During the last two centuries many methods have been proposed to solve the angles-only initial orbit determination problem. As this problem continues to be relevant as an initial estimate is needed before high accuracy orbit determination is accomplished, it is important to perform direct comparisons among the popular methods with the aim of determining which methods are the most suitable (accuracy, robustness) for the most important orbit determination scenarios. The methods tested in this analysis were the Laplace method, the Gauss method (suing the Gibbs and Herrick-Gibbs methods to supplement), the Double R method, and the Gooding method. These were tested on a variety of scenarios and popular orbits. A number of methods for quantifying the error have been proposed previously. Unfortunately, many of these methods can overwhelm the analyst with data. A new method is used here that has been shown in previous research by the author. The orbit error is here quantified by two new general orbit error parameters identifying the capability to capture the orbit shape and the orbit orientation. The study concludes that for nearly all but a few cases, the Gooding method best estimates the orbit, except in the case for the polar orbit for which it depends on the observation interval whether one uses the Gooding method or the Double R method. All the methods were found to be robust with respect to noise and the initial guess (if required by the method). All the methods other than the Laplace method suffered no adverse effects when additional observation sites were used and when the observation intervals were unequal. Lastly for the case when the observer is in space, it was found that typically the Gooding method performed the best if a good estimate is known for the range, otherwise the Laplace method is generally best.

Precision orbit determination (POD) plays a vital role in the success of space-borne laser altimetry missions, such as ICESat (Ice, Cloud, and Land Elevation Satellite). Although current ICESat POD processing standards are achieving remarkable accuracy, new time-varying geopotential models derived from the GRACE (Gravity Recovery And Climate Experiment) mission were investigated as candidates to improve POD performance for the planned ICESat-2 mission. The objective of this research is to examine the effect of these time-varying geopotential models -- which include models of non-tidal atmospheric and ocean variability, seasonal variability caused by water mass motion, and secular variations caused by present-day ice-melt and glacial isostatic adjustment -- on ICESat POD. The quality of the POD solutions produced with the new geopotential models was quantified by examining the usual orbit quality tests -- DDHL (double-differenced high-low) and SLR (satellite laser ranging) observation residuals and orbit overlaps. Although the solutions produced in every test case indicated consistency and high accuracy of 1-2 cm, these metrics were rather insensitive to the small changes in the POD solutions induced by the new geopotential models, and were incapable of identifying any statistically significant improvements in the POD. However, examination of geographically correlated radial orbit perturbations showed that the radial orbit differences exhibited significant variability on the order of several millimeters, and were coherent with the temporal variability of the models implemented. Since radial orbit errors directly relate to the scientific quantities of interest in the ICESat mission -- the altimetry measurements and derived ice-sheet surface elevations -- this result is of obvious importance. The most notable effects included an annual radial orbit variation of up to 4 mm over the Amazon region induced by implementing the GRACE Annual model, and a secular variation of radial orbit differences over Greenland when the GRACE Trend model was applied. The effect of radial orbit error on ice-sheet altimetry was quantified by examining the mean geographically correlated radial orbit differences. Since the ice sheet elevation rates computed by ICESat scientists are on the order of tens of centimeters per year, it was concluded that, although the radial orbit perturbations are readily observable, with magnitudes on the order of a few millimeters they are too small to have a significant impact on the altimetry science. However, depending on the scientific objectives and radial orbit accuracy requirements set for ICESat-2, these effects may be important, and the use of time-varying geopotential models in ICESat-2 POD may be beneficial. / text

Thesis (M.S.)--Missouri University of Science and Technology, 2010. / Vita. The entire thesis text is included in file. Title from title screen of thesis/dissertation PDF file (viewed April 7, 2010) Includes bibliographical references (p. 70-77).

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.

This thesis will focus on the performance of angles only initial orbit determi- nation (IOD) methods on observational data of low Earth orbit (LEO) CubeSats. Using data obtained by Lockheed Martin’s Space Object Tracking (SpOT) facil- ity, four methods: Gauss, Double-R, Gooding and Assumed Circular, will use different amounts of orbital arc to determine which methods perform the best in the short arc regime of less than 10 degrees of orbital arc. Once the best method for estimating the orbit is determined, there will be analysis on whether these IOD methods are accurate enough to predict a secondary observation session. Finally non-linear regression will be performed to determine if the error metrics follow a predictable trend based on how much orbital arc is seen by the observer. It was determined that above a certain amount of orbital arc, angles only IOD methods can reliably predict a secondary observation session to facilitate more observations. Below 4 degrees of orbital arc, which is around 60 seconds of ob- serving time for LEO objects, none of the methods were able to reliably predict a secondary observation session. The Assumed Circular method was the best method for observing LEO CubeSats because it forces the IOD solution to be circular, which limits the error in the shape of the orbit as the amount of orbital arc decreases. Finally, many metrics follow an exponential trend when compared to the orbital arc. Thus, the amount of orbital arc seen is a strong predictor for the accuracy of the angles only IOD solutions.

