Eccentric Reference Orbit Research Articles

Energy- and time-optimal reconfiguration of spacecraft clusters with collision avoidance

Spacecraft cluster reconfiguration is one of the enabling technologies to ensure non-traditional attributes of distributed space systems. This work treats energy- and time-optimal reconfiguration problems for both circular and eccentric reference orbits. Furthermore, typical local and coupling constraints have been taken into account. Particularly, focus is given to two coupling constraints: final configuration constraints and collision avoidance constraints. For final configuration constraints, a parameterization method is applied to ensure that the reconfiguration problem can be solved as only one optimization problem, rather than a large number of optimization problems resulting from the traditional discretization method. A generalized formulation is proposed for non-convex collision avoidance constraints, which are then convexified via linearization and convex restriction technology. This method provides the affine approximation as a special case. After incorporating above constraints, the reconfiguration problem is formulated as an open-loop optimal control problem, which is solved via the Gauss pseudospectral method (GPM). By virtue of elegant features of GPM, those solutions can serve as a counterpart and stepping stone for a distributed implementation of reconfiguration algorithms. Various simulations demonstrate that minimum-energy/time cluster reconfiguration problems with collision avoidance for circular and eccentric reference orbits can be solved effectively and efficiently using GPM.

Optimal periodic relative orbit and rectilinear relative orbits with eccentric reference orbits

The problem of two-body linearized periodic relative orbits with eccentric reference orbits is studied in this paper. The periodic relative orbit in the target-orbital coordinate system can be used in fly-around and formation-flying orbit design. Based on the closed-form solutions to the Tschauner–Hempel equations, the initial condition for periodic relative orbits is obtained. Then the minimum-fuel periodic-orbit condition with a single impulse is analytically derived for given initial position and velocity vectors. When considering the initial coasting time, the impulse position of the global minimum-fuel periodic orbit is proved to be near to the perigee of the target and can be obtained by numerical optimization algorithms. Moreover, the condition for a special periodic orbit, i.e., the rectilinear relative orbit in the target-orbital frame, is obtained. Numerical simulations are used to demonstrate the efficacy of the method, and show the geometry of the periodic relative orbit and the rectilinear relative orbit.

Design of Satellite Formations in Orbits of High Eccentricity withPerformance Constraints Specified over a Region of Interest

The Magnetospheric Multiscale (MMS) mission requires a formation of four satellites in a nearly regular tetrahedron throughout a Region of Interest (RoI), defined near the apogee of a highly eccentric reference orbit. In this paper, an approach for determining the differential mean orbital element initial conditions for formations in highly eccentric orbits to maximize a Quality Factor (QF) in a RoI is presented. Previous works on the design of formations in high- eccentricity orbits have used numerical integration-based approaches for orbit propagation. We present a fast optimization scheme by using the Gim-Alfriend state transition matrix. Two optimization approaches are presented for the long-term MMS formation design: a single-orbit constrained (SOC) optimization, subject to an along-track drift condition, and a multi-orbit unconstrained (MOU) optimization. A new along-track drift condition is proposed to minimize the along-track drift only in certain portions of the orbit and is shown to produce much more stable long-term behavior of the QF than the previous condition. A verification of the results sing the NASA General Mission Analysis Tool is provided. Under ideal conditions, i.e., without errors in the establishment of the initial conditions, the best formation can satisfy the MMS mission QF requirements for over 80 days without corrective action.

Optimal Formation Design for Magnetospheric Multiscale Mission Using Differential Orbital Elements

The Magnetospheric Multiscale Mission requires a formation of four satellites in a nearly regular tetrahedron throughout a region of interest defined near the apogee of a highly eccentric reference orbit. Previous papers have addressed the design of formations in orbits of high eccentricity to maximize a quality factor in a region of interest, including the use of differential mean orbital elements as design variables. In this paper, a robust optimization method is presented to improve formation performance in the presence of formation initialization errors. Several design methods are analyzed by applying differential semimajor axis errors, which have a strong effect on the long-term stability of spacecraft formations. It is shown that large formations can satisfy mission requirements for a longer time than smaller formations, when the same magnitude of errors are considered, and generally exhibit less variation in quality factors due to these errors. The robust optimization method is applied to these smaller formations and produces results that are much more stable when semimajor axis errors are included, at a cost of some performance in the nominal error-free case. The results are verified using the NASA General Mission Analysis Tool and are shown to be reasonably accurate, except in predicting very long-term behavior. A physical analysis of the geometry of several magnetospheric multiscale formation designs is provided, and eight distinct optimal tetrahedron orientations are identified (two configurations, in which the chief satellite can be placed at any of the four vertices).

