Orbit Calculation and Re-Entry Control of Vorsat Satellite

Home , Elliptic orbit, Highly elliptical orbit, High Earth orbit, Hyperbolic trajectory, Parabolic trajectory, Specific orbital energy

FACULDADEDE ENGENHARIADA UNIVERSIDADEDO PORTO

Orbit Calculation and Re-Entry Control of VORSat Satellite

Cristiana Monteiro Silva Ramos

PROVISIONAL VERSION

Master in Electrical and Computers Engineering

Supervisor: Sérgio Reis Cunha (PhD)

June 2011

Resumo

Desde o lançamento do primeiro satélite artiﬁcial para o espaço, Sputnik, mais de 30 000 satélites foram desenvolvidos e foram lançados posteriormente. No entanto, por mais autonomia que o satélite possa alcançar, nenhum satélite construido pelo Homem teria valor se não fosse possível localiza-lo e comunicar com ele. Os olhos humanos foram os primeiros recursos de que os seres humanos dispuseram para observar o Universo. O telescópio surgiu depois e foi durante séculos o único instrumento de exploração do Espaço. Hoje em dia, é possível localizar e seguir o trajecto que o satélite faz no espaço através de programas computacionais que localizam o satélite num dado instante e fazem a previsão do cálculo da órbita. Esta dissertação descreve o desenvolvimento de um sistema de navegação enquadrado no caso de estudo do satélite VORSat. O VORSat é um programa de satélite em desenvolvido na Faculdade de Engenharia da Universidade do Porto (FEUP), Potugal. Este projecto refere-se à construção de um CubeSat, de nome GAMA-Sat, e à reentrada de uma cápsula na Terra (ERC). Os objectivos consistem em determinar os parâmetros que descrevem uma órbita num dado instante (elementos de Kepler) e em calcular as posições futuras do satélite através de observações iterativas, minimizando o erro associado a cada observação. Também se pretende controlar, através de variações do termo de arrastamento, o local de reentrada. Dois métodos estimativos foram abordados: Filtro de Kalman e Mínimos quadrados. Estes foram adaptados ao caso de estudo por forma a avaliar a melhor performance num sistema não-linear tendo em consideração três aspectos: (1) O consumo de bateria no microcontrolador; (2) A capacidade de espaço limitada no microcontrolador; e (3) A precisão das medidas. Apesar deste projecto estar a ser desenvolvido no âmbito do VORSat, utilizou-se dois satélites que se encontram em órbita para validação de resultados. Optou-se pelo PoSAT e o International Space Station (ISS) por ambos se encontrarem em órbita baixa (LEO), terem uma órbita estável e ainda permanecem em órbita passado tantos anos, o que não é usual em satélites de órbita baixa.

i ii Abstract

Since the first artificial satellite was launched into space, Sputnik, more than 30,000 satellites have been developed and followed him. Regardless of the level of autonomy that the satellites can reach, any man-made spacecraft would have no value if it was unable to locate it and communicate with it. Human eyes were the first resources that humans had to observe the universe. The telescope came later and was used for centuries for space exploration. Nowadays, it is possible to locate and follow satellites in space through computer programs that compute the location of the satellites at any time and predict their orbits. This thesis describes the development of a navigation system that was carried out having in mind a specific case study, VORSat. VORSat is a small satellite program being developed at the Faculty of Engineering of the University of Porto (FEUP), Potugal. This project regards both the development of a CubeSat (named GAMA-Sat) and an Earth Reentry Capsule (ERC). The objective to determine the parameters that describe an orbit at a given moment (Kepler elements) and, through iterative observations to calculate the future positions of the satellite. Er- rors associated with each observation were to be minimized. Another objective was control the location of de-orbiting, through variations of the drag term. Two methods were discussed for the estimator technique: Kalman Filter and Least Squares. They were adapted to the case study in order to evaluate the best performance in a nonlinear system taking into account three aspects: (1) Battery consumption in the microcontroller; (2) limited storage space in the microcontroller; and (3) the accuracy of the measurements. Within the scope of this thesis, it was used two satellites: International Space Station (ISS) and PoSAT, in order to obtain results of the developed algorithm. The choice felt upon these two satellites due the fact that both of them are in LEO orbit, have a stable orbit and remain in orbit for many years.

iii iv Acknowledgments

To my supervisor, Prof. Sérgio Reis Cunha, for allowing me to enter into the VORSat world with which I strongly identiﬁed myself. To Prof. for his guidance, endless support, contagious enthusiasm and for always trusting in me and my capabilities. I also want to thank all VORSat team for helping me integrating into the project, the long conversations we had on this work possibilities and for their patience while explaining me every detail of VORSat project. To the European Space Agency for sponsoring my participation in the ﬁrst Conference in the IAA University CubeSat Satellites and Missions of the Winter Workshop Session in Rome. Conference in which I learned a lot, that allowed me to learn about various projects increasing my motivation to develop this thesis. They were classmates, but in these last six months have become more than colleagues. For the games and the support, thank you, Alexandre Gomes, Mariana Magalhaes and Paulo Pereira. To all my dear Friends for their support and assistance over the years. Throughout the university time I acquired knowledge, soft skills but nothing compares to the friendships that I acquired during my stay at FEUP. A very special thanks to my closest ones, Sónia, Hugo, Tiago, Filipe, Pedro and Miguel. Although I have already been grateful to my closest friends, I want to show my special thanks to one of the people who was always there when I needed throughout this years. To you, Teresa, for your patience with me during all this years and while reviewing and advising me about this thesis. This thesis would not be a thesis if it had not been for your love and support, I can not thank you enough! Lastly, however more important, I wish to thank my parents, my sister and Luís, for instill the values and principles that have been guiding my life and have made me the person I am today. It all starts and ends at family and it is impossible to thank properly for everything they represent and do for me. Thank you one and all.

v vi "It is difﬁcult to say what is impossible, for the dream of yesterday is the hope of today and reality of tomorrow."

Robert Goddard

vii viii Contents

1 Introduction 1 1.1 Motivation ...... 1 1.2 Objectives and Work Organization ...... 3 1.3 Structure ...... 3

2 Case Study Presentation 5 2.1 CubeSat ...... 5 2.1.1 QB50 ...... 6 2.2 Project ...... 7 2.2.1 Project’ Vision and Objectives ...... 7 2.2.2 GAMA-Sat ...... 7 2.2.3 VORSat ...... 10 2.3 Summary ...... 12

3 Literature Review 13 3.1 Satellites ...... 13 3.1.1 Satellites Features ...... 14 3.1.2 Satellites Classiﬁcation by Altitude ...... 15 3.2 Navigation System ...... 18 3.2.1 Principles of Orbital Motion ...... 18 3.2.2 Orbital Motion as a Two-Body Problem ...... 19 3.2.3 Orbit Determination Methods ...... 26 3.3 Perturbed Orbits ...... 29 3.3.1 Atmospheric models ...... 31 3.4 Interaction Station - Satellite ...... 33 3.4.1 Measuring from the Earth ...... 33 3.4.2 Measuring from Space ...... 34 3.4.3 Communication ...... 37 3.5 Summary ...... 38

4 Orbit and Location of De-orbiting Parameters Determination 39 4.1 Problem Description ...... 39 4.1.1 Case study Parameters ...... 40 4.1.2 Kalman Filter Algorithm ...... 42 4.1.3 Least-Squares Estimation Algorithm ...... 44 4.1.4 Methods Assessment ...... 46 4.2 Prediction of the Decay ...... 47 4.3 Summary ...... 49

ix x CONTENTS

5 Implementation and Results 51 5.1 Orbit Prediction ...... 51 5.1.1 Position and Velocity ...... 52 5.1.2 Kalman Filter ...... 54 5.2 Test Cases ...... 60 5.2.1 Unkown Inputs Parameters Values ...... 60 5.2.2 Accuracy of Kalman ﬁlter ...... 62 5.2.3 Observations only with positions ...... 64 5.3 De-orbiting Location Prediction ...... 64 5.4 Summary ...... 65

6 Conclusion and Future Work 67 6.1 Main Results ...... 67 6.2 Work’s Assessment ...... 67 6.3 Future Work ...... 68

A Time and Coordinate Systems 69 A.1 Time ...... 69 A.1.1 Sidereal Time ...... 69 A.1.2 Universal Time ...... 69 A.2 Coordinates ...... 71 A.2.1 Geographic Coordinates ...... 71 A.2.2 Cartesian Coordinates ...... 71 A.2.3 Datum ...... 73

B Futher Analysis of the Results 75 B.1 TLE ﬁle used in the simulations ...... 75 B.2 Kalman Filter Performance ...... 75 B.3 Unkown Inputs Parameters Values ...... 76

C Futher Analysis of the Results - Continue 91 C.1 Accuracy of Kalman Filter ...... 91 C.2 Observations only with positions ...... 91

References 97 List of Figures

1.1 Work Plan ...... 3

2.1 First draft of VORSat/GAMA-Sat architecture [1] ...... 8 2.2 GAMA-Sat Structure [1] ...... 9 2.3 Capsule phases. Courtesy of João Gomes ...... 10 2.4 Capsule Structure. Courtesy of João Gomes ...... 11 2.5 Interior of the Capsule. Adapted from [2] ...... 11 2.6 Batery Charge [1] ...... 11

3.1 Evolution of satellites over the years ...... 14 3.2 The two-boy motion [3] ...... 20 3.3 Motion of an orbiting body [4] ...... 23 3.4 Keplerian Elements ...... 24 3.5 Procedure of the Kalman Filter ...... 28 3.6 Ground Station. [5] ...... 34 3.7 Equipment ON/OFF state depending on battery charge. [5] ...... 35

5.1 Loop to perform the propagation ...... 53 5.2 Calculated position and velocity ECI ...... 53 5.3 Convert position and velocity coordinates ...... 53 5.4 ISS Orbit computed with position and velocity ...... 55 5.5 PoSAT Orbit computed with position and velocity ...... 55 5.6 Iteration of the observation matrix ...... 57 5.7 Innovation ...... 58 5.8 Posteriori state and its covariance matrix ...... 58 5.9 Application of Kalman filter with PoSAT satellite - Positions errors ...... 59 5.10 Application of Kalman filter with PoSAT satellite - Velocity errors ...... 59 5.11 Application of Kalman filter with ISS satellite - Positions errors ...... 59 5.12 Application of Kalman filter with ISS satellite - Velocity errors ...... 59 5.13 PoSAT error positions with random initial inputs ...... 60 5.14 PoSAT error velocity with random initial inputs ...... 60 5.15 ISS error positions with random initial inputs ...... 61 5.16 ISS error velocity with random initial inputs ...... 61 5.17 Value that the position error converges with random initial inputs - PoSAT . . . . 61 5.18 Value that the velocity error converges with random initial inputs - PoSAT . . . . 61 5.19 Value that the position error converges with random initial inputs - ISS ...... 61 5.20 Value that the velocity error converges with random initial inputs - ISS ...... 61 5.21 Position accuracy of Kalman filter with simulations steps of 10 minutes - PoSAT 62 5.22 Velocity accuracy of Kalman filter with simulations steps of 10 minutes - PoSAT 62

xi xii LIST OF FIGURES

5.23 Position accuracy of Kalman filter with simulations steps of 10 minutes - ISS . . 62 5.24 Velocity accuracy of Kalman filter with simulations steps of 10 minutes - ISS . . 62 5.25 Position accuracy of Kalman filter with simulations steps of 45 minutes - PoSAT 63 5.26 Velocity accuracy of Kalman filter with simulations steps of 45 minutes - PoSAT 63 5.27 Position accuracy of Kalman filter with simulations steps of 45 minutes - ISS . . 63 5.28 Velocity accuracy of Kalman filter with simulations steps of 45 minutes - ISS . . 63 5.29 Position accuracy of Kalman filter with simulations steps of 60 minutes - PoSAT 63 5.30 Velocity accuracy of Kalman filter with simulations steps of 60 minutes - PoSAT 63 5.31 Position accuracy of Kalman filter with simulations steps of 60 minutes - ISS . . 64 5.32 Velocity accuracy of Kalman filter with simulations steps of 60 minutes - ISS . . 64 5.33 Kalman filter performance with observations based only with positions - PoSAT . 64 5.34 Kalman filter performance with observations based only with positions - ISS . . . 64 5.35 Point where it should initiate the control of the drag-term ...... 65

A.1 Function of the Greenwich Sidereal Time ...... 70 A.2 Function of the JD given the year, month, day, and time ...... 71 A.3 Converts ECEF into LLH ...... 72 A.4 Geodetic WGS84 ...... 73

B.1 How long it takes to converges with random initial inputs - PoSAT ...... 76 B.2 How long it takes to converges with random initial inputs - ISS ...... 76 List of Tables

3.1 Two-line Element - Line 1 ...... 25 3.2 Two-line Element - Line 2 ...... 25

5.1 Input Data SPG4 Algorithm ...... 52 5.2 Input Data Kalman Filter Algorithm ...... 54

B.1 The real values of the ISS position ...... 77 B.2 The estimate values of ISS position ...... 79 B.3 New ISS TLE values calculated using Kalman filter ...... 81 B.4 The real values of the PoSAT position ...... 83 B.5 The estimate values of the PoSAT position ...... 85 B.6 New PoSAT TLE values calculated using Kalman filter...... 87 B.7 New PoSAT TLE values calculated using Kalman filter with Unkown Inputs Pa- rameters Values ...... 89 B.8 New ISS TLE values calculated using Kalman filter with Unkown Inputs Parame- ters Values ...... 90

C.1 Kalman Filter Performance with steps of 10 to 10 minutes - ISS ...... 91 C.2 Kalman Filter Performance with steps of 45 to 45 minutes - ISS ...... 92 C.3 Kalman Filter Performance with steps of 60 to 60 minutes - ISS ...... 92 C.4 Kalman Filter Performance with steps of 10 to 10 minutes - PoSAT ...... 92 C.5 Kalman Filter Performance with steps of 45 to 45 minutes - PoSAT ...... 93 C.6 Kalman Filter Performance with steps of 60 to 60 minutes - PoSAT ...... 93 C.7 New ISS TLE values calculated using Kalman ﬁlter with Observations only with positions ...... 94 C.8 New ISS TLE values calculated using Kalman ﬁlter with Observations only with positions ...... 95

xiii xiv LIST OF TABLES Abbreviations and Symbols

ECEF Earth Centered Equatorial-Fixed ECI Earth Centered Inertial FEUP Faculty of Engineering of University of Porto GEO Geostationary Orbit GMST Greenwich Mean Sidereal Time GPS Global Positioning System HEO Highly Elliptical Orbit ISS International Space Station JD Julian Date LEO Low Earth Orbit LLH Latitude, Longitude, Height MEO Medium Earth Orbit SI International System of Units SGP Simpliﬁed General Perturbations models TLE Two Line Elements UT Universal Time WGS Word Geodetic System

xv xvi ABREVIATURAS E SÍMBOLOS Chapter 1

Introduction

This chapter is a small introduction to a satellite program in which the work described in this thesis is included. Additionally this work objectives and organization is presented, as well as a brief description of the structure of this document.

1.1 Motivation

"The space age began when the cientific community began to accept the extraterrestrial origin of meteorites. However, it took more one hundred and fifty years before technology could support the engineering part of the space age: the design and construction of an artifact, its launch and recovery from orbit" [6]. Despite the first man-made spacecraft was only launched in 1957, Sputnik-1, satellite orbits had already been studied. Starting from Newton’s formulation of the law of gravity, scientists sought continuously to develop and refine analytical theories describing the motion of the Earth’s only natural satellite, the Moon. But, only at the 20th century was possible the physical (and not only theoretical) exploration with the construction of first rockets. Common reasons for exploring space include advancing scientific research, communications, uniting different nations, ensuring the future survival of humanity and developing military and strategic advantages against other countries. In the past few years much of the attention of the space industry has shifted towards the development of small satellites. These satellites, often called picosats, nanosats, or microsats are generally less than 200 kg and, in many cases, are as little as 1 - 5 kg. Such satellites, which range in size from refrigerators to small soda cans, offer many potential benefits over traditional space satellites. The development of small satellites and the use of commercial off-the-shelf components has dropped the price of satellite launches to as low as a few million dollars for light satellites, and a few tens of millions for heavy satellites. This fact led to a significant increase in the number

1 2 Introduction of satellites developed and launched both by companies and universities . According to data provided by National Aeronautics and Space Administration (NASA), since the first artificial satellite, Sputinik-1, thousands of satellites have already been launched. Currently there are approximately 3,000 satellites operating in Earth orbit, out of the roughly 8,000 man-made objects, together with countless pieces of space debris[7]. When satellites reach the end of their mission, the majority of these have fallen into unstable orbits and incinerated during reentry. Regardless of the level of autonomy that the satellites can reach, any man-made spacecraft would have no value if it was unable to locate it and communicate with it. Furthermore, the navigation system is concerned with the mathematical and physical description of artificial satellite orbits, as well their control. Many of the satellites previously mentioned, need an active control of their orbit in accordance with the specific mission requirements. Navigation is, therefore, an essential part of satellite operations. It comprises the planning, the determination,the prediction and the correction of satellite’s trajectory in line with the established mission goals. In the present project, a literature review on Satellites and Navigation was carried out having in mind a specific case study, concerning a small satellite program being developed at the Faculty of Engineering of the University of Porto (FEUP), Potugal. This project regards both the development of a CubeSat in partnership with the Portuguese company TEKEVER (named GAMA-Sat) and an Earth Reentry Capsule (ERC). The cubesat has as main mission targets the determination of the satellite’s attitude from the ground and from other near cubesats, the communication between satellites and the measurement at the differential arm between them to obtain the differential drag affecting neighbour satellites. The ERC main goals are to successfully achieve the survival of the reentry capsule, with a desired reentry path and in a desired landing site. The integration of a cork composite material in the ablative Thermal Protection System (TPS) of a nanosatellite/picosatellite and the use of ARGOS system inside ERC as the primary communication channel after reentry are being considered. This MSc Thesis will focuses on the development of an onboard navigation and guidance system capable of predicting the satellite path and control the location of de-orbiting. Concerning the capsule, the mission success may depend on using trajectory planning algorithms for the determination of the time and place of landing for its recovery. In this case, instead of having multiple stations trying to track the satellite, they must continuously estimates its evolution from present and past Global Position System (GPS) data and from model of the atmosphere and transmit it to the mission control . Because of this, the algorithm must be able to estimate how many transmi- tions can be made, depending on the stage at which the orbit lies and battery status. The objective is to maximize the number of useful transmissions while minimizing the battery usage. Although the subject is widely studied internationally, the project itself can be considered as innovative because even though Portugal had a successful operation of a satellite developed by a consortium of mainly national entities (PoSAT-1), the fact is that the Portuguese aerospace industry is very small and its activity is primarily focused on supplying high quality parts for the foreign market. The Portuguese experience in fully designing and building a satellite and reentry capsule is rather small[8]. 1.2 Objectives and Work Organization 3

1.2 Objectives and Work Organization

Based on the satellite operating architecture, the main objectives of this work are the following:

• To develop algorithms to calculate the evolution of both satellites in their Low Earth Orbit (LEO), from their release until the time of reentry, by using present and past GPS data and a the model of the atmosphere. It is particularly important to determine the time and the place of reentry;

• To develop a methodology to update satellite ephemeris information obtained through sporadic GPS observations (omnidirectional antenna from array of patch antennas on different faces).

• To develop a control algorithm to adaptively adjust the point of re-acting on the drag coefficient of the satellite. The actuation of the drag coefficient is yet to be defined, however something like, opening and closing light spoilers on the surfaces of the satellite is foreseen.

The objectives mentioned above were undertaken during a period of approximately eight months, and the work load was divided as shown in the Figure 1.1 :

Figure 1.1: Work Plan

1.3 Structure

This document is organized in three chapters. Each chapter starts with a small introductory text describing the chapter’s intent and to the exception of the present one, ends with a brief summary. The current introduction chapter presents the background motivation for Satellites in general and the project. It also enumerates the proposed objectives as well as the work plan for the duration of the project. Finally, it describes the document’s organization. In the second chapter, the Case Study is presented and its architecture described. The present development stage is also discussed while contextualizing the relevance of this work within the project. 4 Introduction

The third chapter offers a view on the state-o f -art related with satellites and its navigation system. The intent of this chapter is to introduce the different scientiﬁc themes and trends relevant to this thesis and the developed work. Chapter 4, Orbit Determination , formally describes the way to determine the trajectory of VORSat and presents the parameters to be considered in the implementation later. Afterwards, mathematical formulations are adapted to the case study features. The chapter ends with a conclusion of the best reentry based on the results from the literature instances. Chapter 5, Case Study Results, implements the solution methods presented in the previous chapter by using the case study instance and exposes the results along with their analysis. Finally, chapter 6, Conclusion and Future Work, infer on the work’s achieved objectives. Futhermore, considerations are made on open new paths for the work developed. Chapter 2

Case Study Presentation

This chapter aims to provide a general overview of the case in study, presenting and describing its vision and objectives. The ﬁrst section will provide a brief description of CubeSats and the following presents the satellites that are being developed by FEUP CubeSat Team.

2.1 CubeSat

The CubeSat concept arose in 1999, through a collaboration between Dr. Bob Twiggs (Stanford University) and Dr. Jordi Puig-Suari (California Polytechnic State University San Luis Obispo- Cal Poly)[9]. Cubesats were developed to help universities worldwide to perform space science and exploration. A CubeSat standard is a miniaturized satellite with the restricted dimensions and weight of, 10 cm cube and 1 kg, respectively. It offers the following standard functions of a normal satellite:

• Attitude determination and control;

• Uplink and downlink telecommunications;

• Power subsystem including a battery and body-mounted solar panels;

• On-board data handling and storage by a CPU ;

• Plus either a technology package or a small sensor or camera.

To keep it simple this is achieved by using commercial off-the-shelf components. Typically, they are launched and deployed from a mechanism called a Poly-Picosatellite (P-POD). During the launch phase the signal is turned off, the Cubesat are put into a Low Earth Orbit (LEO) and deployed once the proper signal is received from the launch vehicle. Approximately 90% of CubeSats were launched with P-POD [10].

5 6 Case Study Presentation

The bottleneck for small satellites remains in the physical space available, so the payloads are very limited due to the extremely restricted size and mass of a CubeSat satellite. This seri- ously restricts payloads requiring large optics or bulky components. The limited surface area of a CubeSat restricts the amount of solar power that may be generated, restricting power available for computation, communications, and payloads. Despite the limitations of CubeSats, in the past few years much of the attention of the space industry has shifted towards the development of small satellites because it represent a cost-effective independent mean of getting a payload into orbit. They offer many potential beneﬁts over traditional space satellites, such as, carry new scientiﬁc instruments, unproven technologies in low prices but also greatly increase the available functionality. Nevertheless, several companies but mostly universities have already developed CubeSats and launched them with a mixed record of successfully orbited and failed missions. With almost thirty CubeSats in orbit and over one hundred teams actively designing CubeSats, the CubeSat concept has been greatly successful in providing access for space research.

2.1.1 QB50

A single CubeSat is too small to carry many or specialized sensors for significant scientific research. However, when combining in a network a large number of CubeSats with identical sensors, besides the addition educational value, fundamental scientific questions can be addressed, which otherwise would be inaccessible. The launch of a set of CubeSats have been under discussion in the CubeSat community for several years, but until now no university, company or space agency took the initiative to create and coordinate a network that big. The problem of reliability of the CubeSat is not an issue because the network can still achieve the objectives of his mission, even if any CubeSats fail. QB50 is an iniative of the Von Karman Institute, ESA and NASA and has the scientific objective of studying in-situ the temporal and spatial variations of a number of key constituents and parameters in the lower thermosphere (90-320 km), using a network of 50 double CubeSats, sepa- rated by a few hundred kilometres and carrying identical sensors. QB50 will also study the re-entry process by measuring a number of key parameters during re-entry and by comparing predicted and actual CubeSat trajectories and orbital lifetimes [11]. Since the atmospheric network mission for in-situ measurements is a pioneering experience, one of the low-cost solutions and an acceptable risk would be the use of CubeSats or very low- cost satellites. The cost of such a network for industry standards, would be extremely high and not justified due to the limited lifetime in orbit and traditionally, universities do not have means of funding and the development of CubeSats are for educational purposes. Among the 50 satellites of the network, there are 38 CubeSats from 22 European countries. Portugal have submitted a CubeSat proposal to be one of the satellites to be launched in QB50. 2.2 Project 7

2.2 Project

A small satellite program has started in the Faculty of Engineering of the University of Porto (FEUP) regarding the development of a CubeSat in partnership with the Portuguese company TEKEVER, GAMA-Sat, and an Earth Reentry Capsule (ERC). It is an academic project composed of graduate and PhD students and professors from different engineering ﬁelds (Electrical, Industrial, Mechanical, etc.), none of which is directly related to aerospace sector. The idea came with the challenge made by the Education Ofﬁce of ESA, that worked previously with FEUP in the STRAtospheric PLatform EXperiment (STRAPLEX) program. The aim was to carry European students’ experiments to the stratosphere using atmospheric balloons[12]. As state previously, this it can be considered an innovative project since, in the Portuguese context the Portuguese aerospace industry is residual, and there has only been one successful Portuguese satellite operation, PoSAT-1[8].

2.2.1 Project’ Vision and Objectives

The main goal is to show the capability of a Portuguese student team to perform the reentry of a capsule trough the Earth atmosphere and the design and implementation of a CubeSat. This offers the opportunity for students to apply their engineering qualiﬁcations to their common interest in space. Each of these missions and its goals will be described below in more detail.

2.2.2 GAMA-Sat

The name GAMA is a tribute to the famous Portuguese navigator Vasco da Gama who, during the Fifteenth Century, discovered the maritime route from Europe to India. GAMA-Sat will serve for both technology demonstration and scientific purposes. The technology demonstration will focus on the usage of Software Defined Radio (SDR) to establish inter satellite links and support adhoc networking, range and attitude determination applications. These capabilities will be used to serve the scientific purpose of calculating the differential evolution of atmospheric drag between CubeSats [2]. The team will develop an innovative CubeSat transceiver module based on SDR technology, capable of performing communications and navigation by GNSS (Global Navigation Satellite Sys- tem). This device will be simultaneously a telecommunications transceiver and a GNSS receiver, supporting multiple capabilities and applications, such as [2] :

• VHF, S-band and GNSS waveforms in a unique HW platform;

• Inter satellite adhoc networking capabilities, allowing each CubeSat to become a node in a mobile adhoc network;

• GNSS, for receiving both GPS and GALILEO SIS (Signal in Space); 8 Case Study Presentation

• Range and attitude determination through the VHF Omni-directional radio Range (VOR) principle (generating VHF waveforms that will be transmitted by patch antennas on each face of the CubeSat).

As previously mentioned, the project represents the work of different people, integrated into a single architecture. It has evolved from a simple control into a multilevel control architecture depicted in Figure 2.1.

