Thesis (M.Sc.). / Written for the Dept. of Physics. Title from title page of PDF (viewed 2008/12/04). Includes bibliographical references.

A wide variety of scientifically interesting missions could be enabled by orbital flight altitudes of 150 – 250 km. For the present work, this range of altitudes is defined as extremely Low Earth Orbit (eLEO). The use of low-cost nanosatellites (mass < 10 kg) has reduced the cost barrier to orbital flight over the last decade and the present study investigates the feasibility of using primarily commercial, off-the-shelf (COTS) hardware to build a nanosat specifically to allow extended mission times in eLEO. CubeSats flying in the lower thermosphere have the potential to enable close monitoring of the Earth’s surface for scientific, commercial, and defense-related missions. The results of this research show that the proper selection of primary and attitude control thrusters combined with precise control techniques result in significant extension of the orbital life of a CubeSat in eLEO, thus allowing detailed explorations of the atmosphere. In this study, the orbit maintenance controller is designed to maintain a mission-averaged, mean altitude of 244 km. An estimate is made of the primary disturbance torque due to aerodynamic drag using a high-fidelity calculation of the rarefied gas drag based on a Direct Simulation, Monte-Carlo simulation. The primary propulsion system consists of a pair of electrospray thrusters providing a combined thrust of 0.12 mN at 1 W. Results of a trade study to select the best attitude control option indicate pulsed plasma thrusters operating at 1 W are preferable to reaction wheels or mangetorquers at the selected altitude. An extended Kalman filter is used for orbital position and spacecraft attitude estimations. The attitude determination system consists of sun sensors, magnetometers, gyroscopes serving as attitude sensors. The mission consists of two phases. In Phase I, a 4U CubeSat is deployed from a 414 km orbit and uses the primary propulsion system to deorbit to an initial altitude within the targeted range of 244 +/- 10 km. Phase I lasts 12.73 days with the propulsion system consuming 5.6 g of propellant to deliver a ∆V of 28.12 m/s. In Phase II the mission is maintained until the remaining 25.2 g of propellant is consumed. Phase II lasts for 30.27 days, corresponding to a ∆V of 57.22 m/s with a mean altitude of 244 km. The mean altitude for an individual orbit over the entire mission was found to vary from a maximum of 252 km to a minimum of 236 km. Using this approach, a primary mission life of 30.27 days could be achieved, compared with 3.1 days without primary propulsion.

The problem of observing a given set of Earth target sites within an assigned time frame is examined. Attention is given mainly to visiting these sites as sub-satellite nadir points. Solutions to this problem in the literature require thrusters to continuously maneuver the satellite from one site to another. A natural solution is proposed. A natural solution is a gravitational orbit that enables the spacecraft to satisfy the mission requirements without maneuvering. Optimization of a penalty function is performed to find natural solutions for satellite orbit configurations. This penalty function depends on the mission objectives. Two mission objectives are considered: maximum observation time and maximum resolution. The penalty function poses multi minima and a genetic algorithm technique is used to solve this problem. In the case that there is no one orbit satisfying the mission requirements, a multi-orbit solution is proposed. In a multi-orbit solution, the set of target sites is split into two groups. Then the developed algorithm is used to search for a natural solution for each group. The satellite has to be maneuvered between the two solution orbits. Genetic algorithms are used to find the optimal orbit transfer between the two orbits using impulsive thrusters. A new formulation for solving the orbit maneuver problem using genetic algorithms is developed. The developed formulation searches for a mini mum fuel consumption maneuver and guarantees that the satellite will be transferred exactly to the final orbit even if the solution is non-optimal. The results obtained demonstrate the feasibility of finding natural solutions for many case studies. The problem of the design of suitable satellite constellation for Earth observing applications is addressed. Two cases are considered. The first is the remote sensing missions for a particular region with high frequency and small swath width. The second is the interferometry radar Earth observation missions. In satellite constellations orbit's design, a new set of compatible orbits, called the "Two-way orbits",whose ground track path is a closed-loop trajectory that intersects itself, in some points, with tangent intersections is introduced. Conditions are derived on the orbital elements such that these Two-way Orbits exist and satellites flying in these orbits pass the tangent intersection points at the same time. Finally, the recently proposed concept of observing a space object from onboard a spacecraft using a star tracker is considered. The measurements of the star tracker provide directions to the target in space and do not provide range measurements. Estimation for the orbit of the target space object using the measurements of the star tracker is developed. An observability analysis is performed to derive conditions on the observability of the system states. The Gaussian Least Squares Differential Correction Technique is implemented. The results obtained demonstrate the feasibility of using the measurements of the star tracker to get a good estimate for the target orbit within a period of measurements ranging from about 20 percent to 50 percent of the orbital period depending on the two orbits.

