Spacecraft Formation Flying Research Articles

Flyaround Orbit Design for Autonomous Rendezvous Based on Relative Orbit Elements

I N RECENT years, there has been considerable interest in spacecraft formation flying, due to the advantages of autonomy, flexibility, and relatively low cost, which is mostly required in the future space missions. One important application is autonomous rendezvous and proximity operation to a cooperative or noncooperative target in space [1–4], where two spacecraft should be ensured to fly in close formation in order to accomplish inspection, servicing, or capture at the specified time [5]. To implement autonomous rendezvous, it is essential to study the relative motion of one spacecraft (referred to as the chaser) with respect to another reference spacecraft (referred to as the target). The relative motion problems have been treated in many references. For instance, early in 1960, the first study on the relative motion of satellite clusters was performed by Clohessy and Wiltshire [6]. The authors presented the linearized equations of relative motion under circular-orbit assumption, well known as Clohessy–Wiltshire equations. Thereafter, Lawden [7] and Tschauner and Hempel [8] first derived the relative motion equations for eccentric orbits. The effects of the reference orbit eccentricity, the gravity perturbation J2, drag, and solar radiation pressure on the relativemotion are shown in [9–13]. Additionally, in these studies, along with developing the relative motion dynamics, the relative trajectory design is also presented to specify six initial conditions for relative position and velocity components, also seen in [14–17]. However, further studies should be developed on two issues. On one hand, these studies mostly used relative position and velocity components to describe the analytic solution and to design the relative orbit. However, it is difficult to intuitively know the size, location, and orientation of the relative trajectory with respect to the target orbit from the relative position and velocity components. Lovell and Tragesser [18] first introduced a set of relative orbit elements to replace the position and velocity components through a simple coordinate transformation. Nevertheless, especially for the description of the orientation of relative trajectory plane, the use of the relative orbit elements is not intuitive enough. Hence, we introduce one new relative orbit element to describe intuitively the orientation of the relative trajectory plane. On the other hand, most of these studies seldom provided the practical considerations such as the requirements of the vehicle’s solar power generation and communications subsystems and thus cannot exploit the relationship of the relative orbit elements and the specified mission design. Reference [19] first investigates the relative orbit design while considering the relative navigation pointing and sun pointing constraints, especially optimizing the solar collection. However, the author illustrates the problem mainly based on one simple example and its simulation. Instead, the objective of our research is to provide thorough theoretical derivation for a general case. In this Note, flyaround orbit design during autonomous rendezvous for a cooperative target or noncooperative target is considered. The rest of this Note is organized as follows. In Sec. II, the relative dynamics is characterized and the relative orbit elements are introduced to describe intuitively the relative trajectory. Section III provides the chaser’s desired attitude constraints to satisfy relative navigation pointing and sun pointing requirements. Then the flyaround orbit design is mainly investigated to specify the initial conditions of relative orbit elements, especially the initial phase angle on the consideration of maximizing the solar power collection of the chaser. Conclusions and future work are given in Sec. IV.

Fault Tolerant Reconfigurable Satellite Formations Using Adaptive Variable Structure Techniques

This paper proposes two adaptive nonlinear control algorithms based on a variable-structure control design for multiple spacecraft formation flying. The nonlinear dynamics describing the motion of the follower spacecraft relative to the leader spacecraft are considered for the case in which the leader spacecraft is in an elliptical reference orbit, and the stability of such a formation in the presence of external perturbations is investigated. This paper presents fault-tolerant control schemes to account for accidental or degradation faults in spacecraft sensors and thrusters. The nonlinear analytical model describing the system is used to develop two adaptive fault-tolerant control laws (continuous sliding mode control and nonsingular terminal sliding mode control) that guarantee global asymptotic convergence of the position tracking error in the presence of unknown follower spacecraft mass and external disturbances. Several numerical examples are presented to demonstrate the efficacy of the proposed controllers to maintain the relative motion by correcting for initial offsets and external perturbation effects that tend to disperse the formation. Simulation results confirm that the suggested methodologies yield submillimeter formation, keeping precision and effectiveness in ensuring formation maneuvering. In addition, an abrupt blockage of the relative position sensors, thruster failure for a period of time, and thruster degradation (amidst formation keeping and reconfiguration maneuvers) are simulated to demonstrate the fault recovery capability of the controllers. The numerical results clearly establish the robustness of the proposed reconfigurable adaptive control scheme for precise formation keeping in the event of sensor and thruster faults.