Figure 2.1: First draft of VORSat/GAMA-Sat architecture [1]

The satellite can be functionally broken down into the blocks described in 2.1. The four main parts are the Electrical Power Subsystem, the Command Subsystem, the CW Beacon and the communication system, one of the subjects of this thesis, more speciﬁcally the GPS. This methodology was taken into consideration for the CubeSat to determine its position when were necessary to adjust its trajectory due to the drag. The signal transmission depends on the available energy and satellite location, and therefore the satellites will have a combination of solar panels and antennas to maximize energy. The Electrical Power Subsystem is composed by two power sources (the solar cells and the secondary batteries), a MPPT unit and auxiliary circuits to perform power regulation and battery charging (not included in the drawing). The Command Subsystem is composed by a microcontroller, by its battery (primary battery) and by a set of switches this microcontroller actuates to turn operation of the CW Beacon and Communications Subsystems on and off. 2.2 Project 9

The CW Beacon subsystem is composed by a frequency synthesizer, a power ampliﬁer and the respective antennas. It will operate in the Amateur Radio band of the 1296.075 MHz reserved for CW beacons. The Communications Subsystem is the heart of the satellite. Its microcontroller is switched on only when enough power is available to sustain its operation for a signiﬁcant amount of time. Once in operation, one of its tasks is to control de operation of the GPS receiver. The microcontroller will seek position and velocity data from time to time in order to maintain and update the estimate of its orbit (ephemeris). This ephemeris is stored in non-volatile memory and update periodically. To measure the satellite attitude from the ground, the system is based on a set of RF signals transmitted from orbit. The idea is to combine multiple signals and antennas in such a way that certain modulated information depends on the direction from which the signals are received. Such information is coded in the form of signal phases, allowing to compute the satellite attitude with one degree of expected accuracy. (A similar approach is used in the VOR - VHF the name VORSat.) The satellite will be travelling along its orbit and transmitting signals in all directions. Know- ing the position of the satellite relatively to a ground station, by measuring the phase differences of signals transmitted by different antennas on the faces of the satellite and their evolution as the satellite passes by the ground station coverage area, it is possible to compute the satellite attitude from the ground. Three orthogonal rings of antennas are necessary for this purpose, which implies having 3 individual antennas per face,(see Figure 2.2 ). Other antennas are also required, for GPS (which must be omnidireccional) and for a simple localization beacon.

Figure 2.2: GAMA-Sat Structure [1]

The transmitted signals are planned for the 2.4 GHz ISM band, using a bandwidth of just 15 KHz. The satellite will be completely solid-state, as the direction dependent signals will be obtained by applying beam-forming techniques, through a combination of the several antennas. 10 Case Study Presentation

The ground-station requirements are particularly simple: a 2 DOF parabolic dish with a 2.4 GHz feed and LNA,which is a combination of a tunable down-converter to base-band, a sound card and a computer running software to demodulate the signals. There will be a study for the adjustment of the drag coefﬁcient, using the control loop with feedback from GPS, in order to make the re-entry into the desired spot. In this particularly case the re-entry will over Portugal, splashdown on the North Atlantic. The idea is to adjust the drag coefﬁcient by extending and retracting light pannels.

2.2.3 VORSat

This mission main goal is to successfully accomplish the reentry and recovery of an ERC. Atmo- spheric reentries have been achieved since the mid Twentieth century, but it has not been attempted at such a small scale. Reentry is divided into three sub steps: adjusting friction in the higher atmosphere to adjust expected area of impact, surviving reentry during the deceleration phase and having a safety landing and recovery. Figure 2.3 depicts the three steps described.

Figure 2.3: Capsule phases. Courtesy of João Gomes

To ensure the surviving reentry during the decelaration phase, it is needed to obtain a light weight vehicle, Figure 2.4, therefore the structure will be mainly composed of Carbon Fiber Re- inforced Polymer. Due to the high costs and use restrictions, the option for a non-ablative thermal protection system (TPS), such as a Reinforced Carbon Carbon, has been discarded. It was decided to use a light ablative material, preferably composed of cork, a considerably important product in the Portuguese economy [2]. For the recovery of the ERC it is necessary to know the entry point and its flight path. The prediction of the flight path will be achieved through a specially developed algorithm that uses sporadic GPS observations. After the entry phase, the position of the capsule will be sent to the ground stations (at least one stationary and several mobile) using, as the main communications 2.2 Project 11 channel, the ARGOS system, due to its low cost and high reliability. This system is commonly used to track animals in their migrations[2]. The landing point is an essential part of the mission as it can determine the success or failure of the recovery operation. Two main options are currently under analysis: splashing down in an ocean or landing in a desert. These options are being considered due to the low population and building density, reducing the possibilities of collateral damages. In order to minimize the force acting upon landing, parachutes will be deployed by the capsule [2]. Due the small size of the capsule there are some constrains on the equipament placed onboard. This will, in turn, depend on the maximum dimensions allowed, the state of development of the technology that is to be tested, space conflicts between different payloads, energy storage, oscillation damping system, etc., Figure 2.5, [2].

Figure 2.4: Capsule Structure. Figure 2.5: Interior of the Capsule. Adapted Courtesy of João Gomes from [2]

The battery also has some obstacles to the project due to its limited capacity. The algorithms mentioned above running on the microcontroller, the GPS, the communications and the ﬂash used to facilitate the location of the capsule, consume energy. Therefore the algorithms must be able to adjust its communications within the ground based on the values of the available battery. In the following ﬁgure we can observe the level of consumption of each component:

Figure 2.6: Batery Charge [1] 12 Case Study Presentation

As in the CubeSat, the capsule will have a system of energy generation to maximize the signal transmission.

2.3 Summary

The chapter’s main intent was to brieﬂy present the project that is being developed in FEUP, integrating it in the context of CubeSats and then introducing its objectives and its problems, being this the motivational basis for the work described throughout this thesis. The project’s current architecture was described in a brief manner as well as the different modules that make it up. The project’s work description was complemented with ﬁgures and citations taken from the project’s available literature. Moreover, some considerations on the project’s current development status and its evolutional path were made. The following chapters will describe the research of the literature and the process used to design, implement and integrate the navigation system. Chapter 3

Literature Review

The intent of this chapter is to introduce the different scientific themes and trends relevant to this thesis as well as the work developed in the fields of artificial satellites and their navigation systems. The chapter is divided into 3 sections, each are presenting the state of art, works and concepts related to a specific field.

3.1 Satellites

As already stated, the space exploration began after World War II. Both the United States and the Sovietic Union began researching the feasibility of attaching warheads with long-range rockets capable of crossing half way round the world. These weapons were eventually called intercontinental ballistic missiles or ICBMs. They could be equipped with conventional or nuclear warheads. At the end of World War II, the U.S. had introduced the nuclear warfare age by dropping atomic bombs on Japan [13]. However, only in 1957 the Sovietic Union launched the first artificial satellite, the Sputnik 1 [14]. Sputnik was launched on October 4th and only with the issuance of a "bip-bip", tuned by any amateur radio in the frequencies between 20.005 and 40.002 MHz. It marked the beginning of the space age of communications. Placed at a altitude about 250 km (150 miles) above Earth level, Sputnik orbited for six months before falling. It helped to identify the density of high atmospheric layers through measurement of its orbital change and provided data on radio-signal distribution in the ionosphere[14]. Since the first satellite was put into orbit, numerous countries launched their own artificial satellites in the last years and several of them are currently under development. In Figure 3.1 it is possible to see the development of satellites in the early stages [15]. Among the several satellites presented in the Figure 3.1 , we highlight: the satellite that carried for the first time a living passenger into an orbit (Sputnik -2), the first human spaceflight, the first Portuguese satellite, PoSat-1 and the emergence of small satellites.

13 14 Literature Review

Figure 3.1: Evolution of satellites over the years

3.1.1 Satellites Features

About 75% of satellites launched into space since 1957 had military purposes, but nowadays satellites are, in generally, used for radio communication and television signal. Beyond Earth observation and communications, satellites can be applied to scientiﬁc purposes, including observation of the weather, the exploration of the universe and data collection of Earth.

• Earth observation Satellites - With Earth observation satellites in orbit, it is possible to analyse the environmental conditions, through the processes and interactions between land masses, oceans and global atmosphere, in order to ensure our safety and quality of life. Among other features, they are also capable of detecting environmental risks in a timely manner as well as monitor and manage the Earth’s natural resources. These types of satellites can also be used for defense and security purposes. Their main applications are the generation of maps, the monitoring and tracking of targets and, if necessary, the destruction of enemy warheads, satellites and other space assets. The ﬁrst Earth observation satellite was launched in 1959, Explorer VII, [16].

• Communication Satellites - Communication satellites are used in the transmission of digital information, specifically in the civilian world. Several artificial satellites are stationed in space with the purpose of telecommunications. The first and historically most important application of satellite communication was intercontinental long-distance telephony. As television became the main market, characterized by the simultaneous delivery of relatively few signals of large bandwidth to many receivers the era of geosynchronous comsats was born. After the 1990s, the technology of satellite communications has been used as a way to connect to the Internet via broadband data links. This can be very useful for users who 3.1 Satellites 15

are located in very remote areas, without access to a broadband connection. Communica- tion satellites are also used for military communications applications, such as the Global Command and Control Systems[16].

• Navigation Satellites - Satellite navigation is a space-based radio positioning system, which includes one or more satellite constellations, augmented as necessary to support the intended operation, providing 24 hours and three-dimensional position, velocity and time information to suitably equipped users almost anywhere on the surface of the Earth. Users have nowadays sufﬁcient accuracy and completeness of information to be usable for critical navigation applications. The GPS system is the central element of the ﬁrst satellite navigation system widely available to civilian users. Satellites send out radio signals to mobile receivers on Earth, allowing the determination of the accurate location. The direct reception of the signal from GPS satellites, combined with an increasingly better receives technology, allows the GPS system to determine the position with an error of a few meters in real time [16].

• Weather Forecast Satellites - Weather forecast uses a variety of observations to analyse the current state of the atmosphere. Since the launch of the ﬁrst weather satellite in 1960, TIROS-1, global observations have been possible, even in the remotest areas. [16].

• Scientiﬁc Research Satellites - Scientiﬁc research satellites provide applications such as earth science, marine science, and atmospheric research [16].

3.1.2 Satellites Classiﬁcation by Altitude

To perform the functions mentioned in the previous subsection, these satellites need to be semi computer-controlled systems. They have to be able to realize many tasks, such as power generation, thermal control, telemetry, attitude control and orbit control. Depending on the purpose of the satellite and based on its height above the Earth’s surface, there are several altitude classiﬁcations for geocentric orbits:

• Low Earth Orbit (LEO) satellites operate in orbits of around 500 km to 1,500 km above the Earth’s surface – much lower than traditional communications satellites – which brings them into frequent radio contact with ground stations. LEOs are used for a variety of civil, scientiﬁc and military roles including Earth observation, radar, optical, telecoms and demon- strator [6].

• Medium Earth Orbit (MEO) refers to an altitude below 35,786 km (geostationary orbit) and above the altitude of Low Earth Orbit (LEO). Medium Earth Orbit enables a satellite provider to cover the earth with fewer satellites than Low Earth Orbit, but requires more satellites to do so than geostationary orbit. The most common use for satellites in this region is navigation, such as the GPS and Glonass constellations (Russian counterpart to the United States GPS system) [6]. 16 Literature Review

• Geostationary Orbit (GEO) are situated above the altitude of 35,786 km. Their name is due to the fact that these satellites are placed in an orbit over the equator, having a rotation period equal to Earth’s. Objects in Geostationary orbit revolve around the Earth at the same speed as the Earth rotates, meaning a 24 hours rotation period. Communications and weather satellites are often given geostationary orbits, so that the satellite antennas that communicate with them not having to move to track them, but can be pointed permanently at the position in the sky where they stay [6].

• Highly Elliptical Orbit (HEO) is a geocentric orbit whose apogee lies above the geostationary orbit (35,786 km). HEO is mainly perturbed by the Earth’s oblateness and by gravitational attraction of the Sun and Moon. They are popular orbits for Earth magentospheric measurements and astronomical observatories. [6]

Based on the applications, satellites contain some restrictions in the range of launch and in the need to have or not a powerful ampliﬁer for successful transmissions. This leads to a systematic choice of the orbit where the satellites are placed. The great majority of satellites are launched into lower orbits at altitudes of 500-1500 km. Below that level, a satellite’s orbit would be decay quickly due to the resistance of the Earth’s atmosphere, which extremely restricts low-altitude orbits to short-term ballistic missions or powered trajectories. In any circular orbit, the centripetal force required to maintain the orbit is provided by the gravitational force on the satellite. To calculate the geostationary orbit altitude, one begins with this equivalence, and uses the fact that the orbital period is one sidereal day.

Fc = Fg (3.1)

By Newton’s second law of motion, we can replace the forces F with the mass m of the object multiplied by the acceleration felt by the object due to that force:

m × ac = m × g (3.2)

We note that the mass of the satellite m appears on both sides — geostationary orbit is independent of the mass of the satellite. So calculating the altitude simpliﬁes into calculating the point where the magnitudes of the centripetal acceleration required for orbital motion and the gravitational acceleration provided by Earth’s gravity are equal. The centripetal acceleration’s magnitude is:

2 ac = w × r (3.3) 3.1 Satellites 17

where w is the angular speed, and r is the orbital radius as measured from the Earth’s mass center. The magnitude of the gravitational acceleration is as follows:

2 v KME KME = 2 = 2 (3.4) d d (RE + h)

From the expression 3.1.2 the satellite travelling speed in GEO orbit can easily be obtained.

s KME v = 2 (3.5) (RE + h)

The angular speed w is found by dividing the angle travelled in one revolution by the orbital period (the time it takes to make one full revolution: one sidereal day).

2 × π days w =≈ × solar (3.6) T dayssideral

2 × π 365.24 w =≈ × (3.7) 3600 × 24 366.24

The resulting orbital radius is the sum of the height of GEO satellite, 35786 km, and the mean radius of the earth, 6371 km. The GEO orbital speed (how fast the satellite in GEO is moving through space) is calculated by multiplying the angular speed by the orbital radius:

v = w × (RE + h) = 42213000 = 3.057km/s (3.8)

Therefore, the velocity of a LEO satellite is:

√ r RE + hGEO 42213 vLEO = vGEO × √ = 3061 × = 7.719km/s (3.9) RE + hLEO 6650

LEO satellites travel at 7.7 km/s while Earth travels at 3 km/s, making a complete revolution around the Earth in about 90 minutes. On the other hand, above 1500 km altitude, the satellite launchers are more complex due to the distance to be achieve. This causes a higher cost compared to the launch of LEO. More powerful ampliﬁers are required for a successful transmission and with the increasing distance, a delay of about 0.5 s in the signal is expected. This delay brings more complex problems in the protocols 18 Literature Review of veriﬁcation and correction of incorrect data. The system has to stop transmitting data while awaiting the response of the receiver so that there is no error in the sent data. [17] However, the LEO systems have a few disadvantages. Once they are closer to Earth, LEO satellites have to compensate for Doppler shifts cause by their relative movement and a higher number of satellites are needed to cover the Earth’s surface. Due to the atmospheric drag, causing gradual orbital deterioration, LEO satellites have much shorter life span than MEO and GEO. Typically values for LEO’s life span range from 5 to 8 years [17].

3.2 Navigation System

Regardless of the level of autonomy that the satellites can reach, any man-made spacecraft would have no value if it was not possible to locate and establish communication. The navigation system is concerned with the mathematical and physical description of the artificial satellite orbits, as well as their control. Many of the satellites previously mentioned, need an active control of their orbit according with the specific mission requirements. Navigation is, therefore, an essential part of satellite operations. It comprises the planning, determination, prediction and correction of satellite’s trajectory in line with the established mission goals. Therefore, this chapter is subdivided in three main subjects, definition of an orbit, the description and parameters to describe an orbit and finally the common estimation techniques employed for purposes of orbit determination.

3.2.1 Principles of Orbital Motion

Several models have been developed to compute the motion of artiﬁcial Earth satellites to achieve the high level of accuracy for many and sundry applications nowadays. However the main features of their orbits may still be described by a reasonably simple approximation. This is due to the fact that a body travelling around a point in space is under the inﬂuence of some gravitational force. Such force results from the Earth’s central mass and overrules all other forces by several order of magnitude, in the same way as the attraction of the Sun governs the motion of the planets. The laws of planetary motion were found empirically by Johannes Kepler, demonstrating that the motion of planets, known as geocentric model of the solar system, were in fact, elliptical motions, and not perfect spheres or rings, as previously suspected.

1 The orbit of every planet is an ellipse and the Sun is not located at the center of the orbits, but at one of the two foci;

2 A line joining a planet and the Sun sweeps out equal areas during equal intervals of time, in other words, the orbital speed of each planet is not constant, depending on the planet’s distance from the sun; 3.2 Navigation System 19

3 The third law, is a universal relationship between the orbital properties of all planets orbiting the sun. The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit.

Afterwards, Isaac Newton solidiﬁed Kepler’s laws by showing that they were a natural con- sequence of his inverse square law of gravity and that, in general, the orbits of bodies subject to gravity were conic sections, if the force of gravity propagated instantaneously. Newton showed that, for a pair of bodies, the orbits’ sizes are in inverse proportion to their masses, and that the bodies revolve about their common center of mass. When one body is much more massive than the other, it is a convenient approximation to take the center of mass as coinciding with the center of the more massive body. Further, Newton extended Kepler’s laws in a number of important ways such as allowing the calculation of orbits around other celestial bodies and proved that relation- ships like Kepler’s laws would apply exactly under certain ideal conditions. These laws are a good approximation fulﬁlled in the solar system, as consequences of Newton’s own laws of motion and law of universal gravitation. Because of the nonzero planetary masses and resulting perturbations, Kepler’s laws apply only approximately and not exactly to the motions in the solar system. Current understanding of the mechanics of orbital motion is based on Albert Einstein’s general theory of relativity, which was able to show that gravity exists because of the curvature of space- time. In the theory of relativity, the orbits follow geodesic trajectories, which are very close to the Newtonian predictions. However, there are differences and these can be used to determine which of the two theories agree with reality. Essentially all experimental evidence agrees with the theory of relativity. The laws of planetary motion may, therefore, equally well be applied to a satellite’s orbit around the Earth.

3.2.2 Orbital Motion as a Two-Body Problem

The motion of a spacecraft under the inﬂuence of a celestial body in the solar system is usually ap- proximated as a two-body problem by ignoring the gravitation caused by the other objects, as well as, the actual, nonspherical shapes of the two bodies. This subsection follows the developments found in the book [4] and [18]. Since the distances between the bodies are generally large in comparison with their dimensions, it is a standard assumption to treat the two-body problem as collection of two particles, mutually attracted by the Newton’s inverse-square law of gravitation. As the net external force on the system of total mass, m, is zero, we can write the equation of motion of the system’s center of mass

d2r µ + r = 0 (3.10) dt2 r3

where r is the position of m2 relative to m1 and µ = G(m1 + m2). 20 Literature Review

Once the Eq.(3.10) is a nonlinear vector differential equation, we do not expect it to have a closed-form solution. However, we can analytically determine the necessary integral of motion governing the problem, which allow us to represent the trajectory either as an inﬁnite series ex- pansion, or by related numerical approximations. Let us begin by taking the vector product of Eq. (3.10) with r:

d dr dr dr (r × ) − × = 0 (3.11) dt dt dt dt

d dr which implies that dt (r × dt ) = 0.

Figure 3.2: The two-boy motion [3]

Therefore, it follows that the speciﬁc angular momentum of m2 relative to m1, deﬁned by

dr h = r × = r × v (3.12) dt

is conserved, where v is the relative velocity. Since the time derivative of r × r˙ equals zero, the quantity itself must be a constant. Geometrically, the cross product of two vector is a vector at right angles to both of them. Therefore, the position vector r, as well as, the velocity vectorr ˙ are always perpendicular to h, that is, the orbit conﬁned to a plane. Hence, h is constant implies Kepler’s second law of areas. Considering the satellite’s motion as linear over a small time step ∆t, then

dr h = r × = r × v (3.13) dt 3.2 Navigation System 21

is just the are swept by radius vector during the time ∆t. The magnitude value |h| is therefore known as areal velocity and since h remains constant, the radius vector sweeps over equal areas in equal time intervals (Kepler’s second law). The two body trajectory can be classiﬁed according to the magnitude and direction of a constant h. The case of h = 0 represents rectilinear motion along the line joining the two bodies, while h 6= 0 represents the more common trajectory involving rotation of m2 about m1. A further classiﬁcation of the trajectories is possible taking the vector product Eq. (3.10) with h:

d2 µ × h + (r × h) = 0 (3.14) dt2 r3

We note that,

d2r d × h = (v × h) (3.15) dt2 dt

since h is constant. Another interesting elementary identity can be obtained by differentiating r2:

dr2 ∂r = 2r (3.16) dt ∂t d(r · r) = (3.17) dt = 2r · dotr = 2r · v, (3.18)

from which it follows that r · v = rr˙ (dot represents time derivative). We use this and another vector identity to evaluate the second term in Eq (3.14) as follows:

µ µ (r × h) = (r × (r × v)) (3.19) r3 r3 d r = −µ ( ). (3.20) dt v

Substituting Eqs. (3.15) and (3.14), we obtained

d µr (v × h − ) = 0. (3.21) dt r

µr Therefore we can deﬁne a constant vector e, known as eccentricity, such that µe = v × h − r . Since e is a constant vector, we expected provide three more scalar integrals of motion; however since e and h are related bye·h = 0, we get a total of only ﬁve scalar integrals from the two constant 22 Literature Review vectors. It is also clear that e, being perpendicular to h, lies in the plane of motion formed by r and v. An important insight into the two-body motion is obtained by writing the magnitude of the eccentricity vector as:

1 2 e2 = e.e = (v × h) · (v × h) − r · (v × h) + 1. (3.22) µ2 µr

Since v and h are mutually perpendicular, it follows that (v × h) · (v × h) = v2h2. Futhermore, its true that r · (v · h) = (r × v) · h = h2.

v2h2 2h2 h2 2 v2 1 − e2 = − + 1 = ( − ). (3.23) µ2 µr µ r µ

Deﬁning the variable p (which denotes the distance of the satellite from Earth’s center at right angles to perigee and apogee, is called semi-latus rectum) and a (the mean value of the minimum and maximum distance is the semi-ma jor axis), both having units of length, by

h2 1 2 v2 p = ; = − (3.24) µ a r µ

which are related by p = a(1 − e2), it is possible use the deﬁnition of a, to deﬁne another integral of motion:

µ v2 µ ε = − = − . (3.25) 2a 2 r

Last but not least, another interesting law of Keplerian motion may be derived, which relates v2 the satellite’s velocity to the distance from the center of Earth. Note that 2 is the specific kinetic µ energy and r is the specific potential energy of relative motion of m2 about m1. Hence, we call ε the energy integral of the relative motion, called as the vis-viva law. The energy integral can be used to describe the various kinds of trajectories of the two-body motion. For a bound orbit of m2 about m1, the magnitude of potential energy must be greater than the kinetic energy, which implies 1 ε < 0 and a > 0. For m2 to escape the gravity of m1, the relative kinetic energy must be greater 1 than or equal to the magnitude of potential energy, implying ε > 0 and a < 0. As seen from the differential Eq. (3.10) it was possible solve mathematically the two body case and the resulting orbit, which follows Kepler’s laws of planetary motion, is known as "Kepler orbit". And finally, since, by definition p > 0, the eccentricity e, can also be used to determine if the orbit is closed or open. That means, the Kepler orbit can describes the motion of an orbiting body, as well as, an ellipse, parabola, or hyperbola: 3.2 Navigation System 23

Figure 3.3: Motion of an orbiting body [4]

We also have the circular orbit, that is a special case of a Elliptic Orbit, it has eccentricity equal to zero.

3.2.2.1 Keplerian Elements Description

In the previous section, we were able to derive a special trajectory of the two-body motion and despite having several different ways to mathematically describe the same orbit, certain schemes each consisting of a set of six parameters are commonly used in astronomy and orbital mechanics. These parameters deﬁne an ellipse, orient it about the earth, and place the satellite on the ellipse at a particular time. [19] The main two elements that deﬁne the shape and size of the ellipse are:

• Eccentricity(e) - shape of the ellipse, describing how ﬂattened it is compared with a circle.

• Semimajor axis (a) - the sum of the periapsis and apoapsis distances divided by two. For circular orbits the semimajor axis is the distance between the bodies, not the distance of the bodies to the center of mass. Two elements deﬁne the orientation of the orbital plane in which the ellipse is embedded:

• Inclination - The orbit ellipse lies in a plane known as the orbital plane. The orbital plane always goes through the center of the Earth, but may be tilted any angle relative to the equator. Inclination is the angle between the orbital plane and the equatorial plane. (angle i in diagram).

• Longitude of the ascending node - The Earth is spinning, therefore it is not possible to use the common latitude/longitude coordinate system to specify where the line of nodes points. Instead, it is use an astronomical coordinate system, known as the right ascension / declination coordinate system, which does not spin with the Earth. So right ascension is horizontally orients the ascending node of the ellipse (where the orbit passes upward through the reference plane) with respect to the reference frame’s vernal point (angle Ω in diagram). 24 Literature Review

And ﬁnally:

• Argument of periapsis - deﬁnes the orientation of the ellipse (in which direction it is ﬂattened compared to a circle) in the orbital plane, as an angle measured from the ascending node to the semi-major axis. (angle ω in diagram).

• Mean anomaly - Now that we have the size, shape, and orientation of the orbit firmly established, the only thing left to do is specify where exactly the satellite is on this orbit ellipse at some particular time. Our very first orbital element (Epoch) specified a particular time, so all we need to do now is specify where, on the ellipse, our satellite was exactly at the Epoch time.

Figure 3.4: Keplerian Elements

In the diagram of ﬁgure 3.4, the orbital plane intersects a reference plane. For earth-orbiting satellites, the reference plane is usually the Earth’s equatorial plane, and for satellites in solar orbits it is the ecliptic plane. The intersection is called the line of nodes, as it connects the center of mass with the ascending and descending nodes. This plane, together with the Vernal Point, establishes a reference frame.

