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

The system of sparse gridpoints was used to propagate uncertainty forward in time through orbital mechanics simulations. Propagation of initial uncertainty through a nonlinear dynamic model is examined in regards to the uncertainty of orbit estimation. The necessary underlying mechanics of orbital mechanics, probability, and nonlinear estimation theory are reviewed to allow greater understanding of the problem. The sparse grid method itself and its implementation is covered in detail, along with the necessary properties and how to best it to a given problem based on inputs and desired outputs. Three test cases were run in the form of a restricted two-body problem, a perturbed two-body problem, and a three-body problem in which the orbiting body is positioned at a Lagrange point. It is shown that the sparse grid method shows sufficient accuracy for all mean calculations in the given problems and that higher accuracy levels allow for accurate estimation of higher moments such as the covariance.

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.

For the upcoming formation flying technology demonstration mission NetSat a relative orbit propagator as well as a relative orbit controller were developed. The formation will consist of four equal nano-satellites with an electric propulsion system for orbit correction manoeuvres. This demands the use of continuous low-thrust models for relative orbit control, which is a novel field. A software framework was developed which allows orbit simulations of the whole fleet in a fully non-linear environment. The final on-board relative propagator is based on the Gim-Alfriend STM and incorporates eccentricity and the non-spherical shape of the Earth. The controller uses control Lyapunov function-based design and model predictive control, depending on the task. The guidance and control system is able to safely govern the relative motion for one-, two and three-dimensional formation configurations with inter-satellite distances as low as 50m. Based on these results, a complete mission plan is proposed.

Dissertation (Ph. D.)--University of Stellenbosch, 1997. / Bibliography. Also available via the Internet

This graduate project compares legacy simplified general perturbation model (SGP4) code developed by NASA Johnson Space Center, Orbital Debris Program Office, to a recent public release of SGP4 code by David Vallado. The legacy code is a subroutine in a larger program named PREDICT, which is used to predict the location of orbital debris in GEO. Direct comparison of the codes showed that the new code yields better results for GEO objects, which are more accurate by orders of magnitude (error in meters rather than kilometers). The public release of SGP4 also provides effective results for LEO and MEO objects on a short time scale. The public release code was debugged and modified to provide instant functionality to the Orbital Debris Program Office. Code is provided in an appendix to this paper along with an accompanying CD. A User’s Guide is presented in Chapter 7.

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.

<pre>Because of the increasing population of space objects, there is an increasing necessity to monitor and predict the status of the near-Earth space environment, especially of critical regions like geosynchronous Earth orbit (GEO) and low Earth orbit (LEO) regions, for a sustainable future. Space Situational Awareness (SSA), however, is a challenging task because of the requirement for dynamically insightful fast orbit propagation models, presence of dynamical uncertainties, and limitations in sensor resources. Since initial parameters are often not known exactly and since many SSA applications require long-term orbit propagation, long-term effects of the initial uncertainties on orbital evolution are examined in this work. To get a long-term perspective in a fast and efficient manner, this work uses analytical propagation techniques. Existing analytical theories for orbital perturbations are investigated, and modifications are made to them to improve accuracy. While conservative perturbation forces are often studied, of particular interest here is the orbital perturbation due to non-conservative forces. Using the previous findings and the developments in this thesis, two SSA applications are investigated in this work. In the first SSA application, a sensor tasking algorithm is designed for the detection of new classes of GEO space objects. In the second application, the categorization of near-GEO objects is carried out by combining knowledge of orbit dynamics with machine learning techniques.</pre>

Thesis (PhD (Electrical and Electronic Engineering))--University of Stellenbosch, 1997. / This dissertation considers the feasibility of using atmospheric drag to control constellations of micro-satellites in low Earth orbits. The constellation control requirements include an acquisition phase and a maintenance phase. Optimal strategies are designed to control the relative positions of the satellites during these two phases. It is shown that the feasibility and success of the strategies depend on many factors, including the satellite properties and orbital configuration. A nominal test constellation is presented and used as a generic example for the application of the control strategies. The dissertation also focuses on the accurate modelling and simulation of a typical low Earth orbit satellite, moving under the influence of a variety of significant orbit perturbation forces. The simulations form an integral part of the study and are used to verify the application of all the proposed control strategies.

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.