Universal Angles-Only Cislunar Initial Orbit Determination Using Sparse Grid Collocation

Space-based initial orbit determination (IOD) is crucial for space situational awareness. However, classical algorithms often fail to converge or become trapped in trivial solutions when applied to very short arc scenarios with angles-only measurements. To overcome this challenge, this paper proposes a simple and fast algorithm based on the online sequential extreme learning machine (OSELM). First, the dynamic model and angles-only measurement model of the space target were constructed. Then, an OSELM-based orbit determination algorithm is designed, including the generation of the training dataset, data preprocessing, and network structure design, which focuses on establishing a nonlinear mapping model from angles-only measurements to the orbital state of space targets using the OSELM. Finally, numerical simulations for low Earth orbit (LEO) scenarios demonstrate the algorithm's high accuracy in estimating the target initial state, along with strong noise resistance and generalization capability.

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.

The following thesis regards the use of small aperture telescopes for space domain awareness efforts. The rapidly populating space domain was motivation for the development of a new operation scheme to conduct space domain awareness feasibility studies using small telescopes. Two 14-inch Schmidt-Cassegrain Telescopes at the California Polytechnic State University and the Air Force Research Lab in Kirtland AFB, NM, in conjunction with a dedicated CCD camera and a commercial DSLR camera, were utilized to conduct optical observations on satellites in Earth orbit. Satellites were imaged during August 2019, and from January 2020 to March 2020, resulting in the collection of 77 valid images of 16 unique satellites. These images were used to obtain celestial spherical coordinates, which were used in Gauss and Double-R angles-only initial orbit determination methods. Initial orbit determination methods successfully produced valid results, reaffirming the feasibility of using small aperture telescopes for such methods. These orbit determinations were used to propagate orbit states forward in time to determine the feasibility of future imaging of the targets with the same apparatus. Propagation results demonstrated that initial orbit determinations rapidly decayed in accuracy over distant times and are most accurate for immediate satellite passes. In addition, an attempt to combine multiple initial orbit determinations using Lambert’s problem solutions was made. Combination of these multiple initial orbit determinations resulted in either no orbit state accuracy improvement compared to individual initial orbit determinations, or a decrease in accuracy compared to these methods. Ultimately, efforts demonstrated that small telescope usage is feasible for orbit determination operations, however there may be a need for hardware and operational revisions to improve the ability of the apparatus.

Spatial variability of Darcy velocity is presented due to the heterogeneity of aquifer parameters. The uncertainty qualification of velocity suffers great challenge in the complex porous media. This work focuses on the use of sparse grid collocation method in velocity simulation. Since the sparse grid collocation method provides a non-intrusive way to incorporate any existing deterministic solver, the mixed finite element method is combined as the deterministic solver to retain the local continuity of Darcy velocity. We decompose the error of the velocity into three components, and illustrate that the Karhunen-Loeve truncation brings more error into velocity approximation than into head. The convergence properties of velocity moments restrict the application of sparse grid collocation method in the problems with small correlation lengths. This work provides insights towards the application of sparse grid collocation method to velocity modeling. It is demonstrated that for which problems the sparse grid collocation method is expected to be competitive with the Monte Carlo simulation. Further work about the anisotropic sparse grid collocation method should be extended to circumvent the obstacle of dimensionality.

In recent years, the number of space objects has been increasing rapidly. It has become crucial to utilize multi-source information fusion techniques for the orbit measurement and cataloguing of numerous objects. This report elaborates on the utilization of ground-based radar, optical, and space-based optical measurements in joint precision orbit determination. Regarding the real-time surveillance of a vast amount of space debris, investigations have been conducted concerning the initial orbit determination upon the first detection of targets, data association, orbit refinement, and the generation of catalogued orbital elements.Space-based technologies assume a particularly crucial role owing to their remarkable advantages in temporal and spatial coverage. To fulfill the data processing requisites for orbit determination of space-based measurement platforms, the onboard GNSS precision orbit determination software, SODA, has been developed, attaining centimeter-level accuracy in orbit determination. In the scenario of long-arc tracking via ground single-station astronomical optical cameras, an orbit determination accuracy within the range of several tens of meters can be accomplished. In the situation where multiple satellites are equipped with optical cameras, a substantial enhancement in the monitoring performance of space debris can be achieved both temporally and spatially.