Periodic Orbits of Nonlinear Relative Dynamics Along an Eccentric Orbit

This paper is concerned with formation acquisition and reconfiguration problems with an eccentric reference orbit. As a first step, the characterization problem is considered for all initial conditions that constitute periodic solutions of the nonlinear equations of relative motion. Under the condition that the inertial orbits of the leader and a follower are coplanar, initial conditions of all periodic relative orbits are generated in terms of parameters of their orbits and their initial positions. Then the inertial orbit of the follower is rotated successively around the two axes of its perifocal reference system, and the initial conditions of all noncoplanar relative orbits are derived. Based on these periodic relative orbits, formation acquisition and reconfiguration problems by a feedback controller are formulated. The main performance index of a feedback controller is the total velocity change during the operation. Using the property of null controllability with vanishing energy of the Tschauner-Hempel equations, suboptimal controllers are designed via the differential Riccati equation of the linear regulator theory of periodic systems.

An alternative representation of the relative motion: The local orbital elements

The study of the relative motion between two very-close non-massive objects has been usually done through Hill equations. More recent approaches use the differences of orbital elements to extrapolate relative motion. In this paper, we present another possible representation of the relative motion. This representation takes advantage of the very particular most common relative trajectory: the ellipse. Our representation induces a better temporal comprehension of the motion but it also provides the topology of the natural relative motions. Our approach is useful for circular and slightly eccentric reference orbits.

Effects of Perturbation on Formation Flights around an Eccentric Reference Orbit

In this study, we propose a basic method for the construction of a formation flight around an eccentric reference orbit under the influence of gravitational disturbances (J2). The relative motion between two spacecraft in the formation flight is affected by J2. The effects of this perturbing force are analyzed, and the initial conditions of the formation flight for the suppression of these effects are derived on the basis of these analytical results.

Satellite Formation Design and Optimal Stationkeeping Considering Nonlinearity and Eccentricity