Design and Performance Analysis of GPS Based Precise Relative Navigation for Rendezvous and Formation Flying Missions in Low Earth Orbit

Precise relative GPS navigation is essential technology for rendezvous and formation flying of spacecraft in Low Earth Orbit. Key design issues of precise relative GPS navigation software are studied and a novel formulation is proposed for the mission which requires high accuracy when the separation distance is up to several km. The navigation filter estimates float navigation solutions by extended Kalman filter with elaborate dynamics models, and resolves ambiguities of integer carrier phase biases to achieve high accuracy fix navigation solutions. A relative navigation software with the proposed formulation is implemented and evaluated in two different ways. One is a test using spacebourne GPS receivers and a GPS signal simulator to evaluate the performance and sensitivity of the software against variation of parameters. The other is a test using actual telemetry data from Gravity Recovery and Climate Experiment (GRACE) to demonstrate the software performance. The design, implementation, and results of the evaluation are presented and discussed on this paper.

Hardware-in-the-Loop Simulation for Spacecraft Formation Flying

This paper presents a hardware-in-the-loop (HITL) simulation approach for multiple spacecraft formation flying. Considering a leader-follower formation flying configuration, a Fuzzy Logic controller is developed first to maintain the desired formation shape under external perturbations and the initial position offsets. Cold-gas on/off thrusters are developed to be introduced to the simulation loop, and the HITL simulations are conducted to validate the effectiveness of the proposed simulation configuration and Fuzzy Logic control.

Two-Point Boundary Value Problem Solutions to Spacecraft Formation Flying

The two-point boundary value problem of a leader-follower spacecraft formation flying in unperturbed elliptical reference orbits is studied. The initial and final relative positions and times and the orbit of the leader are known, and the orbit of the follower must be determined. This problem will be called the relative Lambert's problem. First, we will show this relative Lambert's problem can be solved like the classical Lambert's problem. Then, a set of approximate analytic solutions are obtained through linearizing the Lagrange's time equation. Meanwhile, a simplified Newton-Raphson algorithm is applied to obtain numerical solutions, and the relevant quantities of the leader are used as initial conditions. Here, our special efforts are dedicated to the study of the periodic relative orbits of the follower. In particular, a set of first-order analytic solutions to the relative Lambert's problem are derived from the periodic solutions of Lawden's equations, and a constraint on the leader's true anomalies (implicitly in time) and relative positions is obtained. From that constraint on the radial/in-track plane of the leader local-vertical-local- horizontal frame, we found that, for specified initial and final times, the locus of final positions of the follower with fixed initial position is a straight line, and so is the locus of initial positions with fixed final position. Furthermore, for fixed initial and final positions, the transfer times with either specified initial time or final time can be expressed as the real roots of a cubic equation, for which there are at most three solutions. Several examples will be given to support these conclusions.

Terminal Sliding Mode Control for Spacecraft Formation Flying

This paper considers the problem of relative motion control for spacecraft formation flying (SFF). Using terminal sliding mode technique, a relative position/velocity tracking control based on the nonlinear model is developed. The presented controller enables rapid formation reconfiguration with feasible fuel cost and strong robustness in the presence of uncertain but bounded disturbances. A nonlinear model with J <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> disturbance and bounded uncertainties is used for dynamic simulation.

