Mean Orbital Elements Research Articles

Formation Establishment and Reconfiguration Using Differential Elements in J2-Perturbed Orbits

A computationally efficient algorithm is developed for onboard planning of -impulse fuel-optimal maneuvers for establishment and reconfiguration of spacecraft formations. The method is valid in circular and elliptic orbits and includes first-order secular effects. The dynamics are expressed in terms of differential mean orbital elements, and relations are provided to allow the formation designer to transform these into intuitive geometric quantities for visualization and analysis. The maneuver targeting problem is formulated as an optimal control problem in both continuous and discrete time. The continuous-time formulation cannot be solved directly in an efficient manner, and the discrete-time formulation, which has an analytical solution, does not directly yield the optimal thrust times. Therefore, a new flight-suitable algorithm is designed by iteratively solving the discrete-time formulation while using the continuous-time necessary conditions to refine the thrust times until they converge to the optimal values. Simulation results illustrate the performance for a variety of reconfiguration maneuvers and reference orbits, including examples for the NASA CubeSat Proximity Operations Demonstration mission.

未知の低推力加速運動を行う人工衛星の軌道推定（第2報：接触軌道要素の推定）

As low-thrust propulsion technology becomes increasingly popular, orbital estimation for low-thrust spacecraft may become an area of increasing interest. More frequent use of low-thrust propulsion to place satellites in orbit gives more opportunities for collisions and radio frequency interference as these spacecraft travel slowly through altitude ranges. The purpose of this paper is to develop a method for estimation of the osculating orbital elements for low-thrust spacecraft. To overcome the instability of the estimation problem with low-thrust acceleration, we estimate the mean elements instead of osculating elements. By use of the averaging technique, Hudson and Scheeres proposed an analytical model of secular variations of orbital elements under thrust acceleration. The resulting averaged equation has a nice property in which only a finite number of Fourier coefficients of the thrust acceleration appear because of the orthogonality of the trigonometric function. Based on the nonlinear state equation representation for the extended state variables which include not only orbital elements but also unknown Fourier coefficients, mean orbital elements and thrust history are estimated from perturbed observation data of mean orbital elements. Then, the mapping from mean to osculating elements which is derived from the perturbation theory is used to estimate the osculating elements. The proposed method is demonstrated through numerical simulations.

未知の低推力加速運動を行う人工衛星の軌道推定（第1報：平均軌道要素の推定）

未知の低推力加速運動を行う人工衛星の軌道推定（第1報：平均軌道要素の推定）

解析的な平均軌道要素を用いた静止衛星の高精度軌道保持

Precise position keeping of a geostationary satellite using an electric propulsion system is investigated in this paper. Analytical mean orbital elements of the satellite which represent a long-term motion of the satellite by perturbation forces such as the gravitational effects of the earth’s oblateness, the sun, and the moon, and the solar pressure force are introduced. Then, a control law of the position keeping by using the analytical mean orbital elements is derived where four electrical thrusters are assumed and the control forces are properly distributed into the thrusters. The effectiveness of the proposed control law is demonstrated by numerical simulations.

Satellite cluster flight using on-off cyclic control

Nano-satellite clusters and disaggregated satellites are new concepts in the realm of distributed satellite systems, which require complex cluster management – mainly regulating the maximal and minimal inter-satellite distances on time scales of years – while utilizing simple on-off propulsion systems. The simple actuators and long time scales require judicious astrodynamical modeling coupled with specialized orbit control. This paper offers a satellite cluster orbit control law which works for long time scales in a perturbed environment while utilizing fixed-magnitude thrusters. The main idea is to design a distributed controller which balances the fuel consumption among the satellites, thus mitigating the effect of differential drag perturbations. The underlying methodology utilizes a cyclic control algorithm based on a mean orbital elements feedback. Stability properties of the closed-loop cyclic control system do not adhere to the classical Lyapunov stability theory, so an effort is made to define and implement a suitable stability theory of noncompact equilibria sets. A state selection scheme is proposed for efficiently establishing a low Earth orbit cluster. Several simulations, including a real mission study, and several comparative investigations, are performed to show the strengths of the proposed control law.

