Extending propagation with user-defined equations, applications to optimization and partial derivatives computation

With the growing number of resident space objects (RSOs) precise orbit determination and prediction plays an increasingly important role. In the scope of protecting o perational spacecraft very accurate orbit prediction is a prerequisite. Predicting close conjunctions and a ssessing collision probabilities well in advance on a reliable confidence level enables planning and performing o f collision avoidance manoeuvres. With sophisticated post-processing methods, when all information is available accuracies in the order of centimeters can be achieved. Post-processing of surveillance data (optica l, radar) is still assumed to be in the order of meters. The accuracy of orbit prediction rather than a post-fit of dat a of common operational products is inferior by orders of magnitudes. This paper addresses the task of orbit propagation based on angle-only observations of RSOs in geostationary and high eccentricity orbits. The focus is on objects which do not perform manoeuvres, to clearly separate the effects of manoeuvre data ‐ which is probably unavailable and/or inaccurate ‐ from modeling the natural forces. The angle-only observations stem from the one-meter ZIMmerwald Laser and Astrometry Telescope (ZIMLAT), the 18-cm Zimmerwald SMall Robotic Telescope (ZimSMART), both located in Zimmerwald, close to Bern, Switzerland, and from the one-meter ESA Space Debris Telescope (ESASDT) located on Tenerife, Spain. The one-meter telescopes have been used for RSO surveys and for so-called follow-up observations over the past decade by the Astronomical Institute of the University of Bern, which is a prime contractor of the European Space Agency (ESA) in the fields of Surveillance and Tracking. Supplementary observations are provided by the courtesy of the International Scientific Optical Network (ISON). Orbit determination and propagation is performed with an enhanced version of the CelMech tool (Beutler, Methods of Celestial Mechanics , Springer 2001). The force model used includes Earth’s pote ntial coefficients up to order and degree 12, perturbations due to earth tides, and corrections due to general relativity. Earth shadow passages are modeled. In addition, a model for estimating the direct radiation pressure (DRP) is used, which allows an estimate of the area to mass ratio as a scaling factor, as well as the osculating elements. Optionally, biases, which account inter alia for asymmetries in the observed object, e.g. misalignment of solar panels, can be estimated. Orbit propagation is perfor med using the determined osculating elements, the estimated area to mass ratio, and the reflection coefficie nt within the force model mentioned above. The estimated biases are not available for orbit propagation. T he ephemerides of the propagated orbits are compared to observations of the same object, which were not used for orbit determination. Those additional angle-only observations serve as ground truth. The additional observations that belong to the same object are validated by a further orbit determination including all ob servations. The influence of the temporal spacing of observations and the effect of fusing data of different ob servation sites is investigated. As a reference, two line element data (TLEs) of the US Strategic Command (USSTRATCOM) catalogue are propagated with the SDP4 propagator and the resulting ephemerides are compared to the angle-only observations. Covariances for the TLE data are then estimated.

Orbit determination is possible for a chaotic orbit of a dynamical system, given a finite set of observations, provided the initial conditions are at the central time. In a simple discrete model, the standard map, we tackle the problem of chaotic orbit determination when observations extend beyond the predictability horizon. If the orbit is hyperbolic, a shadowing orbit is computed by the least squares orbit determination. We test both the convergence of the orbit determination iterative procedure and the behaviour of the uncertainties as a function of the maximum number $n$ of map iterations observed. When the initial conditions belong to a chaotic orbit, the orbit determination is made impossible by numerical instability beyond a computability horizon, which can be approximately predicted by a simple formula. Moreover, the uncertainty of the results is sharply increased if a dynamical parameter is added to the initial conditions as parameter to be estimated. The uncertainty of the dynamical parameter decreases like $n^a$ with $a<0$ but not large (of the order of unity). If only the initial conditions are estimated, their uncertainty decreases exponentially with $n$. If they belong to a non-chaotic orbit the computational horizon is much larger, if it exists at all, and the decrease of the uncertainty is polynomial in all parameters, like $n^a$ with $a\simeq 1/2$. The Shadowing Lemma does not dictate what the asymptotic behaviour of the uncertainties should be. These phenomena have significant implications, which remain to be studied, in practical problems of orbit determination involving chaos, such as the chaotic rotation state of a celestial body and a chaotic orbit of a planet-crossing asteroid undergoing many close approaches.