This dissertation discusses research that focuses on understanding transneptunian objects (TNOs) using a variety of techniques and approaches. In Chapter 1, I introduce the main concepts used throughout this dissertation and discuss the current understanding of the transneptunian region. In Chapter 2, I discuss my efforts to understand how Neptune's late stages of migration affect the Haumea family, the only known collisional family in the transneptunian region. Using advanced simulations of Neptune migration, I find that the Haumea family can plausibly form before the termination of giant planet migration and show that this extensively mixes the family. The simplest explanation for the formation of Haumea and its family is a slow disruption of a large, primordial binary system. In Chapter 3, I examine the detectability of non-Keplerian effects in the mutual orbits of transneptunian binaries. I find non-Keplerian effects are common, with 20% of TNBs best explained by a non-Keplerian orbit. I also demonstrate that one of the components of TNB (66652) Borasisi-Pabu is a contact binary. In Chapter 4, I examine the non-Keplerian orbits of Hi'iaka and Namaka, the satellites of Haumea, showing that they are strongly affected by both inter-satellite gravitational interactions and precession caused by Haumea's nonspherical gravitational field. Future observations of the Haumea system, combined with non-Keplerian fitting, will sensitively probe Haumea's interior. Lastly, in Chapter 5, I explore the mutual orbits of Cold Classical TNO binaries using non-Keplerian orbit fitting. Out of a sample of 18 binaries, 6 have significantly non-Keplerian orbits, allowing detailed characterization of their system architecture. I find that 3 of these systems are best explained as hierarchical systems, while the remaining 3 are consistent with precession due to the Sun's gravitational influence. The hierarchical systems I find strongly support the streaming instability theory of planetesimal formation.

A new method of numerical integration is presented here, the variable-step Stormer-Cowell method. The method uses error control to regulate the step size, so larger step sizes can be taken when possible, and is double-integration, so only one evaluation per step is necessary when integrating second-order differential equations. The method is not variable-order, because variable-order algorithms require a second evaluation. The variable-step Stormer-Cowell method is designed for space surveillance applications,which require numerical integration methods to track orbiting objects accurately. Because of the large number of objects being processed, methods that can integrate the equations of motion as fast as possible while maintaining accuracy requirements are desired. The force model used for earth-orbiting objects is quite complex and computationally expensive, so methods that minimize the force model evaluations are needed. The new method is compared to the fixed-step Gauss-Jackson method, as well as a method of analytic step regulation (s-integration), and the variable-step variable-order Shampine-Gordon integrator. Speed and accuracy tests of these methods indicate that the new method is comparable in speed and accuracy to s-integration in most cases, though the variable-step Stormer-Cowell method has an advantage over s-integration when drag is a significant factor. The new method is faster than the Shampine-Gordon integrator, because the Shampine-Gordon integrator uses two evaluations per step, and is biased toward keeping the step size constant. Tests indicate that both the new variable-step Stormer-Cowell method and s-integration have an advantage over the fixed-step Gauss-Jackson method for orbits with eccentricities greater than 0.15. / Ph. D.