Orbital Elements Evolution Due to a Perturbing Body in an Inclined Elliptical Orbit

This paper intends to highlight the effect of the third-body in an inclined orbit on a spacecraft orbiting the primary mass. To achieve this goal, a new origin of coordinate is introduced in the primary and the X-axis toward the node of the spacecraft. The disturbing function is expanded up to the second order using Legendre polynomials. A double-averaged analytical model is exploited to produce the evolutions of mean orbital elements as smooth curves.

Nanosatellite Cluster Keeping Under Thrust Uncertainties

One of the emerging topics in the realm of distributed space systems is cluster flight of nanosatellites. As opposed to formation flight, cluster flight does not dictate strict limits on the geometry of the cluster, and is hence more suitable for implementation in nanosatellites, which usually do not carry highly accurate sensors and actuators. The actuators are usually simple fixed-magnitude thrusters, which are prone to many sources of errors, such as attitude determination and control errors. In this context, the purpose of this paper is to develop a cluster-keeping control law that is capable of long-term operation under thrust uncertainties, assuming fixed-magnitude thrust provided by a simple cold-gas thruster. To that end, mean orbital elements are used for designing an inverse-dynamics controller. It is shown that, in the differential mean elements space, this controller is time-optimal. An adaptive enhancement is developed to mitigate the thrust pointing errors and restore the original optimal performance, thus saving much fuel. Several simulations and comparative studies are performed to validate the analytical results.

Stability and Performance of Orbital Elements Feedback for Cluster Keeping Using Differential Drag

Differential drag (DD) as a means for fuelless satellite cluster keeping is an old idea, but so far using DD-based cluster keeping while relying on mean orbital elements feedback has not been proposed. This paper develops a DD-based maximum distance keeping method that uses Brouwer-Lyddane differential mean elements feedback for long-term control of the secular drift among satellites. The stability of the maximum distance keeping controller is proven using finite-time stability theory, and high-precision simulation results confirm that the new controller is able to arrest satellite relative drift for mission lifetimes exceeding a year. The maximum distance controller is automatically activated, and does not require a pre-determined activation time. Moreover, as a part of a complete DD-based solution for cluster keeping, a collision-avoidance method based on the same controller structure, albeit with differential osculating elements feedback, is developed and validated. Finally, the possibility to regulate cross-track drift with DD is examined, but it is shown that DD can only provide weak controllability in this case.

The orbits and masses of satellites of Pluto

We present the numerically integrated orbits of Pluto’s satellites. The orbits have been fit to a data set that includes Earth-based and Hubble Space Telescope (HST) astrometry of Charon, Nix, Hydra, Kerberos, and Styx, as well as the lightcurves from the Pluto–Charon mutual events. We also report new, 2010–2012 HST astrometry of all satellites including recently discovered Styx plus a pre-discovery detection of Kerberos in 2006. Pluto-relative data sets have been corrected for the center-of-light vs. center-of-mass offsets with the Pluto albedo model. The results are summarized in terms of the postfit residuals, state vectors, and mean orbital elements. Orbits of Charon, Styx, Nix, and Kerberos are nearly circular, while Hydra’s shows a small eccentricity. All satellites are in near-resonance conditions, but we did not uncover any resonant arguments. Our model yields 975.5±1.5km3s−2, 869.6±1.8km3s−2, and 105.9±1.0km3s−2 for the system’s, Pluto’s, and Charon’s GM values. The uncertainties reflect both systematic and random measurement errors. The GM values imply a bulk density of 1.89±0.06gcm−3 for Pluto and 1.72±0.02gcm−3 for Charon. We also obtain GMNix=0.0030±0.0027km3s−2 GMHydra=0.0032±0.0028km3s−2, GMKerberos=0.0011±0.0006km3s−2, and an upper bound on Styx’s GM of 0.0010km3s−2. The 1σ errors are based on the formal covariance from the fit and they reflect only measurement errors. In-orbit (or along the track), radial, and out-of-plane orbital uncertainties at the time of New Horizons encounter are on the order of few tens of km or less for Charon, Nix, and Hydra. Kerberos and Styx have their largest uncertainty component of ∼140km and ∼500km respectively in the in-orbit direction.