Orbit determination (OD) is a fundamental problem in space surveillance and tracking, crucial for ensuring the safety of space assets. Real-world ground-based optical tracking scenarios often involve challenges such as limited measurement time, short visible arcs, and the presence of outliers, leading to sparse and non-Gaussian observational data. Additionally, the highly perturbative and nonlinear orbit dynamics of resident space objects (RSOs) in low Earth orbit (LEO) add further complexity to the OD problem. This article introduces a variant of the higher order unscented Kalman estimator (HOUSE) called <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$w$</tex-math></inline-formula>-HOUSE, which employs a square-root formulation and addresses the challenges posed by nonlinear and non-Gaussian OD problems. The effectiveness of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$w$</tex-math></inline-formula>-HOUSE was demonstrated through synthetic and real-world measurements, specifically outlier-contaminated angle-only measurements collected for the Sentinel 6 A satellite flying in LEO. Comparative analyzes are conducted with the original HOUSE (referred to as <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\delta$</tex-math></inline-formula>-HOUSE), unscented Kalman filters (UKF), conjugate unscented transformation (CUT) filters, and precise OD solutions estimated via onboard global navigation satellite system measurements. The results reveal that the proposed <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$w$</tex-math></inline-formula>-HOUSE filter exhibits greater robustness when dealing with varying values of the dependent parameter compared to the original <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\delta$</tex-math></inline-formula>-HOUSE. Moreover, it surpasses all other filters in terms of positioning accuracy, achieving 3-D root-mean-square errors of less than 60 m in a three-day scenario. This research suggests that the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$w$</tex-math></inline-formula>-HOUSE filter represents a viable alternative to UKF and CUT filters, offering improved positioning performance in handling the nonlinear and non-Gaussian OD problems associated with LEO RSOs.

: This paper represents orbit propagation and determination of Low Earth Orbit(LEO) satellites, which is immediate orbit satellite positioning. Beiseds, satellite global positioning system (GPS) configured satellite receiver provides position and velocity measures by navigating the filter to get the coordinates of the orbit propagation (OP), while the satellite orbit coordinates is an important basis for the task operation. The main contradictions in real time orbit which is determined by the problem are: orbit positioning accuracy and the amount of calculating two indicators; how to balance these two indicators. This article is dedicated to solving the problem of tradeoffs. To plan to use a non-linear filtering method for immediate orbit tasks requires more precise satellite orbit state parameters in a short time. Although the traditional extended Kalman filter (Extended Kalman Filter, EKF) method is widely used, its linear approximation of the drawbacks in dealing with nonlinear problems was especially evident, without compromising Kalman filter (Unsented Kalman Filter, UKF). As a new non-linear estimation method, it is measured at the estimated measurements on more and more applications. This paper will be the first study to UKF microsatellites in LEO orbit in real time, trying to explore the real-time precision orbit determination techniques.

The solution of the initial relative orbit determination problem using angles-only measurements is important for orbital proximity operations, satellite inspection and servicing, and the identification of unknown space objects in similar orbits. In this paper, a preliminary relative orbit determination performance analysis is conducted utilizing the linearized relative orbital equations of motion in cylindrical coordinates. The relative orbital equations of motion in cylindrical coordinates are rigorously derived in several forms included the exact nonlinear two-body differential equations of motion, the linear-time-varying differential equations of motion for an elliptical orbit chief, and the linear-time-invariant differential equations of motion for a circular orbit chief. Using the nonlinear angles-only measurement equation in cylindrical coordinates, evidence of full-relative-state observability is found, contrary to the range observability problem exhibited in Cartesian coordinates. Based on these results, a geometric approach to assess initial relative orbit determination performance is formulated. To facilitate a better understanding of the problem, the focus is on the 2-dimensional initial orbit determination problem. The results clearly show the dependence of the relative orbit determination performance on the geometry of the relative motion and on the time-interval between observations. Analysis is conducted for leader-follower orbits and flyby orbits where the deputy passes directly above or below the chief.

On the accuracy of covariance matrix: Hessian versus Gauss–Newton methods in atmospheric remote sensing with infrared spectroscopy

Spacecraft orbit propagator integration with GNSS in a simulated scenario

Adaptive square-root cubature–quadrature Kalman particle filter via KLD-sampling for orbit determination

TLE로부터 SGP4/SDP4 모델을 이용하여 인공위성의 가상의 위치 정보를 얻은 후 Gauss 방법을 사용하여 인공위성의 예비궤도를 결정해보았다. 예비궤도 결정에 필요한 임의의 세 점 사이의 시간간격을 변화시켜 얻은 결과를 위성의 위치 참값과 비교하여 최소의 차이를 가지는 관측 시간 간격을 찾아보았으며, Gauss 예비궤도 결정법의 성능을 비교, 분석하였다. 실제 인공위성 관측 결과와의 비교를 위해서 한국천문연구원의 광시야 망원경을 사용하여 GPS위성(PRN 26)과 무궁화 2호의 광학관측 데이터를 얻은 후 같은 방법으로 예비궤도를 결정해 보았다. 인공위성의 정밀궤도결정을 위하여 시뮬레이션에서 얻어진 가상의 광학관측 데이터를 가지고 정밀케도결정을 수행하였으며, 관측 데이터의 오차와 관측 시간 간격에 따라 정밀궤도결정을 수행하여 원하는 정밀도를 얻기 위한 관측 시스템의 조건에 대해서 알아보았다. Gauss method for the initial orbit determination was tested using angle-only data obtained by orbit propagation using TLB and SGP4/SDP4 orbit propagation model.. As the analysis of this simulation, a feasible time span between observation time of satellite resulting the minimum error to the true orbit was found. Initial orbit determination is performed using observational data of GPS 26 and Koreasat 2 from 0.6m telescope of KAO(Korea Astronomy Observatory) and precise orbit determination is also performed using simulated data. The result of precise orbit determination shows that the accuracy of resulting orbit is related to the accuracy of the observations and the number of data.

