Shadowing Lemma and chaotic orbit determination

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 presents a new way to deal with transition matrices handling in variational equations. While working simultaneously on several problems dealing with orbit propagation: low-thrust trajectories and orbit determination, we implemented a feature allowing to add user equations to a propagator in order to solve the first problem. Then it appeared that this feature could be reused to deal with the second one. Indeed, the orbit determination problem is based on variational equations involving transition matrices, which can also be considered as additional parameters, propagated at the same time as the original state vector. This method allows a very modular implementation of both problems, with looser coupling in the equations. It has been successfully integrated in the Orekit open-source library.

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.

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

Initial orbit determination is highly enabling for operational awareness and safety in space. Reliable orbit determination in non-Keplerian systems, such as multi-body orbital mechanics, is challenging due to chaotic and unpredictable motion of these systems over long time scales. Typical approaches to non-Keplerian orbit determination use explicit integration with numerical shooting methods to propagate candidate trajectories. These methods often break down for measurement gaps on the order of days or even hours. This work presents a novel algorithm for non-Keplerian orbit determination using sparse grid collocation methods. The approach provides many benefits, including a large region of convergence without a close initial guess, as well as increased stability and performance over long observing time scales. A further advancement is made for estimating periodic orbits, in which a repeating orbit and its natural period are determined simultaneously from angles-only measurements. Orbit determination using collocation may significantly reduce sensor tasking requirements and enhance capabilities in complex orbital environments such as cislunar space.

A new method of orbit prediction for LEO satellites is proposed by using empirical accelerations to compensate for mismodelling of deterministic force models. The position, velocity and dynamical parameters such as &amp;lt;em&amp;gt;C&amp;lt;/em&amp;gt;&amp;lt;sub&amp;gt;D&amp;lt;/sub&amp;gt;, &amp;lt;em&amp;gt;C&amp;lt;/em&amp;gt;&amp;lt;sub&amp;gt;R&amp;lt;/sub&amp;gt;, and empirical accelerations are calculated in the orbit determination using reduced dynamic batch Least-Squares technique and GPS pseudorange observations. Therefore the position, velocity, &amp;lt;em&amp;gt;C&amp;lt;/em&amp;gt;&amp;lt;sub&amp;gt;D&amp;lt;/sub&amp;gt;, and &amp;lt;em&amp;gt;C&amp;lt;/em&amp;gt;&amp;lt;sub&amp;gt;R&amp;lt;/sub&amp;gt; can be used for prediction directly. Furthermore, the empirical accelerations are showed quasi-periodic and cosine characteristics with respect to time variable, in terms of which Fourier series were used to interpolate empirical accelerations. In this way, the fitted tangential empirical accelerations curve, which is added into deterministic force models to compensate for mismodelling of atmospheric drag model, forms enhanced-accuracy dynamic models that are used for orbit prediction. The GRACE-A real flight GPS pseudorange data and IGS Ultra-rapid products have been used to orbit determination and then the proposed method has been used to orbit prediction. The orbit determination results show that initial positioning accuracy is about 0.2m and velocity accuracy is about 1.0×10&amp;lt;sup&amp;gt;-4&amp;lt;/sup&amp;gt;m·s&amp;lt;sup&amp;gt;-1&amp;lt;/sup&amp;gt;. For a 72-hour orbit pass, the prediction accuracy is better than 60m, which is averagely improved about 2.3 times when compared to conventional dynamic models without considering empirical acceleration series fitting models. The proposed orbit determination and prediction scheme are beneficial to establish advanced, even onboard, satellite autonomous navigation system.

Contribution analysis of inter-satellite ranging observation to BDS-2 satellite orbit determination based on regional tracking stations

