Cooperative Angles-Only Relative Navigation Algorithm for Multi-Spacecraft Formation in Close-Range

As to solve the collaborative relative navigation problem for near-circular orbiting small satellites in close-range under GNSS denied environment,a novel consensus constrained relative navigation algorithm based on the lever arm effect of the sensor offset from the spacecraft center of mass is proposed.Firstly,the orbital propagation model for the relative motion of multi-spacecraft is established based on Hill-Clohessy-Wiltshire dynamics and the line-of-sight measurement under sensor offset condition is modeled in Local Vertical Local Horizontal frame.Secondly,the consensus constraint model for the relative orbit state is constructed by introducing the geometry constraint between the spacecraft,based on which the consensus unscented Kalman filter is designed.Thirdly,the observability analysis is done and the necessary conditions of the sensor offset to make the state observable are obtained.Lastly,digital simulations are conducted to verify the proposed algorithm,where the comparison to the unconstrained case is also done.The results show that the estimated error of the relative position converges very quickly,the location error is smaller than 10m under the condition of 10−3 rad level camera and 5m offset.

Robust Adaptive Gain Higher Order Sliding Mode Observer Based Control-constrained Nonlinear Model Predictive Control for Spacecraft Formation Flying

This work deals with the development of a decentralized optimal control algorithm, along with a robust observer,for the relative motion control of spacecraft in leader-follower based formation. An adaptive gain higher order sliding mode observer has been proposed to estimate the velocity as well as unmeasured disturbances from the noisy position measurements.A differentiator structure containing the Lipschitz constant and Lebesgue measurable control input, is utilized for obtaining the estimates. Adaptive tuning algorithms are derived based on Lyapunov stability theory, for updating the observer gains,which will give enough flexibility in the choice of initial estimates.Moreover, it may help to cope with unexpected state jerks. The trajectory tracking problem is formulated as a finite horizon optimal control problem, which is solved online. The control constraints are incorporated by using a nonquadratic performance functional. An adaptive update law has been derived for tuning the step size in the optimization algorithm, which may help to improve the convergence speed. Moreover, it is an attractive alternative to the heuristic choice of step size for diverse operating conditions. The disturbance as well as state estimates from the higher order sliding mode observer are utilized by the plant output prediction model, which will improve the overall performance of the controller. The nonlinear dynamics defined in leader fixed Euler-Hill frame has been considered for the present work and the reference trajectories are generated using Hill-Clohessy-Wiltshire equations of unperturbed motion. The simulation results based on rigorous perturbation analysis are presented to confirm the robustness of the proposed approach.

GPS/VISNAV integrated relative navigation and attitude determination system for ultra-close spacecraft formation flying

For the improvement of accuracy and better fault-tolerant performance, a global position system （GPS）/vision navigation （VISNAV） integrated relative navigation and attitude determination approach is presented for ultra-close spacecraft formation flying. Onboard GPS and VISNAV system are adopted and a federal Kalman filter architecture is used for the total navigation system design. Simulation results indicate that the integrated system can provide a total improvement of relative navigation and attitude estimation performance in accuracy and fault-tolerance.

Design and control of multiple spacecraft formation flying in elliptical orbits

Spacecraft formation flying is an attractive new concept in international aeronautic fields because of its powerful functions and low cost. In this paper, the formation design and PD closed-loop control of spacecraft formation flying in elliptical orbits are discussed. Based on two-body relative dynamics, the true anomaly is applied as independent variable instead of the variable of time. Since the apogee is considered as the starting point, the six integrating constants are calculated. Therefore, the algebraic solution is obtained for the relative motion in elliptical orbits. Moreover, the formation design is presented and both circular formation and line formation are provided in terms of an algebraic solution. This paper also discusses the PD-closed loop control for precise formation control in elliptical orbits. In this part, the error-type state equation is put forward and the linear quadratic regulator （LQR） method is used to calculate PD parameters. Though the gain matrix calculated from LQR is time-variable because the error-type state equation is time variable, the PD parameters are also considered as constants because of their small changes in simulation. Finally, taking circular formation as an example, the initial orbital elements are achieved for three secondary spacecraft. And the numerical simulation is analyzed under PD formation control with initial errors and J2 perturbation. The simulation results demonstrate the validity of PD closed-loop control scheme.

