Solar radiation pressure (SRP) is one of the major non-gravitational forces acting on spacecraft. Acceleration by radiation pressure depends on the radiation flux; on spacecraft shape, attitude, and mass; and on the optical properties of the spacecraft surfaces. Precise modeling of SRP is needed for dynamic satellite orbit determination, space mission design and control, and processing of data from space-based science instruments. During Earth penumbra transitions, sunlight is passing through Earth's lower atmosphere and, in the process, its path, intensity, spectral composition, and shape are significantly affected. This dissertation presents a new method for highly physical SRP modeling in Earth's penumbra called Solar radiation pressure with Oblateness and Lower Atmospheric Absorption, Refraction, and Scattering (SOLAARS). The fundamental geometry and approach mirrors past work, where the solar radiation field is modeled using a number of light rays, rather than treating the Sun as a single point source. This dissertation aims to clarify this approach, simplify its implementation, and model previously overlooked factors. The complex geometries involved in modeling penumbra solar radiation fields are described in a more intuitive and complete way to simplify implementation. Atmospheric effects due to solar radiation passing through the troposphere and stratosphere are modeled, and the results are tabulated to significantly reduce computational cost. SOLAARS includes new, more efficient and accurate approaches to modeling atmospheric effects which allow us to consider the spatial and temporal variability in lower atmospheric conditions. A new approach to modeling the influence of Earth's polar flattening draws on past work to provide a relatively simple but accurate method for this important effect. Previous penumbra SRP models tend to lie at two extremes of complexity and computational cost, and so the significant improvement in accuracy provided by the complex models has often been lost in the interest of convenience and efficiency. This dissertation presents a simple model which provides an accurate alternative to the full, high precision SOLAARS model with reduced complexity and computational cost. This simpler method is based on curve fitting to results of the full SOLAARS model and is called SOLAARS Curve Fit (SOLAARS-CF). Both the high precision SOLAARS model and the simpler SOLAARS-CF model are applied to the Gravity Recovery and Climate Experiment (GRACE) satellites. Modeling results are compared to the sub-nm/s^2 precision GRACE accelerometer data and the results of a traditional penumbra SRP model. These comparisons illustrate the improved accuracy of the SOLAARS and SOLAARS-CF models. A sensitivity analyses for the GRACE orbit illustrates the significance of various input parameters and features of the SOLAARS model on results. The SOLAARS-CF model is applied to a study of penumbra SRP and the Earth flyby anomaly. Beyond the value of its results to the scientific community, this study provides an application example where the computational efficiency of the simplified SOLAARS-CF model is necessary. The Earth flyby anomaly is an open question in orbit determination which has gone unsolved for over 20 years. This study quantifies the influence of penumbra SRP modeling errors on the observed anomalies from the Galileo, Cassini, and Rosetta Earth flybys. The results of this study prove that penumbra SRP is not an explanation for or significant contributor to the Earth flyby anomaly. / Ph. D.

The Virginia Tech Optical Satellite Tracking Telescope (VTOST) aims to test the feasibility of a commercial off-the-shelf (COTS) designed tracking system for Space Situational Awareness (SSA) data contribution. A novel approach is considered, combining two COTS systems, a high-powered telescope, built for astronomy purposes, and a larger field of view (FOV) camera. Using only publicly available two-line element sets (TLEs), orbital propagation accuracy degrades quickly with time from epoch and is often not accurate enough to task a high-powered, small FOV telescope. Under this experimental approach, the larger FOV camera is used to acquire and track the resident space object (RSO) and provide a real-time pointing update to allow the high-powered telescope to track the RSO and provide possible resolved imagery. VTOST is designed as a remotely taskable sensor, based on current network architecture, capable of serving as a platform for further SSA studies, including unresolved and resolved imagery analysis, network tasking, and orbit determination. Initial design considerations are based on the latest Raven class and other COTS based telescope research, including research by the Air Force Research Lab (AFRL), ExoAnalytic Solutions, and other university level telescope projects. A holistic system design, including astronomy, image processing, and tracking methods, in a low-budget environment is considered. Method comparisons and results of the system design process are presented. / Master of Science

