SIMULATION OF GPS CARRIER PHASE OBSERVABLES AND FEASIBLE ATTITUDE DETERMINATION ALGORITHMS FOR UNIVERSITETSKIY-TATYANA SATELLITE Москва, МГУ

2 Precise attitude determination - Precise attitude determination has an application in navigation, geodesy, gravimetry, magnetometry, construction and so on - Carrier phase measurements provided by GPS hardware have potential accuracy of millimeters in linear distance and several advantages compared to inertial navigation systems; this allows using them for attitude determination - Some special properties of carrier phase measurements require a stack of different mathematical methods and computational techniques to be processed instead of one algorithm - Observability and consistency would be checked and evaluated at every stage of calculations - Different methods and its effectiveness could be objectively compared using actual survey data - Precisely known distances between antennas are available as an additional a priori information that could be involved in computational process

3 Satellite navigation systems - 24-hour all-weather operation - Common standards - Up to 14 visible GPS satellites - Up to 5 visible GLONASS satellites - Redundant measurements - Several carrier frequencies - Emitted radio-signals are available to receive worldwide

4 Carrier phase measurements - Commonly used carrier phase measurements equation (n is a number of visible satellites, m is number of antennas, N is time span in epochs): p is carrier frequency number, all items are measured in cycles of a carrier The wanted quantity is The rest of components are errors and instrumental noise

5 Single- and double-differencing - Single-differencing cancels errors due to radio-signal propagation through ionosphere and troposphere, satellite clock errors: - Double-differencing cancels errors due to receiver clock errors; remaining irremovable errors are of less magnitude than others: - Number of measurements is reduced by 2

6 Attitude determination by processing carrier phase measurements characterization (page 1) - The matter of attitude determination is relative positioning of several antennas (not less than 3) - Differencing is quite natural: the equation of measurement is nearly linear - Precise absolute positioning is not necessary - Additional information (distances between antennas) is available to use

7 Attitude determination by processing carrier phase measurements characterization (page 2) - When antennas are close to each other (in dozens of meters) errors due to signal propagation through ionosphere and troposphere, linearization and clock errors are completely canceled by double-differencing - Integer ambiguities introduce a significant complexity in estimation; the estimated vector is not observable at every particular epoch; observability is possible when a set of measurements for several epochs is available

8 Formal statements for different approaches to attitude determination by processing carrier phase measurements (page 1) - Weighted least-squares method (errors in measurements are correlated) - Least-squares method with special restrictions given by known distances between antennas; certain algorithms deliver the solution

9 Formal statements for different approaches to attitude determination by processing carrier phase measurements (page 2) - Reduction to static case using additional velocity measurements: - Reduction to static case and introducing the additional information as extra- measurements:

10 Formal statements for different approaches to attitude determination by processing carrier phase measurements (page 3) - Triple-differencing free of integer ambiguities and using special properties of conditionality

11 Least-squares method (LSM) compared to LSM with restrictions - Least-squares method is well formulated in variety of books - LSM with restrictions is produced as iterative algorithm with linearized restrictions of equation type; least-squares solution is the initial value

12 Least-squares method (LSM) compared to LSM with restrictions; actual measurements processing

13 Dynamics of observability during constellation evolution - Let H 1 and H 2 be observation matrices for two epochs for m carrier phase measurements containing integer ambiguities - Necessary and sufficient condition for observability is as follows: - The condition states that direction vectors to satellites are not transformed by a single linear operator; furthermore the difference from any that transformation would be enough to satisfy the condition - The more suitable equivalent condition is the m-dimensional basis at the bottom n – m rows of matrices: H 1 =U 1 S 1 V 1 T is singular value decomposition, H 1 =Q 1 R 1 is QR-decomposition

Field surveys - Field surveys have been conducted several times in Moscow region and once near Tver - Topcon hardware and software - Two antennas - Static and dynamic series - Lengthy series of measurement

15 References (page 1) 1. Вавилова Н.Б., Голован А.А., Парусников Н.А., Трубников С.А., Математические модели и алгоритмы обработки измерений спутниковой навигационной системы GPS, Москва, Шебшаевич В.С., Дмитриев П.С., Иванцевич Н.В. и др., Сетевые спутниковые радионавигационные системы, 2-е издание, Радио и связь, Москва, Степанов О.А., Кошаев Д.А. Исследование методов решения задачи ориентации с использованием спутниковых систем. Гироскопия и навигация.1999, 2, Несенюк Л.П. и др., Интегрированная инерциально-спутниковая система ориентации и навигации с разнесенными антеннами, Гироскопия и навигация, 2000, Leick A., GPS Satellite Surveying, 2 nd edition, John Wiley & Sons, Inc., Leick A., GPS Satellite Surveying, 3 rd edition, John Wiley & Sons, Inc., Saad Y., Iterative methods for sparse liner systems, 2 nd edition, Бейко И.В., Бублик Б.Н., Зинько П.Н., Методы и алгоритмы решения задач оптимизации, «Вища школа», Киев, Голуб Дж., Ван Лоун Ч., Матричные вычисления, «Мир», Москва, Лоусон Ч., Хенсон Р., Численное решение задач методом наименьших квадратов, 1986

16 References (page 2) 11. Maybeck P.S., Stochastic Models – Estimation and Control, Acad. Press, New York, Арутюнов А.В., Условия экстремума – анормальные и вырожденные задачи, «Факториал», Москва, Химмельблау Д., Прикладное нелинейное программирование, «Мир», Москва, J. Chris McMillan, G. Lachapelle, G. Lu, Dynamic GPS Attitude Performance Using INS/GPS Reference 15. Ruiz S., Font J., Griffiths G., Castellon A., Estimation of heading gyrocompass error using a GPS 3DF system: Impact on ADCP measurements, Scientia Marina, 66(4), Favey E., Cerniar M., Cocard M., Geiger A., Sensor attitude determination using GPS antenna array and INS, ISPRS WG III/1 Workshop, Direct versus indirect methods of sensor orientation, Barcelona, Schleppe J., Development of a real-time attitude system using a quaternion parameterization and non-dedicated GPS receivers, UCGE reports, University of Calgary, Поваляев Е., Хуторной С., Системы спутниковой навигации ГЛОНАСС и GPS. Часть 1 и 2, электронная версия журнала CHIP News, Украина, 2000 (