Multi-modes control method for spacecraft formation establishment and reconfiguration near the libration points

This paper studies Multi-modes control method for libration points formation establishment and reconfiguration. Firstly, relations between optimal impulse control and Floquet modes are investigated. Method of generating modes is proposed. Characteristics of the mode coefficients stimulated at different time are also given. Studies show that coefficients of controlled modes can be classified into four types, and formation establishment and reeonfiguration can be achieved by multi-impulse control with the presented method of generating modes. Then, since libration points formation is generally unstable, mutli-modes keeping control method which can stabilize five Floquet modes simultaneously is proposed. Finally, simulation on formation establishment and reconfiguration are carried out by using method of generating modes and mutli-modes keeping control method. Results show that the proposed control method is effective and practical.

Spacecraft formation reconfiguration with multi-obstacle avoidance under navigation and control uncertainties using adaptive artificial potential function method

In this paper,an adaptive artificial potential function(AAPF)method is developed for spacecraft formation reconfiguration with multi-obstacle avoidance under navigation and control uncertainties.Furthermore,an improved Linear Quadratic Regular(ILQR)is proposed to track the reference trajectory and a Lyapunov-based method is employed to demonstrate the stability of the overall closed-loop system.Compared with the traditional APF method and the equal-collision-probability surface(ECPS)method,the AAPF method not only retains the advantages of APF method and ECPS method,such as low computational complexity,simple analytical control law and easy analytical validation progress,but also proposes a new APF to solve multi-obstacle avoidance problem considering the influence of the uncertainties.Moreover,the ILQR controller obtains high control accuracy to enhance the safe performance of the spacecraft formation reconfiguration.Finally,the effectiveness of the proposed AAPF method and the ILQR controller are verified by numerical simulations.

Autonomous attitude coordinated control for spacecraft formation with input constraint,model uncertainties, and external disturbances

To synchronize the attitude of a spacecraft formation flying system, three novel autonomous control schemes are proposed to deal with the issue in this paper. The first one is an ideal autonomous attitude coordinated controller, which is applied to address the case with certain models and no disturbance. The second one is a robust adaptive attitude coordinated controller, which aims to tackle the case with external disturbances and model uncertainties. The last one is a filtered robust adaptive attitude coordinated controller, which is used to overcome the case with input con- straint, model uncertainties, and external disturbances. The above three controllers do not need any external tracking signal and only require angular velocity and relative orientation between a spacecraft and its neighbors. Besides, the relative information is represented in the body frame of each spacecraft. The controllers are proved to be able to result in asymptotical stability almost everywhere. Numerical simulation results show that the proposed three approaches are effective for attitude coordination in a spacecraft formation flying system.

Delay Depending Decentralized Adaptive Attitude Synchronization Tracking Control of Spacecraft Formation

This paper deals with the problem of cooperative attitude tracking with time-varying communication delays as well as the delays between inter-synchronization control parts and self-tracking control parts in the spacecraft formation flying. First, we present the attitude synchronization tracking control algorithms and analyze the sufficient delay-dependent stability condition with the choice of a Lyapunov function when the angular velocity can be measured. More specifically, a class of linear filters is developed to derive an output feedback control law without having direct information of the angular velocity, which is significant for practical applications with low-cost configurations of spacecraft. Using a well-chosen Lyapunov-Krasovskii function, it is proven that the presented control law can make the spacecraft formation attitude tracking system synchronous and achieve ex- ponential stability, in the face of model uncertainties, as well as non-uniform time-varying delays in communication links and different control parts. Finally, simulation results are presented to demonstrate the effectiveness of the proposed control schemes.