<p dir="ltr">Object tracking in cislunar space has become an area of interest within many communities where cislunar space domain awareness (SDA) is critical to operations. Due to the influence of both the Earth and Moon on objects in this domain, the classical two body problem does not accurately describe the dynamics of the state. Legacy tracking capabilities fall short in providing accurate state estimates due to the large volume of space and the highly non-linear dynamics involved. In order to advance SDA in cislunar space, tracking capabilities must be updated for this domain. </p><p dir="ltr">Both the Extended Kalman Filter (EKF) and Gaussian Mixture Extended Kalman Filter (GM-EKF) are used for orbit determination in this thesis along side the Circular Restricted Three Body Problem (CR3BP) to model the non-linear dynamics. The filters are utilized to determine the best estimate of the state as well as its covariance. The two filter's performances are compared to highlight areas in which the assumptions surrounding the EKF are violated resulting in failed tracking, as well as to highlight the power of the GM-EKF for non-linear systems using splitting and merging techniques. </p><p dir="ltr">This thesis presents single and multiple object tracking of objects in a multitude of cislunar orbits using a Moon ground-based sensor. Multiple object tracking is accomplished using a novel Lyapunov-based scheduler in order to reduce the total system uncertainty. The environment is modeled to include exclusion zones which preclude measurements. These zones consist of conjunction from the Earth and Sun, brightness constraints, and camera field of regard (FOR). When measurements are unavailable the uncertainty in the state estimation rises significantly.</p><p dir="ltr">An investigation of varied sensor placements and Sun-Earth-Moon geometries provides results to inform locations and trends which are able to confidently track both single and multiple objects in cislunar orbits. </p>

Earth science observations are crucial for our understanding of the Earth’s climate, water cycle, land, and atmosphere. Signals of Opportunity (SoOp) has recently emerged as an innovative method for producing these observations. SoOp reuses existing satellite communication signals for science measurements. A key factor in the accuracy of SoOp measurements, is the accuracy with which the transmitting satellite’s position can be determined. This thesis developed a distributed network of receivers, which performed time delay of arrival (TDOA) measurements, to solve for the position of a transmitting satellite, using their existing signals. These results were used to characterize the sensitivity of the calculated satellite position, to the TDOA measurement error.

The expansion of space activities has led to an increase in congestion. With this increase, there has been a growing emphasis on the importance of space situational awareness. Without it, the space environment can become hazardous, with increasing threats of collision and debris generation. As the utilization of space continues to grow, effective methods for tracking objects to maintain situational awareness must also be considered. One approach to tracking objects in space involves a series of steps, including optical observations and orbit estimations using initial orbit determination methods, followed by additional observations and continuous tracking. However, a challenge with this tracking method is the low quality and quantity of observational data, which can impact the accuracy achieved from these methods. This thesis will study two new and two traditional methods of initial orbit determination, analyzing improvements in accuracy with limited data for each method. Additionally, the impact of observation elevation will be analyzed to assess its effects on the quality of data, and how this, along with a limited amount of data, can affect the overall initial orbit determination accuracy.

