Equinoctial Orbital Elements Research Articles

Методика балістичного планування місій низькоорбітального сервісного обслугову-вання з малою постійною тягою рушійних систем

The current stage of space exploration is characterized by an increased interest in the development, deployment, and operation of low-orbit satellite constellations (LOSC) for Earth and near-Earth space remote sensing for military and civilian purposes and for global and regional satellite communications. Reusable space launch vehicles have significantly reduced the orbital injection cost. As a result, satellite operators are developing and deploying large-scale LOSCs of various orbital structures with a large number of spacecraft. According to current estimates, more than 70% of all the operating satellites operate in low-Earth orbits (LEOs) at altitudes between 160 km and 2,000 km. Since LEO satellites are generally much cheaper than satellites in geostationary orbits, the possibility of their on-orbit servicing (OOS) has not been the focus of research. However, the use of LEO OOS has prospects for growth. Techniques for ballistic planning of LEO OOS missions have been and are being developed. The disadvantages of approximate techniques include the use of simplified flight dynamics models. Most of the existing exact techniques are based on the use of full mathematical models of flight dynamics and the shooting method to solve the boundary value problem of an orbit transfer. Using the shooting method requires a sufficiently accurate initial guess, which is difficult to determine. To obtain a second approximation, use is mainly made of optimization methods, which do not always find a global minimum. In this regard, there is a need to develop new techniques that would be free from the above disadvantages. The goal of the article is to develop a ballistic planning technique for low-orbit servicing missions with low constant thrust propulsion systems. The technique includes the identification of LEO areas promising for OOS, a mathematical model of the dynamics of perturbed OOS orbit transfers in modified equinoctial orbital elements, and a solution algorithm for the boundary value problem of determining the control parameters of perturbed OOS low-orbit transfers. The problem is solved using methods of statistical analysis, flight dynamics, shooting, genetic optimization, and mathematical simulation. The novelty lies in the identification of LEO areas promising for OOS and the development of a mathematical model of orbit transfer dynamics in modified equinoctial orbital elements and a solution algorithm for determining the control parameters of perturbed OOS low-orbit transfers. The results of the work may be used in the justification and planning of LEO OOS missions and the formulation of requirements for LEO OSS mission propulsion systems.

Optimal Guidance for Heliocentric Orbit Cranking with E-Sail-Propelled Spacecraft

In astrodynamics, orbit cranking is usually referred to as an interplanetary transfer strategy that exploits multiple gravity-assist maneuvers to change both the inclination and eccentricity of the spacecraft osculating orbit without changing the specific mechanical energy, that is, the semimajor axis. In the context of a solar sail-based mission, however, the concept of orbit cranking is typically referred to as a suitable guidance law that is able to (optimally) change the orbital inclination of a circular orbit of an assigned radius in a general heliocentric three-dimensional scenario. In fact, varying the orbital inclination is a challenging maneuver from the point of view of the velocity change, so orbit cranking is an interesting mission application for a propellantless propulsion system. The aim of this paper is to analyze the performance of a spacecraft equipped with an Electric Solar Wind Sail in a cranking maneuver of a heliocentric circular orbit. The maneuver performance is calculated in an optimal framework considering spacecraft dynamics described by modified equinoctial orbital elements. In this context, the paper presents an analytical version of the three-dimensional optimal guidance laws obtained by using the classical Pontryagin’s maximum principle. The set of (analytical) optimal control laws is a new contribution to the Electric Solar Wind Sail-related literature.