F ORMATION flying has received much attention in recent years because of the possible advantages of replacing a single, complex satellite with a cluster of smaller ones. Flying a formation of satellites offers improved flexibility and redundancy and the ability to construct much large virtual sensors than can be flown on a single, monolithic satellite. Several missions have identified formation flying as an enabling technology for increasing the science return and reducing the total mission costs [1,2]. A critical problem in the formation-flying application is the formation design. The purpose of formation design is twofold. First, various flying missions have different requirements for formation design, such as the Aperture Synthesis Radar mission, the Earth Observer-1 mission, the Laser Interferometer Space Antenna mission, etc. The primary purpose of formation design is searching the proper formation array that fulfills the formation mission requirement. The scientific or other need of the mission is the main constraint for the formation design and can be defined as the mission constraint for the formation design. Second, formation-flying satellites operate on orbits with long time spans, in general, from several months to several years, and frequent thruster firings to keeping formation will consume so much propellant and cannot be accepted bymany consideringmissions. And so, it is more important to design nature periodic relative motion orbits and avoid the secular drift. This can be defined as the orbit constraint for formation design. The problem of satellite formation design has been studied by many researchers and many advances have been made. Sabol et al. [3] used the Hill’s equations to design formation and proposed four special formations for various formation-flying missions. Hill’s equations are the constant coefficient differential equations and have simple analytical solutions with an in-track secular term in the solutions. The drift can be avoided by a proper initialization. These simple analytical expressions permit the use of intuitive methods to design formations. However, Hill’s equations can only design valid formations for circular reference orbits and linearized relative motion. Carter et al. [4–6] studied the relative motion of two vehicles in nearby elliptical orbits and presented an analytical solution under assumptions of linearization and without perturbations, and developed an initialization procedure to obtain initial conditions for the formation satellites period relative motion at reference orbit perigee. But, the solutions of Carter’s have a definite integral and are complex in form and cannot be conveniently applied in formation design. Using orbital element differences and Cartesian coordinates, respectively, Lane and Axelrad [7] and Xing et al. [8] derived the simple periodic analytic solutions in eccentric orbits under assumptions of linearization. Theseworks investigated the linearized problem of relative motion of formation flying. Considering nonlinearity and eccentricity, Gurfil [9] and Xing et al. [10,11] independently proposed generalized periodic relative motion conditions (GPRMC) for formation flying on arbitrary Keplerian elliptic orbits. This paper focuses on the formation design for nonlinear relative motion and eccentric reference orbits under GPRMC. Because there are some linearized errors for formation design usingHill’s equations and solutions of Xing’s [8], a new method is proposed to design formation considering nonlinearity and eccentricity which use GPRMC to correct the initial conditions derived from Hill’s equations or Xing’s solutions and get the periodic relative motion orbits for formation flying. This approach is attractive for two reasons. First, the process of formation design uses the GPRMC and removes the secular drift of relative motion orbit designed by Hill’s equations or Xing’s solutions and avoids frequent thruster firings to keep formation. More important, the approach adequately uses the previous research results which were derived from Hill’s equations and could use the intuitive methods to design formations for various formation-flying missions. It is assumed that the formation designed by Hill’s equations fulfilled the requirement of the reality formation-flying mission. Considering nonlinearity and eccentricity, the designed formations were not perfectly consistent with Hill’s. The remainder of this paper developed an optimal impulse formation-keeping maneuver based on the orbit constraint (GPRMC) and the mission constraint (Hill’s solutions). We used the orbit constraint and the mission constraint to keep periodic relativemotion and fulfill the formation-flyingmission requirement, respectively.

Design and verification of a Robust formation keeping controller

Formation keeping is a key technology of distributed satellite system. In order to deal with the satellite formation keeping problem including perturbations, normalized linear equations of satellite relative motion including J 2 perturbation are derived. Then a sliding mode variable structure control with matching uncertainty and disturbance is introduced, the method using a reaching law to design VSC controller directly for the discrete time system. Then control law for formation keeping of satellite formation in LEO with small eccentricity reference orbit is designed, so the modeling error and disturbance is included. A simulation environment with layer control architecture and nonlinear dynamics including perturbations are developed to verify the control algorithms. Simulations for various system parameters are conducted for both circular reference orbit and eccentric reference orbit. Simulation results established the robustness and performance of the controller.

RESILIENT STRUCTURED PERIODIC H2 SYNTHESIS FOR A SPACECRAFT IN ELLIPTICAL ORBITS

Station-keeping of a small spacecraft is studied and a solution based on resilient periodic state-feedback control laws is proposed. Linearized equations of the relative motion of the satellite near an eccentric reference orbit are derived in the presence of the second zonal gravitational harmonic J2 and atmospheric drag perturbations. After a discretization of the model, a resilient H2 state-feedback control law is computed by a linear matrix inequality-based algorithm. Illustrative nonlinear simulations show the efficiency of the approach.

