Orbit Propagation Research Articles

Too-Short-Arc Association and Clustering Method for Space-Based Optical Observations

Space-based optical observation is assuming an increasingly significant role in space surveillance and tracking. However, it often yields tracks that are too short, which are known as too short arcs (TSAs). A single TSA lacks sufficient information for reliable initial orbit determination of a resident space object. Therefore, it is typically necessary to perform tracklet association during catalog building and maintenance. This study presents a novel tracklet association method that improves upon existing techniques by combining admissible regions and nonlinear orbital uncertainty propagation. Proceeding by constructing TSA association matrices and TSA clustering matrices, the bond energy algorithm, traditionally employed in the design of distributed database systems, is applied to cluster TSAs based on the association results between two arbitrary TSAs. Furthermore, the proposed splitting algorithm reduces the computational load associated with clustering multiple tracklets and effectively handles erroneous association results. To assess the effectiveness of this approach, it was tested in a space-based optical-survey scenario, accounting for practical observation uncertainties. The simulation results demonstrate that the method achieves a high level of accuracy in both TSA association and clustering.

An accurate and efficient second-order J2 model for the draper semianalytic satellite theory

Semi-analytical orbit propagation methods combine the advantages of analytical and numerical methods to achieve an excellent balance between efficiency and accuracy. The increasingly congested space has led to the growing prevalence of semi-analytical methods, driven by the requirements for efficient and accurate orbit propagation in Space Situational Awareness (SSA). However, the accuracy of semi-analytical methods is only retained by considering the contribution of the J2 zonal harmonic second order terms, namely the J22 term, which has not been completely modeled in semi-analytical methods due to the complex mathematical formulations. This paper proposes a complete semi-analytical solution for the J22 contribution consistent with the Draper Semianalytic Satellite Theory (DSST), including the fully closed-form and nonsingular mean element rate equations and computationally efficient short-period terms. Firstly, for the direct application of DSST in the construction of the J22 solution, the difficulties of high computational costs induced by multiplying infinite Fourier series together are analyzed. Next, an improved construction process for the J22 solution is developed by introducing a new general form of the partial derivatives of osculating elements rates to eliminate these difficulties. Considering the complexity of the construction process, the computation details for each term in this process are then presented. Numerical experiments demonstrate that the proposed J22 solution achieves remarkable accuracy and efficiency compared to previous works.

Characterization and verification of the optimal feedback gain of a satellite magnetorquer-based attitude control system

In spacecraft mission planning and operation, the attitude determination and control subsystem (ADCS) of a satellite provides information about the orientation of the satellite in the inertial reference frame. Furthermore, this subsystem produces the control actions required to adjust the orientation of the satellite, especially in the low-Earth orbit (LEO) regime. This paper focuses on the satellite’s three-axis attitude control problem within the context of active and passive control, which includes detumbling control, pointing control, magnetic control, and attitude stabilization after solar panel wing deployment using magnetorquers as the primary actuators. The objective is to stabilize and reduce the angular rate while orienting the satellite to the desired attitude. The proposed satellite attitude control system (ACS) strategies are designed, developed, characterized, and verified. These strategies encompass the B-dot control algorithm for detumbling control along with pointing control and attitude stabilization after solar panel wing deployment. hardware-in-the-loop simulation (HiLs) tests are conducted to assess the performance of the satellite magnetorquer-based ACS in the presence of noise. These tests involve a relative Earth’s magnetic field (EMF) generator in conjunction with SGP-4-based satellite orbital propagator high-level control software. Additionally, cascade proportional-integral-derivative (PID) and state-dependent Riccati equation (SDRE) controllers are implemented to generate sufficient torque using three-axis magnetorquers on a frictionless air-bearing platform. The platform is balanced to closely simulate the dynamic motion of a spacecraft in space. The testing includes a single initial condition and three inertia conditions for stabilization after solar panel wing deployment. Finally, the effectiveness of the cosimulation as a primary experiment through an integrated HiLs process is validated. This comprehensive approach confirms the control system’s performance and its ability to meet mission requirements.