Perturbed low-thrust geostationary orbit transfer guidance via polynomial costate estimation

This paper proposes an optimal, robust, and efficient guidance scheme for the perturbed minimum-time low-thrust transfer toward the geostationary orbit. The Earth's oblateness perturbation and shadow are taken into account. It is difficult for a Lyapunov-based or trajectory-tracking guidance method to possess multiple characteristics at the same time, including high guidance optimality, robustness, and onboard computational efficiency. In this work, a concise relationship between the minimum-time transfer problem with orbital averaging and its optimal solution is identified, which reveals that the five averaged initial costates that dominate the optimal thrust direction can be approximately determined by only four initial modified equinoctial orbit elements after a coordinate transformation. Based on this relationship, the optimal averaged trajectories constituting the training dataset are randomly generated around a nominal averaged trajectory. Five polynomial regression models are trained on the training dataset and are regarded as the costate estimators. In the transfer, the spacecraft can obtain the real-time approximate optimal thrust direction by combining the costate estimations provided by the estimators with the current state at any time. Moreover, all these computations onboard are analytical. The simulation results show that the proposed guidance scheme possesses extremely high guidance optimality, robustness, and onboard computational efficiency.

Orbital Uncertainty Propagation Based on Adaptive Gaussian Mixture Model under Generalized Equinoctial Orbital Elements

The number of resident space objects (RSOs) has been steadily increasing over time, posing significant risks to the safe operation of on-orbit assets. The accurate prediction of potential collision events and implementation of effective and nonredundant avoidance maneuvers require the precise estimation of the orbit positions of objects of interest and propagation of their associated uncertainties. Previous research mainly focuses on striking a balance between accurate propagation and efficient computation. A recently proposed approach that integrates uncertainty propagation with different coordinate representations has the potential to achieve such a balance. This paper proposes combining the generalized equinoctial orbital elements (GEqOE) representation with an adaptive Gaussian mixture model (GMM) for uncertainty propagation. Specifically, we implement a reformulation for the orbital dynamics so that the underlying state and the moment feature of the GMM are propagated under the GEqOE coordinates. Starting from an initial Gaussian probability distribution function (PDF), the algorithm iteratively propagates the uncertainty distribution using a detection-splitting module. A differential entropy-based nonlinear detector and a splitting library are utilized to adjust the number of GMM components dynamically. Component splitting is triggered when a predefined threshold of differential entropy is violated, generating several GMM components. The final probability density function (PDF) is obtained by a weighted summation of the component distributions at the target time. Benefiting from the nonlinearity reduction caused by the GEqOE representation, the number of triggered events largely decreases, causing the necessary number of components to maintain uncertainty realism also to decrease, which enables the proposed approach to achieve good performance with much more efficiency. As demonstrated by the results of propagation in three scenarios with different degrees of complexity, compared with the Cartesian-based approach, the proposed approach achieves comparable accuracy to the Monte Carlo method while largely reducing the number of components generated during propagation. Our results confirm that a judicious choice of coordinate representation can significantly improve the performance of uncertainty propagation methods in terms of accuracy and computational efficiency.

The Oblate Lambert Problem: Geometric Formulation and Solution of an Unperturbed, Generalized Lambert Problem Governed by Vinti’s Potential

Numerous methods exist for solving the Lambert problem, the two-point boundary value problem (BVP) governed by two-body dynamics. Many applications would benefit from a solution to a perturbed Lambert problem; a few studies have attempted to solve one. Establishing a larger pool of alternative solution methods gives practitioners greater latitude in choosing the solution that best suits their needs. To that end, a novel Lambert-type BVP is constructed in this work that includes oblateness by way of Vinti’s potential, rendering the problem mathematically unperturbed. This BVP is first defined and then converted to a system of equations that is amenable to an iterative solution. The formulation, which is valid for both the zero- and multiple-revolution problems, couples oblate spheroidal (OS) universal variables and OS equinoctial orbital elements together to sow robustness across all orbital regimes, only excepting orbits that are sufficiently rectilinear. For the first time, the solution space is broadly explored, exposing multiple new insights of significant practical use. Initial guess and root-solve techniques are offered to solve the system of equations. When assessed at Earth for robustness, accuracy, and computational efficiency, the zero-revolution algorithm excels across all three performance metrics, with runtimes averaging only about 15 times slower than a typical two-body Lambert solver. The multiple-revolution algorithm, while not yet evaluated as extensively, also exhibits high levels of performance, the formulation generally characterizing the existence of solutions around oblate bodies more accurately than its Keplerian counterpart.