Effect of Kinematic Rotation-Translation Coupling on Relative Spacecraft Translational Dynamics

A CCURATEmodeling of the differential translation and rotation between two spacecraft is essential for cooperative distributed space systems, spacecraft formation flying (SFF), rendezvous, and docking. High-fidelity relative motion modeling, as opposed to absolute motion modeling, is particularly important for autonomous missions [1]. Point-mass models for relative spacecraft translational motion have been extensively studied over the past 50 years, since Clohessy and Wiltshire (CW) presented a rendezvous model for a circular reference orbit and a spherical Earth [2]. Following the work of Clohessy and Wiltshire, variants on the point-mass model were developed, such as generalizations to elliptic reference orbits [3–5] and an oblate Earth [6,7]. The growing interest in SFF motivated the research of relative spacecraft motion modeling, yielding more accurate and complete equations and solutions for perturbed relative motion [8–10]. However, most of the works focused on point-mass, 3 degrees-offreedom (DOF) spacecraft. Obviously, performing a space mission that consists of several cooperative space vehicles requires modeling the relative rotational motion in addition to the relative translation, that is, 6-DOF models. Models for the relative motion of 6-DOF spacecraft have gained attention in the literature only in recent years. Among the first to suggest treating the spacecraft relative angular velocity in an SFF control problem were Pan and Kapila [11], who addressed the coupled translational and rotational dynamics of two spacecraft. By defining two body-fixed reference frames, one attached to the leader and the other attached to the follower, it was proposed [11] to use a two-part relative motion model: one that accounts for the relative translational dynamics of the body-fixed coordinate frame origins, and another that captures the relative attitude dynamics of the two body-fixed frames. A similar modeling approach was used for relative motion estimation [1]. In addition, tensorial equations of motion for a formation consisting ofN spacecraft, each modeled as a rigid body, were derived [12]. However, only the absolute equations of motion were developed [12]; a relative version of these equations was not given. Moreover, a clear mathematical relationship between the developed models and the traditional nonlinear point-mass relative motion and CW models was not provided. The coupling between the translational and rotational motion in the aforementioned models [1,11] was induced by gravity torques. The kinematic coupling, which is essentially a projection of the rotational motion about the center of mass (c.m.) onto the relative translational configuration space, was neglected. It is this kinematic coupling that the current paper is concerned with. In general, rigid-body dynamics can be represented as translation of the c.m. and rotation about the c.m. [13]. Thus, spacecraft relative motion must be composed by combining the relative translational and rotational dynamics of arbitrary points on the spacecraft. Whenever one of these points does not coincide with the spacecraft’s c.m., a kinematic coupling between the rotational and translational dynamics of these points is obtained. The purpose of this paper is to quantify the kinematic coupling effect and to show that this effect is key for high-precision modeling of tight SFF, rendezvous, and docking. This effect is also important in vision-based relative attitude and position control, where arbitrary feature points on a target vehicle are to be tracked. Given two rigidbody spacecraft, the model presented herein is formulated in a general manner that describes the motion between any two arbitrary points on the spacecraft. The relative translational motion is then generated by both the spacecraft orbitalmotion and the rotation about the c.m. In addition, this paper provides a CW-like approximation of the relative motion that includes the kinematic coupling. This new approximation is aimed at alleviating an apparent contradiction in linearized relative motion theories: to obtain linear equations of motion, the spacecraft are assumed to operate in close proximity. However, if the spacecraft are close to each other, then they can no longer be treated as point masses, because the spacecraft shape and size affects the relative translation between off-c.m. points. This effect is accentuated as the distances between spacecraft decrease. The remainder of this paper is organized as follows. First, a background on the relative position and attitude dynamics is given. Then, a new coupled relative spacecraft motion model is presented. The newly developed model is then examined in a simulation.

Maneuver planning optimization for spacecraft formation flying missions

Formation flying of spacecraft in orbit around the Earth and in deep space is becoming an enabling technology for a number of space mission concepts. One such concept is separated spacecraft interferometry, a technique to be used by ESA’s Darwin planet finding mission. In this paper, a maneuver planning algorithm is introduced designed to find the post-maneuver spacecraft positions that satisfy interferometry science goals whilst optimizing for fuel use or fuel balancing considerations. Compared to a benchmark planning algorithm gains of up to 23% less fuel for fuel minimizing maneuvers and a significant increase in fuel balancing for fuel balancing maneuvers are observed using Darwin-like mission parameters.