3.2.2.2 Trajectory Data Format: The Two-line Elements

The trajectory data, Kepler’s elements, which describes the motion of a satellite, can be transmitted in a variety of formats. The most common of them is the NASA/NORAD "two-line el- 3.2 Navigation System 25 ements"(TLE) set that consists of two 69-character lines of data. The only valid characters in a two-line element set are the numbers 0-9, the capital letters A-Z, the period, the space, and the plus and minus signs, no other characters are valid. [20] Two-Line Element (TLE) assume the following format:

1 AAAAAU YYLLLA BBBBB.BBBBBBBB .CCCCCCCC DDDDD-D EEEEE-E F GGGZ 2 AAAAA HH.HHHH III.IIII JJJJJJJ KKK.KKKK MMM.MMMM NN.NNNNNNNNRRRRRZ

Table 3.1: Two-line Element - Line 1

Columns Example Description 1 1 Line number 3-7 AAAAA Satellite number 8 U Classiﬁcation (U=unclassiﬁed) 10-11 YY International Designator (last two digits of launch year) 12-14 LLL International Designator (launch number of the year) 15-17 A International Designator (piece of the launch) 19-20 BB Epoch Year (last two digits of year) 21-32 BBB.BBBBBBBB Epoch (day of the year and fractional portion of the day) 34-43 .CCCCCCCC First time Derivative of the Mean Motion 45-52 DDDDD-D Second Time Derivative of Mean Motion (decimal point assumed) 54-61 EEEEE-E BSTAR drag term (decimal point assumed) 63-63 F Ephemeris type (number zero) 65-68 GGG Element number 69-69 Z Checksum

Table 3.2: Two-line Element - Line 2

Columns Example Description 1 2 Line number 3-7 AAAAA Satellite number 9-16 HHH.HHHH Orbit Inclination (degrees) 18-25 III.IIII Right Ascension Node (degrees) 27-33 JJJJJJJ Eccentricity (decimal point assumed) 35-42 KKK.KKKK Argument of Perigee (degrees) 44-51 MMM.MMMM Mean Anomaly (degrees) 53-63 NN.NNNNNNNN Mean Motion (revolution per day) 64-68 RRRRR Revolution Number at Epoch 69 Z Checksum

The TLE of a satellite can be obtained by the program Orbitron [21], which is a satellite tracking system. This program, together with the Simpliﬁed Perturbations models a set of ﬁve mathematical models can be used to determine the position and velocity of the satellite relatively to the Earth Centered Inertial coordinate system. 26 Literature Review

3.2.3 Orbit Determination Methods

In order to determine the unknown orbit of a man-made satellite, some observations of its motion with time are required. In early modern astronomy, the only available observational data for celestial objects were the longitude ascending node and declination. They were obtained by observing the body while it moved relative to the ﬁxed stars. Nowadays, it is customary use the direct computation of a priori orbital elements from a large set of tracking data.

3.2.3.1 Simpliﬁed Perturbations models

Perturbations models are used for tracking of objects in orbit at all times. The original SGP model was developed by Kozai in 1959, refined by Hilton and Kuhlman in 1966 and was originally used by the National Space Surveillance Control Center (and later the United States Space Surveillance Network). Propagation models are needed because of the lack of telescopes to watch every single object in the sky at all times. After finding the position and velocity of an object once, the propagator is able to determine where the object is in the future in order to be located again. There are five mathematical perturbations models: SGP, SGP4, SDP4, SGP8 and SDP8. The SGP model used was SGP4 because it is for a spacecraft in a typical Low Earth orbit. This model was developed by Ken Cranford in 1970 and it was obtained by simplication of the more extensive analytical theory of Lane and Cranford (1969) which uses the solution of Brouwer (1959) for its gravitational model and a power density function for its atmospheric model. The accuracy obtained with the SGP4 orbit model is on the order of 1 km within a few days of the epoch of the element set.

3.2.3.2 Improvement of a priori orbit

Satellites, encounter disturbances, or perturbations, along its path that complicates their motion. These perturbations are caused by the Earth’s shape (spherical harmonics), drag, radiation, and effects from other bodies (the sun and moon generally). Therefore, there is a need to improve a priori orbit information and it can be obtained by using two techniques: the least-squares estimator or the Kalman ﬁlter. While the least-squares estimator improves an epoch state estimate by processing a whole set of observations in each run, the Kalman ﬁlter processes one measurement at a time and yields subsequent estimates of the state vector at the time of each measurement.

1. Least Square

In order to estimate the parameters of the equations that describes the motion of planets and comets with the data measured by telescopes, Gauss proposed the method of least squares in 1795 (Sorenson, 1970).

The main idea of least-squares estimation applied to orbit determination is ﬁnding the trajectory and model parameters that minimizes the squared difference between the observation 3.2 Navigation System 27

modeled and the actual measurements. In other others, finding the trajectory which best fits the observations in a least-squares of the residual sense. The least-squares estimation method yields an estimate of epoch state vector by processing the complete set of observations in each iteration. Therefore, it requires that all measurements to be considered in an orbit determination are available before the a priori information can be improved. This makes least-squares estimation less convenient for real-time or near- real-time applications that call for a quasi-continuous update of the state information with each observation. In addition, the least-squares method requires the estimate of the epoch state vector to fit the entire data span, which makes it susceptible to dynamical model errors and the assumption of constante measurements biases. An alternative estimation method which copes with these problems, is known as Kalman Filter.

2. Kalman Filter Since its introduction in 1960, by Rudolf E. Kalman, the Kalman filter has become an integral component in thousands of military and civilian navigation systems. This deceptively simple and recursive digital algorithm has been an early-on favorite for conveniently integrating (or fusing) navigation sensor data to achieve optimal overall system performance [22] . The Kalman filter is a set of mathematical equations that provides an efficient computational (recursive) means to estimate the state of the process, in a way that minimizes the squared error of the mean. The filter is very powerful in several aspects: it supports estimations of the past, present, and even future states, and when the precise nature of the modeled system is unknown. Its purpose is to use measurements that are observed over time that contain noise (random variations) and other inaccuracies, and produce values that tend to be closer to the true values of the measurements and their associated calculated values. This procedure is repeated for each time step with the state of the previous time step as initial value. "Predict" and "Update". The predict phase uses the state estimate from the previous timestep to produce an estimate of the state at the current timestep. This predicted state estimate is also known as the a priori state estimate because, although it is an estimate of the state at the current timestep, it does not include observation information from the current timestep. In the update phase, the current a priori prediction is combined with current observation information to refine the state estimate. This improved estimate is termed the a posteriori state estimate. The basic components of the Kalman Filter are the state vector, the dynamic model and the observation model, which are explained below.

State Vector The state vector contains the variables of interest. It describes the observed states of the dynamic system and represents its degrees of freedom for motion, (i.e. translatory 28 Literature Review

Figure 3.5: Procedure of the Kalman Filter

variables distance, velocity and acceleration for location and orientation in e.g. any right handed system of coordinates). In this case of study, our variables will be the elements on the TLE and we can described them such as:

" # p x = (3.26) v

The state vector has two values at the same time, that is the a priori value, the predicted value before the correction (x-), and the a posteriori value (x+), the corrected value after the correction. Dynamic Model The dynamical system concept is a mathematical formalization for any ﬁxed "rule" which describes the time dependence of a point’s position in its ambient space, in other words, it describes the transformation of the state vector over time. And its represented :

xk˙+1 = Axk + Buk + Gµk (3.27)

where A is the transition matrix, the difference between the new elements of the TLE and the oldest, x the state vector, B the control input model which is applied to the 3.3 Perturbed Orbits 29

control vecto and ﬁnally, G is the process noise which is assumed to be drawn from a zero mean multivariate normal distribution with covariance Q.

Observation Model The Observation model represents the relationship between the state and the measurements.

y˙k = Hxk + Fηk (3.28)

having as the observation matrix, H, F is the noise of the measurement process with the covariance matrix R.

On this MSc Thesis the idea is to use old TLE parameters, and at each discrete time increment, a linear operator is applied to the state to generate the new state, with some noise mixed in. Then, another linear operator mixed with more noise generates a new TLE that should be close to the true position values of the measurements.

3.3 Perturbed Orbits

All the discussion of the Keplerian motion in chapter 3 considered ideal conditions, that is, perturbations by external disturbances, leading to deviation from the orbits, were not considered. Examples of such disturbances include a non-uniform gravitational field due to an a spherical planetary mass, atmospheric drag, gravitational influence of a third body (when flying toward the inner planets), thrust of rocket engines, and solar radiation pressure. As an example, a satellite in a high earth orbit experiences: negligible atmospheric drag and planetary gravitational anomaly but an appreciable tug from lunar and solar gravitation, as well as some effects of solar radiation. Meanwhile, a vehicle in a low orbit experiences the largest orbital perturbations due to atmospheric drag and non spherical gravity field. The objective in this section is to present the modeling and simulation of the two-body orbits perturbed by atmospheric drag. The study of perturbed orbits is important to understand the necessary conditions for satellites re-entering and to predict decay point. The equation of perturbed relative motion in the two-body problem (preceding section) can be expressed as follows equation:

d2r µ + r = a (3.29) dt2 r3 d

where ad is the perturbing acceleration on mass m2, whose position relative to m1 is r. The

perturbing force, m2ad, can be either conservative, such as gravitational anomalies due to aspheri- cal shape and other bodies, or non-conservative, such as atmospheric drag, rocket thrust, and solar radiation pressure. 30 Literature Review

Spacecraft in a low-orbit experience a signiﬁcant atmospheric drag, whose magnitude is proportional to the product of atmospheric density, ρ, and square of relative speed, r˙2. Since drag opposes the orbital motion, we can express the perturbed equation of motion by

d2r µ + r = −B∗r˙r˙ (3.30) dt2 r3

where

1 C A B∗ = ρ D (3.31) 2 m2

and CD is the drag coefﬁcient of the spacecraft based upon a reference area, A. The dependence of drag on velocity, rather than on position, renders it a non-conservative force, which results in µ a decline of the orbital energy,ε = − 2a , and hence,the semi-major axis, a. Taking the scalar product of the deceleration due to drag with the relative velocity, we have the rate of change of orbital energy

µ ε˙ = − a˙ = −B∗r˙r˙· r˙ = −B∗v3 (3.32) 2a2

where v = r˙. Clearly, the rate of decline of the orbit increases proportionally with B∗ and diminishes as the orbit size increases. The atmospheric density at orbital altitudes can be approxi- − z a mated as an exponentially decaying function of the altitude, z = r − Rearth, whereby ρ = ρ0e H , and we can write

2 a CDA 3 − z a˙ = − ρov e H (3.33) m2µ

The rate of decline of the orbit due to atmospheric drag is an important factor in determining the life of the satellite in a low orbit. However, the prediction of a satellite’s orbital life involves an accurate estimate of the atmospheric properties at high altitudes over long periods (years and decades), which is seldom possible due to random external disturbances of solar radiation and geomagnetic ﬁeld. Taking the vector product of the deceleration due to drag with the relative velocity, we have the rate of change of angular momentum,

h˙ = −r × B∗r˙r˙ = −B∗rh˙ (3.34) 3.3 Perturbed Orbits 31

Now, it is to be noted that taking the time derivative of the equation h2 = h · h results in the following:

h˙ = −B∗rh˙ (3.35)

Since h = hih, we have

di h˙ = hi˙ + h h . (3.36) h dt

Substituing Eqs 3.3 into 3.3,

di h h = B∗rhi˙ − B∗rh˙ = 0, (3.37) dt h di h = 0. (3.38) dt

Therefore, there is no change in the orbital plane due to atmospheric drag; hence, the Euler angles, i, ω and Ω, defining the orbital plane, remain invariant with time. The above-demonstrated invariance of the orbital plane due to drag can be advantageously utilized in effectively braking a spacecraft as it arrives at a planet from an interplanetary flight. Due to the exponential increase in the density with a decrease in the altitude, an elliptical orbit experiences the largest drag at its periapsis. The planetary atmospheres on the earth and Mars have a negligible density above about 150 km altitude. Thus, a highly elliptical orbit with a periapsis altitude less than 150 km around either the earth or Mars experiences a negative velocity impulse everytime it passes the periapsis, which remains fixed in space. Hence, if a spacecraft’s arrival hyperbola is converted into a highly eccentric elliptical orbit, either by retro-rockets (called orbit insertion burn ) or by an initial pass through the atmosphere (called aerocapture ), the orbital eccentricity as well as the semi-major axis can be reduced by making successive passes through the atmosphere. Such an approach is referred to as aeroassited orbital transfer , or aerobraking, and has been employed in several spacecraft, such as the Magel lan Venus mission, the Mars Global Surveyor (MGS), and the Mars Odyssey mission.

3.3.1 Atmospheric models

For modeling a planetary entry trajectory it is necessary develop a detailed planetary atmosphere model, including nondimensional aerodynamic parameters for modeling aerothermodynamic forces, moment and heat transfer, to provide the atmosphere adequately for computer or analytical modeling what the satellite will surfer during the atmospheric ﬂight. The atmosphere might be described as a thin layer of gases clinging to the planetary surface by gravitational attraction and in hydrostatic equilibrium set by solar radiation. Since solar radiation 32 Literature Review and atmospheric reradiation must vary diurnally and annually, any atmospheric model cannot be more than an approximation. It is expect that the properties of the model varies not only with time, but also as a function of location. Therefore, an Atmospheric Model is a mathematical model constructed around the full set of primitive dynamical equations which govern atmosphere motions and must be able to represent the vertical run of pressure, density, and temperature. The ﬁrst fundamental relationship is the atmospheric equilibrium equation, which relates pressure P and density ρ. Summing the pressure ([P−(P+dP)]A) and gravitational forces (ρgA) gives

−ρgAdZ + [P − (P + dP)]A = 0 (3.39)

By assuming force equilibrium, it is assumed the atmosphere to be a thermodynamic ﬂuid. Therefore, an interesting relation between density, pressure and temperature can be described by the thermal equation of state of an ideal gas,

PV = NR∗T (3.40)

where the parameters of temperature, T and pressure, P can be related to the volume V, the number of moles present N, and the universal gas constant R∗. The density, ρ, is deﬁned as the mass, m, per unit volume, v, and has units of kg/m3. It can be mathematically described as

∆m dm ρ = lim = (3.41) ∆v→0 ∆v dv

The gas can be affected by density in two ways: (1)Being the weight of a given volume gas d,irectly proportional to its density, a stationary column of gas would exert a force directly proportional to its density; (2)The inertia of a ﬂowing gas is determined by its density, therefore, a denser packet of gas is accelerated to a smaller speed when the force is applied on it. 3 The temperature is a measure of the average kinetic energy of a gas particle, exoressed as 2 kT where k is the Boltzmann’s constant. Finally, the pressure can be deﬁned as the net rate of change of normal momentum of the gas particles striking per unit area.

∆ f d f p = lim n = n (3.42) ∆A→0 ∆A dA

Hence, pressure can be regarded as the force per unit area that a hypothetical solid surface would encounter, were it present at the given point. By Pascal’s law, a gas at rest has the same 3.4 Interaction Station - Satellite 33 pressure at all points. By contrast, a ﬂowing gas has pressure varying from point to point. Once, we just want to analyze and regulate operation of the satellite, the atmospheric model used serves as a common reference for the calibration of ﬂight instruments. The atmospheric model used was developed by Ken Cranford in 1970 such model can have utility in the evaluation of the performance of a re-entry satellite or components.

3.4 Interaction Station - Satellite

To obtain measurements related to the instantaneous position of a satellite is necessary to appeal to the using of the telemetry, tracking, command and communication (TTCC) system. It provides meanings for the monitoring and the controlling of the satellite in orbit.

3.4.1 Measuring from the Earth

3.4.1.1 Tracking

The tracking subsystem provides facilities by the satellite orbit which can be determined. The command subsystem provides the meanings from which the satellite is controlled and the communications subsystem provides links between the satellite and the earth station, for the transmission of the necessary signals for these various purposes. A variety of tracking systems may be used to obtain measurements related to the instantaneous position of satellite or its rate of change.

1. Visually Tracking One easy way to view a satellite in orbit is the unaided eye. That’s because, when the satellite is near the Earth, it reﬂects the Sun’s light off their surfaces toward the observer and becomes possible to follow it.

2. Radar Tracking Most of these systems are based on radio signals transmitted to or from a ground antenna 3.6. Common radio tracking systems are able to perform angle measurements by locating the direction of a radio signal transmitted by a satellite. The resolution of these measurements depends on the angular diameter of the antenna cone, which is determined by the ratio of the carrier wavelength to the antenna diameter. Distance and velocity information can be obtained by measuring the turn-around delay or Doppler-shift of a radio signal sent to the spacecraft and returned via transponder.

3. Satellite Laser Ranging Satellite laser ranging (SLR) systems provides highly accurate distance measurements by determining the turn-around light time of laser pulses transmitted to a satellite and returned 34 Literature Review

Figure 3.6: Ground Station. [5]

by a retro-reﬂector. Depending on the distance and the resulting strength of the returned signal, accuracies of several centimetres may be achieved. SLR is mainly used for scientiﬁc and geodetic mission that requires an ultimate precision.

3.4.2 Measuring from Space

3.4.2.1 Telemetry

The telemetry subsystem supplies measurements of various parameters, such as pressures and temperatures, on on-board equipment and converts them into electrical signals for transmission. The data is send to the earth station, which is responsible for satellite management, and receives the earth control station’s commands to perform equipment operation adjustments. This throughput is more that enough for the satellite to continuously send the required telemetry data. This is composed by timing information (derived from the GPS receiver), by computed ephemeris of the satellite and satellite status information.

1. Gobal Positing System (GPS)

The Global Positioning System (GPS), which was developed to meet military needs of the Department of Defense but now is used in every day life, is a radio based navigation system that gives three dimensional coverage of Earth 24 hours a day in any weather condition.

Going even further, GPS ranging signals offer the opportunity to obtain position measurements on-board a satellite, completely independently of a ground station.

It was decided that this system will be used for this mission because it is less expensive, it requires less antennas, is less prone to orbit error, and is less labor intensive for scheduling, 3.4 Interaction Station - Satellite 35

collecting, and transferring data. GPS is the most accurate navigation device that functions within LEO, which is where the cubesat will be operating. A GPS receiver is a high- accuracy navigation device that obtains ampliﬁed signals from a GPS antenna (which obtains signals from GPS satellites) and outputs data in a coordinate format. The solution of the GPS receiver includes the cubesat’s predicted position above the earth’s surface, velocity vector, time, and date.

(a) GPS Limitations i. Available Energy Energy is the most critical factor for the use of GPS. The Electrical Power Subsys- tem is composed by two power sources (the solar cells and the secondary batteries), a MPPT unit and auxiliary circuits to perform power regulation and battery charging (not included in the drawing). This subsystem is autonomous relatively to the others since its operation state does not depend on any command or state of other subsystems. The amount of energy available in the batteries can continuously be estimated by the Command Subsystem. Using an area of 50 cm2 of solar panels with 25% efﬁciency per satellite face and considering that the satellite is 50% under sunlight and 50% under darkness, each face under random orientation, provides an average of 2.5 Watts of power along time. This is clearly not enough to operate continuously the system, which is especially true for the transmission of the radio signals [5]. Therefore, the satellite will selectively switch the most energy demanding devices on and off according the energy level present in the batteries. 3.7 presents the method provided for the energy management. All switching will have hysteresis to avoid short duration cycles [5].

Figure 3.7: Equipment ON/OFF state depending on battery charge. [5]

A minimal energy level is required to switch the Communications Subsystem microcontroller on. Until then, only the Command Subsystem microcontroller is consuming a very limited amount of energy from its own long-life battery. When operating, the main microcontroller decides to switch the GPS receiver 36 Literature Review

only when in need to obtain a position and velocity measurement to update the satellite ephemeris and its clock. The energy consumption of this microcontroller plus the GPS receiver can be designed to be signiﬁcantly less than 2.5 Watts (less than 0.5 Watts, in fact). Therefore, the satellite average energy accumulated in the secondary batteries will have a global tendency to increase until reaching the second level. ii. Doppler Effect The principle of position determination by GPS and the accuracy of the positions strongly depends on the nature of the signals. When Sputnik was been monitoring through radio transmissions, a team of U.S. scientists led by Dr. Richard B. Kershner discovered that,the frequency of the signal being transmitted by Sput- nik was higher as the satellite approached, and lower as it continued away from them. They realized that because they knew their exact location on the globe, they could pinpoint where the satellite was along its orbit by measuring the Doppler distortion [14] . In other words, the Doppler effect causes a change in the frequency of emitted waves produced by motion of an emitting source relative to an observer. There- fore, when we want to track the satellite it is necessary to take account the Doppler effect. The change in frequency will be given by:

v v v fd = = c = f × (3.43) λ f c

where, • v is the velocity of the satellite; • c is the speed of wave; • λ is the wavelength of the transmitted wave in the reference frame of the source. The satellite will transmit at a frequency of 1.575 GHz and will have a velocity on Earth of 3 km/s 3.1.2 and approximately 9 km/s between the GPS and LEO (moving in opposite directions).3.1.2. The change in frequency on earth due the doppler effect it will be :

3 × 103 f = 1.575 × 102 = = 15KHz (3.44) d 3 × 108

And on LEO:

9 × 103 f = 1.575 × 102 = ≈ 50KHz (3.45) d 3 × 108 3.4 Interaction Station - Satellite 37

As stated earlier, we can observe that the Doppler effect will be 3 times more on LEO satellites than GEO. Therefore the maximum dynamic environment which the carrier will be subject to, in LEO, is a Doppler shift of 100 Hz/s. iii. COCOM restrictions CoCom is an acronym for Coordinating Committee for Multilateral Export Con- trols. CoCom was established by Western bloc powers in the ﬁrst ﬁve years after the end of WWII, during the Cold War, to put an arms embargo on COMECON (Warsaw Pact) countries [23]. Immediate access to satellite measurements and navigation results is disabled when the receiver’s velocity is computed to be greater than 1000 knots (approximately 1,151 mph or 1,852 km/h), or its altitude is computed to be above 60 thousand feet (18,000 meters). The receiver continuously resets until the CO- COM situation is cleared [23]. In GPS technology, the phrasing "COCOM Limits" is also used to refer to a limit on how high and how fast a GPS will operate and a limit placed to GPS tracking devices that should disable tracking when the device realizes the restriction mentioned above. This was intended to avoid the use of GPS in ICBM-like applications. These restrictions make the prices of receivers become more competitive, which means the price increases. Depending of the manufacturers they can apply this limit literally (disable when both limits are reached), other manufacturers disable tracking when a single limit is reached. iv. Noise When a GPS satellite passes across the sky, the signal that is sent out has to travel through the atmosphere, and occasionally it bounces off the land before getting to a GPS instrument. Both the atmosphere and the land create problems called multi-path noise. This noise is suppressed as much as possible by the instruments, but it is never completely eliminated, and it leads to error in high-precision GPS positioning data. One solution is the use of Kalman Filter.

3.4.3 Communication

Communication is an essential part of navigation. Without the communication it was not possible to monitor the satellite in its LEO orbit. The microcontroller is used as the brain of the satellite. The main tasks of this board are to handle the commands received from the ground station on the earth, to perform tasks such as run the algorithms, processing power, reading and storing the sensor values, and to send data back to the ground station. Its microcontroller is switched on only when enough power is available to sustain its operation for a signiﬁcant amount of time. Once in operation, one of its tasks is to control de operation of 38 Literature Review the GPS receiver. The microcontroller will seek position and velocity data from time to time in order to maintain and update the estimate of its orbit (ephemeris). This ephemeris is stored in a non-volatile memory and is updated periodically. This allows operating the GPS in low power mode most of the time (or even switching it off, this detail to be deﬁned later; nevertheless, the GPS receiver is turned off together with the main microcontroller)[5].

3.5 Summary

The chapter’s objective was to present the different fields relevant to this thesis. It began with a brief description of satellites, presenting various types of satellites according to their functions and their disadvantages and advantages. The discussion was then redirected to the satellite navigation system, the main topic of this thesis. Regarding the satellites navigation system, the following studies were mentioned: the general proprieties of the two-body problem, description of the Kepler parameters, models to compute the position and velocity of a satellite, two techniques to correct the orbit, the Kalman filter and the classical least-squares estimator, and in which format can the parameters be received. Due to the second part of the thesis, a study was done about Earth reentry. The study focused on the procedure and the parameters that vary in orbit that lead to the satellite decay until the time of reentry. Moreover, the middleware technology currently used to enable communications between satellites and ground station was also presented. The description was focused in GPS Systems, although other technologies that implement solutions for devices and software were also presented. Throughout this document references will be made to this chapter and to the concepts and systems presented. Therefore this chapter can be perceived as the underlying basis for the work developed, linking it to the advancements in the described fields. Chapter 4

Orbit and Location of De-orbiting Parameters Determination

This chapter aims to model and present parameters and different solution methods to determinate the orbit of a satellite in space. In the same line of the literature review, it exposes multiple solutions adapted to the parameters and objectives of the case study. The solution methods considered try to explore the potential of these formulations in order to conclude the best compromise for the prediction of the orbit.

4.1 Problem Description

The problem concerns the determination of the path traversed by the VORSat satellite. All strategic decisions are already taken and known in advance, such as type of orbit, height that will be achieved. The mass of the satellite is negligible comparable with the mass of Earth and the parameters that compose the ephemeris of the satellite change. The problem is therefore reduced to a tactical level and four simultaneous decisions are in- volved: (1) How does each of the parameters vary the orbit?; (2) Which estimation method minimizes the error between observations?; (3) What is the spacing between observations necessary to accurately determine the orbits?; (4) By changing the drag term variable of the TLE, will be possible to determine the re-entry point? The objective of the problem is to minimize the operation of the microcontroller, in order to minimize energy consumption, converging the error associated with the iteration of observations, and predicted the orbit decay, taking the following characteristics into consideration:

• The precision with which the orbit is obtained;

• The robustness of convergence of the algorithm given unknown initial parameters;

39 40 Orbit and Location of De-orbiting Parameters Determination

4.1.1 Case study Parameters

As been shown, a total of six independent parameters are required to define the motion of a satellite in its orbit. Two of these orbital elements (a and e) describe the form of the orbit, the element Ma defines the position along the orbit and the three others, Ω, i and ω finally define the orientation of the orbit in space. Given the position and velocity vector, it is always possible to uniquely determinate exactly one set of orbital elements that corresponds the place that the satellite is on the ellipse at a particular time. Part of the answer is already evident from the solution of the two body problem presented on ??. In the solar system, the inclination of the orbit of a planet is ecliptic — which is the plane containing Earth’s orbital path. First of all to compute the inclination we must have the orbital momentum vector, h, and its modulus can be obtained from the position and velocity. Hence the inclination is given by

h i = acos( z ) (4.1) |h|

By convention, inclination is a number between 0 and 180 degrees. The longitude of the ascending node, also known as Right Ascension of the Ascending Node, represents the angle from a reference direction, called the origin of longitude, to the direction of the ascending node, measured in a reference plane. Is a number in the range 0 to 360 degrees, and it follows that

n = kh = [−hy,hx,0] (4.2) n Ω = acos( x )(n > 0) (4.3) |n| y n Ω = 2πacos( x )(n < 0) (4.4) |n| y

Here, n is a vector pointing towards the ascending node. The reference plane is assumed to be the xy-plane, and the origin of longitude is taken to be the positive x-axis. k is the unit vector (0, 0, 1), which is the normal vector to the xy reference plane. For non-inclined orbits (with inclination equal to zero), longitude is undeﬁned. For computation it is then, by convention, set equal to zero. That is, the ascending node is placed in the reference direction, which is equivalent to letting n point towards the positive x-axis. The eccentricity may take the following values:

• Circular orbit: e=0

• Elliptic orbit:0

• Parabolic trajectory: e=1 4.1 Problem Description 41

• Hyperbolic trajectory: e>1

In the case of gravitational force, it is possible to calculate the eccentricity of an orbital through:

s 2εh2 e = 1 + ( ) (4.5) µ2

where ε, is the speciﬁc orbital energy (total energy divided by the reduced mass), µ the standard gravitational parameter based on the total mass. The argument of periapsis (also called argument of perifocus or argument of pericenter) is the angle between the orbit’s periapsis (the point of closest approach to the central point) and the orbit’s ascending node (the point where the body crosses the plane of reference from South to North). The angle is measured in the orbital plane and in the direction of motion. By convention, ARGP is an angle between 0 and 360 degrees and can be compute as:

n.e ω = arccos( ) (4.6) |n||e|

But if

ez < 0 ⇒ ω = 2π − ω (4.7)

where, n is the vector pointing towards the ascending node (i.e. the z-component of n is zero) and e is the eccentricity vector (the vector pointing towards the periapsis). When ω = 0, the perigee occurs at the same place as the ascending node. That means that the satellite would be closest to earth just as it rises up over the equator. When ω = 180 degrees, apogee would occur at the same place as the ascending node. That means that the satellite would be farthest from earth just as it rises up over the equator. In the case of equatorial orbits, though the argument is strictly undeﬁned. Finally, relating position and time for a body moving in an orbit we have the mean anomaly. The mean anomaly can be compute through the mean motion. The mean motion give us the measurement of how fast a satellite progresses around its elliptical orbit (Typically speciﬁed in revolutions per day). Unless the orbit is circular, the mean motion is only an average value, and does not represent the instantaneous angular rate.

rGMm mo = (4.8) a3 rGMm Ma = mo ∗t = ∗t (4.9) a3 42 Orbit and Location of De-orbiting Parameters Determination

where:

• G is the gravitational constant

• M is the mass of Earth and m are the mass of the orbiting body (it can be despised)

• a is semi-major axis.