Determination of the spacecraft’s spin axis orientation. Photometric patterns method

Near-Earth space is becoming increasingly congested; the number of satellites in low Earth orbit (LEO), where there is already the greatest spatial density of objects (including man-made debris), is increasing rapidly. Knowing the status of non-cooperative space objects is one of the requirements of space situational awareness (SSA). Assessing the dynamic properties of large inactive satellites and rocket bodies, such as their rotation period and the spatial location of the rotation axis, is necessary to predict their orientation. This information is critical to both the success of active debris removal (ADR) missions and improved orbital propagation of objects in LEO. Monitoring the state of RSO is carried out by various means, including using ground-based optical sensors by collecting photometric data, processing it and analyzing light curves. This paper presents a new method for estimating the orientation of the RSO rotation axis in space. This method relies on structural analysis of RSO light curves and the search for similar fragments, called ”photometric patterns,” in observations obtained from one or several sites simultaneously or over a short period of time. The method does not require knowledge of the RSO shape and does not impose strict requirements on the quality of photometric observations.

SPRISS: Scalable and precise resistive impact sensor for smallsats, architecture description and tests

Space debris are a tangible risk for satellites in Earth orbits. Indeed, in the last decades fragmentation events have generated a large number of uncontrolled objects that have made the debris population grow. Modelling the space debris environment is becoming a fundamental task to evaluate the vulnerability of operational satellites, the probability of accidental collisions with uncontrolled objects, and the evolution of the debris population. With this aim, remote and in-situ measurements provide valuable data to tune the space debris population models and improve their reliability. While large satellites can be observed and tracked from ground, the sub-millimeter debris population requires in-situ measurements. In this context, in-orbit impact sensors are a key technology to obtain information about the sub-mm space debris environment. In this framework, a small-scale impact sensor sized to be integrated in a 2U CubeSat is being developed at the University of Padova. The sensor consists of a multitude of thin, conductive stripes arranged on a thin film of non-conductive material. When a debris hits the sensor, one or more stripes are severed, and the impact is detected. Moreover, the sensor design ensures low power consumption, making it feasible for CubeSat space missions. This work presents the latest outcomes obtained from the development of the sensor. Specifically, structural analyses are performed to assess that the sensor can withstand the launch loads, as well as thermal analyses to confirm its endurance capability with in-orbit temperatures. The number of expected impacts during the mission is predicted through orbital propagation, using state-of-the-art debris environment modelling tools. Finally, the paper presents a functional shooting test and vibration tests, executed onto a development model of the sensor, which verify its functionality and validate a Technology Readiness Level (TRL) of 4.

Preliminary Exploration of Coverage for Moon-Based/HEO Spaceborne Bistatic SAR Earth Observation in Polar Regions

To address the challenge of achieving both temporal consistency and spatial continuity in Earth observation data of polar regions, this paper proposes an innovative concept of Moon-based/Highly Elliptical Orbit (HEO) Spaceborne Bistatic Synthetic Aperture Radar (MH-BiSAR), with transmitters on the Moon and receivers on HEO satellites. By utilizing ephemeris data and an orbit propagator, this study explores MH-BiSAR’s geometric coverage capabilities in polar regions and conducts a preliminary analysis of its characteristics. The findings reveal that MH-BiSAR could provide continuous multi-day revisit observations of polar regions within each sidereal month, presenting a significant advantage for monitoring high-dynamic and large-scale scientific phenomena, such as polar sea ice observations. This innovative observational method offers a new perspective for polar monitoring and is expected to deepen our understanding of polar phenomena.

Configuration uncertainty propagation of gravitational-wave observatory using a directional state transition tensor

Configuration stability is essential for a space-based Gravitational-Wave (GW) observatory, which can be impacted by orbit insertion uncertainties. Configuration uncertainty propagation is vital for investigating the influences of uncertainties on configuration stability and can be potentially useful in the navigation and control of GW observatories. Current methods suffer from drawbacks related to high computational burden. To this end, a Radial-Tangential-Ddirectional State Transition Tensor (RT-DSTT)-based configuration uncertainty propagation method is proposed. First, two sensitive directions are found by capturing the dominant secular terms. Considering the orbit insertion errors along the two sensitive directions only, a reduced-order RT-DSTT model is developed for orbital uncertainty propagation. Then, the relationship between the uncertainties in the orbital states and the uncertainties in the configuration stability indexes is mapped using high-order derivatives. The result is a semi-analytical solution that can predict the deviations in the configuration stability indexes given orbit insertion errors. The potential application of the proposed RT-DSTT-based method in calculating the feasible domain is presented. The performance of the proposed method is validated on the Laser Interferometer Space Antenna (LISA) project. Simulations show that the proposed method can provide similar results to the STT-based method but requires only half of the computational time.