Revisiting Universal Variables for Robust, Analytical Orbit Propagation Under the Vinti Potential

To meet the growing complexity and demands of modern spacecraft missions, analytical solutions to initial value problems see continued use, typically supporting global searches of large trajectory design spaces. These efforts often employ universal two-body orbit propagators for their recognized speed and robustness, but many applications, like active space debris removal, would benefit from a comparable propagator with greater accuracy. Vinti propagators, which consider planetary oblateness, may serve this purpose, but existing Vinti solutions possess computational difficulties in certain orbital regimes. To mitigate these deficiencies, the present study develops and validates an analytical, third-order universal Vinti propagator free of computational difficulties by leveraging standard, oblate spheroidal (OS) universal variables and OS equinoctial orbital elements. Accuracy of the third-order approximation is assessed for multiple examples across an array of orbital regimes. Computational runtime is also evaluated, and performance is directly compared to the benchmark Vinti6 algorithm. On average, the Vinti propagator implemented in this work is only slower than a typical universal Kepler propagator by a factor of 4.0 and slower than Vinti6 by a factor of 1.8, but with greater robustness than the benchmark. The new form of the equations of motion also has favorable implications for the associated boundary value problem.

Near-Linear Orbit Uncertainty Propagation Using the Generalized Equinoctial Orbital Elements

This paper addresses the problem of minimizing the impact of nonlinearities when dealing with uncertainty propagation in the perturbed two-body problem. The recently introduced generalized equinoctial orbital elements set is employed as a means to reduce nonlinear effects stemming from and higher-order gravity field harmonics. The uncertainty propagation performance of the proposed set of elements in different Earth orbit scenarios, including low-thrust orbit raising, is evaluated using a Cramér–von Mises test on the Mahalanobis distance of the uncertainty distribution. A considerable improvement compared to all sets of elements proposed so far is obtained.

Nonlinear Model Predictive Control of J2-perturbed impulsive transfer trajectories in long-range rendezvous missions

This paper develops a Nonlinear Model Predictive Control (NMPC) strategy for robust tracking of multi-impulse smooth transfer trajectories in J2-perturbed orbital environments. The reference trajectories are designed for long-range rendezvous of servicing satellites with orbiting targets. In the proposed NMPC, the control signals are velocity increments at the impulse times, and the prediction horizon is variable due to the impulsive nature of the reference trajectories. The reference control input and the impulse times are pre-specified by the reference trajectory. Then, the NMPC calculates the optimal correction control vector to track the trajectory with minimum deviation and control effort, during the time between two consecutive impulses. To arrive near the target at the end of the transfer, the optimization in the last horizon is modified to be free-final-time, with an added soft constraint to minimize the final distance between the servicer and the target satellite. To avoid singularities in the mean dynamics of the J2-perturbed servicer, the equinoctial orbital elements are used in the process model. We also include the first-order long-periodic effects in this model. We investigate the robustness of the proposed NMPC in a simulation environment that additionally models: (i) the first-order short-periodic effects, (ii) a bounded uncertain acceleration to capture unmodelled dynamics, and (iii) burning time for satellite thrusters. Finally, an evolutionary optimization algorithm is implemented and embedded in the controller to decrease the computational complexity and to handle impulsive control inputs. Simulation results are provided to illustrate the tracking effectiveness of the developed controller.

Kernel-based ensemble gaussian mixture filtering for orbit determination with sparse data

In this paper, a modified kernel-based ensemble Gaussian mixture filtering (EnGMF) is introduced to produce fast and consistent orbit determination capabilities in a sparse measurement environment. The EnGMF is based on kernel density estimation (KDE) to combine particle filters and Gaussian sum filters. This work proposes using Silverman’s rule of thumb to reduce the computational burden of KDE. Equinoctial orbital elements are used to improve the accuracy of the KDE bandwidth parameter in the modified EnGMF. A bi-fidelity approach to propagation and an adaptation algorithm for selecting the appropriate number of particles are also applied to the EnGMF to reduce the computational burden with an acceptable loss in accuracy for long time propagation. Through numerical simulation, the proposed implementation is compared to state-of-the-art approaches in terms of accuracy, consistency, and computational speed.