In celestial mechanics, the mean anomaly is a parameter based on the fact that equal areas are swept at the focus in equal intervals of time. It increases uniformly from 0 to 2π radians during each orbit. However, it is not an angle. Due to Kepler’s second law, the mean anomaly is proportional to the area swept by the focus-to-body line since the last periapsis.

4.1.2 Kalman Filter Algorithm

So far only linear systems have been considered. But in practice the dynamic or the observation model can be nonlinear. One approach to the Kalman filter for such nonlinear problems is the so called Extended Kalman filter, which was discovered by Stanley F. Schmidt. This version of Kalman filter linearises about the current estimated state. Thus the system must be represented by continuously differentiable functions. One disadvantage of this problem is that the implications of its members individually are random and unpredictable drops to linear systems. These systems evolve in time domain with an unbalanced and aperiodic behaviour, where the future state is extremely dependent on its current state, and can be changed radically from small changes in the present.

4.1.2.1 Prediction

As stated before, the ﬁrst step of the Kalman Filter is the prediction. The predict state, is calculated by neglecting the dynamic noise and solving the differential equations that describes the dynamic model. We have a choice of elements to use in the state. A choice of position and velocity vectors is sometimes envisioned, however the parameters all change rapidly (over the course of a revolution), and this is undesirable for a differential correction process. Keplerian orbital elements (a, e, i, ω, Ω, m) can be used, but there can be singularities with some types of orbits. Notice the use of the semi-major axis instead of the mean motion. This can make a change with some implementations with respect to numerical precision.

xk+1 = Adxk + Bkuk (4.10)

A is called the state transition matrix, which transforms any initial state xk to its corresponding state xk+1 at time t. In nonlinear case A, is a function of the state to be estimate. So the predicted state is calculate by solving differentials equations; The matrix B is the control-input model which is applied to the control vector and we assumed that it equals to zero. 4.1 Problem Description 43

The a priori estimate error covariance, P−, is obtained with the law of error propagation

− T Pk = E(ekek ) (4.11)

and the a posteriori estimate error covariance is

− T Pk+1 = E(ek+1ek+1) (4.12)

− T T Pk+1 = E(ek+1ek+1) = E[(Adek − Gd µk)(Adek − Gd µk) ] (4.13) T T T T T T T T = E[Adekek Ad − Adekµk Gd − Gd µkek) Ad + Gd µkµk Gd ] (4.14)

In the more generalized form, where also the covariance matrix of the noise Q is the function of time, the covariance matrix is

T T T T = AdE(ekek )Ad + GdIGd = AdPkAd + Qd (4.15)

4.1.2.2 Correction

Then, moving to the second step, the correction step, the predicted state vectorx ˆ(k+1) is improved with observations made at the epoch k + 1,

y = Hx + Rη (4.16)

Like in the state transition matrix, H, the observation matrix will be computed through solving differentials equations and higher order terms are neglected; R will be the error associated with each term of y, in our case, position and velocity. Futher on, the a posteriori state has the form

xˆk+1 = xˆk + ∆xˆk (4.17)

with convariance matrix

+ − Pk+1 = Pk + ∆Pk (4.18) 44 Orbit and Location of De-orbiting Parameters Determination

As said before, the Kalman ﬁlter is an optimal ﬁlter, this means that the state variances in the state covariance matrix P+ are minimized. As P− is already known from the prediction step it follows that

+ (+) (+)T Pk+1 = E(ek+1ek+1 ) (4.19) (+) (−) (−) ek+1 = xk+1 + kk+1[H(xk+1 − xk+1 + Fη+1] − xk+1 (4.20) (−) T T = (I − kK+1H)PK+1(I − kK+1H) + kK+1FIFT kK+1 (4.21)

where,

FIFT = R (4.22)

Using a mathematical formula of Joseph, due applying the formula for the optimal gain the error covariance is numerically unstable we get

(+) (−) T T Pk+1 = (I − Kk+1H)Pk+1(I − Kk+1H) + Kk+1RT Kk+1 (4.23)

with

∗ (−) T (−) T −1 Kk+1 = Pk+1H (HPk+1H + R) (4.24)

K is know as the gain matrix. The difference (y − yˆ) is called the Innovation ( or residual). − − It reﬂects the discrepancy between the predite measurementy ˆ = Hxˆk+1 and the actual measurement. Finally the corrected state is obtained by

xˆk+1 = xˆk + Kk(y − yˆ) (4.25)

4.1.3 Least-Squares Estimation Algorithm

The Least Squares method is a mathematical optimization technique which seeks to determinate the value for a vector variable trying to minimize the sum of the squares of the differences between the value estimated and observed data (such differences are called residuals). This data is a linear function of the vector variable. For a non-linear function, the basis of the method is to approximate the model by a linear one and to reﬁne the parameters by successive iterations. The trajectory of a satellite can be described by solving the equation 4.1 Problem Description 45

y = f (x) ≈ f0 + Ax (4.26)

under this mathematical formulation, y is the position and velocity vector of the satellite at a number of discrete instants

  px   py     p   z  y =   (4.27) vx    v   y  vz,

the vector x are the TLE parameters that is intended to optimize.

  x1   x2    x =  ..  (4.28)      ..  xN

and f0, the intial value of y at epoch t0. In practice, there are always error measurements due to the disturbance in the environment and an uncertainty in the knowledge of the orbit of the satellite. Therefore, assuming n as the measurement noise, with zero-mean white Gaussian with the standard deviation of σ, one gets

y = yˆ+ n (4.29) = f o + A(x + ∆x) (4.30)

The least-squares orbit determination problem may now be deﬁned as ﬁnding the next state, that minimizes the loss function

N H = ∑ | f (xˆ) − f (xˆ+ ∆x)|2 (4.31) i=1

In order to ﬁnd the minimum of the function y, it is required to equalize the partial derivative to zero ( 5y = 0) 46 Orbit and Location of De-orbiting Parameters Determination

H = 5y = 0 (4.32)

Furthemore,

h ∂y ∂y ∂y i Hi = (4.33) ∂∆x10 ∂∆x20 ∂∆x..N0 ∆x10=∆x20=∆x..0=∆xN=0

gives the partial derivatives of the observation measurements with respect to the state vector at the reference epoch t0. Linearizing all quantities around a reference state, the residual vector is approximately given by

r = Hx (4.34)

where r is the difference between the real position and the computed one. The orbit determination problem is now reduced to the linear least-squares problem of ﬁnding x such that

x = (HT H)−1HT r (4.35)

Due to linearization, the optimal value of x used on the next iteration, sometimes is not the optimum value of the linearization, possibly leading the algorithm not to converge. In order to increase the probability of convergence, a correction factor is used:

x = x − 0.5r (4.36)

The Eq. (4.36) turns the convergence slower but makes it more robust. The process is repeated iteratively since Eq. (4.33) to ensure an optimum convergence.

4.1.4 Methods Assessment

As mentioned in the previous chapter, the common estimation techniques employed for purposes of determining the orbit are closely related and a smooth transition is possible from the method of least squares for the various forms of Kalman ﬁlter. Each estimator has inherent advantages and disadvantages and there must be a trade-off to select the most appropriate estimation method for a particular application: 4.2 Prediction of the Decay 47

• Measurements processing and state correction : The classical least squares method calculates the epoch state estimate after processing the complete set of observation. If improved epoch state estimates are required after each measurement, there is a need to use another type of least squares. The Kalman ﬁlter, on the other hand, processes a single scalar or vector measurement at a time and yields sequential state estimates at the measurement time.

• Treatment of non-linearities : If the system is nonlinear, there will be a nonlinear relationship between the epoch state vector and the measurements, therefore, it will be necessary multiple iterations in the least-squares method to compute a state estimate that actually minimizes the loss function. Using the Extended Kalman ﬁlter, once the solution is changed with each observation, the number of iteration will be reduced.

• Numerical stability : Both the techniques may be subject to numerical problems in the case of bad computation observability matrix, the state transition matrix, or, covariance matrix. This problem can be minimized in both methods, using numerically stable algorithms em- ploying different types of matrix factorizations.

• Divergence : Due to the factors mentioned in the previous point, a bad initial state estimate and high non-linearities, both methods are subject to divergence. And the divergence of the state estimate from true solutions tends to aggravate in the use of the Kalman filter when the covariance becomes small and the filter gets insensitive to new observations. However, the Kalman filter, incorporates process noise into the estimation process, so these divergences are avoid, making more realistic predictions of covariance in the presence of unmodeled accelerations.

After this brief analysis, the method to be implemented will be the Kalman ﬁlter since we are dealing with a dynamic system.

4.2 Prediction of the Decay

During the life of a satellite, it passes through three major phases: the launch, orbiting and re-entry into Earth’s atmosphere. The phases framed in this thesis are the second and third. Previously, it has been approached the second phase, therefore in this subchapter it will discuss the re-entry. In the phase of reentry into Earth’s atmosphere, the satellite suffer the action of some forces, including gravity and drag. The force of gravity naturally pulls the object to Earth. However, gravity alone only would cause the satellite falls dangerously fast. Fortunately, the Earth’s atmosphere contains few air molecules that are present at the altitudes. The interaction of this molecules with the drag force that such a satellite experiences creates friction. This friction causes the satellite to experience drag, or air resistance, which slows the object down to a safer entry speed. For SGP4 users, it is possible to determine atmospheric drag, by a static method to approximate the effect of atmospheric drag on satellite’s motion. It models the density of the Earth’s upper 48 Orbit and Location of De-orbiting Parameters Determination atmosphere using the fourth power of the orbital altitude. The drag coefficient (B∗) provided in TLE data is determined empirically as a continually updated fit to the changes in revolution per day observed over the long term, with the pseudo-drag term being ignored. Thus, the drag term is the first time derivative of the mean motion (which has the dimension revolutions per day) and is derived by the following equation:

1 C A B∗ = ρ D (4.37) 2 m

The mean motion and the semi-major axis are ﬁrst recovered from its altered form and the drag effect is obtained from the SGP4 drag term. The secular effects of atmospheric drag and gravitation are included through the equations

2 2 2 4 3k2(−1 + 3θ ) 3k2 (13 − 78θ + 137θ ) 00 MDF = M0 + [1 + 002 3 + 004 7 ]n0(t −t0) (4.38) 2a0 β0 16a0 β0 2 2 2 4 2 4 3k2(1 − 5θ ) 3k2 (7 − 114θ + 395θ ) 5k4(3 − 36θ + 49θ ) 00 ωDF = w0 + [− 002 4 + 004 8 + 004 8 ]n0(t −t0) 2a0 β0 16a0 β0 4a0 β0 (4.39) 2 3 2 3k2θ 3k2 (4θ + 19θ ) 5k4θ(3 − 7θ ) 00 ΩDF = Ω0 + [− 002 4 + 004 8 + 004 8 ]n0(t −t0) (4.40) a0 β0 2a0 β0 2a0 β0

From the values calculated of the mean anomaly, argument of perigee and longitude of the ascending node, altered by variations of the atmosphere, are compute the other parameters of the

TLE, where (t −t0) is time since epoch.

∗ δω = B C3(cosw0)(t −t0) (4.41)

2 4 ∗ 4 aE 3 3 δM = − (q0 − s) B ξ [(1 + ηcosMDF ) − (1 + ηcosM0) ] (4.42) 3 e0η

In this formula q0 − s are parametrs for the SGP4 density function. 4.3 Summary 49

Mp = MDF + δω + δM (4.43)

ω = ωDF − δω − δM (4.44) 00 21 n0k2θ 2 Ω = ΩDF − 002 2 C1(t −t0) (4.45) 2 a0 β0 ∗ ∗ e = e0 − B C4(t −t0) − B C5(sinMp − sinM0) (4.46) 00 2 3 4 2 a = a0[1 −C1(t −t0) − D2(t −t0) − D3(t −t0) − D4(t −t0) ] (4.47) q β = (1 − e2) (4.48) µ n = 3 (4.49) a 2

In the previous formulas, the constants C, D and k2 represents parameters of the atmosphere.

3 IL = M + ω + Ω + n00[ C (t −t )2 + (D + 2C2)(t −t )3 (4.50) p 0 2 1 0 2 1 0 1 + (3D + 12C D + 10C3)(t −t )4 (4.51) 4 3 1 2 1 0 1 + (3D + 12C D + 6D2 + 30C2D + 15C4)(t −t )5] (4.52) 5 4 1 3 2 1 2 1 0

It should be noted that when epoch perigee is less than 220 km, the equations containing parameters of C above C1 are truncated and the terms involving C5 δω, δM are discarded [24].

4.3 Summary

This chapter intent was to identify and describe the requirements of the case study problem. The different solution methods presented in the literature review were adapted and afterwards developed with a set of instances available in the literature in order to evaluate the best compromise for the case study. Two different approaches were considered: The estimation using the Kalman filter and the least-squares. Both techniques are applied to a discrete system and therefore were evaluated according to this requirement. The least squares estimator is more robust and easier to implement than Kalman filter due that all data point are processed using a common reference trajectory, this facilitates the handling of bad data point. The Extended Kalman Filter requires a balancing between a priori covariance, measurement weighting and process noise to allow a rejection of bad data points. The formulation with most potential of achieving better results is therefore the Extend Kalman filter method due to the incorporating process noise, which provides more realist estimates of the error near Earth. Similar conclusions have been obtained by Oliver Gill, [?]. 50 Orbit and Location of De-orbiting Parameters Determination

A second part of this chapter was directed as will be developed to predict the point of the satellite reentry. The parameter B∗, that comes from reading the TLE, is directly related to the drag term. Since the SGP4 model assumes a static atmospheric model and a deﬁnition of perigee height that does not depend on the actual argument of perigee, whenever the real drag term changes, the B∗ term changes by the same factor. Chapter 5

Implementation and Results

This chapter aims to present optimized orbits for satellites in orbit. The implementation is described with some illustrative sections of the Matlab source code. Four plans are presented, corresponding to the results of which orbit is better described applying the method with the Kalman ﬁlter or just with position and velocity, the variation of results with observations based on positions or observations based in positions and velocity, the space between observations to achieve an optimize orbit and ﬁnally, from unknown input parameters it possible that the error between observations eventually converge . A second part of this chapter is based on orbit decay. A comparison is made in order to conclude which is the best implementation of this problem.

5.1 Orbit Prediction

Within the scope of this study thesis, two satellites were used, International Space Station - Zarya (ISS) and PoSAT, to be possible to obtain results of the developed algorithm. The choice fell upon these two satellites due to the fact that of both of them are in LEO orbit, both have a stable orbit and still remain in orbit after so many years. ISS ([7]) is a joint project of ﬁve space agencies (the National Aeronautics and Space Administration (United States), the Russian Federal Space Agency (Russian Federation), the Japan Aerospace Exploration Agency (Japan), the Canadian Space Agency (Canada) and the European Space Agency (Europe)), has been in orbit since 1998 and maintains an altitude of 350km, PoSAT, as already mentioned was the ﬁrst satellite developed by Portugueses, and has been in orbit since 1993 and is 380km from the Earth. Typically, satellites in LEO orbits lasting just over two months, however these two satellites have something unusual. Slight though the number of atmospheric molecules is at the ISS orbital altitudes, it’s still enough to cause drag on the ISS and pulls it toward Earth. So, to keep ISS orbiting around 350km, the altitude required for the proper operation of everything from labs and living quarters to logistics and launch vehicles, ISS gets periodically boosts (propulsion). What

51 52 Implementation and Results keeps PoSAT-1 in orbit its it size. The larger the dimensions of the satellite the longer it keeps in orbit because the drag depends on the size

1 F = C ρAv2 (5.1) 2 x

since the LEO satellites keep an orbital velocity of 7 km/s, we consider that the v ≈ constant,

F a = (5.2) m 1 A = C ρv2 (5.3) 2 x m

where, A (area) is proportional to L2 and m (mass) to L3,

1 ρ 1 = C v2 (5.4) 2 x d L

5.1.1 Position and Velocity

As stated on State o f Art, to determine the position and velocity of a satellite in orbit was used the algorithm SGP4. Space propagation models use current state information of a satellite to predict a future state of the satellite. The first step of the program is to fill in the inputs requirements, reading the required data (TLEs) from text file that the user has to prepare before using the program.

5.1.1.1 Input Requirements

In this way the following data is read in:

Table 5.1: Input Data SPG4 Algorithm

the initial time (when the TLE was generated) startmfe the ﬁnal time (duration of the simulation) stopmfe the time steps of the observations deltamin TLE with the format presented in Appendix B Name.TLE

5.1.1.2 SGP4 algorithm

After the algorithm reads the text ﬁle .TLE it executes the function twoline2rv. It converts the two line element set character string data to variables and initializes the sgp4 variables, the main SGP4 routine. Several intermediate variables and quantities are determined, as well as, the time of element set. The reference time frame is universal time coordinate (UTC). After the initial variables, the algorithm enters into a loop to perform the propagation 5.1 Orbit Prediction 53

Figure 5.1: Loop to perform the propagation

At each iteration of time that passes until the end of the cycle imposed by the user, lunar-solar periodics re added, extra mean quantities are calculated, an update for short period periodics and orientation vectors are computed to calculate the position and velocity (in km and km/sec) by function sgp4.m

Figure 5.2: Calculated position and velocity ECI

This function is called by eci2ecef, who received the ECI position and velocity vectors and converted to ECEF, which returns to the initial cycle, where the results are stored in a vector v.

Figure 5.3: Convert position and velocity coordinates 54 Implementation and Results

We cannot, however, talk about position and velocity without discussing the coordinate system that these values are measured relative to. The ECI coordinate system is typically defined as a Cartesian coordinate system, where the coordinates (position) are defined as the distance from the origin along the three orthogonal (mutually perpendicular) axes. The z axis runs along the Earth’s rotational axis pointing North, the x axis points in the direction of the vernal equinox (more on this in a moment), and the y axis completes the right-handed orthogonal system. But this coordinate system does not take into account the rotation of the earth, therefore its necessary to transform coordinates to ECEF. ECEF is represented by x,y,z coordinate. The z-axis is pointing towards the north but it does not coincide exactly with the instantaneous Earth rotational axis. The slight "wobbling" of the rotational axis is known as polar motion. The x-axis intersects the sphere of the earth at 0’ latitude (Equator) and 0’ longitude (Greenwich). This means that ECEF rotates with the earth and therefore, coordinates of a point fixed on the surface of the earth do not change. Futhermore, is possible to view attachments, how to convert from ECI to ECEF coordinates, as well as,how to compute the greenwich sidereal time from UTC time. For a LEO satellite the revolution time is approximately T=90 minutes. Only one revolution is observed. Every 10 minutes the distance from origin to the satellite is observed. The following figure (5.4 and 5.5) shows the calculated trajectory computes only through position and velocity. As we can see on figures 5.4 and 5.5 the orbit described by ISS, once is orbiting near Earth, is circular and PoSAT describes an elliptic orbit.

5.1.2 Kalman Filter

Through the study developed in chapter 4, the best approach for the improvement of an orbit was the use of the Extended Kalman Filter. This Kalman ﬁlter linearises about the current estimated state. Thus the system must be represented by continuously differentiable functions.

5.1.2.1 Input Parameters

Table 5.2: Input Data Kalman Filter Algorithm

the initial time (when the TLE was generated) startmfe the ﬁnal time (duration of the simulation) stopmfe the time steps of the observations deltamin initial value of the state vector x inital value of the convariance matrix P covariance matrix of the dynamic noise Q dynamic noise G transitions matrix A observations matrix of the dynamic noite etar

Due to their dependency on variables of the state vector, much of this data is symbolic expres- sions. 5.1 Orbit Prediction 55

Figure 5.4: ISS Orbit computed with position and velocity

Figure 5.5: PoSAT Orbit computed with position and velocity 56 Implementation and Results

5.1.2.2 Prediction

Each row in the solution array x correspond to a parameter of kepler elements.

    mok+1 mok+1 1 0 0 0 0 0 0  no   no   k+1    k+1    0 1 0 0 0 0 0   ndotk+1    ndotk+1    0 0 1 0 0 0 0  nodeo   nodeo   k+1   k+1   = 0 0 0 1 0 0 0  (5.5)  eccok+1    eccok+1    0 0 0 0 1 0 0   inclo    inclo   k+1    k+1    0 0 0 0 0 1 0  argpok+1 argpok+1   0 0 0 0 0 0 1   bstark+1 bstark+1   √  q dt3  µmo smo dt sno 3 0 0 0 0 0    √  µno   0 sno dt 0 0 0 0 0     µ    ndot   0 0 snodeo 0 0 0 0    µnodeo + 0 0 0 secco 0 0 0   (5.6)   µ    ecco   0 0 0 0 sinclo 0 0     µinclo   0 0 0 0 0 sargpo 0    µargpo 0 0 0 0 0 0 sbstar   µbstar

where equation (5.5) is relative to the transition matrix, A, and (5.6) to the state vector ﬁrst derivative, G.

1. A Priori State

As described in chapter 4, the continuous state from the time span tk − 1 to t is obtained by solving the differential equation.

x = A × x; (5.7)

2. Covariance of the Priori State

The covariance Q of the dynamic noise may depend on the state variable as weel, so it must be calculated at the predicted state xd−, before the priori Covariance matrix can be computed.

Q = GGT ; (5.8)

P = APA0 + Q; (5.9) 5.1 Orbit Prediction 57

5.1.2.3 Correction

Before the gain matrix can be calculated, the observation matrix must be determined. In the linear case this is just the Jacobian matrix of the observation equation h, but in nonlinear case it still depends on the state variables. Therefore within the following iteration the state, at which the observation matrix H is evaluated, is quickly improved with adjustment that minimizes the improvements of the observations. So the H matrix that is determinate at this improved state is closer to the real observations.

Figure 5.6: Iteration of the observation matrix

1. Gain Matrix And then the gain is computed with the observation matrix, which emanates from the ﬁgure above.

0 0 K = PH (HPH + R)−1 (5.10) 58 Implementation and Results

where R is the error associated to each observation parameter.

R = etar × etar; (5.11)

and etar is the observation noise

  (10/4.0075e + 00) ∗ 360 0 0 0 0 0    0 (10/4.0075e + 00) ∗ 360 0 0 0 0     0 0 15/1000 0 0 0    etar =    0 0 0 0.01/1000 0 0     0 0 0 0 0.01/1000 0    0 0 0 0 0 0.015/1000 (5.12)

2. Innovation Matrix

And ﬁnally to predict the corrected state its necessary to calculate the Innovation, which is the difference of what was observed and what was anticipated

Figure 5.7: Innovation

3. Posteriori State and Covariance

The posteriori state and its covariance matrix can be computed with the corresponding formulas,

Figure 5.8: Posteriori state and its covariance matrix

The next step will be the initial state x, until the cycle time, imposed by the user, ﬁnishes. 5.1 Orbit Prediction 59

Figure 5.9: Application of Kalman ﬁlter with PoSAT Figure 5.10: Application of Kalman ﬁlter with satellite - Positions errors PoSAT satellite - Velocity errors

Figure 5.11: Application of Kalman ﬁlter with ISS Figure 5.12: Application of Kalman ﬁlter with ISS satellite - Positions errors satellite - Velocity errors 60 Implementation and Results

Figure 5.13: PoSAT error positions with random ini- Figure 5.14: PoSAT error velocity with random initial inputs tial inputs

The next figures, 5.9, 5.10, 5.11 and 5.12, is possible to see the application of Kalman filter with duration of 2 days with step 30 minutes. The purpose of applying the Kalman filter is the error associated with observations converge to an acceptable value. In the figures 5.9, 5.10, 5.11 and 5.12, is shown the error associated with the position and velocity measurements for each satellite in coordinates NED (north, east, down), in meters.It is also observed, that the error relative to positions eventually converge to approximate value of 40 meters on both satellites and the velocity error converge to 5e−3m/s and 4e−5m/s in PoSAT and ISS satellite, respectively. It is possible to consult Appendix B to obtain the numerical values of the previous simulation and the new values of the TLEs obtained by Kalman Filter.

5.2 Test Cases

In order to test the robustness of convergence of the algorithm, three experiments were made. As seen before, all ﬁgures will be described as the error associated with measurements at each observation NED (north, east, down) in meters.