Effects of Earth’s Oblateness on Black Hole Imaging through Earth–Space and Space–Space Very Long Baseline Interferometry

Earth-based very long baseline interferometry (VLBI) has made rapid advances in imaging black holes. However, due to the limitations imposed on terrestrial VLBI by the Earth’s finite size and turbulent atmosphere, it is imperative to have a space-based component in future VLBI missions. This paper investigates the effect of the Earth’s oblateness, also known as the J 2 effect, on orbiters in Earth–space and space–space VLBI. The paper provides an extensive discussion on how the J 2 effect can directly impact orbit selection for black hole observations and how, through informed choices of orbital parameters, the effect can be used to a mission’s advantage, a fact that has not been addressed in previous space VLBI investigations. We provide a comprehensive study of how the orbital parameters of several current space VLBI proposals will vary specifically due to the J 2 effect. For black hole accretion flow targets of interest, we demonstrate how the J 2 effect leads to a modest increase in shorter-baseline coverage, filling gaps in the (u, v) plane. Subsequently, we construct a simple analytical formalism that allows isolation of the impact of the J 2 effect on the (u, v) plane without requiring computationally intensive orbit propagation simulations. By directly constructing (u, v) coverage using J 2-affected and J 2-invariant equations of motion, we obtain distinct coverage patterns for M87* and Sgr A* that show extremely dense coverage on short baselines as well as long-term orbital stability on longer baselines.

Multivariate Attention-Based Orbit Uncertainty Propagation and Orbit Determination Method for Earth–Jupiter Transfer

Current orbit uncertainty propagation (OUP) and orbit determination (OD) methods suffer from drawbacks related to high computational burden, limiting their applications in deep space missions. To this end, this paper proposes a multivariate attention-based method for efficient OUP and OD of Earth–Jupiter transfer. First, a neural network-based OD framework is utilized, in which the orbit propagation process in a traditional unscented transform (UT) and unscented Kalman filter (UKF) is replaced by the neural network. Then, the sample structure of training the neural network for the Earth–Jupiter transfer is discussed and designed. In addition, a method for efficiently generating a large number of samples for the Earth–Jupiter transfer is presented. Next, a multivariate attention-based neural network (MANN) is designed for orbit propagation, which shows better capacity in terms of accuracy and generalization than the deep neural network. Finally, the proposed method is successfully applied to solve the OD problem in an Earth–Jupiter transfer. Simulations show that the proposed method can obtain a similar estimation to the UKF while saving more than 90% of the computational cost.

Thermospheric Density Estimation Method Using a First-Order Gauss–Markov Process

Low-Earth-orbit (LEO) spacecraft are significantly influenced by atmospheric drag. Accurately estimating thermospheric density is pivotal for the precise calculation of drag acceleration. However, thermospheric density along a specific orbit, computed using existing thermospheric models, has certain inaccuracies. In this work, a first-order Gauss–Markov process is used to model the deviation of atmospheric drag acceleration. With the Markov parameter of the initial state iteratively computed through sequential estimation and the smoothing method, the thermospheric density is derived from high-precision GPS measurements. In simulation scenarios, the root-mean-square error and relative error of the estimated thermospheric density reduce by about 45 and 50% relative to the prior density, respectively. Using the estimated density for orbit propagation, satellite trajectories’ one-day position and velocity error are, respectively, within 100 m and 0.1 m/s, and an average improvement in orbit precision is over 80%. The proposed method has been applied to the real Tsinghua Science Satellite (Q-SAT) GPS measurements for effectiveness verification. It shows strong adaptability under extreme space weather and during the occurrence of geomagnetic storms. Due to the estimated Markov parameter of the initial state obeying the Langevin dynamics properties, the proposed method also offers short-term thermospheric density forecasting potential.

Low-Order Automatic Domain Splitting Approach for Nonlinear Uncertainty Mapping

This paper introduces a novel method for the automatic detection and handling of nonlinearities in a generic transformation. A nonlinearity index that exploits second-order Taylor expansions and polynomial bounding techniques is first introduced to estimate the Jacobian variation of a nonlinear transformation. This index is then embedded into a low-order automatic domain splitting algorithm that accurately describes the mapping of an initial uncertainty set through a generic nonlinear transformation by splitting the domain whenever nonlinearities grow above a predefined threshold. The algorithm is illustrated in the critical case of orbital uncertainty propagation, and it is coupled with a tailored merging process that limits the growth of the domains in time by recombining them when nonlinearities decrease. The low-order automatic domain splitting algorithm is then combined with Gaussian mixture models to accurately describe the propagation of a probability density function.