Decentralized Formation Flying Control in a Multiple‐Team Hierarchy

In recent years, formation flying has been recognized as an enabling technology for a variety of mission concepts in both the scientific and defense arenas. Examples of developing missions at NASA include magnetospheric multiscale (MMS), solar imaging radio array (SIRA), and terrestrial planet finder (TPF). For each of these missions, a multiple satellite approach is required in order to accomplish the large-scale geometries imposed by the science objectives. In addition, the paradigm shift of using a multiple satellite cluster rather than a large, monolithic spacecraft has also been motivated by the expected benefits of increased robustness, greater flexibility, and reduced cost. However, the operational costs of monitoring and commanding a fleet of close-orbiting satellites is likely to be unreasonable unless the onboard software is sufficiently autonomous, robust, and scalable to large clusters. This paper presents the prototype of a system that addresses these objectives-a decentralized guidance and control system that is distributed across spacecraft using a multiple team framework. The objective is to divide large clusters into teams of "manageable" size, so that the communication and computation demands driven by N decentralized units are related to the number of satellites in a team rather than the entire cluster. The system is designed to provide a high level of autonomy, to support clusters with large numbers of satellites, to enable the number of spacecraft in the cluster to change post-launch, and to provide for on-orbit software modification. The distributed guidance and control system will be implemented in an object-oriented style using a messaging architecture for networking and threaded applications (MANTA). In this architecture, tasks may be remotely added, removed, or replaced post launch to increase mission flexibility and robustness. This built-in adaptability will allow software modifications to be made on-orbit in a robust manner. The prototype system, which is implemented in Matlab, emulates the object-oriented and message-passing features of the MANTA software. In this paper, the multiple team organization of the cluster is described, and the modular software architecture is presented. The relative dynamics in eccentric reference orbits is reviewed, and families of periodic, relative trajectories are identified, expressed as sets of static geometric parameters. The guidance law design is presented, and an example reconfiguration scenario is used to illustrate the distributed process of assigning geometric goals to the cluster. Next, a decentralized maneuver planning approach is presented that utilizes linear-programming methods to enact reconfiguration and coarse formation keeping maneuvers. Finally, a method for performing online collision avoidance is discussed, and an example is provided to gauge its performance.

타원궤도의 위성편대비행을 위한 초기조건 결정

The initial conditions that generate bounded motion in eccentric reference orbit are determined for satellite formation flying. Because Hill`s equations cannot describe the relative motion between two satellites in eccentric orbit, a new relative dynamics utilizing the nonlinearity and eccentricity correction for Hill`s initial conditions is implemented. The constraint that matches angular rates of chief and deputy satellites is used to obtain the bounded motion between them. The constraint can be applied to satellite formation motions in eccentric orbit, since it implicates J2 perturbation due to the central body`s aspherical gravitational forces. The periodic bounded motions are analyzed for the orbit with the eccentricity of less than 0.05 and about 0.5 km relative distance between chief and deputy satellites. It is mainly illustrated that the satellite formations in small eccentric orbits can have hounded motions; consequently, the formation can be kept by matching angular rates of the satellites. These results demonstrate an useful method that reduces the cost for operating satellites by providing effective initial conditions for satellite formation flying in eccentric orbit.

Relative Dynamics and Control of Spacecraft Formations in Eccentric Orbits

Formation eying is a key technology for both deep-space and orbital applications that involve multiple spacecraft. Many future space applications will beneet from using formation e ying technologies to perform distributed observations (e.g., synthetic apertures for Earth mapping interferometry) and to provide improved coverage for communication and surveillance. Previous research has focused on designing passive apertures for these formation e ying missions assuming a circular reference orbit. Those design approaches are extended and a complete initialization procedure for a large e eet of vehicles with an eccentric reference orbit is presented. The main result is derived from the homogenous solutions of the linearized relative equations of motion for the spacecraft. These solutions are used to end the necessary conditions on the initial states that produce T-periodic solutions that have the vehicles returning to the initial relative states at the end of each orbit, that is, v(t0)=v(t0+T). This periodicity condition and the resulting initialization procedure are originally given (in compact form) at the reference orbit perigee, butthis is alsogeneralized to enable initialization atanypoint around thereference orbit. In particular, an algorithm is given that minimizes the fuel cost associated with initializingthe vehicle states (primarily the in-track and radial relative velocities) to values that are consistent with periodic relative motion. These algorithms extend and generalize previously published solutions for passive aperture forming with circular orbits. The periodicity condition and the homogenous solutions can also be used to estimate relative motion errors and the approximate fuel cost associated with neglecting the eccentricity in the reference orbit. The nonlinear simulations presented clearlyshowthatignoringthereferenceorbiteccentricitygeneratesanerrorthatiscomparabletothedisturbances caused by differential gravity accelerations.