A generalization of the equinoctial orbital elements

We introduce six quantities that generalize the equinoctial orbital elements when some or all the perturbing forces that act on the propagated body are derived from a disturbing potential. Three of the elements define a non-osculating ellipse on the orbital plane, other two fix the orientation of the equinoctial reference frame, and the last allows one to determine the true longitude of the body. The Jacobian matrices of the transformations between the new elements and the position and velocity are explicitly given. As a possible application we investigate their use in the propagation of Earth's artificial satellites showing a remarkable improvement compared to the equinoctial orbital elements.

STORM: A semi-analytical orbit propagator for assessing the compliance with Mars Planetary Protection Requirements

STORM is a general semi-analytical propagation code developed at Airbus Defence and Space and originally designed to perform long-term Mars orbit simulations, for mission analysis purposes. Since the 1960’s, humans have sent 18 spacecraft to study the Red Planet, 8 of which are active as of February 2021. While space-debris surveillance and mitigation have been gathering momentum in the past decades in the case of Earth, the issue of artificial objects left in orbit around Mars is another cause for concern in view of the Committee of Space Research’s Planetary Protection Policy. In that perspective, STORM is introduced as a potential tool for assessing the compliance of spacecraft with orbital lifetime requirements. The programme makes it possible to study the evolution of orbits over several decades by propagating a set of mean equinoctial orbital elements, with a comprehensive set of gravitational and non-gravitational perturbations. Particular emphasis is laid on atmospheric entry cases as the programme is interfaced with the European Mars Climate Database, which is the current best knowledge of the state of the Martian atmosphere. This work is focused on the semi-analytical propagator’s implementation and its actual use for predicting entry dates. A sensitivity analysis is conducted to ascertain the influence of atmosphere modelling and initial conditions, providing insight into how to account for uncertainties in a probabilistic way.

On a set of J$$_2$$ equinoctial orbital elements and their use for uncertainty propagation

On a set of J$$_2$$ equinoctial orbital elements and their use for uncertainty propagation

Extended analytical formulae for the perturbed Keplerian motion under low-thrust acceleration and orbital perturbations

This paper presents a collection of analytical formulae that can be used in the long-term propagation of the motion of a spacecraft subject to low-thrust acceleration and orbital perturbations. The paper considers accelerations due to: a low-thrust profile following an inverse square law, gravity perturbations due to the central body gravity field and the third-body gravitational perturbation. The analytical formulae are expressed in terms of non-singular equinoctial elements. The formulae for the third-body gravitational perturbation have been obtained starting from equations for the third-body potential already available in the literature. However, the final analytical formulae for the variation of the equinoctial orbital elements are a novel derivation. The results are validated, for different orbital regimes, using high-precision numerical orbit propagators.

Multi-revolution low-thrust trajectory optimization using symplectic methods

Optimization of low-thrust trajectories that involve a larger number of orbit revolutions is considered a challenging problem. This paper describes a high-precision symplectic method and optimization techniques to solve the minimum-energy low-thrust multi-revolution orbit transfer problem. First, the optimal orbit transfer problem is posed as a constrained nonlinear optimal control problem. Then, the constrained nonlinear optimal control problem is converted into an equivalent linear quadratic form near a reference solution. The reference solution is updated iteratively by solving a sequence of linear-quadratic optimal control sub-problems, until convergence. Each sub-problem is solved via a symplectic method in discrete form. To facilitate the convergence of the algorithm, the spacecraft dynamics are expressed via modified equinoctial elements. Interpolating the non-singular equinoctial orbital elements and the spacecraft mass between the initial point and end point is proven beneficial to accelerate the convergence process. Numerical examples reveal that the proposed method displays high accuracy and efficiency.