Autonomous navigation is the determination of satellites position and velocity vectors onboard the satellite, using the measurements available onboard. The orbital information of a satellite needs to be obtained to support different house keeping operations such as routine tracking for health monitoring, payload data processing and annotation, orbit manoeuver planning, and prediction of intrusion in various sensors' field of view by celestial bodies like Sun, Moon etc. Determination of the satellites orbital parameters is done in a number of ways using a variety of measurements. These measurements may originate from ground based systems as range and range rate measurements, or from another satellite as in the case of GPS (Global Positioning System) and TDUSS (Tracking Data Relay Satellite Systems), or from the same satellite by using sensors like horizon sensor^ sun sensor, star tracker, landmark tracker etc. Depending upon the measurement errors, sampling rates, and adequacy of the estimation scheme, the navigation accuracy can be anywhere in the range of 10m - 10 kms in absolute location. A wide variety of tracking sensors have been proposed in the literature for autonomous navigation. They are broadly classified as (1) Satellite-satellite tracking, (2) Ground- satellite tracking, (3) fully autonomous tracking. Of the various navigation sensors, it may be cost effective to use existing onboard sensors which are well proven in space. Hence, in the current thesis, the Horizon scanner is employed as the primary navigation sensor-. It has been shown in the literature that by using horizon sensors and gyros, a high accuracy pointing of the order of .01 - .03 deg can be achieved in the case of low earth orbits. Motivated by such a fact, the current thesis deals with autonomous orbit determination using measurements from the horizon sensors with the assumption that the attitude is known to the above quoted accuracies. The horizon scanners are mounted on either side of the yaw axis in the pitch yaw plane at an angle of 70 deg with respect to the yaw axis. The Field Of View (FOV) moves about the scanner axis on a cone of 45 deg half cone angle. During each scan, the FOV generates two horizon points, one at the space-Earth entry and the other at the Earth-space exit. The horizon points, therefore, lie• on the edge of the Earth disc seen by the satellite. For a spherical earth, a minimum of three such horizon points are needed to estimate the angular radius and the center of the circular horizon disc. Since a total of four horizon points are available from a pair of scanners, they can be used to extract the satellite-earth distance and direction.These horizon points are corrupted by noise due to uncertainties in the Earth's radiation pattern, detector mechanism, the truncation and roundoff errors due to digitisation of the measurements. Owing to the finite spin rate of the scanning mechanism, the measurements are available at discrete time intervals. Thus a filtering algorithm with appropriate state dynamics becomes essential to handle the •noise in the measurements, to obtain the best estimate and to propagate the state between the measurements. The orbit of a low earth satellite can be represented by either a state vector (position and velocity vectors in inertial frame) or Keplerian elements. The choice depends upon the available processors, functions and the end use of the estimated orbit information. It is shown in the thesis that position and velocity vectors in inertial frame or the position vector in local reference frame, do result in a simplified, state representation. By using the f and g series method for inertial position and velocity, the state propagation is achieved in linear form. i.e. Xk+1 = AXK where X is the state (position, velocity) and A the state transition matrix derived from 'f' and 'g' series. The configuration of a 3 axis stabilised spacecraft with two horizon scanners is used to simulate the measurements. As a step towards establishing the feasibility of extracting the orbital parameters, the governing equations are formulated to compute the satellite-earth vector from the four horizon points generated by a pair of Horizon Scanners in the presence of measurement noise. Using these derived satellite-earth vectors as measurements, Kalman filter equations are developed, where both the state and measurements equations are linear. Based on simulations, it is shown that a position accuracy of about 2 kms can be achieved. Additionally, the effect of sudden disturbances like substantial slewing of the solar panels prior and after the payload operations are also analysed. It is shown that a relatively simple Low Pass Filter (LPF) in the measurements loop with a cut-off frequency of 10 Wo (Wo = orbital frequency) effectively suppresses the high frequency effects from sudden disturbances which otherwise camouflage the navigational information content of the signal. Then Kalman filter can continue to estimate the orbit with the same kind of accuracy as before without recourse to re-tuning of covariance matrices. Having established the feasibility of extracting the orbit information, the next step is to treat the measurements in its original form, namely, the non-linear form. The entry or exit timing pulses generated by the scanner when multiplied by the scan rate yield entry or exit azimuth angles in the scanner frame of reference, which in turn represents an effective measurement variable. These azimuth angles are obtained as inverse trigonometric functions of the satellite-earth vector. Thus the horizon scanner measurements are non-linear functions of the orbital state. The analytical equations for the horizon points as seen in the body frame are derived, first for a spherical earth case. To account for the oblate shape of the earth, a simple one step correction algorithm is developed to calculate the horizon points. The horizon points calculated from this simple algorithm matches well with the ones from accurate model within a bound of 5%. Since the horizon points (measurements) are non-linear functions of the state, an Extended Kalman Filter (EKF) is employed for state estimation. Through various simulation runs, it is observed that the along track state has got poor observability when the four horizon points are treated as measurements in their original form, as against the derived satellite-earth vector in the earlier strategy. This is also substantiated by means of condition number of the observability matrix. In order to examine this problem in detail, the observability of the three modes such as along-track, radial, and cross-track components (i.e. the local orbit frame of reference) are analysed. This difficulty in observability is obviated when an additional sensor is used in the roll-yaw plane. Subsequently the simulation studies are carried out with two scanners in pitch-yaw plane and one scanner in the roll-yaw plane (ie. a total of 6 horizon points at each time). Based on the simulations, it is shown that the achievable accuracy in absolute position is about 2 kms.- Since the scanner in the roll-yaw plane is susceptible to dazzling by Sun, the effect of data breaks due to sensor inhibition is also analysed. It is further established that such data breaks do not improve the accuracy of the estimates of the along-track component during the transient phase. However, filter does not diverge during this period. Following the analysis of the' filter performance, influence of Earth's oblateness on the measurement model studied. It is observed that the error in horizon points, due to spherical Earth approximation behave like a sinusoid of twice the orbital frequency alongwith a bias of about 0.21° in the case of a 900 kms sun synchronous orbit. The error in the 6 horizon points is shown to give rise to 6 sinusoids. Since the measurement model for a spherical earth is the simplest one, the feasibility of estimating these sinusoids along with the orbital state forms the next part of the thesis. Each sinusoid along with the bias is represented as a 3 state recursive equation in the following form where i refers to the ith sinusoid and T the sampling interval. The augmented or composite state variable X consists of bias, Sine and Cosine components of the sinusoids. The 6 sinusoids together with the three dimensional orbital position vector in local coordinate frame then lead to a 21 state augmented Kalman Filter. With the 21 state filter, observability problems are experienced. Hence the magnetic field strength, which is a function of radial distance as measured by an onboard magnetometer is proposed as additional measurement. Subsequently, on using 6 horizon point measurements and the radial distance measurements obtained from a magnetometer and taking advantage of relationships between sinusoids, it is shown that a ten state filter (ie. 3 local orbital states, one bias and 3 zero mean sinusoids) can effectively function as an onboard orbit filter. The filter performance is investigated for circular as well as low eccentricity orbits. The 10-state filter is shown to exhibit a lag while following the radial component in case of low eccentricity orbits. This deficiency is overcome by introducing two more states, namely the radial velocity and acceleration thus resulting in a 12-state filter. Simulation studies reveal that the 12-state filter performance is very good for low eccentricity orbits. The lag observed in 10-state filter is totally removed. Besides, the 12-state filter is able to follow the changes in orbit due to orbital manoeuvers which are part of orbit acquisition plans for any mission.