5.2.1 Unkown Inputs Parameters Values

As the initial input values of the satellite will not be known, one of the test fells on study how the Kalman filter behave with random initial values. The tests were made with duration of 2 days with step 30 minutes. As expected, the Kalman filter converged after 5 hours in PoSAT and 2,4 hours in ISS satellite. The chart from where this value was taken can be found in Appendix B . Comparing the values of the convergence of these figures (5.17, 5.22, 5.19 and 5.20) with the previous test (5.9, 5.10, 5.11 and 5.12), we found that both errors converge to the values given above. 5.2 Test Cases 61

Figure 5.15: ISS error positions with random initial Figure 5.16: ISS error velocity with random initial inputs inputs

Figure 5.17: Value that the position error converges Figure 5.18: Value that the velocity error converges with random initial inputs - PoSAT with random initial inputs - PoSAT

Figure 5.19: Value that the position error converges Figure 5.20: Value that the velocity error converges with random initial inputs - ISS with random initial inputs - ISS 62 Implementation and Results

Figure 5.21: Position accuracy of Kalman ﬁlter with Figure 5.22: Velocity accuracy of Kalman ﬁlter with simulations steps of 10 minutes - PoSAT simulations steps of 10 minutes - PoSAT

Figure 5.23: Position accuracy of Kalman ﬁlter with Figure 5.24: Velocity accuracy of Kalman ﬁlter with simulations steps of 10 minutes - ISS simulations steps of 10 minutes - ISS

5.2.2 Accuracy of Kalman ﬁlter

This study was made to determine the distance between observations so that the filter works correctly. That is, how many observations are needed to avoid loss of information while not under- charging the program? For this purpose it were made simulations with steps of 10, 45 and 60 minutes. With a step of 10 minutes, the error converges to a value lower than in the case of the step to 30 minutes (40m to 20m), however, the program becomes slower. The variation of values of x (TLE) is little, which makes the information too excessive. In case of step 45 minutes, there is a delay in convergence and a slight increase in the final error, especially in the case of satellite ISS (see figure 5.27). With increasing step for 60 minutes, more information is lost which causes the increase of the error, and the filter converges slowly as expected. Observing the four cases, it is concluded that the best step for a duration of 2 days is 30 minutes. 5.2 Test Cases 63

Figure 5.25: Position accuracy of Kalman ﬁlter with Figure 5.26: Velocity accuracy of Kalman ﬁlter with simulations steps of 45 minutes - PoSAT simulations steps of 45 minutes - PoSAT

Figure 5.27: Position accuracy of Kalman ﬁlter with Figure 5.28: Velocity accuracy of Kalman ﬁlter with simulations steps of 45 minutes - ISS simulations steps of 45 minutes - ISS

Figure 5.29: Position accuracy of Kalman ﬁlter with Figure 5.30: Velocity accuracy of Kalman ﬁlter with simulations steps of 60 minutes - PoSAT simulations steps of 60 minutes - PoSAT 64 Implementation and Results

Figure 5.31: Position accuracy of Kalman ﬁlter with Figure 5.32: Velocity accuracy of Kalman ﬁlter with simulations steps of 60 minutes - ISS simulations steps of 60 minutes - ISS

Figure 5.33: Kalman ﬁlter performance with obser- Figure 5.34: Kalman ﬁlter performance with observations based only with positions - PoSAT vations based only with positions - ISS

The change in the value of x can be found in Appendix C. With numerical values of x is possible to obtain the precise moment that the input converge to the right values.

5.2.3 Observations only with positions

As seen in the previous section the observations are calculated using the position and velocity of the satellite. Therefore, the aim is to study what happens if the accuracy of observations decreases, which are based only in the positions of the satellite. As expected, with a decrease in accuracy in the observation matrix, the ﬁlter takes longer to converge and converges to a very high error (2000 meters) in both cases.

5.3 De-orbiting Location Prediction

Earlier in the chapter 3 and 4, it was demonstrated how the drag term affects the trajectory. In this chapter, we intend to study through a series of Iridium-79 satellite TLE’s, between which positions 5.4 Summary 65 are worth acting on the drag term, for the satellite re-enter into the point we decide. The Iridium 79 satellite is part of a large group of satellites providing voice and data coverage to satellite phones, pagers and integrated transceivers over Earth’s entire surface. The choice fell on this satellite because it was in a LEO orbit in 2000 when it reentered in Earth atmosphere. Iridium-79 was launched on 8 September 1998, reaching an altitude approximately of 781 km. Having access to a TLE historical at the position corresponding to the altitude between 250 and 150 km, is possible to study the variation of drag term, obtaining the uncertainty (error) on the altitude at which the satellite is. As an output we will have the Figure 5.35.

Figure 5.35: Point where it should initiate the control of the drag-term

The ideal would be an error as low as possible, but on the other hand, the altitude can not get too low values because the altitudes below 120 km the satellite goes into free fall.

5.4 Summary

After in the previous chapter, having been made the choice of the estimator to use, Kalman Filter, this chapter will explain how the algorithm is developed and the results obtained. Within the scope of this study thesis, it was used two satellites, International Space Station (ISS) and PoSAT, to be possible to obtain results of the developed algorithm. This chapter began with a demonstration of the algorithm for calculating the positions and speed, SGP4, with a duration of one orbit (90 minutes) and as output we get the orbit of each satellite. After checking that the positions were correct through the program Orbitron, it was done the study of the robustness of the Kalman ﬁlter and the prediction of the location of de-orbiting. In order to test the robustness of convergence of the algorithm, three experiments were made: (1) 66 Implementation and Results

Apply the Kalman Filter when we have unknown input parameters values; (2) Determine the distance between observations so that the filter works correctly; and (3) What happens when we apply the observations only with positions. From the first experiment it was possible to conclude that the filter converges rapidly when the inputs are unknown. By varying the steps of the observations, with 10 and 60 minutes, it was possible to conclude that it takes longer to converge in more than 30 minutes observations but with 10 minute observations the simulation converges earlier but takes longer, increasing battery consumption. On the third step, as expected, with the reduction of the precision of the observations (based only on position) the filter converged worse than before. Finally, as studied on chapter 4, with the variation of the parameter B∗ we achieved the reentry point of the satellite. However, for lack of a history of TLE the study was simply theoretical. Chapter 6

Conclusion and Future Work

This chapter’s intent is to infer on the proposed objectives. It will also open new paths for the work developed.

6.1 Main Results

As a validation support, the algorithm developed to predict orbits was subjected to tests by using two satellites: International Space Station (ISS) and PoSAT. The achieved results prove that the main ideas and concepts that lay behind the design of the system are really effective, providing a minimum error and converging as expected.

6.2 Work’s Assessment

When assessing the work carried out during the thesis, two major topics must be discussed: Orbit Calculation and Re-Entry Control of the VORSat Satellite. Regarding the orbit calculation, the following studies were approached: the general proprieties of the two-body problem, the description of the Kepler parameters, models to compute the position and velocity of a satellite, two techniques to correct the orbit, the Kalman filter and the classical least-squares estimator, and in which format can the parameters be received. In this thesis the aim was to decide which was the best estimator for the case study. Two different approaches were considered: Kalman filter and least-squares. Both techniques are applied to a discrete system and therefore they were evaluated according to this requirement. The least squares estimator is more robust and easier to implement than the Kalman filter due to the fact that all data points are processed using a common reference trajectory. This facilitates the handling of bad data points. The Extended Kalman Filter requires a balancing between a priori covariance, measurement weighting and process noise to allow a rejection of bad data points.

67 68 Conclusion and Future Work

Therefore, the implementation and the results are based on the Extend Kalman filter method, due to the incorporating process noise, which provides more realist estimates of the error near Earth. In order to test the convergence robustness of the algorithm, three experiments were made: (1) Apply the Kalman Filter when we have unknown input parameters values; (2) Determine the distance between observations so that the filter works correctly; and (3) Verify what happens when we apply the observations only with positions. All results were satisfactory since the filter converged quickly in all the studied cases. Due to the second part of the thesis, a study was done about Earth re-entry. The study focused on the procedure and the parameters that vary in orbit and that lead to the decay of the satellite until the time of re-entry. The parameter B∗, that comes from reading the TLE, is directly related to the drag term. Since the SGP4 model assumes a static atmospheric model and a definition of perigee height that does not depend on the actual argument of perigee, whenever the real drag term changes, the B∗ term changes by the same factor. By varying the drag term, in a long simulation, it was possible to obtain positions of de-orbiting. However, for lack of information has not been possible to obtain results.

6.3 Future Work

Due to the reduced existing time, not all objectives were applied in practice and theoretical research was also limited. In the following items, a set of changes and further analysis proposals are presented:

• Demonstrate in computational experiments the Least-Squares Estimation, stated in the literature review, to better assess which of the methods would achieve better results.

• The developed algorithm depends of the epoch of the satellite. The time was calculated in order to be viable only until by the year of 2056. Once this algorithm will be applied on VORSat, the release date of this CUBESat is planned for 2013 and was not designed to stay in orbit for years.

• One of the objectives of this MSc Thesis was the control of the location of the de-orbit but due to reduced time this objective was not studied properly due to lack of information.

• One of the most important phases of this project is the location of the landing. A future work may be coupler to the this program an algorithm to predict the landing.

Furthermore, the project, during its development phase, was faced with numerous and different adversities. Some of these were more severe than others. Amongst these are the following: small compiler bugs, the use of SGP4 algorithm that calculates the position and velocity can make the algorithm slow because they contain many variables not used in our case. Although the existence of a great amount of features and functions that would add value to this work, they could not be implemented in time for the project’s deadlines. Appendix A

Time and Coordinate Systems

Both time and reference systems have traditionally been based on the rotational and translational motion of the Earth. As the physical theories and technology evolved, more time and reference systems emerged towards ideally uniform atomic time scales and ideally non-rotating quasar-tied celestial reference frames. This appendix aims to present some systems relevant for the thesis and follows the chapter with the same name of [4].

A.1 Time

Time is traditionally measured in solar days. However, due to the eccentricity of the orbital motion of the Earth around the sun, solar days are longer than the period of Earth’s rotation, known as sidereal days. Because of the resulting seasonal variations of the Sun’s apparent motion, solar days are not well suited for time reckoning purposes. Different time scales overcome this fact, and two of them are relevant for this thesis: Universal Time UT and Sidereal Time.

A.1.1 Sidereal Time

Greenwich Mean Sidereal Time (GMST) is a direct measure of the Earth’s rotation through the angle between the mean vernal equinox and the Greenwich meridian. It is measured either in angular units or units of time with 360o corresponding to 24h. In SI seconds, the length of a sidereal time is about four minutes shorter than a 24h solar day. This time scale cannot the computed from other time scales because of the variations in the length of the day of several milliseconds.

A.1.2 Universal Time

Greenwich Mean Time (GMT) or Universal Time, uses the concept of mean Sun with the purpose of achieving a constant average length of the solar day of 24h. As a result, the length of one second is not constant because the actual mean length of a day depends on the rotation of the Earth and the apparent motion of the Sun.

69 70 Time and Coordinate Systems

Universal time is deﬁned as a function of sidereal time. UT1 is the principal form of universal time. For an arbitrary time of a particular day, UT1 is deﬁned as the instant at which GMST has the value:

2 3 GMST = 24110.54841+8640184.812866T0 +1.002737909350795UT1+0.093104T −0.0000062T (A.1) In this expression, the time argument T0 and T denotes the number of Julian centuries of Universal Time that have elapsed since 2000Jan. 1.5 UT1 at the beginning of the day (0h) and at that time of the day.

JD(0hUT1) − 2451545 T = (A.2) 0 36525 JD(UT1) − 2451545 T = (A.3) 36525

Figure A.1: Function of the Greenwich Sidereal Time

A.1.2.1 Julian Date

The Gregorian calendar, internationally accepted civil calendar, measures time in term of years, months and days, can not be suited in the astronomy community, due to the difﬁculty to compute the time difference between two dates or advancing a data by certain time increment. To overcome this problem, a continuous day count is often used, known as Julian Date. The Julian date (JD) is the interval of time in days and fractions of a day since January 1, 4713 BC Greenwich noon, Julian proleptic calendar. The astronomical computation is showed in the next ﬁgure: A.2 Coordinates 71

Figure A.2: Function of the JD given the year, month, day, and time

A.2 Coordinates

In this MSc Thesis a variety of coordinate frames are used. Begins calculating the position and velocity of the satellite in Earth-Centered Inertial (ECI) coordinate which in turn are converted to Earth-Centered, Earth-Fixed (ECEF) to take account of the Earth’s rotation and ultimately the evaluate of the positions are in Latitude, Longitude and heigh (LLH).

A.2.1 Geographic Coordinates

The Geographic Coordinates system differs from the Cartesian coordinates system, since the ﬁrst one is constructed to match the Earth’s surface and the origin of the latter is the center of Earth. In the Geographic Coordinates system the Earth consists in 360 divisions that intersect at the North Pole and the South Pole and parallel lines that circle the Earth and are parallel to the equator. The lines that intersect the poles are called lines of longitude or meridians. Those parallel to the equator are called lines of latitude or parallels. Medians are perpendicular to the equator. Every location on the Earth’s surface can be described in terms of its latitude and longitude, counting degrees north or south of the equator and east or west of Greenwich. Since the surface of the Earth is not a perfect ellipse it is need to make an adaptation of the ellipsoid to the Earth. That is, the latitude, longitude and heigh will be relative to a datum, an ellipsoid. In the ﬁgure (A.3) is shown how to convert cartesian coordinates to LLH.

A.2.2 Cartesian Coordinates

As mentioned previously, the Cartesian coordinates and geographical differ on the point of origin. Within the Cartesian coordinate system there are two coordinates: ECEF and ECI. Both have their point of origin at the center of the Earth, but ECI are called inertial frames in contrast to the Earth- centered, Earth-ﬁxed (ECEF) frames which rotate in inertial space in order to remain ﬁxed with respect to the surface of the Earth.

A.2.2.1 ECEF

ECEF stands for Earth-Centered, Earth-Fixed, and is a Cartesian coordinate system, and it represents positions as an X, Y, and Z coordinate. The point (0,0,0) is deﬁned as the center of mass of 72 Time and Coordinate Systems

Figure A.3: Converts ECEF into LLH

the earth, hence the name Earth-Centered. Its axes are aligned with the International Reference Pole (IRM) and International Reference Meridian (IRM) that are ﬁxed with respect to the surface of the Earth, hence the name Earth-Fixed.

The z-axis is pointing towards the north but it does not coincide exactly with the instantaneous Earth rotational axis. The slight "wobbling" of the rotational axis is known as polar motion. The x-axis intersects the sphere of the earth at 0o latitude (Equator) and 0o longitude (Greenwich). This means that ECEF rotates with the earth and therefore, coordinates of a point ﬁxed on the surface of the earth do not change. Conversion from a WGS84 Datum to ECEF can be used as an intermediate step in converting velocities to the North East Down coordinate system.

A.2.2.2 ECI

Earth-Centered Inertial (ECI) is the name given to a group of coordinate frames that moves with the center of mass of the Earth but is free of rotation. It is convenient to represent the positions and velocities of terrestrial objects in ECEF coordinates or with latitude, longitude, and altitude. However, for objects in space, the equations of motion that describe orbital motion are simpler in a non-rotating frame such as ECI. The ECI frame is also useful for specifying the direction toward celestial objects.

The intersection of the ecliptic and Earth’s equatorial plane can be used as a principal direction for ECI frames and is called the vernal equinox. The Sun lies in the direction of the vernal equinox around March 21. The fundamental plane for ECI frames is usually either the equatorial plane or the ecliptic. A.2 Coordinates 73

A.2.3 Datum

Previously, it was mentioned that all position were concerning to an datum. Being the "datum" an ellipsoid that translate positions indicated on their products to their real position on Earth. Common to all systems is the goal of establishing a global coordinate system that originates at the Earth’s center of mass and is closely aligned with the Greenwich meridian and adopted pole. A variety of global geodetic datums are in widespread use. The World Geodetic System used was the WGS84 that has been developed by the United States Department of Defense (DoD) and the Defence Mapping Agency (DMA) for use with the TRANSIT and GPS satellite navigation system. WGS84 constants are shown in ﬁgure (A.4)

Figure A.4: Geodetic WGS84

where mu is the earth gravitational parameter, radiusearthkm represents the radius of the earth in km, tumin is minutes in one time unit, xke the reciprocal of tumin and the constants j are the un-normalized zonal harmonic values. 74 Time and Coordinate Systems Appendix B

Futher Analysis of the Results

To complement the test results carried out in chapter 5, it is present the following tables and ﬁgures:

B.1 TLE ﬁle used in the simulations

POSAT (PO-28) 1 22829U 93061G 11176.62834215 -.00000081 00000-0 -14757-4 0 8054 2 22829 98.5155 122.3447 0009032 207.6785 152.3894 14.30302813925856 ISS (ZARYA) 1 25544U 98067A 11176.86731656 .00011327 00000-0 13182-3 0 6010 2 25544 51.6418 116.9474 0011156 288.5621 176.2010 15.62179605722263

B.2 Kalman Filter Performance

To obtain the ﬁgures 5.9, 5.10, 5.11 and 5.12, it was used an algorithm, ang2plan that converts a vector with coordinates given by Lattitude, Longitude and Altitude into planar coordinates (North, East and Depht), for the plane tangent to the WGS84 ellipsoide at point 0. The output of this function is the difference between real positions and the estimated one. The follow tables, B.1 and B.2, present the values obtained so we could have the error between measurements. In the following table, B.3 , it is shown the new values of TLE that comes as output of the Kalman ﬁlter. The tables B.4 and B.5 refers to the measurement of real and estimated positions of the satellite PoSAT. As the ISS satellite, the table B.6 shows the new Posat TLE values.

75 76 Futher Analysis of the Results

Figure B.1: How long it takes to converges with ran- Figure B.2: How long it takes to converges with random initial inputs - PoSAT dom initial inputs - ISS

B.3 Unkown Inputs Parameters Values

This section are presented the table B.7 and B.8 showing the new values of the TLE of PoSAT and ISS satellites, respectively. Also provided, is a new image, B.1, B.2 where it is possible to observe how long it takes the ﬁlter to converge to the correct values of x. B.3 Unkown Inputs Parameters Values 77

Table B.1: The real values of the ISS position

Latitude Longitude Altitude Vx Vy Vz 15h34m48s -31,77640314 92,62216045 388,0011734 -5,46332496 -3,014190019 -4,475729786 16h04m48s -16,22263017 -137,416611 380,8458153 2,15673262 -4,757773248 5,635986513 16h34m48s 51,33225144 -31,36584488 394,6932761 4,681818913 6,044189423 -0,68182657 17h04m48s -25,91986699 63,05577698 386,8745551 -5,815446661 0,236184385 -5,009292138 17h34m48s -22,48924528 -166,0712094 381,9864961 -0,864898515 -5,525722865 5,269038249 18h04m48s 51,76455905 -67,48941946 394,4419316 6,998174671 3,148474966 0,18529819 18h34m48s -19,78048922 34,13236863 385,8660331 -4,451384581 3,094673504 -5,438653169 19h04m48s -28,52484768 164,7419783 383,4546603 -3,924785661 -4,545855752 4,792324418 19h34m48s 50,71051502 -103,434962 393,7275431 7,587771863 -0,506755248 1,048670105 20h04m48s -13,4484453 5,661591052 385,0723589 -1,847354838 4,737342009 -5,754674734 20h34m48s -34,22796962 134,818549 385,153945 -6,0921138 -2,033762598 4,215841728 21h04m48s 48,28336054 -138,4580433 392,5494612 6,308408926 -3,946250023 1,89057214 21h34m48s -6,996001598 -22,52073359 384,6507268 1,165362847 4,711772525 -5,950790734 22h04m48s -39,4651164 103,9260461 386,8971455 -6,678602429 1,338669743 3,55181706 22h34m48s 44,70660815 -172,148285 391,016599 3,535727353 -6,261077628 2,693468079 23h04m48s -0,484421608 -50,55855425 384,6427284 3,645675307 3,066425559 -6,022851989 23h34m48s -44,06176014 71,82752795 388,5177312 -5,440894674 4,634219759 2,81411048 24h04m48s 40,23222931 155,5531004 389,1636445 0,054999427 -6,865246742 3,440808375 24h34m48s 6,030852276 -78,58594179 384,9767006 4,818621376 0,333957646 -5,969390811 01h04m48s -47,79721515 38,34959375 389,9894421 -2,65514162 6,918326418 2,018239635 01h34m48s 35,08686035 124,492677 387,1770936 -3,164180234 -5,659523211 4,116915689 02h04m48s 12,49371504 -106,7362787 385,7817086 4,303300583 -2,626333851 -5,791544225 02h34m48s -50,42049224 3,507934566 391,1082128 0,95775869 7,527543171 1,180784453 03h04m48s 29,45134564 94,4375345 385,136152 -5,244963117 -3,0555839 4,707686386 03h34m48s 18,84445106 -135,1515153 386,844264 2,22516167 -4,881079359 -5,493114915 04h04m48s -51,69796378 -32,34574397 391,8579392 4,434851921 6,260766847 0,319029415 04h34m48s 23,46501055 65,15311081 383,236641 -5,652993962 0,146415346 5,200657983 05h04m48s 25,01343433 -163,9908605 388,2162511 -0,818862085 -5,697408546 -5,08051689 05h34m48s -51,49503769 -68,5047061 392,1711889 6,84366773 3,438395872 -0,549109173 06h04m48s 17,23339876 36,42990708 381,601445 -4,3465958 2,991487306 5,58543937 06h34m48s 30,91376626 166,5590476 389,6355705 -3,932594563 -4,759577581 -4,562355594 07h04m48s -49,83487804 -104,1366636 392,1085261 7,546196063 -0,187024995 -1,405727854 07h34m48s 10,83838639 8,089225238 380,3420184 -1,784375685 4,647553802 5,85378361 08h04m48s 36,43321791 136,2828248 391,108474 -6,180407937 -2,264898462 -3,949615397 08h34m48s -46,88806976 -138,6207103 391,6176582 6,37878298 -3,643968971 -2,2330986 09h04m48s 4,347425488 -20,02348937 379,5273148 1,212165379 4,646394981 5,99997416 09h34m48s 41,42342976 104,9421173 392,4565195 -6,859374339 1,128794781 -3,255039657 10h04m48s -42,90032884 -171,7080682 390,7800072 3,696923225 -6,016819112 -3,014109433 10h34m48s -2,180725628 -48,04788882 379,2583855 3,701095566 3,01868874 6,020927053 11h04m48s 45,69191754 72,31503102 393,5091939 -5,705220857 4,489074646 -2,493139767 11h34m48s -38,1182871 156,5498711 389,7638269 0,271525651 -6,704461692 -3,7326107 12h04m48s -8,690561227 -76,11651785 379,4909618 4,897087407 0,283171786 5,916157878 12h34m48s 49,00333238 38,28605924 394,2211833 -2,972393163 6,874573069 -1,679730887 13h04m48s -32,74946389 125,9515411 388,609123 -2,932914346 -5,588210692 -4,37358923 13h34m48s -15,12486385 -104,3661268 380,2303783 4,402604636 -2,706252847 5,687894102 14h04m48s 51,10616472 2,990486278 394,5230743 0,634136081 7,604266338 -0,831500952 14h34m48s -26,95737902 96,25735971 387,4310338 -5,034670883 -3,060978435 -4,923735565 15h04m48s -21,41985848 -132,9443336 381,4023362 2,326210092 -5,011428342 5,340972693 15h34m48s 51,7970281 -33,0696942 394,3064583 4,156884325 6,453162627 0,033996606 16h04m48s -20,8641589 67,2414338 386,3524674 -5,486452018 0,090256033 -5,371596946 16h34m48s -27,49895818 -162,0222565 382,8936593 -0,746938292 -5,885704723 4,882822407 17h04m48s 50,99934162 -69,09989997 393,6270903 6,656129627 3,717733781 0,898881738 17h34m48s -14,5646014 38,70827979 385,5469676 -4,229456329 2,915253215 -5,707657051 18h04m48s -33,2651881 168,2020491 384,5847963 -3,923274437 -4,994769771 4,323144172 18h34m48s 48,80024999 -104,3191275 392,447477 7,4754127 0,132992125 1,745400371 19h04m48s -8,133239866 10,48998784 385,0813079 -1,705891909 4,577010525 -5,924966042 19h34m48s -38,59183152 137,5000854 386,3548018 -6,259334934 -2,517987633 3,6737412 20h04m48s 45,40898612 -138,2512659 390,9093841 6,426712881 -3,335472033 2,556001219 78 Futher Analysis of the Results

20h34m48s -1,63239474 -17,55944861 384,9792877 1,27363134 4,59485319 -6,018951908 21h04m48s -43,31246291 105,6329247 388,1298902 -7,035767519 0,898719836 2,948253164 21h34m48s 41,07727886 -170,7835906 389,0915083 3,841494738 -5,765278773 3,313928904 22h04m48s 4,881298319 -45,57550457 385,2400631 3,769251616 2,983287159 -5,987599603 22h34m48s -47,21467079 72,40897055 389,6863323 -5,967379406 4,324978668 2,161889134 23h04m48s 36,03806191 157,9569437 387,0885397 0,473937933 -6,537880636 4,003399975 23h34m48s 11,35300489 -73,6901464 385,9195464 4,988409792 0,246469378 -5,831686277 24h04m48s -50,05002738 37,79963275 390,9344603 -3,288883065 6,810897227 1,330964048 24h34m48s 30,48094677 127,7445694 384,9989066 -2,716932336 -5,511775399 4,609992143 01h04m48s 17,7241411 -102,0436767 386,8921848 4,519060092 -2,769904858 -5,554526334 01h34m48s -51,57774683 2,089829276 391,8502169 0,308335014 7,658176986 0,472727067 02h04m48s 24,55143737 98,34215788 383,0538096 -4,842635291 -3,058602447 5,120928937 02h34m48s 23,92729257 -130,7907616 388,1429462 2,452391794 -5,126358811 -5,161972082 03h04m48s -51,64011546 -34,06837524 392,3294047 3,870398647 6,619842026 -0,395025169 03h34m48s 18,35974042 69,53553607 381,3287864 -5,339643681 0,048595371 5,52548412 04h04m48s 29,87942769 -160,112092 389,5403383 -0,640936069 -6,065024914 -4,662322621 04h34m48s -50,2310908 -69,84067174 392,3403704 6,450452295 3,971270836 -1,254419804 05h04m48s 11,99149869 41,14073808 380,0154595 -4,129096776 2,862371125 5,81497711 05h34m48s 35,47326073 169,7821181 390,918424 -3,873960295 -5,233098403 -4,066049493 06h04m48s -47,49884969 -104,5506966 391,9506696 7,375896165 0,432368845 -2,087732139 06h34m48s 5,516291305 13,00022272 379,1507782 -1,635828977 4,537529134 5,983322692 07h04m48s 40,56876236 138,6564079 392,2142518 -6,29888429 -2,789599739 -3,385684613 07h34m48s -43,68134795 -137,8839225 391,2575402 6,43767221 -3,037019572 -2,877744604 08h04m48s -1,006376977 -15,02780142 378,8014071 1,339028104 4,577504496 6,026862583 08h34m48s 44,98165802 106,281275 393,3180145 -7,181154332 0,635635123 -2,635302048 09h04m48s -39,02846746 -169,849162 390,2856302 3,946489004 -5,509633283 -3,608092622 09h34m48s -7,52060949 -43,07621437 378,9752315 3,854564425 2,978806543 5,94464155 10h04m48s 48,48352744 72,51249748 394,0083247 -6,213100654 4,118768295 -1,830411718 10h34m48s -33,75625884 159,3692774 389,1907799 0,641000977 -6,353067246 -4,263693009 11h04m48s -13,96988524 -71,28007218 379,6629508 5,10706643 0,23153306 5,73850385 11h34m48s 50,82297108 37,42850699 394,3183586 -3,605806395 6,703929177 -0,987741924 12h04m48s -28,03463561 129,5335177 387,9958707 -2,526263851 -5,406912699 -4,83079164 12h34m48s -20,29220065 -99,78417349 380,8181591 4,667392317 -2,824198753 5,412770991 13h04m48s 51,7816911 1,461425234 394,1252193 -0,03250139 7,67527529 -0,124552306 13h34m48s -21,99282288 100,4143313 386,9665468 -4,663316323 -3,02386071 -5,297667358 14h04m48s -26,41391103 -128,7541836 382,2967925 2,608905911 -5,24377576 4,974385799 14h34m48s 51,25533358 -34,63111262 393,4738043 3,56020755 6,761045685 0,741323317 15h04m48s -15,72945937 71,81047338 386,089933 -5,194420758 0,036973894 -5,654520646 B.3 Unkown Inputs Parameters Values 79