The cubed-sphere model provides rapid evaluation of the gravity field for more efficient orbit propagation. This paper characterizes the improved computational efficiency of sequential orbit determination, specifically the extended and unscented Kalman filters, when using this new model instead of the common spherical harmonic model. To use the new gravity model with the extended Kalman filter, capabilities to represent the Jacobian of the gravity acceleration are added to numerically integrate the state transition matrix. Filter tests consider improvements for several simulated satellite scenarios with several combinations of measurements provided for estimation. Since cubed-sphere models of higher degree require only a slight change in computation time, orbit propagation and determination systems may now use this model to improve fidelity without any significant change in cost. Using the cubed-sphere model reduces the computational burden of the orbit determination process, with larger benefits found for high-degree filter models. Differences in the estimated trajectories when using the disparate gravity models remain several orders of magnitude less than the absolute filter error for the cases examined.

In this paper we present a new result on rejection of unmatched external disturbances on port-Hamiltonian systems using Control by Interconnection (CbI). The PHS structure is used to design a controller that rejects unmatched constant disturbances from non-passive outputs. In the PHS framework, the disturbance rejection problem has been addressed adding integral action and using a change of coordinates. In our approach, we avoid a change of coordinates keeping the original state vector, which contains variables with physical interpretation. The methodology proposed in this paper is illustrated on an electrical circuit and on a permanent magnet synchronous motor. Simulation of the later example shows the performance of the control design.

In this contribution, we present the LUH-GRACE2018 time series of monthly gravity field solutions covering the period January 2003–March 2016. The solutions are obtained from GRACE K-Band Range Rate (KBRR) measurements as main observations. The monthly solutions are computed using the in-house developed GRACE-SIGMA software. The processing is based on dynamic orbit and gravity field determination using variational equations and consists of two main steps. In the first step, 3-hourly orbital arcs of the two satellites and the state transition and sensitivity matrices are dynamically integrated using a modified Gauss-Jackson integrator. In this step, initial state vectors and 3D accelerometer bias parameters are adjusted using GRACE Level-1B reduced-dynamic positions as observations. In the second step, normal equations are accumulated and the normalized spherical harmonic coefficients up to degree and order 80 are estimated along with arc-wise initial states, accelerometer biases and empirical KBRR parameters. Here KBRR measurements are used as main observations and reduced-dynamic positions are introduced to solve for the low frequency coefficients. In terms of error degree standard deviations as well as Equivalent Water Heights (EWH), our gravity field solutions agree well with RL05 solutions of CSR, GFZ and JPL.

During re-entry objects with low-eccentricity orbits traverse a large portion of the dense atmospheric region almost every orbital revolution. Their perigee decays slowly, but the apogee decays rapidly. Because ballistic coefficients change with altitude, re-entry predictions of objects in low-eccentricity orbits are more difficult than objects in nearly circular orbits. Problems in orbit determination, such as large residuals and non-convergence, arise for this class of objects, especially in the case of sparse observations. In addition, it might be difficult to select suitable initial ballistic coefficient for re-entry prediction. We present a new re-entry prediction method based on mean ballistic coefficients for objects with low-eccentricity orbits. The mean ballistic coefficient reflects the average effect of atmospheric drag during one orbital revolution, and the coefficient is estimated using a semi-numerical method with a step size of one period. The method is tested using Iridium-52 which uses sparse observations as the data source, and ten other objects with low-eccentricity orbits which use TLEs as the data source. We also discuss the performance of the mean ballistic coefficient when used in the evolution of drag characteristics and orbit propagation. The results show that the mean ballistic coefficient is ideal for re-entry prediction and orbit propagation of objects with low-eccentricity orbits.

Application of Semi-analytical Satellite Theory orbit propagator to orbit determination for space object catalog maintenance

The Aeronomy of Ice in the Mesosphere (AIM) mission is a NASA Small Explorer observatory that was launched on April 25, 2007. Due to its coarse on-board orbit knowledge requirements, AIM was not designed with any orbit determination sensors or software except for an orbit propagator which was to be re-seeded by ground command at regular intervals. Several days after launch the spacecraft’s communications receiver began to suffer from an intermittent failure. The uncertainty in performance of the spacecraft’s receiver created a desire to develop, with urgency, a way for the spacecraft to navigate for very long periods of time without ground-provided updates. This paper presents the solution that was developed, loaded, and has been successfully operating in flight. A ground-loaded set of parameters define the orbit elements which are readily predictable over the long term (such as orbit plane evolution and radius) while an on-board observation of a discrete event – occultation of the Sun behind the Earth each orbit – provides the final in-track location fix. This unique software satisfied the on-board navigation requirement of < 73 km error through several years of nominal and extended mission operations without the operations team needing to send a single additional navigation command.