Application of Analytic Solution in Relative Motion to Spacecraft Formation Flying in Elliptic Orbit

The current paper presents application of a new analytic solution in general relative motion to spacecraft formation flying in an elliptic orbit. The calculus of variations is used to analytically find optimal trajectories and controls for the given problem. The inverse of the fundamental matrix associated with the dynamic equations is not requiredfor thesolution in the currentstudy. Itis verifiedthat the optimal thrust vector is a function of the fundamental matrix of the given state equations. The cost function and the state vector during the reconfiguration can be analytically obtained as well. The results predict the form of optimal solutions in advance without having to solve the problem. Numerical simulation shows the brevity and the accuracy of the general analytic solutions developed in the current paper.

Robust Control of Spacecraft Formation Flying

In this paper, a robust control scheme for two spacecraft in formation subjected to time-variant external disturbances in the space environment was developed. The proposed controller consists of two parts, the first part is for the nominal system without disturbances and the second part is to compensate for effects of system disturbances. A dynamic relative motion error model was established to design the second part of the controller and to analyze the stability of the closed-loop system using Lyapunov stability theory. Furthermore, the robustness of proposed control method to the system disturbances is analyzed based on robust control theory. It is proven that the relative motion error of two spacecraft in formation is uniformly ultimately bounded under the proposed controller for the assumed disturbances in the dynamic model.

Autonomous Formation Flying for the PRISMA Mission

PRISMA is a technology demonstration mission for satellite formation flying and in-orbit servicing. The space segment comprises the fully maneuverable Main minisatellite and the smaller Target satellite in a low Earth orbit at 700-km altitude. A key mission objective is to demonstrate onboard, fully autonomous, robust, safe, and precise formation flying of spacecraft. This is accomplished by spaceborne global positioning system navigation, guidance, and control functionalities for the maintenance of the relative motion between the two spacecraft. An innovative estimation approach employs a common Kalman filter for the absolute states of Main and Target, which accounts for the interdependency of absolute and relative navigation without the need for an explicit relative state. As a result, the onboard navigation system provides absolute and relative orbit information in real time with a position accuracy of 2 and 0.1 m, respectively. The formation control achieves accuracies of a few tenths of meters with minimum usage of thrusters. The guidance and control concept is detailed with emphasis on a relative eccentricity and inclination vector separation strategy. The paper derives estimates of the expected relative orbit control performances based upon realworld simulations using typical global positioning system receiver and propulsion system characteristics.

Formation flying of multiple spacecraft with autonomous rendezvous and docking capability

The idea of incorporating autonomous rendezvous and docking (ARVD) capability into spacecraft flying in formation is explored. The situations where it is advantageous to incorporate ARVD capability into spacecraft formations are identified. Attention is focused on various potential problems pertaining to the control of multiple spacecraft with ARVD capability. Multiple-model-based control laws for formation acquisition and ARVD are developed using generic spacecraft models. In particular, the use of such control laws for simultaneous docking of multiple spacecraft in formation acquisition and reconfiguration is studied. Various optimal formation reconfiguration problems are briefly discussed. The effectiveness of the developed control laws is studied via computer simulation using a six-spacecraft formation model.

Adaptive Synchronization Control of Multiple Spacecraft Formation Flying

An adaptive nonlinear synchronization control approach is developed for multiple spacecraft formation flying with elliptical reference orbits. It can guarantee that both the tracking errors and the synchronization errors of the relative positions converge to zero globally, even in the presence of uncertain parameters. The generalized synchronization concept allows us to design various synchronization errors so that different synchronization performance can be obtained. Simulation results of a leader-follower spacecraft pair and the maneuvering of multiple spacecraft in formation flying are presented to verify the effectiveness of the proposed control technique.