Table B.2: The estimate values of ISS position

Latitude Longitude Altitude Vx Vy Vz 15h34m48s -31,77645144 92,62199571 388,0350561 -5,463333582 -3,014193207 -4,475710171 16h04m48s -16,22211951 -137,4159929 380,7365446 2,15681672 -4,757690459 5,635995831 16h34m48s 51,33210385 -31,365585 394,6803052 4,681809786 6,044204929 -0,681854055 17h04m48s -25,91961546 63,05534688 386,8889574 -5,815384788 0,236220578 -5,009339992 17h34m48s -22,4893097 -166,0714589 381,9741129 -0,864894336 -5,525752649 5,269028707 18h04m48s 51,76446399 -67,48999287 394,4740994 6,99818163 3,14847378 0,185319796 18h34m48s -19,78029833 34,13210285 385,8565092 -4,451322796 3,094691533 -5,438662301 19h04m48s -28,52483791 164,7419217 383,4686233 -3,924790749 -4,545852991 4,792315601 19h34m48s 50,71047208 -103,4349008 393,7463825 7,587766003 -0,506740027 1,04867844 20h04m48s -13,44864163 5,661397547 385,079327 -1,847378993 4,737341838 -5,754687609 20h34m48s -34,22772603 134,8187549 385,1255846 -6,09211907 -2,033816285 4,215861235 21h04m48s 48,2835551 -138,4583137 392,5737889 6,308367543 -3,946254008 1,890580998 21h34m48s -6,996064909 -22,52060081 384,6575193 1,165355812 4,711801136 -5,950790381 22h04m48s -39,46510586 103,9263093 386,8656724 -6,678603089 1,338643618 3,551817023 22h34m48s 44,7068681 -172,1481483 390,9910148 3,535752059 -6,261092855 2,693467772 23h04m48s -0,484213761 -50,55867905 384,6381999 3,645669494 3,066423396 -6,022845128 23h34m48s -44,06177832 71,82725167 388,5390637 -5,440860264 4,634231444 2,814090742 24h04m48s 40,23217634 155,5533965 389,1459918 0,055020851 -6,865265614 3,440791402 24h34m48s 6,030751688 -78,58595835 385,0104634 4,81862756 0,333933923 -5,969375351 01h04m48s -47,79721948 38,35042819 389,9581503 -2,655157774 6,918308164 2,018302014 01h34m48s 35,08618295 124,4924382 387,1706889 -3,164181891 -5,659479639 4,116939062 02h04m48s 12,49368083 -106,7365437 385,787972 4,303297532 -2,626355536 -5,791522231 02h34m48s -50,42057579 3,507547107 391,1118506 0,957785486 7,527553779 1,180740973 03h04m48s 29,45138036 94,43739939 385,1390414 -5,24495447 -3,055560252 4,707704463 03h34m48s 18,84451362 -135,1514272 386,8713356 2,225157399 -4,881054838 -5,493126047 04h04m48s -51,69788663 -32,34576884 391,8532939 4,434862277 6,260755543 0,319039231 04h34m48s 23,46519873 65,15325433 383,2262369 -5,653010584 0,146388249 5,200663419 05h04m48s 25,01339789 -163,9909414 388,1836463 -0,818833658 -5,697405265 -5,080507968 05h34m48s -51,49500548 -68,50496347 392,1995853 6,84368127 3,438409078 -0,549130391 06h04m48s 17,2333683 36,42981649 381,5963249 -4,346606029 2,991493747 5,585429741 06h34m48s 30,91386388 166,5591618 389,6503282 -3,932585388 -4,759581185 -4,562377035 07h04m48s -49,83467548 -104,1371374 392,107618 7,546182336 -0,187042554 -1,405736808 07h34m48s 10,83835419 8,089107834 380,3447095 -1,784374425 4,647532356 5,853780034 08h04m48s 36,43313484 136,2827965 391,1044243 -6,180397176 -2,264916755 -3,949619095 08h34m48s -46,88798928 -138,6211543 391,624848 6,378765754 -3,643975565 -2,233116884 09h04m48s 4,347372071 -20,02373395 379,5400967 1,212183014 4,646402608 5,999989855 09h34m48s 41,42345531 104,9422929 392,4250053 -6,85938275 1,128782689 -3,255067029 10h04m48s -42,90003481 -171,7084914 390,8229815 3,696875627 -6,016810082 -3,014155638 10h34m48s -2,180708696 -48,0479037 379,250302 3,701114572 3,018706519 6,020930691 11h04m48s 45,6921142 72,31487438 393,5072629 -5,705190134 4,489089415 -2,493147888 11h34m48s -38,11808611 156,550107 389,7558132 0,271543564 -6,704465316 -3,732585554 12h04m48s -8,690742894 -76,11673829 379,505134 4,897077708 0,28314972 5,916147782 12h34m48s 49,00338557 38,28595133 394,2220948 -2,972402737 6,874553843 -1,679688169 13h04m48s -32,75021355 125,9519608 388,5738182 -2,932869011 -5,58827012 -4,373548505 13h34m48s -15,12491731 -104,3660063 380,2228223 4,402632919 -2,706255408 5,687868043 14h04m48s 51,10616207 2,99044203 394,5022941 0,634157605 7,604266165 -0,831505749 14h34m48s -26,95728916 96,25725755 387,437244 -5,034666398 -3,060983981 -4,923746088 15h04m48s -21,4200536 -132,9446949 381,4020074 2,326183291 -5,011438393 5,340961869 15h34m48s 51,79687206 -33,06998688 394,310913 4,156908677 6,453146706 0,034013921 16h04m48s -20,86417572 67,24163349 386,3811545 -5,486455137 0,090254654 -5,371580537 16h34m48s -27,4988325 -162,0216214 382,8892789 -0,746916936 -5,885685706 4,882858368 17h04m48s 50,99920864 -69,10012504 393,6117833 6,656130036 3,717716226 0,898886306 17h34m48s -14,56469126 38,70826242 385,5777316 -4,229459682 2,915255928 -5,707653405 18h04m48s -33,26512168 168,2023637 384,5975847 -3,923236571 -4,994777286 4,32316735 18h34m48s 48,80027804 -104,3189889 392,4502066 7,475422804 0,133033236 1,745362478 19h04m48s -8,133308442 10,49010118 385,0851214 -1,705887377 4,577010437 -5,924950903 19h34m48s -38,59183642 137,5002384 386,3837731 -6,25932174 -2,517981985 3,673779358 80 Futher Analysis of the Results

20h04m48s 45,40910502 -138,2506843 390,9003926 6,426734865 -3,335442054 2,555946135 20h34m48s -1,632809265 -17,55917669 384,9505222 1,273572467 4,594884925 -6,018929463 21h04m48s -43,31268532 105,6331915 388,1218866 -7,035772655 0,898711503 2,948280302 21h34m48s 41,07710754 -170,7837555 389,0782151 3,84147139 -5,765295576 3,313925536 22h04m48s 4,881568564 -45,57557339 385,2486191 3,769279049 2,983277032 -5,987598805 22h34m48s -47,21469977 72,40873841 389,68812 -5,967352912 4,325005369 2,161871018 23h04m48s 36,03826501 157,9569249 387,0500709 0,473939858 -6,537877313 4,003398709 23h34m48s 11,35337608 -73,69039212 385,9151502 4,988418347 0,246449923 -5,831671226 24h04m48s -50,05005807 37,79909291 390,9550984 -3,288856005 6,810913189 1,330924502 24h34m48s 30,4811707 127,7446956 384,990595 -2,716923943 -5,511797526 4,609988605 01h04m48s 17,72436273 -102,043468 386,8991661 4,519066939 -2,769917687 -5,554514325 01h34m48s -51,57779108 2,090020686 391,83103 0,308337052 7,658182357 0,472775801 02h04m48s 24,55098436 98,34180174 383,0804253 -4,842632767 -3,058534452 5,120952444 02h34m48s 23,92727586 -130,7907819 388,1401478 2,452405154 -5,126367716 -5,16197009 03h04m48s -51,6402673 -34,06853719 392,3089341 3,870414824 6,619825762 -0,395027075 03h34m48s 18,35935403 69,53537536 381,3740238 -5,339620743 0,048652826 5,52547605 04h04m48s 29,87899973 -160,111663 389,5267681 -0,640902515 -6,065000145 -4,66235009 04h34m48s -50,23092089 -69,84094926 392,3430856 6,450429812 3,971274073 -1,254447782 05h04m48s 11,99161609 41,14083297 380,0362606 -4,129102596 2,862369026 5,814985125 05h34m48s 35,47320727 169,7821086 390,9185721 -3,873964358 -5,233092458 -4,0661019 06h04m48s -47,4986224 -104,5506111 391,9738545 7,375901392 0,43237983 -2,087737052 06h34m48s 5,516401091 13,00025282 379,1477074 -1,635863749 4,537514646 5,983325887 07h04m48s 40,56861707 138,6564289 392,2241314 -6,298886492 -2,789599615 -3,38568651 07h34m48s -43,68161958 -137,8833999 391,2445783 6,437706126 -3,037003512 -2,877713376 08h04m48s -1,006304233 -15,02760119 378,7901489 1,338999714 4,57751532 6,02686732 08h34m48s 44,981651 106,2812092 393,2918976 -7,181146374 0,635651111 -2,635276861 09h04m48s -39,02872175 -169,8490369 390,2786175 3,946505597 -5,50963757 -3,608087783 09h34m48s -7,520634295 -43,07617416 378,9770971 3,854538828 2,978823716 5,944661358 10h04m48s 48,48367945 72,51184036 394,0206519 -6,21309101 4,118791978 -1,83037236 10h34m48s -33,75633505 159,3694188 389,153377 0,640999848 -6,353073385 -4,263645366 11h04m48s -13,970052 -71,27980709 379,6747305 5,107060558 0,231497951 5,738495763 11h34m48s 50,82286214 37,4285259 394,3179711 -3,605796619 6,703936521 -0,987736307 12h04m48s -28,03476033 129,5336907 388,006516 -2,526278858 -5,406923497 -4,830781867 12h34m48s -20,29216663 -99,7839345 380,7962279 4,667392354 -2,824188702 5,41278067 13h04m48s 51,78170956 1,461402809 394,1468637 -0,032483191 7,675275201 -0,124548335 13h34m48s -21,99309216 100,4145473 386,9321427 -4,663316735 -3,023906778 -5,297656519 14h04m48s -26,41405457 -128,7544153 382,3081738 2,608897279 -5,243812537 4,974356006 14h34m48s 51,25553172 -34,63099908 393,4793982 3,560194963 6,761036811 0,741334451 15h04m48s -15,72980436 71,81047888 386,080716 -5,194459975 0,036945106 -5,654490816 B.3 Unkown Inputs Parameters Values 81

Table B.3: New ISS TLE values calculated using Kalman ﬁlter

mo no nodeo ecco inclo argpo bstar 15h34m48s 0,4890848 15,622006 0,3248545 0,0011113 0,1434494 0,8019181 0,000131792 16h04m48s 0,4892730 15,621781 0,3248541 0,0011145 0,1434492 0,8017363 -0,000713583 16h34m48s 0,4892078 15,621754 0,3248542 0,0011151 0,1434495 0,8018025 -0,000886144 17h04m48s 0,4893299 15,621778 0,3248540 0,001115 0,1434487 0,8016801 -0,000331447 17h34m48s 0,4895852 15,621760 0,3248540 0,0011156 0,1434494 0,8014267 -0,00049338 18h04m48s 0,4893076 15,621780 0,3248536 0,0011158 0,1434496 0,8017024 -0,00065472 18h34m48s 0,4894785 15,621774 0,3248537 0,0011163 0,1434494 0,8015336 -0,00019092 19h04m48s 0,4893835 15,621794 0,3248537 0,0011163 0,1434495 0,80162542 -0,000235081 19h34m48s 0,4893678 15,621824 0,3248533 0,0011162 0,143449 0,80163565 -4,58E-05 20h04m48s 0,4892321 15,621831 0,3248542 0,0011160 0,1434492 0,80176911 -0,00024845 20h34m48s 0,4893998 15,621765 0,3248537 0,0011159 0,1434498 0,80161586 0,000241124 21h04m48s 0,4893277 15,621830 0,3248542 0,0011160 0,1434492 0,80167179 0,000947753 21h34m48s 0,4892035 15,621837 0,3248541 0,0011160 0,1434497 0,80179392 0,000931028 22h04m48s 0,4892270 15,621797 0,3248535 0,0011160 0,1434496 0,80178191 0,000943466 22h34m48s 0,4895344 15,621762 0,3248537 0,0011160 0,1434491 0,80148512 0,000758177 23h04m48s 0,4894962 15,621764 0,3248536 0,0011156 0,1434490 0,80152260 0,000656598 23h34m48s 0,4893796 15,621825 0,3248542 0,0011156 0,1434491 0,80161788 0,00041316 24h04m48s 0,4894266 15,621792 0,324853 0,00111559 0,1434495 0,80158312 0,000190734 24h34m48s 0,4897050 15,621828 0,324854 0,00111580 0,1434498 0,80129094 0,000217787 01h04m48s 0,4896844 15,62177 0,3248544 0,00111578 0,1434490 0,80133115 0,000284199 01h34m48s 0,4893809 15,621752 0,3248531 0,0011156 0,143449 0,801647589 2,93E-05 02h04m48s 0,4895048 15,621762 0,3248538 0,0011156 0,1434499 0,80151918 3,68E-05 02h34m48s 0,4893062 15,621762 0,324853608 0,00111563 0,143449878 0,801719235 5,92E-05 03h04m48s 0,4893155 15,621777 0,32485377 0,001115774 0,143449553 0,801702522 0,000270654 03h34m48s 0,4893071 15,621826 0,32485374 0,001115635 0,143449105 0,801685015 0,000290116 04h04m48s 0,4893211 15,621824 0,32485421 0,001115639 0,143449302 0,801672137 0,00027907 04h34m48s 0,4894986 15,621811 0,32485419 0,001115648 0,143449129 0,801501155 0,000280512 05h04m48s 0,4893216 15,621787 0,32485421 0,001115562 0,143449462 0,80169201 0,000333864 05h34m48s 0,4893091 15,621797 0,32485374 0,001115517 0,143449453 0,801698942 0,00041254 06h04m48s 0,4893355 15,621786 0,32485399 0,001115388 0,143449641 0,801679403 0,00027651 06h34m48s 0,4891550 15,621803 0,32485377 0,001115373 0,143449329 0,801848462 0,00028365 07h04m48s 0,4894841 15,621792 0,32485374 0,001115361 0,143449617 0,801527023 0,00030484 07h34m48s 0,4893494 15,621794 0,32485390 0,001115235 0,14344921 0,801660352 0,000188815 08h04m48s 0,4892156 15,621793 0,32485385 0,00111523 0,143449534 0,801794073 0,000214794 08h34m48s 0,4893928 15,621793 0,32485384 0,001115209 0,143449333 0,801617671 0,00022884 09h04m48s 0,4895446 15,621807 0,32485443 0,001115138 0,143449412 0,801455021 0,00018345 09h34m48s 0,4894028 15,621742 0,32485424 0,001115142 0,143449095 0,801646778 0,00012040 10h04m48s 0,4894610 15,621808 0,32485403 0,001115188 0,143449182 0,80153767 6,00E-05 10h34m48s 0,4896437 15,621794 0,32485401 0,001115239 0,143449617 0,801365869 0,00010292 11h04m48s 0,4894650 15,621806 0,32485468 0,001115264 0,143449059 0,801534201 0,00016053 11h34m48s 0,4896713 15,621779 0,32485400 0,001115195 0,143449648 0,801350822 0,000196954 12h04m48s 0,4895857 15,621806 0,32485432 0,001115035 0,143449506 0,80141326 7,92E-05 12h34m48s 0,4896116 15,621841 0,32485348 0,001115078 0,143449647 0,801357208 0,00016818 13h04m48s 0,4894754 15,621812 0,32485412 0,001115167 0,143448988 0,80151754 0,000167033 13h34m48s 0,4894461 15,621800 0,32485329 0,001115201 0,143449599 0,801558335 0,00019005 14h04m48s 0,4895112 15,621771 0,32485378 0,001115187 0,143449549 0,801521658 0,000172899 14h34m48s 0,4895100 15,621766 0,32485365 0,001115092 0,143449343 0,801527436 0,000187932 15h04m48s 0,4896002 15,621780 0,32485421 0,001114988 0,143449315 0,80142396 0,000105134 15h34m48s 0,4896745 15,621790 0,32485453 0,001114996 0,143449747 0,801339545 0,00011747 16h04m48s 0,4897829 15,621829 0,32485435 0,001115028 0,14345015 0,801190145 0,000100188 16h34m48s 0,4896322 15,621809 0,32485380 0,001115103 0,143449243 0,80136235 0,00016835 17h04m48s 0,4895144 15,621789 0,32485386 0,001115107 0,143449559 0,801502085 0,000169549 17h34m48s 0,4895533 15,621827 0,32485428 0,001115109 0,143449667 0,801420739 0,00015874 18h04m48s 0,4896266 15,621850 0,32485402 0,001115087 0,143449483 0,801319795 0,00013254 18h34m48s 0,4892962 15,621815 0,32485366 0,001114993 0,14344939 0,80168982 4,01E-05 19h04m48s 0,4894298 15,621825 0,32485355 0,001115088 0,143449578 0,80154498 3,50E-05 19h34m48s 0,4895245 15,621862 0,32485410 0,001115085 0,143449145 0,801404685 2,54E-05 82 Futher Analysis of the Results

20h04m48s 0,48937445 15,62185077 0,324854031 0,001115095 0,143449428 0,801567992 3,09E-05 20h34m48s 0,489408212 15,62181699 0,324854037 0,001115114 0,143449792 0,80157455 3,74E-05 21h04m48s 0,489624996 15,62178733 0,324853701 0,00111512 0,143448997 0,801394247 6,35E-05 21h34m48s 0,48961395 15,62176312 0,324853902 0,001115098 0,143449419 0,801436528 4,07E-05 22h04m48s 0,489530202 15,62177185 0,32485347 0,001115068 0,143449621 0,801509882 3,96E-05 22h34m48s 0,489354093 15,62179078 0,324854092 0,001115072 0,143449574 0,801661515 1,92E-05 23h04m48s 0,489501418 15,62176127 0,324854317 0,001115146 0,143449235 0,801553267 5,27E-05 23h34m48s 0,489364945 15,62176427 0,324854115 0,001114971 0,143449334 0,801686445 7,20E-05 24h04m48s 0,489248292 15,62179996 0,324854241 0,001114983 0,143449525 0,80175503 5,81E-05 24h34m48s 0,489402823 15,6217961 0,324854473 0,001115011 0,143449388 0,801605223 7,18E-05 01h04m48s 0,489462704 15,62180338 0,324853412 0,001115038 0,143449324 0,801535966 6,68E-05 01h34m48s 0,489757303 15,62175933 0,324854033 0,001114997 0,143449125 0,801303509 9,91E-05 02h04m48s 0,489524865 15,62178934 0,324853889 0,001114969 0,14344913 0,801493667 9,46E-05 02h34m48s 0,48948514 15,62177886 0,324853664 0,001114996 0,143449424 0,801549103 8,79E-05 03h04m48s 0,489535134 15,62175773 0,324854136 0,001115009 0,14344901 0,801530672 8,36E-05 03h34m48s 0,489146652 15,62181418 0,324854051 0,001114988 0,14344948 0,801834575 8,49E-05 04h04m48s 0,489398954 15,62179599 0,324854339 0,001115028 0,1434493 0,801609152 7,06E-05 04h34m48s 0,489601138 15,62180152 0,324853412 0,001115038 0,143449473 0,80139916 6,74E-05 05h04m48s 0,489571299 15,62183192 0,324853556 0,001115094 0,143449477 0,801380513 9,79E-05 05h34m48s 0,489357586 15,62180517 0,324854254 0,001115127 0,143449086 0,801635934 8,00E-05 06h04m48s 0,489368394 15,62182134 0,324853701 0,001115085 0,143449573 0,801599316 8,82E-05 06h34m48s 0,489442518 15,62181323 0,324853829 0,001114979 0,143449347 0,801537726 4,35E-05 07h04m48s 0,489571061 15,62182457 0,324854063 0,001114978 0,143449591 0,801390349 3,40E-05 07h34m48s 0,489384697 15,62181646 0,324854204 0,001115 0,143449585 0,801589373 3,17E-05 08h04m48s 0,489494955 15,62180138 0,32485374 0,001114994 0,143449566 0,801504541 2,84E-05 08h34m48s 0,48970521 15,62178863 0,324853802 0,001114985 0,143449568 0,801316908 3,90E-05 09h04m48s 0,489530532 15,62179034 0,324854188 0,001115025 0,143449283 0,801487879 3,27E-05 09h34m48s 0,489732234 15,62179376 0,324854153 0,001114912 0,143449008 0,801279347 -1,34E-05 10h04m48s 0,489538509 15,62181206 0,32485404 0,001114923 0,143449129 0,801441244 -1,87E-06 10h34m48s 0,489735092 15,62177376 0,324854179 0,00111495 0,143449616 0,801313831 5,17E-06 11h04m48s 0,489406549 15,62178634 0,324853424 0,001115031 0,143449098 0,80162022 4,30E-05 11h34m48s 0,489587763 15,62179133 0,32485365 0,001115027 0,143449351 0,801430062 4,63E-05 12h04m48s 0,48946658 15,62181671 0,324853991 0,001115113 0,143449768 0,801503337 3,24E-05 12h34m48s 0,489499176 15,62178198 0,324853475 0,001115167 0,143449444 0,801536534 5,82E-05 13h04m48s 0,489621818 15,62181983 0,324854001 0,001115178 0,143449389 0,801341254 7,64E-05 13h34m48s 0,48950706 15,62177891 0,324853867 0,001115202 0,143449378 0,801534563 8,10E-05 14h04m48s 0,489397651 15,62178982 0,324853654 0,001115212 0,143449827 0,801623902 9,20E-05 14h34m48s 0,48945464 15,62181681 0,324853708 0,001115261 0,143449296 0,801513098 0,000136618 15h04m48s 0,489420382 15,62180051 0,324854541 0,001115216 0,143449723 0,801578827 0,000141024 Mo - Mean Anomaly; no - Mean Motion; nodeo - Right Ascension of the Ascending Node; ecco - Eccentricity; inclo - Inclination; argpo - Argument of Perigee; Bstar - Drag Term B.3 Unkown Inputs Parameters Values 83

Table B.4: The real values of the PoSAT position

Latitude Longitude Altitude Vx Vy Vz 15h34m48s 70,95989255 -179,1987382 804,145397 -6,330636565 3,280457786 -2,184395781 16h04m48s -34,32981368 141,8970855 797,2074592 4,063283224 -1,490670391 -6,07257451 16h34m48s -37,96980086 -33,09075242 807,7757164 2,996821045 -3,618489642 5,77767762 17h04m48s 67,3523598 -68,17886485 807,2077557 -5,010054053 4,837091202 2,65633018 17h34m48s 3,792554028 125,7999952 787,7216298 0,611141195 1,044748216 -7,364231242 18h04m48s -74,24640728 -30,3985231 816,2218267 3,145631145 -6,529886574 1,701214725 18h34m48s 30,22233841 -74,7430939 800,9893761 -2,200318207 3,222437325 6,349204899 19h04m48s 41,85485227 110,4010395 794,545005 -0,309498297 5,076016472 -5,460056646 19h34m48s -63,89429809 77,53965258 809,9867749 1,079286513 -6,684107086 -3,108068023 20h04m48s -7,836635228 -91,09197512 800,3437114 -1,131568119 -0,97428568 7,29894309 20h34m48s 77,47651158 122,3382906 805,7764625 1,424605914 7,220298689 -1,197173468 21h04m48s -26,53222014 68,46880369 794,1109967 -0,052276037 -3,490903524 -6,591233774 21h34m48s -45,68920361 -105,9914905 810,2232523 -2,944113783 -4,560972944 5,096387999 22h04m48s 59,94689448 -137,1835415 806,2857121 3,027691401 5,797200246 3,574221119 22h34m48s 11,64998131 51,99241776 787,8398451 1,803243472 0,476396856 -7,226585324 23h04m48s -79,9103548 -80,44399663 816,4021452 -5,515273141 -4,951772345 0,703925311 23h34m48s 22,42087495 -148,2904683 799,9398056 1,755046257 2,485674279 6,802590263 24h04m48s 49,57609947 37,80223892 797,0898394 5,400274833 2,04016816 -4,72594209 24h34m48s -56,39419118 7,28155883 806,9422379 -5,668381966 -2,726353643 -3,995554234 01h04m48s -15,66309771 -164,8747159 801,7509041 -2,20533911 0,589478921 7,090874627 01h34m48s 81,4139064 85,76971795 806,9330645 7,424451695 0,651641697 -0,187734882 02h04m48s -18,70725365 -5,159161935 791,4669834 -2,441774047 -0,949305927 -6,985218475 02h34m48s -53,34506817 -178,2233717 812,4779877 -5,832734227 1,659996719 4,319390571 03h04m48s 52,36105026 151,5965492 804,9889948 5,88862587 -1,137227169 4,424592482 03h34m48s 19,49815886 -21,74595865 788,6939781 1,843032815 -1,995907512 -6,951378023 04h04m48s -81,21727938 -106,8430661 815,8372476 -6,315105325 3,933149417 -0,306704137 04h34m48s 14,60056795 138,0078712 799,2981409 2,139642416 -0,393267366 7,127117997 05h04m48s 57,21619603 -33,89888169 799,628005 3,866511093 -5,046174888 -3,902637828 05h34m48s -48,7593839 -64,550026 803,6278229 -3,846747832 4,199675314 -4,80837295 06h04m48s -23,47433259 121,439154 803,7244923 -0,495318118 3,112383292 6,748699083 06h34m48s 79,37140688 57,70460423 807,4550569 2,677445855 -6,908161621 0,824962144 07h04m48s -10,86308655 -78,90738185 789,4251042 -1,368205044 1,135609641 -7,246698836 07h34m48s -60,89735534 110,749184 814,3903702 -0,083232205 6,590814316 3,461253005 08h04m48s 44,67981929 79,26079058 803,5611872 0,56668117 -5,317060682 5,19145012 08h34m48s 27,32907838 -95,35601526 790,1684261 -1,550730231 -3,235521462 -6,543993679 09h04m48s -76,80098597 -146,6932209 814,514145 1,988067867 7,055910001 -1,312029886 09h34m48s 6,768808319 64,21682874 799,1611937 0,62159824 -1,265877976 7,316571203 10h04m48s 64,71306654 -103,8942299 802,0175544 -4,016840797 -5,518855784 -3,005889038 10h34m48s -41,04630994 -137,2214614 800,1623595 2,517819555 4,318834349 -5,531000412 11h04m48s -31,26301954 47,91701336 805,9811702 3,4969632 1,951143569 6,279168157 11h34m48s 73,42996198 5,193399692 807,4569 -5,742754635 -4,389176784 1,821820053 12h04m48s -3,007528854 -152,7194637 788,1017443 -0,16899799 1,157082074 -7,370445091 12h34m48s -68,25103058 42,14969308 815,8296665 6,688541889 2,059924917 2,538297477 13h04m48s 36,94070976 6,298494231 802,0777218 -4,2157499 -1,851236642 5,860145003 13h34m48s 35,13400376 -168,7556603 792,2272106 -4,391888854 0,501495282 -6,012360914 14h04m48s -70,16913369 153,3359325 812,4478732 7,086408198 0,062299007 -2,293123324 14h34m48s -1,067296888 -9,612852955 799,6384847 -0,054175979 -1,109277741 7,367344647 15h04m48s 71,90404626 -170,1576039 804,0658575 -6,748507376 2,419978228 -2,052774018 15h34m48s -33,28120174 149,6177916 796,8373045 4,123529906 -0,889484962 -6,149508456 16h04m48s -39,01955798 -25,34051901 808,3988642 3,541165442 -3,243430732 5,691213607 16h34m48s 66,36281469 -59,66720819 806,8855092 -5,570958674 4,095003122 2,784222107 17h04m48s 4,852136993 133,451806 787,5284434 0,374496467 1,216340112 -7,353944458 17h34m48s -75,12490875 -20,66699078 816,6080507 4,004187917 -6,077243194 1,567705128 18h04m48s 29,16261768 -67,04015278 800,8036617 -2,554768937 2,796450583 6,417983214 18h34m48s 42,90180267 118,1824857 794,565782 -1,01720885 5,083328754 -5,366760577 19h04m48s -62,89435621 85,81080047 809,8077342 1,946650746 -6,424136636 -3,231689693 19h34m48s -8,900875409 -83,43688928 800,6657604 -0,979286086 -1,250259835 7,278541136 84 Futher Analysis of the Results