Essential Thrust-Fourier-Coefficient Set of Averaged Gauss Equations for Orbital Mechanics

By applying an averaging method to the Gauss equations, the perturbing accelerations acting on a satellite can be represented as a function of 14 constant thrust Fourier coefficients. Time rates of change of mean orbital elements due to these thrust Fourier coefficients are analyzed, and the representation of these thrust coefficients as a function of the change in orbit states is studied. A selected minimum set of six thrust Fourier coefficients is able to provide a finite basis representation of arbitrary orbital maneuvers that allow the authors to dynamically interpolate between states across an unknown maneuver. Using this essential thrust-Fourier-coefficient set, different types of solutions are obtained, and a comparison study of these solutions is also conducted.

Analytic characterization of measurement uncertainty and initial orbit determination on orbital element representations

This paper presents an approach to characterize the uncertainty associated with the state vector obtained from the Herrick-Gibbs orbit determination approach using transformation of variables. The approach is applied to estimate the state vector and its probability density function for objects in low Earth orbit using sparse observations. The state vector and associated uncertainty estimates are computed in Cartesian coordinates and Keplerian elements. The approach is then extended to accommodate the $$J_2$$ perturbation where the state vector is written in terms of mean orbital elements. The results obtained from the analytical approach presented in this paper are validated using Monte Carlo simulations and compared with the often utilized similarity transformation for Kepler, mean, and nonsingular elements. The measurement uncertainty characterization obtained is used to initialize conventional nonlinear filters as well as operate a Bayesian approach for orbit determination and object tracking.

Analytical Derivation of Single-Impulse Maneuvers Guaranteeing Bounded Relative Motion Under

Keeping a cluster of satellites within bounded relative distances requires active control, because natural perturbations (the most significant of which is the term in the geopotential) tend to move the satellites apart. Whereas there is abundant literature on controlling the relative drift using multiple impulsive maneuvers and approximate astrodynamical models involving mean orbital elements, the works that attempt to minimize the number of impulses while using the inertial position and velocity vectors of the satellites are scarce. In this paper, single-impulse distance-keeping maneuvers are derived, without approximating the -perturbed dynamics. An analytical derivation of minimum-fuel impulsive maneuvers between equatorial orbits is provided, while relying on radial period and orbital angle matching conditions. Then, a continuation procedure is used to obtain single-impulse relative distance control between inclined orbits. The development of the impulsive maneuvers relies on the inertial position and velocity vectors of the satellites and does not involve mean elements. In each step, necessary and sufficient conditions for the existence of a single-impulse maneuver are provided. The results are illustrated using a number of realistic scenarios, such as a simulation that includes a gravitational model, drag, and lunisolar attraction.

Determination and Estimation of Differential Mean Orbital Elements for Formation-Flying Missions

Differential mean orbital elements are often taken as control states for relative-motion operations in spacecraft formation-flying missions. This paper investigates the problem of estimating differential mean orbital elements for formation-flying missions. Semi-analytical solutions for the perturbed Earth orbits with a dimensionless and near-circular modified form are introduced. The differential mean orbital elements are estimated from the absolute positions and velocities of the chief spacecraft, as well as the relative positions and velocities between the dual spacecraft. Three representative approaches, including numerical computation, extended Kalman filter, and square-root unscented Kalman filter, are presented in detail for this purpose. Numerical simulations are provided to test the performance of each approach.