Lyapunov-based low-energy low-thrust transfers to the Moon

This paper investigates the numerical computation of low-fuel low-thrust Earth-Moon transfers in a full ephemeris model incorporating the gravitational influence of the Sun, the Moon and all planets of the solar system plus the solar radiation pressure perturbation based on a sample spacecraft area and mass. First, an initial velocity increment to be provided by the launcher is computed. This initial impulse puts the spacecraft on the counterpart in the full ephemeris model of a stable invariant manifold defined in the Sun-Earth Circular Restricted Three-Body Problem. Then, after a coast arc, a closed-loop thrust law is applied to bring the spacecraft to the target lunar orbit. This control law is based on Lyapunov control theory. More precisely, a control-Lyapunov function is defined as the weighted quadratic distance between the first five equinoctial orbital elements of the spacecraft in a Moon-centered reference frame and those defining the target lunar orbit. The control is computed in such a way so as to make the time derivative of the control-Lyapunov function as negative as possible. Numerical results are provided first for a transfer with constant maximum thrust. Then, it is shown that unlike in the case of an open loop control, concentrating the thrust in the vicinity of the perilune and the apolune increases the transfer duration but without reducing the fuel consumption. This is largely due to the uncontrolled effect of the perturbations acting on the spacecraft during the coast arcs. Finally, the robustness of the guidance law against unexpected engine shutdown events is demonstrated.

Analytical Solution to the Vinti Problem in Oblate Spheroidal Equinoctial Orbital Elements

Equinoctial orbital elements were recently generalized from spherical geometry to the oblate spheroidal geometry of Vinti theory. For the symmetric Vinti potential, which accounts exactly for oblateness, these nonsingular elements are defined for all nondegenerate orbital regimes and resolve the usual problems found in the classical elements associated with angle ambiguities. In the present work, the generalized equinoctial elements are used to solve Vinti’s initial value problem, leading to a fully nonsingular analytical solution for bounded orbits. The result is akin to deriving the equinoctial form of Kepler’s equation for the two-body problem and then solving it, formally completing the introduction of the new equinoctial element set. Derivations of the equinoctial integrals of the motion are included, as well as techniques to eliminate all polar orbit singularities and solve a generalized Kepler’s equation. Multiple examples are presented. Code for predicting the Vinti orbit in oblate spheroidal equinoctial elements is provided online as supplementary material.

State-Transition Matrix for Satellite Relative Motion in the Presence of Gravitational Perturbations

A state-transition matrix (STM) for the relative motion of artificial satellites is presented in this work that is valid in the presence of the gravitational spherical harmonic perturbations. The perturbed relative motion is modeled as a first-order variation of the equinoctial orbital elements describing the reference orbit. A satellite theory is derived for the analytical propagation of the reference orbit that includes the secular and periodic effects due to the zonal, sectorial, and tesseral harmonic perturbations. In case of the relative motion STM, the two main goals achieved are as follows. The analytic expressions for the elements of the STM are derived in a generalized form valid for an arbitrary spherical harmonic, and these expressions are in closed-form without using any series expansion in either the eccentricity or the ratio of the mean motion to the angular velocity of the gravitational body. The STM includes the short-period effects due to all the tesseral harmonics in addition to the secular, long-period, and short-period effects due to all the zonal harmonics. Avoidance of the series expansions in the eccentricity ensures that the accuracy of the STM does not degrade for satellite formations in medium to highly eccentric orbits. Additionally, the STM is nonsingular in case of the reference orbits that are near the resonant region; however, the long-period effects due to the resonant tesseral harmonics are ignored in this work. The proposed relative motion STM is implemented in MATLAB, and its accuracy is validated against numerical propagation with a Earth gravity field in NASA’s General Mission Analysis Tool.