20h04m48s 78,23576445 133,3803743 805,6545619 0,451392109 7,366688743 -1,061354387 20h34m48s -25,47954243 76,15670806 793,7561759 0,433492039 -3,346765987 -6,651955322 21h04m48s -46,73142427 -98,1702149 810,8492991 -2,337982723 -5,000877275 4,996238163 21h34m48s 58,92397717 -129,0811527 805,9330007 2,200862971 6,087894546 3,69398507 22h04m48s 12,70903401 59,65074253 787,7072037 1,794634581 0,824732654 -7,197512314 22h34m48s -80,44184786 -67,5854955 816,6625442 -4,814249211 -5,650772569 0,567598626 23h04m48s 21,35812808 -140,6131038 799,8413903 1,315863332 2,611487543 6,85432785 23h34m48s 50,61480609 45,6774552 797,1567838 5,1509886 2,790886277 -4,619636279 24h04m48s -55,37076669 15,27132077 806,6852326 -5,189084069 -3,422029389 -4,110051563 24h34m48s -16,72541716 -157,2098108 802,242261 -2,38651722 0,245275884 7,051992104 01h04m48s 81,52235284 100,4069139 806,7199572 7,273938342 1,634875964 -0,050276388 01h34m48s -17,65123212 2,508872498 791,1382889 -2,165273513 -1,254064486 -7,028434642 02h04m48s -54,37577301 -170,2791691 813,0509392 -6,079925847 0,871595513 4,207372512 02h34m48s 51,3214146 159,5027704 804,6723537 5,906170079 -0,329142662 4,533976177 03h04m48s 20,55588803 -14,07389909 788,617877 2,214392865 -1,770546673 -6,904139054 03h34m48s -80,88227161 -92,89758816 816,0016723 -6,776101311 3,054427184 -0,443419172 04h04m48s 13,5356014 145,6696032 799,2973492 2,061203675 -0,034539489 7,160776769 04h34m48s 58,24090582 -25,85481271 799,6616273 4,558593734 -4,531799121 -3,785369528 05h04m48s -47,72341746 -56,7042031 803,3023826 -4,310995315 3,584672234 -4,911521112 05h34m48s -24,5334659 129,1216479 804,2590043 -0,976914407 3,119789554 6,692126809 06h04m48s 78,69858553 69,54343206 807,2154404 3,565768155 -6,475769075 0,961463672 06h34m48s -9,80512319 -71,25139848 789,1265475 -1,460883219 0,817970515 -7,271618136 07h04m48s -61,90799443 118,9253194 814,9516895 -0,973808021 6,581763275 3,339594111 07h34m48s 43,63054558 87,06099209 803,2365142 1,260635567 -5,098001007 5,28830542 08h04m48s 28,38447781 -87,66111717 790,1811636 -1,110773052 -3,533211186 -6,479424933 08h34m48s -75,97673774 -136,3829193 814,5080816 1,026703687 7,232691265 -1,446441123 09h04m48s 5,703019622 71,87042029 799,2798697 0,824594848 -1,042531623 7,33152683 09h34m48s 65,71056509 -95,50793854 801,9999084 -3,269051774 -6,053972563 -2,879894447 10h04m48s -40,00232289 -129,4571058 799,7929435 1,865805395 4,528901762 -5,62084845 10h34m48s -32,31794912 55,62864055 806,5524528 3,27942918 2,487797977 6,205946432 11h04m48s 72,50849755 14,51923374 807,1463085 -5,08182085 -5,091138813 1,954847289 11h34m48s -1,948298484 -145,0682236 787,842116 -0,432426885 1,045801246 -7,376554126 12h04m48s -69,22010776 50,83114756 816,2652086 6,394715973 2,953183989 2,409195213 12h34m48s 35,88474646 14,03620068 801,8250524 -3,829497648 -2,358258412 5,942672905 13h04m48s 36,18587759 -161,0238946 792,2489612 -4,527045702 -0,112113739 -5,931685538 13h34m48s -69,21089397 162,1240346 812,3249425 6,971818463 1,001154612 -2,422789429 14h04m48s -2,132725649 -1,960819029 799,8909348 0,22998185 -1,111804866 7,36335335 14h34m48s 72,83434769 -160,9532061 804,0307205 -7,046918615 1,506127417 -1,920449849 15h04m48s -32,23174646 157,3328061 796,4235847 4,096584769 -0,298920209 -6,224375765 B.3 Unkown Inputs Parameters Values 85

Table B.5: The estimate values of the PoSAT position

Latitude Longitude Altitude Vx Vy Vz 15h34m48s 70,96005571 -179,1988666 804,1268076 -6,330629716 3,280452513 -2,184416101 16h04m48s -34,32927601 141,8968162 797,2359842 4,063291906 -1,490513984 -6,072596721 16h34m48s -37,97033055 -33,09050864 807,8401466 2,996850253 -3,61851453 5,777596259 17h04m48s 67,35253563 -68,17879871 807,2126998 -5,010050521 4,837095815 2,656314936 17h34m48s 3,792482024 125,799888 787,6924767 0,611160565 1,044738456 -7,364249134 18h04m48s -74,24648029 -30,39834041 816,2435029 3,145641237 -6,529873276 1,701199291 18h34m48s 30,2224039 -74,74322545 800,9946907 -2,200314157 3,222441903 6,349187326 19h04m48s 41,8549638 110,4009898 794,5418484 -0,309490956 5,076007601 -5,460061235 19h34m48s -63,89436707 77,53975072 809,989863 1,079290832 -6,684121848 -3,108068793 20h04m48s -7,836800394 -91,09180725 800,3153419 -1,13156403 -0,974310565 7,298928934 20h34m48s 77,47648486 122,338454 805,7880532 1,42460628 7,220302901 -1,19716271 21h04m48s -26,53210582 68,46881203 794,1398679 -0,052272664 -3,49089262 -6,591267097 21h34m48s -45,68932602 -105,9915824 810,2174907 -2,944100864 -4,560978461 5,096393866 22h04m48s 59,94695804 -137,1837121 806,2585649 3,027690263 5,797211117 3,574217412 22h34m48s 11,65040629 51,9924405 787,8503083 1,803284434 0,476454605 -7,226542546 23h04m48s -79,91034319 -80,44363069 816,4110469 -5,515277769 -4,951770637 0,703944419 23h34m48s 22,42086621 -148,2904479 799,9235369 1,755052397 2,485678443 6,802583865 24h04m48s 49,57615803 37,80195802 797,0895222 5,400281588 2,040194218 -4,725922087 24h34m48s -56,39410762 7,281305369 806,9566955 -5,668376679 -2,726363565 -3,995574691 01h04m48s -15,66260648 -164,874651 801,776906 -2,205286335 0,589480755 7,090897346 01h34m48s 81,41384577 85,76746533 806,8968382 7,424477816 0,651619937 -0,187762634 02h04m48s -18,70708652 -5,158923275 791,4800333 -2,441795873 -0,949314298 -6,985218435 02h34m48s -53,34493788 -178,2235341 812,4688182 -5,832714075 1,660014329 4,319392654 03h04m48s 52,36140324 151,5962895 804,9841465 5,888630772 -1,137249007 4,424581434 03h34m48s 19,49777263 -21,74573727 788,6827643 1,843000168 -1,995884515 -6,951386762 04h04m48s -81,21720281 -106,84226 815,832449 -6,315089237 3,933156833 -0,306759716 04h34m48s 14,60052063 138,0077451 799,2993716 2,139647526 -0,393283308 7,127102307 05h04m48s 57,21641584 -33,89872128 799,6574505 3,866549693 -5,046147751 -3,902588823 05h34m48s -48,75949943 -64,55039714 803,6208312 -3,84675967 4,199702751 -4,808386548 06h04m48s -23,47455232 121,4391148 803,7225789 -0,495334427 3,112407117 6,74872016 06h34m48s 79,3712518 57,70599214 807,4651895 2,677454148 -6,908152325 0,824982466 07h04m48s -10,86322998 -78,90740939 789,4379078 -1,368233597 1,135612123 -7,246674764 07h34m48s -60,89733785 110,748793 814,3977808 -0,083233305 6,59079506 3,461297895 08h04m48s 44,68001963 79,26071772 803,5396714 0,566657856 -5,317099904 5,191428335 08h34m48s 27,32953889 -95,35562724 790,2046145 -1,550746974 -3,235573603 -6,543957541 09h04m48s -76,80141285 -146,6941936 814,4713148 1,98807955 7,055918025 -1,311968663 09h34m48s 6,768537197 64,21680448 799,1800757 0,621580468 -1,265840406 7,316565165 10h04m48s 64,71304627 -103,8942319 802,020185 -4,016836117 -5,518856518 -3,005899144 10h34m48s -41,04622792 -137,2215268 800,1684787 2,517814433 4,318844854 -5,531009048 11h04m48s -31,26271255 47,91692999 805,9688088 3,496930538 1,951124629 6,279185155 11h34m48s 73,42976973 5,193572113 807,4448251 -5,742765259 -4,389159573 1,821848743 12h04m48s -3,007468842 -152,7192098 788,0989584 -0,169023099 1,157069231 -7,370431249 12h34m48s -68,25088312 42,14917013 815,8075584 6,688543226 2,059937228 2,538328737 13h04m48s 36,94051557 6,298578401 802,1115604 -4,215727663 -1,851226198 5,860149741 13h34m48s 35,13406441 -168,7556663 792,2048759 -4,391891212 0,501528568 -6,012355429 14h04m48s -70,16936769 153,3358226 812,444512 7,086405593 0,062286336 -2,293090192 14h34m48s -1,067151921 -9,612761067 799,6286489 -0,054194063 -1,109293363 7,367349492 15h04m48s 71,90427159 -170,1572966 804,0804295 -6,74851786 2,419968841 -2,052761075 15h34m48s -33,28101996 149,6176774 796,8120778 4,123524942 -0,889495577 -6,149535902 16h04m48s -39,01999052 -25,3402405 808,4266814 3,541195617 -3,243407486 5,691205707 16h34m48s 66,36271384 -59,66735717 806,9197211 -5,57095647 4,094981894 2,784241045 17h04m48s 4,852434342 133,4516254 787,5135024 0,37447695 1,216349427 -7,353962355 17h34m48s -75,12504909 -20,66619414 816,6134377 4,00419472 -6,07724878 1,567677802 18h04m48s 29,16266541 -67,04016692 800,8206921 -2,55476381 2,796440592 6,4179882 18h34m48s 42,90189587 118,1825685 794,5783776 -1,017204745 5,083328849 -5,366738324 19h04m48s -62,89467175 85,81054989 809,7984295 1,94663941 -6,42416525 -3,231629251 19h34m48s -8,90111042 -83,43670094 800,6722731 -0,97927169 -1,250290799 7,278552579 86 Futher Analysis of the Results

20h04m48s 78,23564545 133,3802468 805,6467436 0,451416209 7,366699889 -1,061384276 20h34m48s -25,4795896 76,15658902 793,7426692 0,433514145 -3,346795259 -6,65193898 21h04m48s -46,73180563 -98,17016798 810,8683541 -2,33797446 -5,000905133 4,996173389 21h34m48s 58,92388525 -129,0810841 805,9187951 2,200878062 6,087900354 3,69398056 22h04m48s 12,70930233 59,65092735 787,7171906 1,794637755 0,824772995 -7,197514957 22h34m48s -80,44184459 -67,58513765 816,6755635 -4,814244907 -5,650774704 0,567586412 23h04m48s 21,35812574 -140,6132578 799,8411129 1,315853284 2,611478318 6,85431417 23h34m48s 50,61474604 45,67742446 797,1277518 5,150983838 2,790874112 -4,619624709 24h04m48s -55,37069038 15,2715131 806,701845 -5,189065053 -3,42202839 -4,110036804 24h34m48s -16,72533085 -157,2096783 802,2335159 -2,386528019 0,245287635 7,052001813 01h04m48s 81,52222741 100,4078056 806,7241189 7,273947386 1,634877584 -0,050255033 01h34m48s -17,65142336 2,508594054 791,1311272 -2,165298387 -1,254048719 -7,028450845 02h04m48s -54,37586199 -170,2791292 813,0832877 -6,079940047 0,871580747 4,207360154 02h34m48s 51,32155587 159,5029363 804,6444463 5,906190533 -0,3291424 4,533953518 03h04m48s 20,5559997 -14,07394096 788,6417093 2,214403809 -1,770542372 -6,904094661 03h34m48s -80,88226726 -92,89920852 815,960265 -6,776106455 3,054434226 -0,443364755 04h04m48s 13,53556755 145,6695375 799,2915039 2,061203013 -0,034535403 7,160788697 04h34m48s 58,24059037 -25,85498877 799,6286344 4,558589297 -4,531806428 -3,785415659 05h04m48s -47,72366984 -56,70411666 803,2959473 -4,310996981 3,58465584 -4,911507258 05h34m48s -24,53337088 129,1215587 804,2396982 -0,976914839 3,119784577 6,692143119 06h04m48s 78,69844353 69,54411287 807,2375422 3,565780358 -6,47575314 0,961486698 06h34m48s -9,804975083 -71,25160506 789,1330294 -1,4609305 0,817958363 -7,271595123 07h04m48s -61,90813827 118,9253614 814,9512905 -0,973789819 6,581738082 3,339585442 07h34m48s 43,63042508 87,06098228 803,2488872 1,260639295 -5,097979299 5,288301965 08h04m48s 28,38433972 -87,66103681 790,1850934 -1,11076629 -3,533207305 -6,479419106 08h34m48s -75,97695621 -136,383825 814,4905447 1,026715102 7,232703433 -1,446380395 09h04m48s 5,703304937 71,87063826 799,3009307 0,824546512 -1,042516085 7,331538447 09h34m48s 65,71016205 -95,50859825 801,9732373 -3,269061725 -6,05396238 -2,87994884 10h04m48s -40,00249536 -129,4572666 799,7932221 1,865797013 4,528929715 -5,620843078 10h34m48s -32,31779644 55,62843462 806,5582784 3,27940103 2,48781029 6,205955165 11h04m48s 72,50848273 14,51819295 807,1371724 -5,081834178 -5,091163841 1,954788497 11h34m48s -1,948100852 -145,068228 787,8547898 -0,432422658 1,045760237 -7,376550532 12h04m48s -69,21992818 50,83051432 816,2753367 6,394705789 2,953182765 2,409230006 12h34m48s 35,88491373 14,03617718 801,8343939 -3,829515144 -2,358238923 5,94265295 13h04m48s 36,18548965 -161,0238934 792,2247267 -4,527009162 -0,112091154 -5,931712011 13h34m48s -69,21108493 162,1239348 812,34678 6,971840484 1,001156866 -2,422769101 14h04m48s -2,132586252 -1,960966752 799,8635488 0,229952695 -1,111823562 7,36333708 14h34m48s 72,83432776 -160,9531695 803,9995627 -7,04692096 1,506115053 -1,920436196 15h04m48s -32,23154069 157,3328632 796,4341644 4,096589737 -0,298902115 -6,224373986 B.3 Unkown Inputs Parameters Values 87

Table B.6: New PoSAT TLE values calculated using Kalman ﬁlter.

mo no nodeo ecco inclo argpo bstar 15h34m48s 0,423979532 14,302904 0,3398472 0,00090632 0,27365720 0,576211923 -0,001477 16h04m48s 0,423657421 14,302952 0,3398463 0,00090669 0,27365437 0,5765339 -0,0015169 16h34m48s 0,423234319 14,3030 0,33984630 0,00090405 0,273654 0,57695327 -0,0015223 17h04m48s 0,423316883 14,303052 0,3398462 0,00090402 0,27365426 0,57686971 -0,001527 17h34m48s 0,423327519 14,303001 0,3398465 0,00090321 0,27365411 0,576864003 -0,0016309 18h04m48s 0,423163825 14,303030 0,3398462 0,00090312 0,27365411 0,577024396 -0,0016494 18h34m48s 0,423201517 14,303047 0,3398465 0,00090287 0,27365404 0,576983985 -0,0016608 19h04m48s 0,423372491 14,303034 0,3398467 0,00090253 0,27365405 0,576815093 -0,0018033 19h34m48s 0,42349537 14,3030295 0,3398466 0,00090239 0,27365418 0,576692895 -0,0017608 20h04m48s 0,423481677 14,302994 0,3398462 0,00090243 0,27365422 0,576714375 -0,0017708 20h34m48s 0,423472886 14,303008 0,3398461 0,00090258 0,27365407 0,576720088 -0,0016933 21h04m48s 0,423407127 14,303033 0,3398462 0,00090245 0,27365410 0,576779951 -0,0016980 21h34m48s 0,42335529 14,3030201 0,3398461 0,000902628 0,27365448 0,576835463 -0,0017917 22h04m48s 0,423328031 14,302979 0,3398461 0,00090272 0,27365420 0,576874483 -0,0017337 22h34m48s 0,423245189 14,303018 0,3398462 0,00090307 0,27365425 0,576946419 -0,001566 23h04m48s 0,423385574 14,303048 0,3398463 0,00090320 0,27365383 0,576795994 -0,0016885 23h34m48s 0,423273899 14,303024 0,3398462 0,00090324 0,27365384 0,576916058 -0,0016986 24h04m48s 0,423155356 14,303031 0,3398462 0,00090344 0,27365452 0,577032203 -0,0015235 24h34m48s 0,423118896 14,303041 0,3398463 0,00090334 0,27365404 0,577064928 -0,0014870 01h04m48s 0,423186377 14,303081 0,3398459 0,00090325 0,27365419 0,576979664 -0,00145038 01h34m48s 0,42273776 14,3030552 0,339846 0,000903266 0,27365421 0,577439128 -0,0014125 02h04m48s 0,423247999 14,303057 0,3398458 0,0009032 0,27365419 0,576927185 -0,00139548 02h34m48s 0,423165009 14,303046 0,339846 0,000903172 0,27365420 0,577014789 -0,00137029 03h04m48s 0,423438167 14,303038 0,3398467 0,00090329 0,27365423 0,576745601 -0,0013030 03h34m48s 0,423422167 14,303027 0,339846 0,000903313 0,27365417 0,576766059 -0,00128579 04h04m48s 0,423246245 14,303001 0,3398465 0,00090334 0,27365394 0,576957578 -0,0012755 04h34m48s 0,423004126 14,303004 0,3398469 0,00090326 0,27365400 0,577197196 -0,0012496 05h04m48s 0,423239035 14,303045 0,3398460 0,00090345 0,27365437 0,576939306 -0,0011241 05h34m48s 0,423367497 14,303010 0,3398466 0,00090322 0,27365398 0,576831499 -0,0010579 06h04m48s 0,423535276 14,302996 0,3398467 0,00090340 0,27365415 0,576673305 -0,0011503 06h34m48s 0,423476925 14,303021 0,33984647 0,00090333 0,27365464 0,576716011 -0,00124850 07h04m48s 0,423399425 14,303055 0,3398464 0,00090350 0,27365402 0,576770353 -0,0011784 07h34m48s 0,423320707 14,303055 0,3398464 0,00090375 0,27365446 0,576848922 -0,0013796 08h04m48s 0,423080589 14,303014 0,3398467 0,00090380 0,27365470 0,577116512 -0,0013498 08h34m48s 0,423158597 14,303063 0,3398462 0,00090392 0,27365410 0,577004232 -0,0012819 09h04m48s 0,423214616 14,303013 0,3398464 0,00090388 0,27365352 0,576984814 -0,00121172 09h34m48s 0,423063487 14,303044 0,3398466 0,00090375 0,27365416 0,577112109 -0,0011798 10h04m48s 0,423058162 14,303045 0,3398465 0,00090371 0,27365417 0,577116342 -0,00122298 10h34m48s 0,423181861 14,303045 0,3398465 0,00090367 0,2736539 0,576993091 -0,00121203 11h04m48s 0,423160622 14,303029 0,3398464 0,00090368 0,27365413 0,577026397 -0,0012097 11h34m48s 0,423114365 14,303013 0,3398470 0,00090372 0,27365386 0,577086913 -0,0011629 12h04m48s 0,422962057 14,303028 0,3398463 0,00090383 0,27365387 0,577227092 -0,0011203 12h34m48s 0,423212607 14,303002 0,3398461 0,00090382 0,2736545 0,576998955 -0,00105818 13h04m48s 0,423186286 14,303059 0,3398461 0,00090368 0,27365467 0,576973485 -0,00111490 13h34m48s 0,423252914 14,303025 0,3398466 0,00090357 0,27365412 0,576938801 -0,00118772 14h04m48s 0,423131116 14,303027 0,3398466 0,00090365 0,27365440 0,577058486 -0,0012291 14h34m48s 0,423190178 14,303009 0,3398464 0,00090372 0,27365401 0,577016169 -0,0012506 15h04m48s 0,423116568 14,303036 0,3398462 0,00090369 0,27365403 0,57706355 -0,00132246 15h34m48s 0,423332601 14,30298 0,3398467 0,00090348 0,27365409 0,576903105 -0,00135016 16h04m48s 0,422960918 14,302992 0,3398459 0,00090366 0,27365437 0,577264027 -0,0014944 16h34m48s 0,423083207 14,303039 0,3398456 0,00090367 0,27365433 0,577092379 -0,0015029 17h04m48s 0,423220462 14,3029 0,3398462 0,000903486 0,27365459 0,577000847 -0,00161157 17h34m48s 0,423234814 14,303002 0,3398462 0,00090349 0,27365425 0,57698204 -0,00162268 18h04m48s 0,423414689 14,303033 0,3398462 0,00090349 0,27365426 0,576768273 -0,001654707 18h34m48s 0,423587106 14,303050 0,3398462 0,00090355 0,27365406 0,576576465 -0,00162858 19h04m48s 0,423713553 14,303041 0,3398464 0,00090367 0,27365413 0,576458863 -0,00168557 19h34m48s 0,423742913 14,303042 0,3398459 0,00090375 0,27365432 0,576429601 -0,0017113 88 Futher Analysis of the Results

20h04m48s 0,423557787 14,30302389 0,339846418 0,000903665 0,273654558 0,576636145 -0,001775591 20h34m48s 0,4238054 14,30299892 0,339846516 0,000903653 0,27365389 0,576418583 -0,001757707 21h04m48s 0,423533233 14,30303044 0,339846424 0,000903465 0,273654083 0,576652478 -0,001680961 21h34m48s 0,423322708 14,30300691 0,33984666 0,000903413 0,273654161 0,576892674 -0,001711835 22h04m48s 0,423309828 14,30301581 0,339846145 0,000903394 0,273654456 0,576894749 -0,001726449 22h34m48s 0,423196659 14,30301394 0,339846084 0,000903388 0,273654431 0,577010624 -0,001719776 23h04m48s 0,423314724 14,30302827 0,339846459 0,000903262 0,273654268 0,576873825 -0,00172476 23h34m48s 0,423427886 14,30299739 0,33984666 0,000903273 0,273654237 0,576802296 -0,001691405 24h04m48s 0,423311783 14,30303579 0,339846233 0,000903478 0,273654451 0,576865811 -0,001783144 24h34m48s 0,423533456 14,30302871 0,339846013 0,00090353 0,273654063 0,576654176 -0,001805958 01h04m48s 0,423442551 14,30302749 0,339846146 0,000903546 0,273653702 0,576747293 -0,001789992 01h34m48s 0,423331588 14,30300819 0,339846883 0,000903385 0,273653887 0,576885706 -0,001843304 02h04m48s 0,423304957 14,3030531 0,339846545 0,000903263 0,273653944 0,576847271 -0,001812836 02h34m48s 0,423203435 14,3030138 0,339846263 0,000903259 0,273654085 0,577006182 -0,001802794 03h04m48s 0,423334961 14,30306591 0,33984632 0,000903551 0,273654186 0,576796781 -0,001600737 03h34m48s 0,423669804 14,30307211 0,33984627 0,000903522 0,273654058 0,576451316 -0,001593392 04h04m48s 0,423728542 14,30305901 0,339846475 0,000903626 0,273653971 0,576412928 -0,001605116 04h34m48s 0,423418519 14,30302136 0,3398466 0,000903585 0,273654112 0,576780891 -0,001569999 05h04m48s 0,42303663 14,303024 0,339846166 0,00090357 0,273654133 0,577158684 -0,001570661 05h34m48s 0,423236181 14,30299856 0,33984628 0,000903672 0,273654319 0,576999781 -0,001606625 06h04m48s 0,42327367 14,30302724 0,339845606 0,000903662 0,273655385 0,576916134 -0,001644402 06h34m48s 0,423541424 14,30303644 0,339846245 0,000903748 0,273654215 0,576633183 -0,001598261 07h04m48s 0,423092729 14,30301626 0,339846593 0,000903939 0,27365422 0,577115533 -0,00174083 07h34m48s 0,423141916 14,30304635 0,339846537 0,000903819 0,273654114 0,577016088 -0,00180847 08h04m48s 0,423045651 14,30306408 0,339846223 0,000903938 0,273654106 0,577081954 -0,001720679 08h34m48s 0,42313213 14,30302976 0,339846783 0,000904023 0,273652802 0,577053767 -0,001743852 09h04m48s 0,423457426 14,30307913 0,33984589 0,000904003 0,273653967 0,576642108 -0,001787536 09h34m48s 0,423207799 14,30305368 0,339846137 0,000904033 0,273654492 0,576935476 -0,001684014 10h04m48s 0,423190218 14,30304595 0,339846125 0,000903943 0,273653943 0,576966528 -0,001667474 10h34m48s 0,423269015 14,30306568 0,339845928 0,000903848 0,273654708 0,576851648 -0,001621536 11h04m48s 0,423149863 14,30302485 0,339846456 0,000903654 0,27365292 0,577044792 -0,001763372 11h34m48s 0,423217912 14,30304543 0,339846394 0,000903731 0,273653671 0,576938869 -0,001725723 12h04m48s 0,423200425 14,30305528 0,339846224 0,000903736 0,273653897 0,576937143 -0,001746917 12h34m48s 0,4232275 14,30307871 0,339846345 0,000903647 0,273654308 0,576865318 -0,001795521 13h04m48s 0,423256801 14,30304777 0,339846412 0,000903586 0,273654051 0,576894227 -0,001818757 13h34m48s 0,423383494 14,3030665 0,339846063 0,000903567 0,273654488 0,576730501 -0,001827112 14h04m48s 0,423328146 14,30302946 0,339846572 0,000903567 0,273654063 0,576857752 -0,001818298 14h34m48s 0,423243106 14,30301896 0,339846337 0,000903669 0,273654435 0,576963499 -0,001687595 15h04m48s 0,423512278 14,30302994 0,339846137 0,000903743 0,273654128 0,576672307 -0,001671569 Mo - Mean Anomaly; no - Mean Motion; nodeo - Right Ascension of the Ascending Node; ecco - Eccentricity; inclo - Inclination; argpo - Argument of Perigee; Bstar - Drag Term B.3 Unkown Inputs Parameters Values 89