Sliding mode control for low-thrust Earth-orbiting spacecraft formation maneuvering

This paper considers the control of entire-formation maneuvering in low-thrust Earth-orbiting spacecraft formation flying (SFF). A coordinated control scheme based on the leader-follower approach is developed to achieve formation maneuvers while keeping the internal formation intact. To implement the control scheme, a low-level sliding mode control (SMC) based on nonlinear dynamics is provided. Using a Lyapunov-based control design and stability analysis technique, we prove that the closed-loop system is globally asymptotically stable. Numerical simulations show the advantage of the controller for fuel cost reductions, tracking accuracy, and effectiveness in ensuring entire-formation maneuvering. An exact nonlinear model with modeled disturbances ( J 2 disturbance and bounded uncertainties) is used for dynamic simulation.

Satellite Orbit Transfer and Formation Reconfiguration via an Attitude Control Analogy

A control law applicable to orbit transfer as well as formation reconfiguration has been developed via a Lyapunov function similar to that used for spacecraft attitude control. It uses feedback of kinematic and kinetic error states between the current and desired orbital frames, which are parameterized by Euler parameters, orbital radius and its time derivative, and orbital angular velocities. The efficacy of the control law is demonstrated with examples, including orbit transfers and formation reconfiguration, both with low-eccentricity and high-eccentricity reference orbits, in the presence of J2 perturbations. Further simplifications in the control law are introduced and it is shown that stability is not affected. A methodology is proposed to reduce control requirement by appropriate gain selection and control initiation at predefined locations. The equivalent impulse requirements obtained for the examples compare very favorably with analytically and numerically optimized figures presented in other published works. HE problem of orbital transfers of satellites by the use of con- tinuous, low-thrust control has been studied in great detail. Re- cent advances in the field of electric propulsion have made low-thrust propulsion an operational reality. Formation flying of spacecraft is also a relatively new area of research wherein control of relative mo- tion is a key element. Control of relative motion requires controlling each satellite's orbit precisely, with high regard for intersatellite dis- tances, collision hazards, and other operational constraints. The primary purpose of this paper is to develop a Lyapunov-based control law that has a large range of applicability, free from the influ- ence of singularities that arise from the more common descriptions of a satellite's orbit, such as orbital elements. Previous studies in the literature have addressed the problem of continuous-thrust orbit transfer using Lyapunov function-based feedback controllers. Ilgen 1 uses orbital element feedback for orbital transfers, using the clas- sical as well as equinoctial elements. Ilgen's quadratic Lyapunov function is based on only five of the orbital elements. Gurfil 2 uses classical orbital element feedback for orbital transfer and explores nonlinear controllability issues of the problem, and is able to prove by using Gauss variational equations that a spacecraft's orbital el- ements can be controlled by continuous control, except when the desired orbit is parabolic. Chang et al. 3 approach the problem of orbital transfer by feeding back the angular momentum and eccen- tricity vectors. References 1-3 provide excellent bases for the de- velopment of Lyapunov-function-based controllers for formation reconfiguration using orbital elements. Schaub and Alfriend 4 use the local Cartesian coordinates of the Chief/leader satellite and the differential orbital elements between the Deputy/follower satellite and the Chief as feedback. Schaub et al. 5 present two control laws, one in terms of mean elements, and another in terms of inertial co- ordinates, for formation control, with the added simplification of using mean elements in the presence of J2 perturbations arising due

Fixed Thrust Optimal-Fuzzy Combined Control for Spacecraft Formation Flying

The fixed thrust optimal-fuzzy combined formation control method is mainly studied for spacecraft formation flying in circular orbits based on two-body relative dynamics. The linear Hill’s equation is used in relative dynamics, and the circular formation in three dimensions is designed. In fuzzy control, relative position, velocity and acceleration errors are selected as fuzzy variables and the multiplication and weighted average principle are introduced into fuzzy inference so as to maximize the reduction of the coupling effect. And since J2 perturbation is known, it will be taken into account effectively in fuzzy control method. In time-fuel optimal control, the simplified relative error model is used and an optimal controller is provided through the minimum principle. Thus, this paper combines both controls above to form the optimal-fuzzy formation control. Optimal control is adopted when relative position errors reach a certain threshold, while fuzzy control works for the small errors lower than that threshold. Finally, to take circular formation as an example, the simulation results demonstrate that not only does the optimal-fuzzy combined control system have a good response performance, but also excels in fuel consumption.