Lunar orbit dynamics and maneuvers for Lunisat missions

Lunisat represents a next-generation microsatellite aimed at orbiting the Moon, and equipped with dispensers for the release of nanosatellites. This research is focused on the orbital dynamics of both the main microsatellite and the nanosatellites after release. Due to the irregular concentrations of mass in the Moon, low altitude, near-circular lunar orbits are affected by a considerable number of harmonics of the Moon gravitational field. Nonsingular equinoctial orbit elements are employed for orbit propagations, in conjunction with numerical averaging, which is a numerical technique that allows substantial computational improvements. The dynamical model considers a large number of harmonics of the lunar gravitational field, as well as the Earth and Sun perturbing influence as third bodies. Low-altitude lunar satellites turn out to impact the lunar surface after a few weeks or months. In case of an unsatisfactory lifetime, two simple orbit maintenance strategies are evaluated, together with the related propellant budget. Nanosatellites will be released from Lunisat by means of springs. The orbit of each nanosatellite depends on the release conditions, i.e. the relative velocity magnitude and direction. This work investigates the nanosatellite lifetimes as functions of these conditions. Lastly, minimum-time low-thrust transfers for reducing the orbit altitude are investigated, both for nanosatellite release and for the conclusive phase of the Lunisat mission, which finally terminates with the impact on the lunar surface.

Exact Delaunay normalization of the perturbed Keplerian Hamiltonian with tesseral harmonics

A novel approach for the exact Delaunay normalization of the perturbed Keplerian Hamiltonian with tesseral and sectorial spherical harmonics is presented in this work. It is shown that the exact solution for the Delaunay normalization can be reduced to quadratures by the application of Deprit’s Lie-transform-based perturbation method. Two different series representations of the quadratures, one in powers of the eccentricity and the other in powers of the ratio of the Earth’s angular velocity to the satellite’s mean motion, are derived. The latter series representation produces expressions for the short-period variations that are similar to those obtained from the conventional method of relegation. Alternatively, the quadratures can be evaluated numerically, resulting in more compact expressions for the short-period variations that are valid for an elliptic orbit with an arbitrary value of the eccentricity. Using the proposed methodology for the Delaunay normalization, generalized expressions for the short-period variations of the equinoctial orbital elements, valid for an arbitrary tesseral or sectorial harmonic, are derived. The result is a compact unified artificial satellite theory for the sub-synchronous and super-synchronous orbit regimes, which is nonsingular for the resonant orbits, and is closed-form in the eccentricity as well. The accuracy of the proposed theory is validated by comparison with numerical orbit propagations.

Equinoctial elements for Vinti theory: Generalizations to an oblate spheroidal geometry

Vinti theory constructs orbits on an oblate spheroidal geometry, naturally encoding the gravitational potential of an oblate spheroid in the coordinates. Classical techniques use spherical geometry. Recent work applied Vinti theory to the relative motion problem by way of a linear dynamical model, which is nonsingular in the oblate spheroidal element space. But as with classical spherical elements, the linear mapping between classical spheroidal elements and inertial rectangular coordinates becomes singular for small eccentricities and/or inclinations in the sense of linear dependence of columns in the Jacobian. To mitigate these practical issues, the standard (spherical) equinoctial elements are chosen to inform in a natural way their generalization to a new nonsingular element set: the oblate spheroidal equinoctial orbital elements. The spherical equinoctial elements can be considered a special case of the spheroidal equinoctial elements in the same way that spherical coordinates can be considered a special case of oblate spheroidal coordinates. The new element set is defined and algorithms for converting between spheroidal equinoctial elements and inertial coordinates are derived. Similarities and differences between spheroidal and spherical equinoctial elements are emphasized for clarity, both in terms of the form of equations and geometrical interpretation. The transformations are valid away from the nearly rectilinear orbit regime and are exact except near the poles. When near the poles, the transformations match the accuracy of the approximate analytical solution, which has been developed to the third order in J2 in the literature. As a result, the singularity on the poles is completely eliminated for the first time.