Orbit determination by genetic algorithm and application to GEO observation

This paper demonstrates an initial orbit determination method that solves the problem by a genetic algorithm using two well-known solutions for the Lambert’s problem: universal variable method and Battin method. This paper also suggests an intuitive error evaluation method in terms of rotational angle and orbit shape by separating orbit elements into two groups. As reference orbit, mean orbit elements (original two-lines elements) and osculating orbit elements considering the J2 effect are adopted and compared. Our proposed orbit determination method has been tested with actual optical observations of a geosynchronous spacecraft. It should be noted that this demonstration of the orbit determination is limited to one test case. This observation was conducted during approximately 70min on 2013/05/15UT. Our method was compared with the orbit elements propagated by SGP4 using the TLE of the spacecraft. The result indicates that our proposed method had a slightly better performance on estimating orbit shape than Gauss’s methods and Escobal’s method by 120km. In addition, the result of the rotational angle is closer to the osculating orbit elements than the mean orbit elements by 0.02°, which supports that the estimated orbit is valid.

Mean Orbital Elements Estimation for Autonomous Satellite Guidance and Orbit Control

Satellite guidance and orbit control often use mean elements as inputs. Whereas traditional missions can use ground-station-based calculation of mean elements, this is not possible in autonomous satellites, which are required to perform onboard estimation of the mean elements. This problem is not trivial, because analytical satellite theories are not robust to modeling errors and cannot easily accommodate thrust. The purpose of this paper is to develop an effective filtering algorithm for onboard estimation of the mean orbital elements in small-eccentricity low Earth orbits. To that end, a semianalytical astrodynamical model that includes zonal/tesseral/sectorial harmonics and drag is formulated to capture the daily, long-periodic, and secular evolution of the mean orbital elements. The mapping from mean to osculating elements is used as a measurement equation by adding the short-periodic terms. This unique formulation is then fed into a spherical-simplex square-root unscented Kalman filter, which serves as the mean-elements estimator. A comprehensive performance evaluation for both controlled and uncontrolled orbits shows the potential applicability of the method and its advantages compared with Brouwer-based approaches.

Onboard Real-time Estimation of Mean Orbital Elements with Atmospheric Drag

摘要: 为了满足卫星长期自主运行的任务需求, 本文同时考虑了大气阻力摄动和非球形摄动带谐项两种摄动对卫星平均轨道根数的影响, 提出了采用滤波算法在轨实时估计平均轨道根数的方法. 基于指数大气密度模型推导了大气阻力摄动下平均轨道根数的变化率, 分析了非球形带谐项摄动势函数对平均轨道根数的影响并推导了该影响下平均轨道根数变化率的通用计算方法, 由此建立了滤波状态方程以及以瞬时轨道根数为观测量的量测方程并分析了测量噪声特性; 为减小计算量以便于在轨实现,提出采用基于球形单边Sigma采样的平方根UKF (Unscented Kalman filter)滤波来估计平均轨道根数. 数值仿真结果表明, 该算法有效、精度较高且鲁棒性好, 能够满足卫星长期自主在轨实时计算平均轨道根数的要求. 关键词: 平均轨道根数 / 大气阻力摄动 / 带谐项摄动 / UKF滤波 / 蒙特卡洛

Formation flying in elliptic orbits with the J2 perturbation

Relative dynamics between the chief satellite and the deputy ones in formation flying is crucial to maintaining the formation. A good choice of the formation usually requires a lower control frequency or less control energy. For formation flying missions in highly elliptic orbits, the well-known C-W equation is not accurate enough. Instead, Lawden's equation is often used. First, the solution to Lawden's equation with a very simple form is deduced. Then the J2 perturbation is added. It is found that Lawden's solution is not necessarily valid when the J2 perturbation is considered. Completely discarding Lawden's solution and borrowing the idea of mean orbit elements, two rules to initialize the formation are proposed. The deviation speed is greatly reduced. Different from previous studies on the J2 perturbation, except for the relatively simple expression for the semi-major axis, the tedious formulae of the long period terms and the short period terms of other orbital elements are not used. In addition, the deviation speed is further reduced by compensation of the nonlinear effects. Finally, a loose control strategy of the formation is proposed. To test the robustness of this strategy, a third body perturbation is added in numerical simulations.