Fixed-time position coordinated tracking control for spacecraft formation flying with collision avoidance

In this paper,the fixed-time stability of spacecraft formation reconfiguration(position tracking)is studied.Firstly,a novel nonsingular terminal sliding mode surface is designed and based on which a fixed-time coordinated controller is designed to keep the closed-loop system states have a finite settling time bounded by some predefined constants.Secondly,another nonsingular terminal sliding mode surface is designed by combining the artificial potential function and the aforementioned sliding surface,which meets the mutual distance constraint during transition process among spacecraft when it is bounded.Then another coordinated controller with fixed-time observer considering mutual distance constraint is presented,which guarantees the closed-loop system states stable also in bounded settling time.Finally,simulation results are shown to validate the correctness of the proposed theorems.It is worth mentioning that the control schemes also work even though there is a properly limit on the control input.

Robust nonlinear control of spacecraft formation flying using constraint forces

A robust nonlinear control method is presented for spacecraft precise formation flying.With the constraint forces and consid-ering nonlinearity and perturbations,the problem of the formation keeping is changed to the Lagrange systems with the holonomic constraints and the differential algebraic equations (DAE).The nonlinear control laws are developed by solving the DAE.Because the traditional numerical solving methods of DAE are very sensitive to the various errors and the resulting con-trol laws are not robust in engineering application,the robust control law designed method is further developed by designing the correct coefficients to correct the errors of the formation array constraints.A numeral study simulated the robustness of this method for the various errors in the formation flying mission,including the initial errors of spacecraft formation,the reference satellite orbit determination errors,the relative perturbation forces model errors,and so on.

Shape control of spacecraft formation using a virtual spring-damper mesh

This paper derives a distance-based formation control method to maintain the desired formation shape for spacecraft in a gravitational potential field. The method is an analogy of a virtual spring-damper mesh. Spacecraft are connected virtually by spring-damper pairs. Convergence analysis is performed using the energy method. Approximate expressions for the distance errors and control accelerations at steady state are derived by using algebraic graph representations and results of graph rigidity. Analytical results indicate that if the underlying graph of the mesh is rigid, the convergence to a static shape is assured, and higher formation control precision can be achieved by increasing the elastic coefficient without increasing the control accelerations. A numerical example of spacecraft formation in low Earth orbit confirms the theoretical analysis and shows that the desired formation shape can be well achieved using the presented method, whereas the orientation of the formation can be kept pointing to the center of the Earth by the gravity gradient. The method is decentralized, and uses only relative measurement information. Constructing a distributed virtual structure in space can be the general application area. The proposed method can serve as an active shape control law for the spacecraft formations using propellantless internal forces.

Floquet-based design and control approach to spacecraft formation flying in libration point orbits

A method for spacecraft formation flying (SFF) design and control near libration point orbits was developed by making use of the Floquet theory for periodic orbits. Firstly, the Floquet theory used in libration point orbits was introduced and the coefficients of four Floquet periodic modes were proved to be nearly constant when the amplitude in z direction of earth-moon L1 halo orbits is less than 20000 km. On this basis, a configuration design approach to SFF in L1 halo orbits was proposed, and several types of special configurations were obtained with periodic mode 3 and mode 5 or mode 4 and mode 6. Then, in order to control the SFF configuration concisely, those coefficients of the 5 modes (except the stable one) must be kept constant. A stationkeeping method for SFF was developed, which controls 5 Floquet modes simultaneously. Finally, simulations showed that the Floquet-based approach of configuration design and control for SFF is effective, simple and convenient. The research may be of value for deep space explorations.

Robust image-based coordinated control for spacecraft formation flying

This paper addresses a coordinated control problem for Spacecraft Formation Flying(SFF). The distributed followers are required to track and synchronize with the leader spacecraft.By using the feature points in the two-dimensional image space, an integrated 6-degree-of-freedom dynamic model is formulated for spacecraft relative motion. Without sophisticated threedimensional reconstruction, image features are directly utilized for the controller design. The proposed image-based controller can drive the follower spacecraft in the desired configuration with respect to the leader when the real-time captured images match their reference counterparts. To improve the precision of the formation configuration, the proposed controller employs a coordinated term to reduce the relative distance errors between followers. The uncertainties in the system dynamics are handled by integrating the adaptive technique into the controller, which increases the robustness of the SFF system. The closed-loop system stability is analyzed using the Lyapunov method and algebraic graph theory. A numerical simulation for a given SFF scenario is performed to evaluate the performance of the controller.