Orbit determination for a space-based gravitational wave observatory

Precise Orbit Determination (POD) is an integral part for analyzing measurements from space geodetic techniques such as Satellite Laser Ranging (SLR) and Global Navigation Satellite Systems (GNSS) such as the Global Positioning System (GPS). In the last two decades, POD based on GPS data has furthermore been established as one of the standard techniques to derive trajectories of satellites in the low Earth orbit (LEO) with highest accuracy. Since the launch of dedicated gravity missions, GPS sensors are not only used as a key tracking system for LEO POD, but also for extracting the long wavelength part of the Earth’s gravity field (together with SLR to spherical satellites). This chapter introduces SLR and GNSS measurements collected by the terrestrial networks of the International Laser Ranging Service (ILRS) and the International GNSS Service (IGS) as the observational basis for the realization of a terrestrial reference frame from satellite data. On this foundation, the basic equations and mathematical methods of orbit determination are introduced and extensively discussed. Pseudo-stochastic orbit modeling techniques are eventually presented as a general and efficient concept to determine satellite trajectories of highest quality even in presence of deficient force models, covering the full range between dynamic and purely kinematic solutions. Selected results from the application of the discussed orbit determination techniques are highlighted for GPS LEO data. Special emphasis is also put to present orbit determination in the context of more general orbit determination problems, where satellite trajectories are simultaneously determined with other parameters encompassing (at maximum) all pillars of geodesy, i.e., the shape, rotation, and gravity field of the Earth.

O RBIT determination algorithms are designed to estimate satellite position and inertial velocity from unbiased sensor measurements. Static and dynamic measurement biases, which are low-frequency systematic errors, cause biases in both position and velocity estimates that can significantly degrade track accuracy. As only the random error statistics are characterized, the filter covariance usually underestimates the biased track error statistics. Track biases and covariance fidelity are important issues for correlation and fusion of multiple sensor tracks [1,2]. Many algorithms have been developed [3–10] for estimation of bias parameters in the system dynamics, in the measurements, or in both. Generally, bias parameters are estimated jointly with the position and velocity states. Depending on the number of bias parameters, this approach can be computationally intensive, the system covariance matrix can become ill conditioned, and bias-state observability and separability can be problematic. Alternatively, two-step techniques extract bias parameters from the track measurement residuals. For current radar systems, measurement biases are calibrated by tracking one metric satellite having an accurate ephemeris, or by tracking many satellites with less accurate ephemerides [9]. Range and angle biases are calibrated after (not during) the data collection (s). A noteworthy exception is a real-time algorithm for angle bias estimation in wideband radars [10]. In this Note, an iterated 12-state extended Kalman filter EKF(12) is formulated for simultaneous (real-time) orbit determination and bias estimation. Orbit position and velocity states are augmented with six dynamic (rather than static) bias states that model radar orientation biases and radar range-angle measurement biases. Biasstate observability and separability are enhanced by physical models for the bias dynamics and by careful characterization of bias effects in the measurement residuals. For example, random and certain environmental (i.e., tropospheric) bias errors are intrinsic to the radar measurements, whereas radar orientation biases corrupt the transformation of the prior position estimate from inertial to radar coordinates. Performance evaluations will demonstrate accurate real-time state estimates and excellent covariance fidelity, which are essential for radar-to-radar track correlation and fusion. II. Sensor Models

While building up a catalog of Earth orbiting objects, if the available optical observations are sparse, not deliberate follow ups of specific objects, no orbit determination is possible without previous correlation of observations obtained at different times. This correlation step is the most computationally intensive, and becomes more and more difficult as the number of objects to be discovered increases. In this paper we tested two different algorithms (and the related prototype software) recently developed to solve the correlation problem for objects in geostationary orbit (GEO), including the accurate orbit determination by full least squares solutions with all six orbital elements. Because of the presence in the GEO region of a significant subpopulation of high area to mass objects, strongly affected by non-gravitational perturbations, it was actually necessary to solve also for dynamical parameters describing these effects, that is to fit between 6 and 8 free parameters for each orbit. The validation was based upon a set of real data, acquired from the ESA Space Debris Telescope (ESASDT) at the Teide observatory (Canary Islands). We proved that it is possible to assemble a set of sparse observations into a set of objects with orbits, starting from a sparse time distribution of observations, which would be compatible with a survey capable of covering the region of interest in the sky just once per night. This could result in a significant reduction of the requirements for a future telescope network, with respect to what would have been required with the previously known algorithm for correlation and orbit determination.