Second order constraints in the theory of invariant relative orbits including relativistic and direct solar radiation pressure effects

Working with the mean orbit elements, the secular drift of the longitude of the ascending node and the sum of the argument of perigee and mean anomaly are set equal between two neighboring orbits. By having both orbits drift at equal angular rates on the average, they will not separate over time due to the influence of the perturbative effects of the asphericity of the Earth. The relativistic corrections and the direct solar radiation pressure to the equations of motion, as is considered in this work. The problem is stated. The expressions for the time rate of change of the longitude of the ascending node and the sum of the argument of perigee and mean anomaly in terms of the augmented Delaunay canonical elements are obtained. The expressions for the second order conditions that guarantee that the drift rates of two neighboring orbits are equal on the average are derived. Keywords: Invariant relative orbits, Spacecraft flying formations, Relativistic corrections, Direct solar radiation pressure.

Satellite formation design in orbits of high eccentricity with performance constraints specified over a region of interest: MMS phase II

The formation design problem for Phase II of the Magnetospheric Multiscale (MMS) mission is addressed using differential mean orbital elements as design variables and the Gim–Alfriend state transition matrix for relative motion propagation. The MMS mission requires a formation of four satellites in a nearly regular tetrahedron throughout a Region of Interest (RoI), defined near apogee of a highly eccentric orbit. Previous papers have addressed the problem of formation design with performance constraints specified over a RoI, and these methods are now applied to the present problem. In addition, the use of a modified along-track drift condition as a constraint to ensure long-term formation stability is investigated. The performance of the designed formations in the presence of differential semimajor axis errors is also examined. Finally, the results are verified using the NASA General Mission Analysis Tool.

Post-aerocapture orbit selection and maintenance for the Aerofast mission to Mars

Aerofast is the abbreviation of “aerocapture for future space transportation” and represents a project aimed at developing aerocapture techniques with regard to an interplanetary mission to Mars, in the context of the 7th Framework Program, with the financial support of the European Union. This paper describes the fundamental characteristics of the operational orbit after aerocapture for the mission of interest, as well as the related maintenance strategy. The final orbit selection depends on the desired lighting conditions, maximum revisit time of specific target regions, and feasibility of the orbit maintenance strategy. A sunsynchronous, frozen, repeating-ground-track orbit is chosen. First, the period of repetition is such that adjacent ascending node crossings (over the Mars surface) have a separation compatible with the swath of the optical payload. Secondly, the sunsynchronism condition ensures that a given latitude is periodically visited at the same local time, which condition is essential for comparing images of the same region at different epochs. Lastly, the fulfillment of the frozen condition guarantees improved orbit stability with respect to perturbations due to the zonal harmonics of Mars gravitational field. These three fundamental features of the operational orbit lead to determining its mean orbital elements. The evaluation of short and long period effects (e.g., those due to the sectorial harmonics of the gravitational field or to the aerodynamic drag) requires the determination of the osculating orbital elements at an initial reference time. This research describes a simple and accurate approach that leads to numerically determining these initial values, without employing complicated analytical developments. Numerical simulations demonstrate the long-period stability of the orbit when a significant number of harmonics of the gravitational field are taken into account. However, aerodynamic drag produces a relatively slow orbital decay at the altitudes considered for the mission. This circumstance implies the progressive loss of the sunsynchronism condition, and therefore corrective maneuvers are to be performed. This work proves that actually only in-plane maneuvers are necessary, evaluates the overall delta-v budget needed in the period of repetition (85 Martian nodal days), and proposes a simple maintenance strategy, making reference to the worst-case scenario, which corresponds to the highest seasonal values of the atmospheric density and to the maximum value of the ballistic coefficient of the spacecraft.