Initial orbit determination (IOD), as a basis for initial orbit association and accurate orbit determination (OD), has a crucial role in the process of obtaining space debris orbit information. Among the traditional methods, the Gooding method has better convergence and stability. In this study, the Gooding method is enhanced to solve the issues discovered. A novel initial orbit determination (IOD) method is developed using the proposed improvement measures of the single-parameter initial value determination (SIVD) method, the fitted-curve noise suppression method, the restricted corrective value solution method, the removal of trivial solutions, etc. The experimental results verify the effectiveness of the improved method. The success rate of the initial orbit determination reached 99%, and the accuracy of the solved orbit parameters was significantly improved, especially the semi-major axis (SMA) error of less than 50 km, accounting for 88% of the total. It can be seen that the method meets the demand of space-based space debris cataloging for initial orbit association and can serve in the field of space situational awareness, which has important practical significance and application potential.

I have constructed and demonstrated a new fast-running algorithm to perform refined orbit determination for any spacecraft in GEO (geosynchronous Earth orbit), without user requirement for an a priori orbit estimate. We now have a refined method for GEO initial orbit determination (IOD). This is enabled by the use of equinoctial orbit elements, by a one-dimensional search in the equinoctial orbit element mean argument of orbit longitude, by the use of sensor locations to reduce the one-dimensional search, and by convergence boundaries of the nonlinear least squares (LS) algorithm. I shall refer herein to the new algorithm as LS_GEO_IOD. Algorithm LS_GEO_IOD uses any desired perturbative acceleration model for numerical trajectory integration, and its success appears to be independent of input measurement type.

In this paper, optical target monitoring orbit determination using the Gooding algorithm based on the 400 km altitude space platform was researched. The measurement error was set to 3″ and 6″ respectively for 800 km, 1500 km altitude low Earth orbits and 36 000 km altitude geosynchronous orbit to determine the initial orbit and precise orbit. Simulation results show that the initial orbit determination accuracy of the 4~15 min arc is about 10 km, and the 1~2 min arc is about 100 km when the measurement data error is between 3″ and 6″. The error of 15 min initial determination arc is 100 m. As the arc is less than 10 min, the improved error accuracy of the orbit is in the order of km scale. As the measurement data error is 3″, the initial orbit determination accuracy of the 15~20 min arc is about tens of kilometers, and that of the 8~10 min arc is 100 km. The improved orbit has an error of km. When the measurement data error is 6″, the accuracy of the initial orbit determination of the 20 min arc is at the magnitude of 10 km, the accuracy of the initial orbit determination of the 8~15 min arc is at the magnitude of 100 km, and the error accuracy of the improved orbit is at the magnitude of 10 km.

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.

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.

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.

Enhanced real-time onboard orbit determination for low-Earth-orbit satellites is essential for autonomous spacecraft operations. However, the accuracy of such systems is often limited by signal propagation delays between GPS satellites and the user spacecraft. These delays, primarily due to Earth’s rotation and ionospheric effects become particularly significant in high-dynamic LEO environments, leading to considerable errors in range and range rate measurements, and consequently, in position and velocity estimation. To mitigate these issues, this paper proposes a real-time orbit determination algorithm that applies Earth rotation correction and dual-frequency (L1 and L2) ionospheric compensation to raw GPS measurements. The enhanced orbit determination method is processed directly in the Earth-centered Earth-fixed frame, eliminating repeated coordinate transformations and improving integration with ground-based systems. The proposed method employs a reduced-dynamic orbit determination strategy to balance model fidelity and computational efficiency. A predictive correction model is also incorporated to compensate for GPS signal delays under dynamic motion, thereby enhancing positional accuracy. The overall algorithm is embedded within an extended Kalman filter framework, which assimilates the corrected GPS observations with a stochastic process noise model to account for dynamic modeling uncertainties. Simulation results using synthetic GPS measurements, including pseudoranges and pseudorange rates from a dual-frequency spaceborne receiver, demonstrate that the proposed method provides a significant improvement in orbit determination accuracy compared to conventional techniques that neglect signal propagation effects. These findings highlight the importance of performing orbit estimation directly in the Earth-centered, Earth-fixed reference frame, utilizing pseudoranges that are corrected for ionospheric errors, applying reduced-dynamic filtering methods, and compensating for signal delays. Together, these enhancements contribute to more reliable and precise satellite orbit determination for missions operating in low Earth orbit.