Robust attitude coordinated control for spacecraft formation with communication delays

In this paper,attitude coordinated tracking control algorithms for multiple spacecraft formation are investigated with consideration of parametric uncertainties,external disturbances,communication delays and actuator saturation.Initially,a sliding mode delay-dependent attitude coordinated controller is proposed under bounded external disturbances.However,neither inertia uncertainty nor actuator constraint has been taken into account.Then,a robust saturated delaydependent attitude coordinated control law is further derived,where uncertainties and external disturbances are handled by Chebyshev neural networks（CNN）.In addition,command filter technique is introduced to facilitate the backstepping design procedure,through which actuator saturation problem is solved.Thus the spacecraft in the formation are able to track the reference attitude trajectory even in the presence of time-varying communication delays.Rigorous analysis is presented by using Lyapunov-Krasovskii approach to demonstrate the stability of the closed-loop system under both control algorithms.Finally,the numerical examples are carried out to illustrate the efficiency of the theoretical results.

Coordinated attitude control for flexible spacecraft formation with actuator configuration misalignment

This paper investigates the coordinated attitude control problem for flexible spacecraft formation with the consideration of actuator configuration misalignment.First,an integral-type sliding mode adaptive control law is designed to compensate the effects of flexible mode,environmental disturbance and actuator installation deviation.The basic idea of the Integral-type Sliding Mode Control(ISMC)is to design a proper sliding manifold so that the sliding mode starts from the initial time instant,and thus the robustness of the system can be guaranteed from the beginning of the process and the reaching phase is eliminated.Then,considering the nominal system of spacecraft formation based on directed topology,an attitude cooperative control strategy is developed for the nominal system with or without communication delay.The proposed control law can guarantee that for each spacecraft in the spacecraft formation,the desired attitude objective can be achieved and the attitude synchronization can be maintained with other spacecraft in the formation.Finally,simulation results are given to show the effectiveness of the proposed control algorithm.

Relative Attitude Dynamics and Control of Spacecraft Formation Flying Considering Flexible Panels

As a key technology for orbital applications, researches on spacecraft formation flying(SFF) attract more attention. However, most of existing researches about dynamics and control of SFF focus on rigid spacecrafts without considering the effect of flexible attachments(such as flexible panels). In this paper, relative attitude dynamics and active control of SFF for a flexible spacecraft(follower spacecraft) and a rigid spacecraft(target spacecraft) are investigated. Firstly, a dynamic model of the flexible spacecraft is established by the principle of angular momentum. Then, the equation of relative attitude dynamics between the flexible spacecraft and the rigid spacecraft is derived by the quaternion to represent the attitude relation of the two spacecrafts. Finally,an attitude feedback controller is designed for the SFF system, and its stability is proved by the Lyapunov stability theory. Simulation results indicate that the panel flexibility has an obvious influence on the dynamic behaviour of the system, the designed controller can effectively control the attitude of the two spacecrafts to achieve synchronization, and the elastic vibration of the panels may be suppressed simultaneously.

An improved nonlinear control strategy for deep space formation flying spacecraft

This paper aims to provide further study on the nonlinear modeling and controller design of formation flying spacecraft in deep space missions. First, in the Sun-Earth system, the nonlinear formation dynamics for the circular restricted three-body problem （CRTBP） and elliptic restricted three-body problem （ERTBP） are presented. Then, with the Floquet mode method, an impulsive controller is developed to keep the Chief on the desired Halo orbit. Finally, a nonlinear adaptive control scheme based on Nonzero set- point LQR and neural network is proposed to achieve high precision formation maneuver and keeping. The simulation results indicate that the proposed nonlinear control strategy is reasonable as it considers not only the orbit keeping of the Chief, but also the formation modeling inaccuracy. Moreover, the nonlinear adaptive control scheme is effective to improve the control accuracy of the formation keeping.