Configuration Design Method of Mega Constellation for Low Earth Orbit Observation

The configuration optimization design of Low Earth Orbit observation mega constellation in complex space environment is a nonlinear problem that is difficult to solve analytically. In this paper, a constellation design method is proposed, considering satellite imaging width, formation flying of subgroup satellites, and global uniform coverage by payloads. Firstly, a configuration of satellites with the same subsatellite trajectory is proposed, and its orbital analytical expression under J2 perturbation is provided. Then, the relative motion feature points are extracted near the orbit of each satellite, and a group of uniform natural accompanying satellites are set to corresponding points. Afterwards, the orbit parameters of satellite and its companions are set as initial values, and the precise orbits under the High Precision Orbit Propagator model are solved in the neighborhood by using the Nondominated Sort Particle Swarm Optimization algorithm. Finally, the correctness of the configuration design method is verified by numerical simulation.

Preliminary Debris Risk Analysis for Mega-Constellation Architectures After Orbit-Raising Fragmentation Event

This paper presents a theoretical examination of the potential debris conjunction dangers faced by mega-constellations in both low Earth orbit (LEO) and medium Earth orbit (MEO). The analysis focuses on the risks posed by debris fields created by breakups occurring during an orbit-raising maneuver for vehicle replacement and/or capability reconstitution, using current telecommunications mega-constellations, such as Starlink and OneWeb, as examples. The mega-constellation designs consist of 750 and 150 satellites arranged using the Walker-Delta design for the LEO and MEO cases, respectively. The research employs physics-based orbital propagation and Monte Carlo simulations to evaluate the potential consequences of a single satellite breakup during orbit raising. The results of the simulations are used to calculate the probability of catastrophic collision and compare the debris risk between the LEO and MEO mega-constellations, with a bimodality analysis conducted for the MEO constellation. Monte Carlo analysis indicates that the LEO mega-constellation features the highest percentage of potential catastrophic collisions between debris fragments and the mega-constellation. Specifically, satellite breakup events starting within an altitude range of 100–199 km below the LEO mega-constellation, or approximately at the midpoint of a Hohmann transfer from a 300-km parking orbit, pose the greatest risk to the constellation.

Peculiar Orbital Characteristics of Earth Quasi-Satellite 469219 Kamo‘oalewa: Implications for the Yarkovsky Detection and Orbital Uncertainty Propagation

469219 Kamo‘oalewa is selected as one of the primary targets of Tianwen-2 mission, which is currently believed to be the most stable quasi-satellite of Earth. Here we derive a weak detection of the Yarkovsky effect for Kamo‘oalewa, giving A 2 = (−1.075 ± 0.447) × 10−13 au day−2, with the available ground-based optical observations from Minor Planet Center and a relatively conservative weighting scheme. Due to the quasi-satellite resonance with Earth, we show that the detection of the Yarkovsky effect by orbital fitting with astrometric observations becomes difficult, as its orbital drift shows a slow oscillatory growth resulting from the Yarkovsky effect. In addition, we extensively explore the characteristics of orbital uncertainty propagation and find that the positional uncertainty mainly arises from the geocentric radial direction in 2010–2020 and then concentrates in the heliocentric transverse direction in 2020–2030. Furthermore, the heliocentric transverse uncertainty is clearly monthly dependent, which can arrive at a minimum around January and a maximum around July as the orbit moves toward the leading and trailing edges, respectively, in 2025–2027. Finally, we investigate a long-term uncertainty propagation in the quasi-satellite regime, implying that the quasi-satellite resonance with Earth may play a crucial role in constraining the increase of uncertainty over time. Such an interesting feature further implies that the orbital precision of Kamo‘oalewa is relatively stable at its quasi-satellite phase, which may also be true for other quasi-satellites of Earth.

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.

Effect of solar free oscillations on TianQin’s range acceleration noise

TianQin is a proposed space-based gravitational-wave detector mission to be deployed and operated in high Earth orbits. As a sequel to Zhang et al (2021 Phys. Rev. D 103 062001), we investigate a type of ‘orbital noise’ in TianQin’s range acceleration that is caused by gravitational perturbation associated with solar free oscillations. Frequencies of such oscillations are typically within TianQin’s measurement band of 0.1 mHz–1 Hz, and the disturbance level needs careful assessment. By using high-precision orbit propagation and adding the Sun’s time-variable oblateness J 2 to detailed gravity-field models, we examine the effect in the frequency domain and show that the solar free oscillation noise is expected to be two orders of magnitude lower than the noise requirement on single links and hence has little impact on the mission.

