Thesis (M.Com.(quantitative Management)) / Satellite Scheduling Problems (SSP) are NP-hard and constraint programming and metaheuristics solution methods yield mixed results. This study investigates a new version of the SSP, the Satellite Constellation Time-Window Optimization Problem (SCoTWOP), involving commercial satellite constellations that provide frequent earth coverage. The SCoTWOP is related to the dual of the Vehicle Routing Problem with Multiple Timewindows, suggesting binary solution vectors representing an activation of time-windows. This representation fitted well with the MatLab® Genetic Algorithm and Direct Search Toolbox subsequently used to experiment with genetic algorithms, tabu search, and simulated annealing as SCoTWOP solution methods. The genetic algorithm was most successful and in some instances activated all 250 imaging time-windows, a number that is typical for a constellation of six satellites. / Quantitative Management

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.

Thesis (MEng)--Stellenbosch University, 2013. / ENGLISH ABSTRACT: Throughout its lifetime in space, a spacecraft is exposed to risk of collision with orbital debris or operational satellites. This risk is especially high within the Low Earth Orbit (LEO) region where the highest density of space debris is accumulated. This study investigates orbital decay of some LEO micro-satellites and accelerating orbit decay by using a deorbitsail. The Semi-Analytical Liu Theory (SALT) and the Satellite Toolkit was employed to determine the mean elements and expressions for the time rates of change. Test cases of observed decayed satellites (Iridium-85 and Starshine-1) are used to evaluate the predicted theory. Results for the test cases indicated that the theory tted observational data well within acceptable limits. Orbit decay progress of the SUNSAT micro-satellite was analysed using relevant orbital parameters derived from historic Two Line Element (TLE) sets and comparing with decay and lifetime prediction models. The study also explored the deorbit date and time for a 1U CubeSat (ZACUBE-01). A proposed orbital debris solution or technology known as deorbitsail was also investigated to gain insight in sail technology to reduce the orbit life of spacecraft with regards to de- orbiting using aerodynamic drag. The deorbitsail technique signi cantly increases the e ective cross-sectional area of a satellite, subsequently increasing atmospheric drag and accelerating orbit decay. The concept proposed in this work introduces a very useful technique of orbit decay as well as deorbiting of spacecraft. / AFRIKAANSE OPSOMMING: Gedurende sy leeftyd in die ruimte word 'n ruimtetuig blootgestel aan die risiko van 'n botsing met ruimterommel of met funksionele satelliete. Hierdie risiko is veral hoog in die lae-aardbaan gebied waar die hoogste digtheid ruimterommel voorkom. Hierdie studie ondersoek die wentelbaanverval van sommige Lae-aardbaan mikrosatelliete asook die versnelde baanverval wanneer van 'n deorbitaal meganisme gebruik gemaak word. Die Semi-Analitiese Liu Teorie en die Satellite Toolkit sagtewarepakket is gebruik om die gemiddelde baan-elemente en uitdrukkings vir hul tyd-afhanlike tempo van verandering te bepaal. Toetsgevalle van waargenome vervalde satelliete (Iridium-85 en Starshine-1) is gebruik om die verloop van die voorspelde teoretiese verval te evalueer. Resultate vir die toetsgevalle toon dat die teorie binne aanvaarbare perke met die waarnemings ooreenstem. Die verloop van die SUNSAT mikrosatelliet se wentelbaanverval is ook ontleed deur gebruik te maak van historiese Tweelyn Elemente datastelle en dit te vergelyk met voorspelde baan- elemente. Die studie het ook ondersoek ingestel na die voorspelde baan-verbyval van 'n 1-eenheid cubesat (ZACUBE-01). Die impak op wentelbaanverval deur 'n voorgestelde oplossing vir die beperking van ruimterommel, 'n deorbitaalseil, is ook ondersoek. So seil verkort 'n satelliet se ruimte- leeftyd deur sy e ektiewe deursnee-area te vergroot en dan van verhoogde atmosferiese sleur en sonstralingsdruk gebruik te maak om die vervalproses te versnel. Hierdie voorgestelde konsep is 'n moontlike nuttige tegniek vir versnelde baanverval en beheerde deorbitalering van ruimtetuie om ruimterommel te verminder.

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.

On-orbit servicing (OOS) of space systems provides immense benefits by extending their lifetime, by reducing overall cost of space operations, and by adding flexibility to space missions. Refueling is an important aspect of OOS operations. The problem of determining the optimal strategy of refueling multiple satellites in a constellation, by expending minimum fuel during the orbital transfers, is challenging, and requires the solution of a large-scale optimization problem. The conventional notion about a refueling mission is to have a service vehicle visit all fuel-deficient satellites one by one and deliver fuel to them. A recently emerged concept, known as the peer-to-peer (P2P) strategy, is a distributed method of replenishing satellites with fuel. P2P strategy is an integral part of a mixed refueling strategy, in which a service vehicle delivers fuel to part (perhaps half) of the satellites in the constellation, and these satellites, in turn, engage in P2P maneuvers with the remaining satellites. During a P2P maneuver between a fuel-sufficient and a fuel-deficient satellite, one of them performs an orbital transfer to rendezvous with the other, exchanges fuel, and then returns back to its original orbital position. In terms of fuel expended during the refueling process, the mixed strategy outperforms the single service vehicle strategy, particularly with increasing number of satellites in the constellation. This dissertation looks at the problem of P2P refueling problem and proposes new extensions like the Cooperative P2P and Egalitarian P2P strategies. It presents an overview of the methodologies developed to determine the optimal set of orbital transfers required for cooperative and non-cooperative P2P refueling strategies. Results demonstrate that the proposed strategies help in reducing fuel expenditure during the refueling process.