PurposeAutonomous orbit determination using geomagnetic measurements is an important backup technique for safe spacecraft navigation with a mere magnetometer. The geomagnetic model is used for the state estimation of orbit elements, but this model is highly nonlinear. Therefore, many efforts have been paid to developing nonlinear filters based on extended Kalman filter (EKF) and unscented Kalman filter (UKF). This paper aims to analyze whether to use UKF or EKF in solving the geomagnetic orbit determination problem and try to give a general conclusion.Design/methodology/approachThis paper revisits the problem and from both the theoretical and engineering results, the authors show that the EKF and UKF show identical estimation performances in the presence of nonlinearity in the geomagnetic model.FindingsWhile EKF consumes less computational time, the UKF is computationally inefficient but owns better accuracy for most nonlinear models. It is also noted that some other navigation techniques are also very similar with the geomagnetic orbit determination.Practical implicationsThe intrinsic reason of such equivalence is because of the orthogonality of the spherical harmonics which has not been discovered in previous studies. Thus, the applicability of the presented findings are not limited only to the major problem in this paper but can be extended to all those schemes with spherical harmonic models.Originality/valueThe results of this paper provide a fact that there is no need to choose UKF as a preferred candidate in orbit determination. As UKF achieves almost the same accuracy as that of EKF, its loss in computational efficiency will be a significant obstacle in real-time implementation.

Recurrence in the phase space of complex systems is a well-studied phenomenon, which has provided deep insights into the nonlinear dynamics of such systems. For dissipative systems, characteristics based on recurrence plots have recently attracted much interest for discriminating qualitatively different types of dynamics in terms of measures of complexity, dynamical invariants, or even structural characteristics of the underlying attractor's geometry in phase space. Here, we demonstrate that the latter approach also provides a corresponding distinction between different co-existing dynamical regimes of the standard map, a paradigmatic example of a low-dimensional conservative system. Specifically, we show that the recently developed approach of recurrence network analysis provides potentially useful geometric characteristics distinguishing between regular and chaotic orbits. We find that chaotic orbits in an intermittent laminar phase (commonly referred to as sticky orbits) have a distinct geometric structure possibly differing in a subtle way from those of regular orbits, which is highlighted by different recurrence network properties obtained from relatively short time series. Thus, this approach can help discriminating regular orbits from laminar phases of chaotic ones, which presents a persistent challenge to many existing chaos detection techniques.

The current state of near-Earth asteroids (NEAs) observations shows an annual increase in the number of newly discovered objects However, the frequency distribution of NEAs by size shows a sharp decrease in the number of objects with size less than 300 m, which contradicts the results of theoretical modeling of the NEA population. Considering definition of potentially hazardous asteroids (PHA), only objects with diameters more than 140 m could pose catastrophic consequences to the Earth and mankind in general. But in the same time, impacts of smaller size objects could lead to significant consequences on local level and their large predicted number increases this probability. Due to their small size which results in faint apparent magnitude, such NEAs are discovered in a short interval of their close approach (CA) to the Earth, when their apparent magnitude are tending to be as bright as possible for a given size. This is not only facilitates the detection of such new objects but also increases their observability by small ground-based telescopes. However, apparent rate of motion during this time might exceed 10 deg d −1 making the observations challenging. The used Rotating-drift-scan CCD (RDS CCD) technique allows to get images of fast-moving objects as a point, that in turn to determine the coordinates of their image centers with sufficient astrometric precision. Obtained in current research project positions show errors in the range ± (0.2″ − 0.3″) in both coordinates with comparison both to JPL's HORIZONS 1 system and NEODyS-2 2 service. The part of observations was obtained around time moment of minimal distance to the Earth during current CA for newly discovered NEAs. Such observations are important to extend observed orbital arc for reliable improvement of their orbit determinations and reducing orbital uncertainty, so it will be possible to recover them in next apparitions.