This thesis investigates the use of the Swedish Allsky Meteor Network (SAMN) for observing, identifying, and determining the orbits of satellites. The overall goal of this project is to determine the feasibility of using such a network for Space Situational Awareness (SSA) purposes, which requires identification and monitoring of objects in orbit. This thesis is a collaboration with the Swedish Defense Research Agency (FOI) to support their efforts in SSA. Within the frame of this project, the author developed software that can take data of observations of an object collected from the all-sky cameras of SAMN and do an Initial Orbit Determination (IOD) of the object. An algorithm that improves the results of the IOD was developed and integrated into the software. The software can also identify the object if it is in a database that the program has access to or, if it could not be identified, make an approximate prediction of when and where the object will be visible again the next time it flies over. A program that analyses the stability of the results of the IOD was also developed. This measures the spread in results of the IOD when a small amount of artificial noise is added to one or more of the observed coordinates in the sky. It was found that using multiple cameras at different locations greatly improves the stability of the solutions. Gauss' method was used for doing the IODs. The advantages and disadvantages of using this method are discussed, and ultimately other methods, such as the Gooding method or Double R iteration, are recommended for future works. This is mostly because Gauss' method has a singularity when all three lines of sight from observer to object lie in the same plane, which makes the results unreliable. The software was tested on a number of observations, both synthetic and real, and the results were compared against known data from public databases. It was found that these techniques can, with some changes, be used for doing IOD and satellite identification, but that doing very accurate position determination required for full orbit determination is not feasible. / Detta examensarbete undersöker möjligheterna att använda ett svenskt nätverk av allskykameror kallat SAMN (Swedish Allsky Meteor Network) för att observera, identifiera och banbestämma satelliter. Det övergripande målet med detta projekt är att bestämma hur användbart ett sådant nätverk skulle vara för att skapa en rymdlägesbild, vilken i sin tur kräver bevakning och identifikation av objekt som ligger i omloppsbana. Detta examensarbete är ett samarbete mellan Uppsala Universitet och FOI (Totalförsvarets Forskningsinstitut). Inom ramen för detta projekt har författaren utvecklat mjukvara som kan ta data från observationer av objekt utförda av SAMN och göra initiala banbestämningar av objekten. En algoritm som förbättrar resultaten av den initiala banbestämningen utvecklades och integrerades i programmen. Programmen kan också identifiera satelliter om de finns med i en databas som programmet har tillgång till eller förutsäga objektets nästa passage över observatören om det inte kunde identifieras. Ett annat program som analyserar känsligheten av resultaten av den initiala banbestämningen utvecklades också. Detta program mäter spridningen i resultat som orsakas av små störningar i de observerade koordinaterna på himlen. Det framkom att stabiliteten av resultaten kan förbättras avsevärt genom att använda flera observatörer på olika orter. I detta projekt användes Gauss metod för att göra banbestämningarna. Metodens för- och nackdelar diskuteras och i slutänden rekommenderas istället andra metoder, som Goodings metod eller Dubbel R-iteration, för framtida arbeten. Detta beror mest på att Gauss metod innehåller en singularitet när alla siktlinjer från observatören till objektet ligger i samma plan som varandra vilket gör resultaten opålitliga i de fallen. Programmen testkördes på ett antal olika observationer, både artificiella och verkliga, och resultaten jämfördes med kända positioner. Slutsatsen av arbetet är att de undersökta teknikerna kan, med vissa modifikationer, användas för att göra initiala banbestämningar och satellitidentifikationer, men att göra de väldigt precisa positionsbestämningarna som krävs för fullständig banbestämning är inte genomförbart.

A perfect duality exists between the problem of space-based orbit determination from line-of-sight measurements and the problem of designing an interplanetary autonomous navigation system. Mathematically, these two problems are equivalent. Any method solving the first problem can be used to solve the second one and, vice versa. While the first problem estimates the observed unknown object orbit using the known observer orbit, the second problem does exactly the opposite (e.g. the spacecraft observes a known visible planet). However, in an interplanetary navigation problem, in addition to the measurement noise, the following "perturbations" must be considered: 1) light-time effect due to the finite speed of light and large distances between the observer and planets, and 2) light aberration including special relativistic effect. These two effects require corrections of the initial orbit estimation problems. Because of the duality problem of space-based orbit determination, several new techniques of angles-only Initial Orbit Determination (IOD) are here developed which are capable of using multiple observations and provide higher orbit estimation accuracy and also they are not suffering from some of the limitations associated with the classical and some newly developed methods of initial orbit determination. Using multiple observations make these techniques suitable for the coplanar orbit determination problems which are the case for the spacecraft navigation using visible planets as the solar system planets are all almost coplanar. Four new IOD techniques were developed and Laplace method was modified. For the autonomous navigation purpose, Extended Kalman Filter (EKF) is employed. The output of the IOD algorithm is then used as the initial condition to extended Kalman filter. The two "perturbations" caused by light-time effect and stellar aberration including special relativistic effect also need to be taken into consideration and corrections should be implemented into the extended Kalman filter scheme for the autonomous spacecraft navigation problem.

Nowadays, Global Positioning System (GPS) which provide global coverage, continuous tracking capability and high accuracy has been preferred as the primary tracking system for onboard real-time precision orbit determination of Low Earth Orbiters (LEO). In this work, real-time orbit determination algorithms are established on the basis of extended Kalman, unscented Kalman, regularized particle, extended Kalman particle and extended H-infinity filters. Particularly, particle filters which have not been applied to the real time orbit determination until now are also performed in this study and H-infinity filter is presented using all kinds of real GPS observations. Additionally, performance of unscented Kalman filter using GRAPHIC (Group and Phase Ionospheric Correction) measurements is investigated. To evaluate performances of all algorithms, comparisons are carried out using different types of GPS observations concerning C/A (Coarse/Acquisition) code pseudorange, GRAPHIC and navigation solutions. A software package for real time orbit determination is developed using recursive filters mentioned above. The software is implemented and tested in MATLAB&copy / R2010 programming language environment on the basis of the object oriented programming schema.