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.

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Resumo: O Sistema de Posicionamento Global oferece um poderoso e relativamente barato processo para se determinar órbitas de satélites artificiais da Terra. Este trabalho apresenta um método de determinação de órbita para satélites com um receptor GPS a bordo. Medidas de pseudo-distância são usadas para estimar o vetor de estado. O estimador considerado é o método dos mínimos quadrados recursivo, através de rotações ortogonais de Givens, com a finalidade de evitar problemas numéricos e de inversão de matrizes. É considerado a modelagem das forças devido ao geopotencial de alto grau e ordem. Resultados indicam que a precisão em posição melhor que 10 m foi obtido usando dados reais do satélite Topex (com um mínimo de duas horas de dados - aproximadamente um período orbital). O resíduo de pseudo-distância teve um desvio padrão cerca de 5 m. / Abstract: The Global Positioning System is a powerful and low cost process to compute orbits for some artificial Earth satellites. This work presents a method of orbit determination for satellites with an onboard GPS receiver. Pseudo-ranges are used in the measurements equations for the orbit estimator. The estimator considered is the recursive least squares method, numerically improved with orthogonal Givens rotations and thus avoiding problems concerning inversion of matrices. Up to high order geopotential perturbations are taken into account. Results indicate that precision better than 10 m is easily obtained using batches of one orbital period for the TOPEX satellite (two hours of orbital period). Standard deviation of about 5 m resulted for the residuals. / Orientador: Rodolpho Vilhena de Moraes / Coorientador: Hélio Koiti Kuga / Mestre