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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.

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This thesis studies orbit equivalence in the almost continuous setting. Recently A. del Junco and A. Sahin obtained an almost continuous version of Dye’s theorem. They proved that any two ergodic measure-preserving homeomorphisms of Polish spaces are almost continuously orbit equivalent. One purpose of this thesis is to extend their result to all free actions of countable amenable groups. We also show that the cocycles associated with the constructed orbit equivalence are almost continuous. In the second part of the thesis we obtain an analogue of Dye’s reconstruction theorem for etale equivalence relations in the almost continuous setting. We introduce topological full groups of etale equivalence relations and show that if the topological full groups are isomorphic, then the equivalence relations are almost continuously orbit equivalent.

This thesis studies orbit equivalence in the almost continuous setting. Recently A. del Junco and A. Sahin obtained an almost continuous version of Dye’s theorem. They proved that any two ergodic measure-preserving homeomorphisms of Polish spaces are almost continuously orbit equivalent. One purpose of this thesis is to extend their result to all free actions of countable amenable groups. We also show that the cocycles associated with the constructed orbit equivalence are almost continuous. In the second part of the thesis we obtain an analogue of Dye’s reconstruction theorem for etale equivalence relations in the almost continuous setting. We introduce topological full groups of etale equivalence relations and show that if the topological full groups are isomorphic, then the equivalence relations are almost continuously orbit equivalent.

Orbit of a spacecraft in three dimensional Inertial Reference Frame is in general represented by a standard set of six parameters like Keplerian Orbital Elements namely semimajor axis, eccentricity, inclination, argument of perigee, right ascension of ascending node, and true anomaly. An orbit can also be represented by an equivalent set of six parameters namely the position and velocity vectors, hereafter referred as orbit-vectors. The process of determining the six orbital parameters from redundant set of observations (more than the required minimum observations) is known as Orbit Determination (OD) process. This is, in general, solved using Least Squares principle. Availability of accurate, almost continuous, space borne observations provide tremendous scope for simplifications and new directions in Autonomous OD (AOD). The objective of this thesis is to develop a suitable scheme for onboard autonomy in OD, specifically for low-earth-orbit-missions that are in high demand in the immediate future. The focus is on adopting a simple orbit model by a thorough study and analysis by considering the individual contributions from the different force models or component accelerations acting on the spacecraft. Second step in this work is to address the application of an onboard estimation scheme like Kalman Filter for onboard processing. The impact of the approximation made in the orbit model for filter implementation manifests as propagation error or estimation residuals in the estimation. The normal procedure of tuning the filter is by getting an appropriate state and measurement noise covariance matrices by some means, sometimes through trial and error basis. Since this tuning is laborious and the performance may vary with different contexts, it is attempted to propose a scheme on a more general footing, with dynamically compensating for the model simplification. There are three parts of this problem namely (i) Analysis of different Orbit Dynamics Models and selection of a simplified Onboard Model (ii) Design of an Estimator Filter based on Kalman Filter approach for Onboard Applications and (iii) Development of a suitable Filter Compensation procedure to ensure best estimates of orbit vectors even with the simplified orbit model. Development of a Numerical Integration scheme (and a software tool) and extensive simulation exercises to justify the conclusion on the simple model to be used in the estimation procedure forms the first part of the thesis. Tables quantify the effect of individual accelerations and demonstrate the effects of various model components on orbit propagation. In general, it is well known that the atmospheric drag is a non-conservative force and reduces energy; it is also known that the effect of first zonal harmonic term is predominant than any other gravity parameters; such anticipated trends in the accuracies are obtained. This particular exercise is carried out for orbits of different altitudes and different inclinations. The analysis facilitates conclusions on a limited model orbit dynamics suitable for onboard OD. Procedures and results of this model selection analysis is published in Journal of Spacecraft Technology, Vol. 16, No.1,pp 8-30, Jan 2006, titled “Orbit Model Studies for Onboard Orbit Estimation” [69]. Design of Estimator based on Kalman Filter There are two steps involved in dealing with the next part of the defined work: • Design and implementation of Extended Kalman Filter Estimation (EKF) scheme • Steps to compensate for approximation made in the reduced orbit dynamics The GPS receivers on board some of the IRS satellites (for example, the Resource-Sat-1), output the GPS Navigation Solutions (GPSNS) namely the position and velocity vectors of the IRS satellite along with the Pseudo-range measurements. These are recorded onboard for about two orbits duration, and are down loaded. An Extended Kalman Filter Algorithm for the estimation of the orbit vectors using these GPSNS observations is developed. Estimation is carried out assuming a Gaussian white noise models for the state and observation noises. The results show a strong dependence on the initial covariance of the noise involved; reconstruction of the observations results only if the assumption of realistic noise characteristics (which are unknown) is strictly adhered. Hence this simple non-adaptive EKF is found inadequate for onboard OD scheme. Development of the Dynamics Filter Compensation (DFC) Scheme In next part of the thesis, the problem of dealing with the un-modeled accelerations has been addressed. A suitable model-compensation scheme that was first developed by D.S Ingram el at [60] and successfully applied to Lunar missions, has been modified suitably to treat the problem posed by the reduced orbit dynamics. Here, the un-modeled accelerations are approximated by the OU stochastic process described as the solution of the Langavin stochastic differential equation. A filter scheme is designed where the coefficients of the un- modeled acceleration components are also estimated along with the system state yielding a better solution. Further augmentation to the filter include a standard Adaptive Measurement Noise covariance update; results are substantiated with actual data of IRS-P6 (Resource–Sat 1, see chapter 4). Classified as the Structured Adaptive Filtering Scheme, this results in a Dynamic Filter Compensation(DFC) Scheme which provides distinctly improved results in the position of the state. First, the estimation is carried out using actual GPS Navigation Solutions as observations. What is to be estimated itself is observed; the State-Observation relation is simple. The results are seen to improve the orbit position five times; bringing down the position error from 40 meters to about 8 meters. However, this scheme superimposes an extra factor of noise in the velocity vector of the GPSNS solutions. It is noted that this scheme deals only with the process noise covariance. To tackle the noise introduced in the velocity components, modifications of the original scheme by introducing an adaptive measurement noise covariance update is done. This improves the position estimate further by about 2 meters and also removes the noise introduced in the velocity components and reconstructs the orbit velocity vector output of the GPSNS. The results are confirmed using one more set of actual data corresponding to a different date. This scheme is shown to be useful for obtaining continuous output –without data gaps- of the GPSNS output. Next, the estimation is carried out taking the actual GPS observations which are the Pseudo Range, Range rate measurements from the visible GPS satellites (visible to the GPS receiver onboard ). Switching over to the required formulation for this situation in the state-measurement relation profile, estimation is carried out. The results are confirmed in this case also. Clear graphs of comparisons with definitive orbital states (considered as actual) versus estimated states show that the model reduction attempted at the first part has been successfully tackled in this method. In this era of space-borne GPS observations, where frequent sampling of the orbiting body is suggestive of reduced orbit models, an attempt for replacement of the conventional treatment of expensive and elaborate OD procedure is proved feasible in this thesis work.