Table B.7: New PoSAT TLE values calculated using Kalman ﬁlter with Unkown Inputs Parameters Values

mo no nodeo ecco inclo argpo bstar 15h34m48s 1,13109 14,33332 0,34178 0,02491 0,27995 -0,12734 -0,00148 16h04m48s 1,12925 14,31335 0,33994 0,01029 0,27362 -0,12706 0,70755 16h34m48s 1,12903 14,15146 0,33989 0,00622 0,27363 -0,11914 1,03941 17h04m48s 1,05173 14,24200 0,33983 0,00402 0,27357 -0,04783 1,39191 17h34m48s 1,06489 14,35210 0,33984 0,00282 0,27368 -0,06964 1,41721 18h04m48s 1,04662 14,29913 0,33988 0,00210 0,27368 -0,04519 1,25718 18h34m48s 1,08264 14,27043 0,33984 0,00180 0,27364 -0,07811 1,36455 19h04m48s 1,09450 14,32646 0,33983 0,00135 0,27366 -0,09848 1,37078 19h34m48s 1,09300 14,31658 0,33985 0,00116 0,27368 -0,09490 1,23845 20h04m48s 1,17685 14,28180 0,33985 0,00096 0,27364 -0,17233 1,25054 20h34m48s 1,21046 14,30960 0,33982 0,00077 0,27353 -0,21201 1,20477 21h04m48s 1,26388 14,31214 0,33985 0,00071 0,27366 -0,26580 1,13488 21h34m48s 1,35500 14,29773 0,33985 0,00067 0,27366 -0,35335 1,13271 22h04m48s 1,38897 14,30061 0,33985 0,00066 0,27365 -0,38814 1,12837 22h34m48s 1,41057 14,30507 0,33985 0,00068 0,27366 -0,41103 1,14604 23h04m48s 1,42186 14,30590 0,33985 0,00069 0,27364 -0,42257 1,13916 23h34m48s 1,41798 14,30100 0,33985 0,00072 0,27365 -0,41708 1,13952 24h04m48s 1,42422 14,30270 0,33985 0,00073 0,27366 -0,42394 1,14741 24h34m48s 1,41667 14,30522 0,33985 0,00074 0,27366 -0,41732 1,14410 01h04m48s 1,42179 14,30181 0,33985 0,00075 0,27365 -0,42107 1,13806 01h34m48s 1,42548 14,30126 0,33985 0,00075 0,27366 -0,42454 1,13878 02h04m48s 1,42249 14,30390 0,33985 0,00077 0,27366 -0,42270 1,14497 02h34m48s 1,42348 14,30307 0,33985 0,00077 0,27366 -0,42327 1,14051 03h04m48s 1,42127 14,30179 0,33985 0,00078 0,27365 -0,42047 1,14373 Mo - Mean Anomaly; no - Mean Motion; nodeo - Right Ascension of the Ascending Node; ecco - Eccentricity; inclo - Inclination; argpo - Argument of Perigee; Bstar - Drag Term 90 Futher Analysis of the Results

Table B.8: New ISS TLE values calculated using Kalman ﬁlter with Unkown Inputs Parameters Values

mo no nodeo ecco inclo argpo bstar 15h34m48s 0,463870221 15,62672775 0,324883188 0,000999821 0,143431904 0,82700274 1,54E-05 16h04m48s 0,487616446 15,62184489 0,324853103 0,001067121 0,143449777 0,803398531 -0,031205791 16h34m48s 0,488518355 15,62126269 0,324854252 0,001081654 0,143449698 0,802522584 -0,039402543 17h04m48s 0,49061139 15,62170077 0,324853308 0,001094039 0,143449859 0,800397706 -0,025599163 17h34m48s 0,489683273 15,6217994 0,324853589 0,001094238 0,143449046 0,801324598 -0,025559934 18h04m48s 0,489423922 15,62180306 0,324853538 0,00109417 0,143449544 0,801584094 -0,025333423 18h34m48s 0,489566116 15,62186683 0,324853741 0,001097778 0,143450065 0,801427208 -0,022109061 19h04m48s 0,489528974 15,62176517 0,324854151 0,001097542 0,14344937 0,801484015 -0,020290299 19h34m48s 0,489493204 15,62185904 0,324854199 0,001097382 0,143449033 0,801502843 -0,01916309 20h04m48s 0,489331569 15,62181244 0,324854197 0,001099504 0,143449633 0,801670187 -0,017676486 20h34m48s 0,489397727 15,6216996 0,324853744 0,001099451 0,143449642 0,801632737 -0,016806393 21h04m48s 0,489418512 15,62188642 0,324853865 0,001099595 0,14344947 0,801566292 -0,014709986 21h34m48s 0,489002522 15,62183356 0,324853664 0,001100389 0,143449575 0,801993943 -0,014336161 22h04m48s 0,489274469 15,62167213 0,324853483 0,00110035 0,143449194 0,801770456 -0,013613204 22h34m48s 0,489228233 15,62187375 0,324853985 0,001100821 0,143449685 0,801753979 -0,011557893 23h04m48s 0,489109911 15,62181986 0,324853929 0,00110145 0,143449457 0,801889594 -0,011355591 23h34m48s 0,489296203 15,62173248 0,324853239 0,001101421 0,143449256 0,801735565 -0,010991731 24h04m48s 0,489322698 15,6218694 0,324854371 0,00110194 0,14344986 0,801656381 -0,009600842 24h34m48s 0,489157635 15,62182698 0,32485395 0,001102244 0,14344927 0,801838008 -0,009561955 01h04m48s 0,489485777 15,62166194 0,324854273 0,001102168 0,143449214 0,801578825 -0,009127155 01h34m48s 0,489268658 15,62182898 0,324854222 0,001102821 0,143449335 0,801723305 -0,007782302 02h04m48s 0,489141292 15,62181674 0,324853834 0,00110287 0,143449676 0,8018574 -0,007793815 02h34m48s 0,489345424 15,62174016 0,324853707 0,001102821 0,143449587 0,801690629 -0,007613051 03h04m48s 0,489256982 15,62182871 0,324853866 0,001103381 0,143449624 0,801733259 -0,006731047 Mo - Mean Anomaly; no - Mean Motion; nodeo - Right Ascension of the Ascending Node; ecco - Eccentricity; inclo - Inclination; argpo - Argument of Perigee; Bstar - Drag Term Appendix C

Futher Analysis of the Results - Continue

C.1 Accuracy of Kalman Filter

This part of the appendix, is intended to support the new TLE values of ISS and PoSAT satellites, with simulations steps of 10, tables C.1 and C.4, 45, tables C.2 and C.5 , and 60 minutes,tables C.3 and C.6 . Table C.1: Kalman Filter Performance with steps of 10 to 10 minutes - ISS

mo no nodeo ecco inclo argpo bstar 15h14m48s 0,3939721 14,3083613 0,33987781 0,00113571 0,27368147 0,606141849 -0,0015117 15h24m48s 0,4097320 14,3058510 0,33984645 0,00100468 0,273654486 0,590423676 -0,0054747 15h34m48s 0,4153698 14,3045826 0,33984992 0,00096940 0,273666331 0,58479681 -0,00572859 15h34m48s 0,4163540 14,3039796 0,33984642 0,0009399 0,273653831 0,583819755 -0,00695409 15h44m48s 0,4191337 14,3033483 0,33984617 0,00092319 0,273653947 0,581050401 -0,0061992 15h54m48s 0,4214634 14,3029504 0,33984649 0,00091846 0,273653862 0,578730109 -0,00462028 16h04m48s 0,4225233 14,3027976 0,33984644 0,00091713 0,273654095 0,577674777 -0,0035678 16h14m48s 0,4226638 14,3028312 0,33984608 0,00091551 0,27365461 0,577532891 -0,002867945 16h24m48s 0,4227971 14,3029379 0,33984639 0,00091370 0,273654099 0,577394616 -0,00222706 16h34m48s 0,4231040 14,3030571 0,33984653 0,00091239 0,273654059 0,577081096 -0,0020076 16h44m48s 0,4235099 14,3031248 0,33984621 0,00091170 0,273654357 0,576671948 -0,0021801 16h54m48s 0,4235721 14,3031699 0,33984655 0,00091120 0,273654378 0,576606585 -0,00254001 Mo - Mean Anomaly; no - Mean Motion; nodeo - Right Ascension of the Ascending Node; ecco - Eccentricity; inclo - Inclination; argpo - Argument of Perigee;Bstar - Drag Term

C.2 Observations only with positions

To support the ﬁnal test of the Kalman Filter performance, table C.7 and C.8 shows the new TLE values of ISS and PoSAT satellites with observations based only with positions.

91 92 Futher Analysis of the Results - Continue

Table C.2: Kalman Filter Performance with steps of 45 to 45 minutes - ISS

mo no nodeo ecco inclo argpo bstar 15h49m48s 0,394228218 14,30685256 0,339845267 0,000819908 0,273653234 0,605860541 -0,001003101 16h34m48s 0,419936385 14,30243295 0,339846354 0,000897286 0,273653987 0,580286012 0,000356225 17h19m48s 0,423346826 14,30296046 0,339846508 0,000898126 0,273654132 0,576846631 0,000230961 18h04m48s 0,4230534 14,30300962 0,339846457 0,000898756 0,273654003 0,577138691 8,88E-05 18h49m48s 0,422909771 14,30303463 0,339846192 0,000899085 0,273653936 0,577276546 0,000172883 19h34m48s 0,423336634 14,30309017 0,339846203 0,000900204 0,273653994 0,576840638 5,83E-05 20h19m48s 0,423420094 14,30301412 0,339846575 0,000900835 0,273654153 0,576771815 4,29E-05 21h04m48s 0,423588865 14,30307714 0,339846092 0,00090129 0,273654048 0,576586941 8,61E-05 21h49m48s 0,423626897 14,30299399 0,339846295 0,000901917 0,273654524 0,576571704 6,62E-06 22h34m48s 0,423502475 14,30300515 0,339845909 0,000902035 0,273654298 0,576692923 3,97E-05 23h19m48s 0,423468091 14,3030102 0,339846682 0,000902144 0,273654171 0,576727074 -1,60E-05 24h04m48s 0,42336269 14,3029967 0,339846262 0,000902261 0,273654261 0,576837498 7,04E-05 Mo - Mean Anomaly; no - Mean Motion; nodeo - Right Ascension of the Ascending Node; ecco - Eccentricity; inclo - Inclination; argpo - Argument of Perigee;Bstar - Drag Term

Table C.3: Kalman Filter Performance with steps of 60 to 60 minutes - ISS

mo no nodeo ecco inclo argpo bstar 16h04m48s 0,593733202 14,3333686 0,340017603 -0,00039182 0,273828658 0,404963103 -0,001564323 17h04m48s 0,583478241 14,28754024 0,33984766 0,000818648 0,273631032 0,41783842 -0,001564238 18h04m48s 0,442813636 14,3001483 0,339846928 0,000460487 0,273654305 0,557747812 0,013895825 19h04m48s 0,427920941 14,30369094 0,339846082 0,000618407 0,273656484 0,572089567 3,83E-02 20h04m48s 0,423424004 14,29969322 0,33984669 0,00068875 0,273653411 0,577487263 0,041176702 21h04m48s 0,422414032 14,3055653 0,339846481 0,000753586 0,273655254 0,577142833 4,42E-02 22h04m48s 0,418628556 14,30184576 0,339847914 0,000789036 0,273649979 0,581882573 5,29E-02 23h04m48s 0,4196957 14,30366877 0,339846287 0,000803968 0,273641268 0,580304315 5,11E-02 24h04m48s 0,418859111 14,30389874 0,339846153 0,00081393 0,273654672 0,580983289 5,65E-02 01h04m48s 0,42342168 14,3019399 0,339847041 0,000825694 0,273653734 0,577232182 5,46E-02 02h04m48s 0,423410435 14,30406047 0,33984583 0,000837547 0,273654635 0,576303391 5,53E-02 03h04m48s 0,42188243 14,30235245 0,339847238 0,000847707 0,273652575 0,578635123 5,84E-02 Mo - Mean Anomaly; no - Mean Motion; nodeo - Right Ascension of the Ascending Node; ecco - Eccentricity; inclo - Inclination; argpo - Argument of Perigee;Bstar - Drag Term

Table C.4: Kalman Filter Performance with steps of 10 to 10 minutes - PoSAT

mo no nodeo ecco inclo argpo bstar 15h14m48s 0,393972156 14,3083613 0,339877812 0,001135715 0,27368147 0,60614184 -0,001511794 15h24m48s 0,409732064 14,3058510 0,339846453 0,001004684 0,27365448 0,59042367 -0,005474752 15h34m48s 0,415369871 14,3045826 0,339849924 0,000969409 0,27366633 0,58479681 -0,005728593 15h34m48s 0,416354091 14,3039796 0,339846428 0,000939961 0,27365383 0,58381975 -0,006954094 15h44m48s 0,419133734 14,3033483 0,339846176 0,000923191 0,27365394 0,58105040 -0,006199296 15h54m48s 0,421463497 14,3029504 0,339846499 0,000918469 0,27365386 0,57873010 -0,004620282 16h04m48s 0,422523331 14,3027976 0,339846442 0,000917131 0,27365409 0,57767477 -0,003567895 16h14m48s 0,422663856 14,3028312 0,339846081 0,000915512 0,27365461 0,57753289 -0,002867945 16h24m48s 0,422797193 14,3029379 0,339846395 0,000913706 0,27365409 0,57739461 -0,002227068 16h34m48s 0,423104094 14,3030571 0,339846531 0,000912399 0,27365405 0,57708109 -0,002007698 16h44m48s 0,423509948 14,3031248 0,339846213 0,000911706 0,27365435 0,57667194 -0,002180106 16h54m48s 0,423572178 14,3031699 0,339846553 0,000911209 0,27365437 0,57660658 -0,002540015 Mo - Mean Anomaly; no - Mean Motion; nodeo - Right Ascension of the Ascending Node; ecco - Eccentricity; inclo - Inclination; argpo - Argument of Perigee;Bstar - Drag Term C.2 Observations only with positions 93

Table C.5: Kalman Filter Performance with steps of 45 to 45 minutes - PoSAT

mo no nodeo ecco inclo argpo bstar 15h49m48s 0,39422821 14,3068525 0,339845267 0,00081990 0,273653234 0,605860541 -0,0010031 16h34m48s 0,41993638 14,3024329 0,339846354 0,00089728 0,273653987 0,580286012 0,0003562 17h19m48s 0,42334682 14,3029604 0,339846508 0,00089812 0,273654132 0,576846631 0,0002309 18h04m48s 0,4230534 14,3030096 0,339846457 0,00089875 0,273654003 0,577138691 8,88E-05 18h49m48s 0,42290977 14,3030346 0,339846192 0,00089908 0,273653936 0,577276546 0,0001728 19h34m48s 0,42333663 14,3030901 0,339846203 0,00090020 0,273653994 0,576840638 5,83E-05 20h19m48s 0,42342009 14,3030141 0,339846575 0,00090083 0,273654153 0,576771815 4,29E-05 21h04m48s 0,42358886 14,3030771 0,339846092 0,00090129 0,273654048 0,576586941 8,61E-05 21h49m48s 0,42362689 14,3029939 0,339846295 0,00090191 0,273654524 0,576571704 6,62E-06 22h34m48s 0,42350247 14,3030051 0,339845909 0,00090203 0,273654298 0,576692923 3,97E-05 23h19m48s 0,42346809 14,3030102 0,339846682 0,00090214 0,273654171 0,576727074 -1,60E-05 24h04m48s 0,42336269 14,3029967 0,339846262 0,00090226 0,273654261 0,576837498 7,04E-05 Mo - Mean Anomaly; no - Mean Motion; nodeo - Right Ascension of the Ascending Node; ecco - Eccentricity; inclo - Inclination; argpo - Argument of Perigee;Bstar - Drag Term

Table C.6: Kalman Filter Performance with steps of 60 to 60 minutes - PoSAT

mo no nodeo ecco inclo argpo bstar 16h04m48s 0,59373320 14,333368 0,34001760 -0,00039182 0,27382865 0,40496310 -0,001564323 17h04m48s 0,58347824 14,287540 0,33984766 0,000818648 0,27363103 0,41783842 -0,001564238 18h04m48s 0,44281363 14,300148 0,33984692 0,000460487 0,27365430 0,55774781 0,013895825 19h04m48s 0,42792094 14,303690 0,33984608 0,000618407 0,27365648 0,57208956 3,83E-02 20h04m48s 0,42342400 14,299693 0,33984669 0,00068875 0,273653411 0,577487263 0,041176702 21h04m48s 0,42241403 14,305565 0,33984648 0,000753586 0,27365525 0,57714283 4,42E-02 22h04m48s 0,41862855 14,301845 0,33984791 0,000789036 0,27364997 0,58188257 5,29E-02 23h04m48s 0,4196957 14,303668 0,33984628 0,000803968 0,27364126 0,58030431 5,11E-02 24h04m48s 0,41885911 14,303898 0,33984615 0,00081393 0,273654672 0,580983289 5,65E-02 01h04m48s 0,42342168 14,301939 0,33984704 0,000825694 0,27365373 0,57723218 5,46E-02 02h04m48s 0,42341043 14,304060 0,33984583 0,000837547 0,27365463 0,57630339 5,53E-02 03h04m48s 0,42188243 14,302352 0,33984723 0,000847707 0,27365257 0,57863512 5,84E-02 Mo - Mean Anomaly; no - Mean Motion; nodeo - Right Ascension of the Ascending Node; ecco - Eccentricity; inclo - Inclination; argpo - Argument of Perigee;Bstar - Drag Term 94 Futher Analysis of the Results - Continue

Table C.7: New ISS TLE values calculated using Kalman ﬁlter with Observations only with positions

mo no nodeo ecco inclo argpo bstar 15h34m48s 0,496227629 15,62190755 0,321757279 0,001193601 0,140849654 0,796704094 -0,000182627 16h04m48s 0,502588749 15,62139621 0,324879409 0,001036606 0,143377621 0,788483131 -0,000170957 16h34m48s 0,502247874 15,62232689 0,32485327 0,000957788 0,143445017 0,788758732 -0,000171337 17h04m48s 0,50445959 15,6228955 0,324817194 0,000978578 0,143444312 0,786403303 -0,000169049 17h34m48s 0,508166258 15,62218066 0,324865787 0,000931279 0,143474844 0,782792808 -0,000156846 18h04m48s 0,509242459 15,6213238 0,324869682 0,001002812 0,143504727 0,781842799 -0,000164492 18h34m48s 0,507169886 15,62099763 0,324847479 0,000998043 0,143510275 0,783914146 -0,000162775 19h04m48s 0,506869623 15,62106895 0,324838607 0,001003444 0,14347944 0,784255265 -0,000164621 19h34m48s 0,504315141 15,62197851 0,324871504 0,000932438 0,14342911 0,786692534 -0,000136895 20h04m48s 0,508467618 15,62255887 0,324861531 0,000934779 0,143424083 0,78234049 -0,000145109 20h34m48s 0,507315941 15,62279454 0,324854315 0,000953402 0,143414216 0,783438089 -0,000156264 21h04m48s 0,505386394 15,62326658 0,324851066 0,000919396 0,143438981 0,785309586 -0,000130682 21h34m48s 0,502573692 15,62290946 0,324831003 0,000924005 0,143446777 0,788105847 -0,00012157 22h04m48s 0,503278605 15,62272218 0,324834971 0,000907953 0,143456092 0,787453725 -0,000109788 22h34m48s 0,502906588 15,62278866 0,324850254 0,00090398 0,143450586 0,787846113 -0,000105092 23h04m48s 0,501792523 15,62264929 0,324875615 0,000907963 0,143468909 0,788880268 -0,00010077 23h34m48s 0,500891469 15,62301284 0,324871926 0,00094014 0,143424136 0,789672657 -0,000130289 24h04m48s 0,5015701 15,62292263 0,324868731 0,000945391 0,143434165 0,789054368 -0,000140047 24h34m48s 0,498400408 15,62255062 0,324820367 0,000959148 0,143432336 0,792297613 -0,000129816 01h04m48s 0,49845917 15,62249031 0,324820357 0,000953745 0,143444169 0,792262287 -0,000123017 01h34m48s 0,50155509 15,62214472 0,324845688 0,000969321 0,143436082 0,789349601 -0,000163091 02h04m48s 0,50256823 15,62226401 0,324801149 0,000964121 0,143427811 0,78821029 -0,000164785 02h34m48s 0,502438251 15,62181894 0,324802392 0,000924887 0,14344957 0,788556678 -9,87E-05 03h04m48s 0,50240343 15,62182155 0,324862017 0,000925023 0,143421909 0,788620887 -9,93E-05 03h34m48s 0,500036477 15,62149837 0,324861752 0,000942987 0,143429329 0,791136516 -0,000106247 04h04m48s 0,50161067 15,62226974 0,324850889 0,001008359 0,143411709 0,789115714 -0,000227887 04h34m48s 0,499526778 15,62245105 0,324857547 0,001004462 0,143407039 0,791130126 -0,000211837 Mo - Mean Anomaly; no - Mean Motion; nodeo - Right Ascension of the Ascending Node; ecco - Eccentricity; inclo - Inclination; argpo - Argument of Perigee;Bstar - Drag Term C.2 Observations only with positions 95

Table C.8: New ISS TLE values calculated using Kalman ﬁlter with Observations only with positions

mo no nodeo ecco inclo argpo bstar 15h34m48s 0,4185561 14,29954951 0,3433178 0,00103857 0,274341228 0,581381159 -0,0014496 16h04m48s 0,419119 14,29994574 0,3403102 0,00105336 0,274275347 0,581250798 -0,0014495 16h34m48s 0,4105275 14,302316 0,33983537 0,00091342 0,273719828 0,589683997 -0,00144941 17h04m48s 0,4111358 14,30255599 0,3398726 0,00089239 0,273655208 0,589073448 -0,00144952 17h34m48s 0,4112399 14,30258385 0,3398669 0,00089463 0,273654136 0,588976301 -0,00144950 18h04m48s 0,4056175 14,30338028 0,3398659 0,00091097 0,273640387 0,594518735 -0,00144999 18h34m48s 0,4057616 14,30323232 0,3398612 0,00092499 0,273642808 0,594370656 -0,00145004 19h04m48s 0,4037086 14,30297324 0,3398792 0,00091771 0,273667111 0,596479621 -0,0014502 19h34m48s 0,4054688 14,30265243 0,3398780 0,00089123 0,273663354 0,594814379 -0,0014498 20h04m48s 0,4092844 14,30200813 0,3398325 0,00093658 0,273645883 0,591056282 -0,0014505 20h34m48s 0,4146213 14,302614 0,33982886 0,00091328 0,273658187 0,585656505 -0,0014512 21h04m48s 0,4147112 14,30307 0,33985797 0,00095779 0,273647953 0,585481439 -0,0014510 21h34m48s 0,4120927 14,30332126 0,3398565 0,00095086 0,273642926 0,588007544 -0,0014508 22h04m48s 0,4156136 14,30402908 0,3398348 0,00089274 0,273675601 0,584274989 -0,0014520 22h34m48s 0,4176089 14,30432974 0,3398216 0,00091465 0,273672321 0,582211699 -0,0014513 23h04m48s 0,4144215 14,30462013 0,3398329 0,00092388 0,273616282 0,585206865 -0,0014518 23h34m48s 0,4145500 14,30455075 0,3398504 0,00093030 0,273612512 0,585085643 -0,0014519 24h04m48s 0,4179256 14,30485753 0,3398627 0,00093594 0,273632533 0,581592263 -0,0014515 24h34m48s 0,4210537 14,30434814 0,3398539 0,00089425 0,273642647 0,578616023 -0,0014504 01h04m48s 0,4219640 14,30423307 0,3398148 0,00090200 0,27362135 0,577714109 -0,001450762 01h34m48s 0,4237670 14,30440977 0,3398140 0,00089513 0,273644918 0,575806731 -0,0014511 02h04m48s 0,4237295 14,30445352 0,3397777 0,00089904 0,273650407 0,57575369 -0,00145115 02h34m48s 0,4251015 14,30433992 0,3397776 0,00090113 0,273649499 0,574427815 -0,00145124 03h04m48s 0,4241064 14,30414367 0,3398014 0,00091712 0,273631874 0,575529466 -0,0014506 03h34m48s 0,4242001 14,30415904 0,3398560 0,00091832 0,273662728 0,575385351 -0,0014505 04h04m48s 0,4294592 14,30370635 0,3398532 0,00089945 0,273677252 0,570357855 -0,0014491 04h34m48s 0,4295329 14,30367584 0,3398834 0,00090217 0,273674881 0,570330065 -0,0014491 Mo - Mean Anomaly; no - Mean Motion; nodeo - Right Ascension of the Ascending Node; ecco - Eccentricity; inclo - Inclination; argpo - Argument of Perigee;Bstar - Drag Term 96 Futher Analysis of the Results - Continue References

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