Space-based orbit determination problem using line-of-sight measurements appears in two equivalent cases: 1) to estimate the observer’s orbit when observing a known planetary body (navigation problem) and 2) to determine the orbit of an unknown observed body when the observer’s orbit is known (surveillance problem). These two problems are mathematically equivalent. Star trackers provide attitude autonomy to interplanetary missions while the estimation of position still relies on communications to Earth’s ground stations. This paper shows that, in interplanetary missions, autonomy for position estimation can be obtained just by observing visible planets using star trackers. In any space mission and, in particular interplanetary missions, autonomy plays a critical role for the success of the mission, especially if temporary or permanent failure occurs in the communication system. This study first solves the position and velocity (state) estimation problem as a single-point problem using the Pn space-based Initial Orbit Determination (IOD) algorithm 8 and then uses this estimation to initiate a filtered estimation of the state using Extended Kalman Filter (EKF). Space observations of line-of-sight directions constitute the measurements of two distinct problems: when estimating the orbit of an unknown observed body (space-based orbit determination problem) and when estimating the observer’s orbit when the observed body is known (navigation problem). There is a perfect duality between these two problems as they share the same mathematics meaning that any method capable of solving one problem can be used to solve the other. However, when the distances between observer and observed become large, as in interplanetary missions, the problem becomes more complicated. In fact, ∗Texas A&M University Affiliate. E-mail: reza.aero@gmail.com †Professor, 746C H.R. Bright Bldg, Aerospace Engineering, Texas A&M University, College Station, TX 77843-3141, AAS Fellow, AIAA Associate Fellow. IEEE Senior member. E-mail: mortari@tamu.edu

This paper is predicated on determining the effectiveness of angles-only initial orbit determination (IOD) methods when limited observational data is available for low-Earth orbit (LEO) rocket body debris. The analysis will be conducted with data obtained from Lockheed Martin Space’s Space Object Tracking (SpOT) facility, focusing on their observational data from 2018 that contains tracking of rocket body debris for less than one minute per overhead pass. After the IOD accuracies are better understood, a feasibility study will follow that investigates the possibility of cataloguing LEO orbital debris from a single optical observation facility with similar observational capabilities as that of the SpOT facility. The IOD accuracy analysis will investigate nine different rocket bodies, with a total of 50 orbital passes of data included in the research. Three main IOD approaches will be tested for each data set to determine the best method in achieving high levels of IOD accuracy: a traditional three-point method, an iterative method, and an assumed-circular orbit method. Application of the iterative approach results in increased accuracy for the resultant initial orbit determination as compared to the three-point IOD method, and an assumed-circular orbit assumption allows for a further increase in accuracy, especially for observed objects in near-circular orbits. The feasibility of cataloguing debris from a singular optical facility shows promise, as subsequent target acquisition after an object’s initial observation is determined to be achievable under the correct circumstances. By choosing a correct telescope pointing angle based on the IOD results from one pass of data, an observed rocket body debris object would pass through the field of view of SpOT’s spotter scope (0.7-degrees) during its next overhead pass for two different test cases. An increase field of view would increase both the likelihood of acquiring the target object and the amount of time the object is visible by the telescope.

Autonomous orbit determination of a navigation constellation is the process by which the orbit parameters of navigation satellites are autonomously calibrated onboard the satellites without the need for external aids. It commonly uses a satellite onboard data processing unit and a filtering method to process the measurements of inter-satellite ranges. The onboard data processing unit is the main module of autonomous navigation systems. In this paper, the two main factors that affect the accuracy of autonomous orbit determination for a navigation constellation are discussed first, and then a distributed onboard algorithm for autonomous orbit determination of navigation satellites is proposed. This method is based on a long-term ephemeris prediction and is suitable for the satellite hardware capability. The main feature of this method is that both the distributed computing method and an onboard analytical state transition matrix are used to process inter-satellite range measurements. One of the main advantages of this approach is high-speed computing since the amount of calculations needed is significantly less than that of the centralised computing method and those distributed methods that need to use an onboard numerical integrator. Another advantage of this approach is that the use of the onboard analytical state transition matrix algorithm can save a great amount of resources for both ground-to-satellite data transmissions and data storage units in satellites’ hardware. This could result in substantial cost reduction for space missions. Finally, a simulation method used for testing the proposed algorithm is presented. Results of tests over a period of 90 days show that the user range error of autonomous orbit determination derived from the proposed method is less than three metres.