As technological advances throughout the field of satellite geodesy improve the accuracy of satellite measurements, numerical methods and algorithms must be able to keep pace. This becomes increasingly important for high precision applications, such as high degree/order gravity field recovery. Currently, the Gravity Recovery and Climate Experiment's (GRACE) dual one-way microwave ranging system can determine changes in inter-satellite range to a precision of a few microns; however, with the advent of laser measurement systems nanometer precision ranging is a realistic possibility. With this increase in measurement accuracy, a reevaluation of the accuracy inherent in the numerical integration algorithms is necessary. This study attempts to quantify and minimize these numerical errors in an effort to improve the accuracy of modeling and propagation of various orbital perturbations; helping to provide further insight into the behavior and evolution of the Earth's gravity field from the more capable gravity missions in the future. The numerical integration errors are examined for a variety of satellite accelerations. The propagation of orbits similar to those of the GRACE satellites using a gravitational model that assumes the Earth is a perfect sphere show integration errors, using double precision numerical representations, on the order of 1 micron in inter-satellite range and 0.1 nanometers per second in inter-satellite range-rate. In addition, when the Earth's gravitational field is formulated in spherical harmonics these numerical integration errors begin to contaminate signals to due harmonics approximately above degree 220, for an orbit at GRACE altitudes. Also, when examining the effect of mass anomalies on the Earth's surface, simulated as point masses, it is apparent that numerical integration methods are easily capable of resolving point mass anomalies as small as 0.05 gigatonnes. Finally, a numerical integration procedure is determined to accurately simulate the effect of numerous, small step accelerations applied to the satellite's center of mass due to misalignment and misfiring of the attitude thrusters. Future studies can then use this procedure as a metric to evaluate the accuracy and effectiveness of an accelerometer in reproducing these non-gravitational forces and how these errors might affect gravity field recovery. / text