The study of relative dynamics for spacecraft formation flying in eccentric orbits

Spacecraft formation flying is a key technology for future astronautics. The relative dynamics of formation flying in eccentric orbits is studied, and the relative motion between spacecrafts is analyzed in this paper. Based on the two‐body problem, the extension of Hill equations is achieved and used in relative dynamics of eccentric orbits. Moreover, the transformation of differential variables is applied, and the algebraic solution of the relative motion is obtained, which can be generally used for spacecraft formation flying in eccentric orbits. In addition, the analysis and numerical simulations are given for the relative motion of spacecraft formation flying. The results demonstrate that each spacecraft in eccentric orbits can run in a periodic motion surrounding the master spacecraft under some conditions. And multiple spacecraft can also set up some special formations according to missions.

Stability and control of spacecraft formation flying in trajectories of the restricted three-body problem

This paper addresses the problem of relative position control of spacecraft formation flying (SFF) utilizing the framework of the circular restricted three-body problem (CR3BP) with the Sun and Earth as the primary gravitational bodies. Particularly, the results are not confined to the close vicinity of the collinear Lagrangian libration points. Rather, a linearization is performed relative to an arbitrary non-Keplerian reference trajectory, so that linear time-varying differential equations result. It is rigorously proved that the open-loop linearized SFF dynamics is unstable but controllable. Uncontrolled formations with bounded relative separation that constitute the stable subspace of the dynamics are found using the dual system method. This constitutes a powerful observation, since bounded formations in three-body trajectories have been found thus far only at the vicinity of the Lagrangian equilibrium points. Assuming ideal state measurements, a time-varying continuous linear-quadratic control law is subsequently developed. A complete internal disturbance model is used, rendering a robust disturbance rejection performance. An illustrative example is used to show that the propulsion for deep-space formationkeeping requires a considerable dynamic range. It is concluded that plasma electric propulsion having a micro-thrusting capability is a most suitable means for deep-space formationkeeping.

Adaptive Neural Control of Deep-Space Formation Flying

A novel nonlinear adaptive neural control methodology is presented for the challenging problem of deep-space spacecraft formation flying. When the framework of the circular restricted three-body problem with the sun and Earth as the primary gravitational bodies is utilized, a nonlinear model is developed that describes the relative formation dynamics. This model is not confined to the vicinity of the Lagrangian libration points but rather constitutes the most general nonlinear formulation. Then, a relative position controller is designed that consists of an approximate dynamic model inversion, linear compensation of the ideal feedback linearized model, and an adaptive neural-network-based element designed to compensate for the model inversion errors. The nominal dynamic inversion includes the gravitational forces, whereas the model inversion errors are assumed to stem from disturbances such as fourth-body gravitational effects and solar radiation pressure. The approach is illustrated by simulations, which confirm that the suggested methodology yields excellent tracking and disturbance rejection, thus, permitting submillimeter formation keeping precision.

Adaptive output feedback tracking control of spacecraft formation

AbstractIn this paper, an adaptive, output feedback control design methodology is presented for a spacecraft formation flying (SFF) system. A Lagrangian derivation of the SFF model is considered to produce position dynamics for follower spacecraft #n relative to follower spacecraft #(n−1), where n is an arbitrary positive integer, assuming that the leader spacecraft in the formation follows a no‐thrust, natural, elliptical orbit. Next, a control law is designed to provide a filtered velocity measurement and a desired adaptive compensation with semi‐global, asymptotic, relative position tracking. To show the efficacy of the control algorithm, all desired trajectories are generated online by numerically solving the unperturbed nonlinear SFF dynamics with initial conditions satisfying a no‐thrust, natural orbit constraint equation. The proposed control law is simulated for the case of two and three spacecraft and is shown to yield semi‐global, asymptotic tracking of the relative position in addition to the convergence of disturbance parameter estimates. Copyright © 2002 John Wiley &amp; Sons, Ltd.