “strictification” process is employed to ensure a separation property between the observer and the controller, which further guarantees almost global asymptotic stability of the closed-loop tracking error dynamics. Numerical simulation results for a prototypical spacecraft pose tracking mission application are presented to illustrate the effectiveness and robustness of the proposed method. [ABSTRACT FROM AUTHOR]

This paper addresses the problem of spacecraft attitude and position synchronization in libration point orbits between a leader and a follower. Using dual quaternion, the dimensionless relative coupled dynamical model is derived considering computation efficiency and accuracy. Then a model-independent dimensionless cascade pose-feedback active disturbance rejection controller is designed to spacecraft attitude and position tracking control problems considering parameter uncertainties and external disturbances. Numerical simulations for the final approach phase in spacecraft rendezvous and docking and formation flying are done, and the results show high-precision tracking errors and satisfactory convergent rates under bounded control torque and force which validate the proposed approach. [ABSTRACT FROM AUTHOR]

Results are presented for a study of dual-quaternion-based fault-tolerant control for spacecraft tracking. First, a six-degrees-of-freedom dynamic model under a dual-quaternion-based description is employed to describe the relative coupled motion of a target-pursuer spacecraft tracking system. Then, a novel fault-tolerant control method is proposed to enable the pursuer to track the attitude and the position of the target even though its actuators have multiple faults. Furthermore, based on a novel time-varying sliding manifold, finite-time stability of the closed-loop system is theoretically guaranteed, and the convergence time of the system can be given explicitly. Multiple-task capability of the proposed control law is further demonstrated in the presence of disturbances and parametric uncertainties. Finally, numerical simulations are presented to demonstrate the effectiveness and advantages of the proposed control method. [ABSTRACT FROM PUBLISHER]

Abstract: This paper addresses the synchronized control problem of relative position and attitude for spacecraft with input constraint. First, using dual quaternion, the kinematic and dynamic models of the six-degree-of-freedom relative motion of spacecraft are introduced. Second, a new adaptive sliding mode control scheme is proposed to guarantee the globally asymptotic convergence of relative motion despite the presence of control input constraint, parametric uncertainties and external disturbances. A detailed stability analysis of the resulting closed-loop system is included. Finally, simulation results are presented to illustrate the validity and effectiveness of the proposed controller, which has the following properties: (1) explicit accounting for the problem of input constraint, (2) fast convergent rate and accurate results can be obtained, (3) no chattering phenomenon is present in the control torque and control force, (4) self-adaptive regulation law is dynamically adjusted to ensure the tracking errors tend to zero asymptotically, (5) the upper bounds of unknown variables are estimated dynamically. [Copyright &y& Elsevier]

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]

This article investigates the tracking control problem for the six-degree-of-freedom motion of a follower spacecraft relative to a leader spacecraft. Employing dual quaternion representation, the model of the six-degree-of-freedom relative motion between the two spacecraft is derived. Taking into account the model uncertainties and external disturbances, two robust fast sliding mode control laws are proposed to guarantee the finite-time convergence of the tracking errors, where the second control law can solve the dual-equilibrium problem that exists in the first one. In order to alleviate chattering phenomenon, continuous saturation function is introduced into the controller and a softening control strategy is developed. It is worth pointing out that the convergence rate of the proposed fast sliding mode control is faster than that of the linear sliding mode control and terminal sliding mode control. Finally, numerical simulations are performed to demonstrate the effectiveness of the proposed approaches. [ABSTRACT FROM AUTHOR]

The monocular vision-based navigation algorithm for spacecraft is addressed in this paper. Based on dual number, coupled dynamics equations of relative position and attitude are derived, which can describe the coupling effect between the rotational motion and translational motion. The measurement model is proposed on the basis of the projection on the 2 - D plane of the line fixed on the target spacecraft. And the Extended Kal-man Filtering method is applied to the estimation problem. Numerical simulations are performed to demonstrate that the proposed approaches can provide accurate and fast estimation of relative position and attitude. [ABSTRACT FROM AUTHOR]