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.

Redesign of high-precision reference orbit for interferometric SAR satellite with injection error

A two-step optimization algorithm is proposed to redesign a precision reference repeating sun-synchronous orbit for interferometric synthetic aperture radar satellites with injection error. This purpose is to design a new reference orbit for the satellite that deviates from the original preset orbit. Manoeuvring satellites to this new orbit consumes less propellant than manoeuvring to the original orbit. In addition, the repeatability of the orbit is a guarantee for obtaining good interference images, which puts higher requirements on the revisit accuracy of the redesign reference orbit. In the first step, the initial guess for reference orbital elements is designed by considering the global coverage of the Earth and the propellant consumption, taking into account the J2 and J4 perturbations. A search table of feasible solutions in the neighbourhood of the injection orbit is established. This approach, which is based on a search table, transforms the optimization of continuous variables in infinite space into the traversal of discrete variables in finite space, thus greatly reducing the optimization time. In the second step, the revisit accuracy of the satellite’s full-dimensional state vector is optimized under the orbit propagation of a high-order gravity field. Unlike traditional methods that only optimize the revisit error of position, this paper also optimizes that of velocity. A new optimization method for full-dimensional revisit error of state vector is proposed, which combines nested differential correction with non-dominated sorting genetic algorithm-II. Compared with the traditional genetic algorithms, the proposed nested differential correction method provides the non-dominated sorting genetic algorithm-II with more accurate initial values, which reduces the iterations. Finally, the simulation results show that the total velocity change from the actual injection orbit to the redesigned reference orbit is halved compared to the original preset orbit. Moreover, the revisit errors of position and velocity are within 1m and 1m/s, respectively.

Relative and absolute on-board optimal formation acquisition and keeping for scientific activities in high-drag low-orbit environment

In this paper, a novel Model Predictive Control (MPC) technique for multi-satellite formation flying geometry acquisition and maintenance in high-drag environment is presented. The proposed MPC relies on a linearized and convexified quasi-nonsingular Relative Orbital Elements (ROE) model based on state transition matrices propagation, allowing to include the effect of perturbations in the prediction to optimize fuel efficiency and tracking accuracy. The formation is controlled with respect to a non-decaying orbiting point to perform absolute and relative station keeping simultaneously. For this purpose, a new dedicated plant matrix to include drag effects on ROE in the propagation is derived and validated with respect to numerical results. In all simulations, the satellites are assumed to be equipped with a single low-thrust propulsion unit, therefore, specific constraints are included in the controller to obtain a feasible solution in a real operational scenario. Moreover, a collision avoidance constraint is added in case of close proximity operations exploiting a linear mapping between the set of ROE and cartesian coordinates expressed in the Local-Vertical-Local-Horizontal (LVLH) reference frame. The controller response is simulated in several realistic mission contexts with a high-fidelity orbital propagator and the results are validated for fuel efficiency by comparing them to similar approaches available in literature and to optimal solutions obtained respectively with a direct single shooting algorithm and with a closed-form impulsive formulation.

De-Orbit Maneuver Demonstration Results of Micro-Satellite ALE-1 with a Separable Drag Sail

ALE-1, a micro-satellite created for the demonstration of artificial shooting stars, required orbital descent before mission execution due to safety aspects in orbit. ALE-1 utilized a drag sail called SDOM (Separable De-Orbit Mechanism) for a passive de-orbit maneuver, which was successfully completed, lowering the orbit from about 500 km down to about 400 km. This paper summarizes the detailed history of satellite operation and the results of the de-orbit maneuver demonstration during the past three years. Although the SDOM sail faced difficulty in keeping the desired deployed shape of the drag sail due to mechanical troubles, by letting the sail be a drag flag instead, it could still deliver a meaningful de-orbit performance to allow the satellite to successfully lower the orbit as planned. The de-orbit effect of the drag flag was evaluated using comparisons between orbit propagation simulations and the actual orbit transition flight data provided in the form of TLE (Two-Line Element) sets. Through this study, it is demonstrated that the SDOM can provide orbit transfer capabilities for satellites. Furthermore, the de-orbit performance of the drag flag can be evaluated, which could be an important reference for the future implementation of de-orbit devices to solve space debris problems.