A simple method to design non-collision relative orbits for close spacecraft formation flying

A set of linearized relative motion equations of spacecraft flying on unperturbed elliptical orbits are specialized for particular cases, where the leader orbit is circular or equatorial. Based on these extended equations, we are able to analyze the relative motion regulation between a pair of spacecraft flying on arbitrary unperturbed orbits with the same semi-major axis in close formation. Given the initial orbital elements of the leader, this paper presents a simple way to design initial relative orbital elements of close spacecraft with the same semi-major axis, thus preventing collision under non-perturbed conditions. Considering the mean influence of J_2 perturbation, namely secular J_2 perturbation, we derive the mean derivatives of orbital element differences, and then expand them to first order. Thus the first order expansion of orbital element differences can be added to the relative motion equations for further analysis. For a pair of spacecraft that will never collide under non-perturbed situations, we present a simple method to determine whether a collision will occur when J_2 perturbation is considered. Examples are given to prove the validity of the extended relative motion equations and to illustrate how the methods presented can be used. The simple method for designing initial relative orbital elements proposed here could be helpful to the preliminary design of the relative orbital elements between spacecraft in a close formation, when collision avoidance is necessary.

Developing Dijik-Primbert Algorithm for Finding Unpredictable Paths over Time-Varying Networks

Cooperation among multiple unmanned vehicles is an intensely challenging topic from a theoretical and practical standpoint, with far reaching indications in scientific and commercial mission scenarios. The difficulty of time coordination for a rapid of multirotor UAVs includes predefined spatial paths according to mission necessities. With the solution proposed, cooperative control is accomplished in the presence of time-varying communication networks, as well as stringent temporal constraints, such as concurrent arrival at the desired final locations. The proposed explanation solves the time-coordination problem under the acceptance that the trajectory-genera- tion and the path-following algorithms meeting convinced cohesion conditions are given. Communication is processed in unpredictable paths by the use of path following and directed communication graph. Dijik-Primbert algorithm for finding the shortest collision free paths is used to avoid and detect collision/congestion in unpredictable paths. Without collision detection, it doesn’t seem agreeable to have collision avoidance because there wouldn’t be everything to avoid. Dijikloyd algorithm is used for finding shortest paths in a weighted directed graph with positive and negative edges. Primloyd algorithm is used for finding shortest paths in a weighted undirected graph for conquering the complexity in matrix coding. In case of conges- tion or collision then the whole network is learned about it to all the communica- tors. Hence, communication is taken place in an unpredictable path in a secured manner.

Decentralized attitude synchronization tracking control for multiple spacecraft under directed communication topology

This paper studies the attitude synchronization tracking control of spacecraft formation flying with a directed communication topology and presents three different controllers. By introducing a novel error variable associated with rotation matrix, a decentralized attitude synchronization controller, which could obtain almost global asymptotical stability of the closed-loop system, is developed. Then, considering model uncertainties and unknown external disturbances, we propose a robust adaptive attitude synchronization controller by designing adaptive laws to estimate the unknown parameters. After that, the third controller is proposed by extending this method to the case of time-varying communication delays via Lyapunov-Krasovskii analysis. The distinctive feature of this work is to address attitude coordinated control with model uncertainties, unknown disturbances and time-varying delays in a decentralized framework, with a strongly connected directed information flow. It is shown that tracking and synchronization of an arbitrary desired attitude can be achieved when the stability condition is satisfied. Simulation results are provided to demonstrate the effectiveness of the proposed control schemes. (C) 2016 Production and hosting by Elsevier Ltd. on behalf of Chinese Society of Aeronautics and Astronautics. This is an open access article under the CC BY-NC-ND license.