The Asteroid Impact and Deflection Assessment (AIDA) is an international collaboration supported by ESA and NASA to assess the feasibility of the kinetic impactor technique to deflect an asteroid, combining data obtained from NASA&amp;#8217;s DART and ESA&amp;#8217;s Hera missions [1, 2]. In 2022, DART will perform a kinetic impact on the secondary of the binary near-Earth asteroid (65,803) Didymos, recently named Dimorphos. After 2 years, Hera will follow-up with a detailed post-impact survey of Didymos, to fully characterize this planetary defense technique. Additionally, Hera will deploy two CubeSats around Didymos once the Early Characterization Phase has completed, to complement the observations of the mother spacecraft and increase the scientific return of the mission. The first Cubesat, called Juventas, will complete a low-frequency radar survey of the secondary, to unveil its interior, while the second one has not yet been selected.One of the main objectives of Hera is to characterize the mass and mass distribution of both Didymos primary and secondary by means of radio science investigations. This paper describes the concept of the gravity science investigations to be jointly carried out by the three mission elements, i.e. Hera, Juventas and CubeSat-2. The experiment will combine classical ground-based radiometric measurements, spacecraft-based optical images of Didymos, and Satellite-to-Satellite radiometric tracking between Hera and the Cubesats. Finally, our results and achievable accuracy for the estimation of the mass and gravity field of Didymos and Dimorphos are presented.1. IntroductionMost of the information about the formation processes of an asteroid lies in its interior structure. One of the very few constraints of the internal mass distribution of a celestial body is given by its gravity field, even if the inversion process is not unique. First, the bulk density can be inferred by measuring the mass of the body, combined to the volume estimated from optical images. In addition, the higher degrees of the gravity field provide information about the internal distribution of mass, such as the moments of inertia.The main scientific goals of the Hera radio science investigations are:Determine the mass and gravity field of Didymos and Dimorphos; Reconstruct the motion of Dimorphos around Didymos; Contribute to the characterization of the energy transfer between DART and Dimorphos. Such objectives are a valuable contribution to the Hera mission objectives, leading to a better understanding of the formation and evolution processes of the Didymos system.2. TechniqueThe determination of the gravity field of a celestial body is an application of the orbit determination process of deep space spacecraft. In particular, the gravity of Didymos can be estimated precisely reconstructing the trajectory of Hera during a selected number of close encounters (about 10 km at closest approach). The classical observables used in the orbit determination are obtained from the X-band radio link between the spacecraft and the Earth. The microwave signal is sent to spacecraft from a ground antenna and coherently retransmitted back to Earth, where Doppler and range measurements are obtained. A previous study performed for the AIM proposed mission [4] demonstrated that gravity science at Didymos is feasible using radio tracking data only, under realistic assumptions on the technological capabilities of the space and ground segment. Shorter pericenter distances increase the attainable accuracy. However, a significant improvement can be obtained even at relatively large distances processing also optical images of Didymos and Dimorphos taken by the spacecraft.In addition, Hera may track Juventas and CubeSat-2 by means of a space-to-space inter-satellite link (ISL), capable of determining the relative distance (ranging) and the relative line-of-sight velocity (Doppler) between the two bodies. In particular, the latter is expected to represent a very nice add-on to the gravity investigation carried out by the Hera mission, as the Doppler shift that affects the inter-satellite link contains the information on the dynamics of the system, i.e. masses and gravity field of Didymos and Dimorphos.The expected accuracy in the estimation of Didymos gravity fields were obtained through numerical simulations of the orbit determination of Hera and the two Cubesats. Conservative assumptions were made in terms of both radiometric and optical measurement noises, and large a-priori uncertainties for the estimated parameters were used. &amp;#160;3. ResultsAs a result of the numerical simulations, the masses of Didymos and Dimorphos are expected to be estimated with relative uncertainties less than 10-4 and 10-3, respectively. The addition of the ISL measurements improves the achievable accuracies but it is not required to estimate the masses. However, given the relatively large distance of Hera from the system, the higher degree gravity of Didymos and Dimorphos can be estimated only adding the ISL Doppler measurements between the Cubesats and the mother spacecraft. In this case, the gravity field of Didymos can be estimated to at least degree 3, depending on the assumptions about the ISL operations and performance. Similarly, ISL Doppler measurements allows to estimate the extended gravity field of Dimorphos up to degree 2, with an uncertainty of about 10%.AcknowledgementsThis project has received funding from the European Union&amp;#8217;s Horizon 2020 research and innovation programme under grant agreement No 870377 (project NEO-MAPP).&amp;#160;MZ, IG, ML, EG, RLM, and PT wish to acknowledge Caltech and the Jet Propulsion Laboratory for granting the University of Bologna a license to an executable version of MONTE Project Edition S/W.