DEGREE PROJECT, IN SPACE TECHNOLOGY , SECOND LEVEL STOCKHOLM, SWEDEN 2014
Unscented Kalman Filters for Attitude and Orbit Estimation of a Low Earth Orbit CubeSat
VLAD GRIGORE
KTH ROYAL INSTITUTE OF TECHNOLOGY
AEROSPACE ENGINEERING, SPACE AND PLASMA PHYSICS
Royal Institute of Technology
Master Thesis
XR-EE-SPP 2014:005
Unscented Kalman Filters for Attitude and Orbit Estimation of a Low Earth
Orbit CubeSat
Author:
Vlad Grigore
Supervisor:
Nickolay Ivchenko
A thesis submitted in fulfilment of the requirements for the degree of Master of Science
at the
Department of Space and Plasma Physics
January 2015
Declaration of Authorship
I, Vlad Grigore, declare that this thesis titled, ’Unscented Kalman Filters for Attitude and Orbit Estimation of a Low Earth Orbit CubeSat’ and the work presented in it are my own. I confirm that:
This work was done wholly or mainly while in candidature for a research degree at this University.
Where any part of this thesis has previously been submitted for a degree or any other qualification at this University or any other institution, this has been clearly stated.
Where I have consulted the published work of others, this is always clearly at- tributed.
Where I have quoted from the work of others, the source is always given. With the exception of such quotations, this thesis is entirely my own work.
I have acknowledged all main sources of help.
Where the thesis is based on work done by myself jointly with others, I have made clear exactly what was done by others and what I have contributed myself.
Signed:
Date:
iii 2015.01.06
Grigore Vlad
ROYAL INSTITUTE OF TECHNOLOGY
Abstract
Faculty of Mechanical Engineering
Master of Science
Unscented Kalman Filters for Attitude and Orbit Estimation of a Low Earth Orbit CubeSat
by Vlad Grigore
In this paper two Unscented Kalman Filters (UKF) are implemented to solve the estimation of a satellite’s position in orbit and its orientation relative to Earth’s Centered Inertial frame. Their aim is to see if an absolute position accuracy of less than 1 km and an orientation estimation to less than 10 are possible for a 3U CubeSat with GPS, sun sensors, magnetometers and a star tracker as sensors.
The orbit UKF is based on a Runge Kutta 7(8)th-order integration method for orbit propagation. The dynamic model uses perturbational accelerations due to Earth’s geo- potential, the gravity of the Moon and the Sun, atmospheric drag and solar pressure.
The state of the filter is identical to the observation vector and consists of position and velocity vectors of the satellite. The GPS receiver is used for measurements. Emphasis has been put on the sampling rate of the GPS receiver as its availability is constrained by the satellite’s power limitations. The state of the attitude UKF consists of an error quaternion, angular velocities and magnetometer bias. The sun sensors, magnetometers and the star tracker are used for measurements. Different sensor combinations have been simulated with the purpose of determining their influence on the estimated attitude.
The results of the simulations indicate that an absolute position error of less than 1 km is feasible for sampling times of 15 minutes or less (approximately 6 per orbit).
Furthermore an accuracy of less than 10 is achieved by using the star tracker for attitude determination either alone or in combination with the other sensors.
Acknowledgements
I would like to express my great appreciation to Nickolay Ivchenko for his guid- ance, suggestions and useful feedback during the planning, development and writing of the present thesis. I would also like to acknowledge the support and congeniality of the people working at GomSpace. I am particularly grateful for the opportunity to work on a real space project alongside a dynamic and diverse team.
I would like to thank Gunnar Tibert for his lectured course at KTH which aroused my interest and gave me the confidence to pursue the subject of spacecraft dynamics.
My special thanks are extended to my fellow colleagues and students at KTH in particular to Amin Nezami for his moral and support and encouragement during the writing of this thesis.
I would also like to acknowledge the support provided by my family during the preparation of this thesis and throughout my studies abroad.
vii
Contents
Declaration of Authorship iii
Abstract v
Acknowledgements vii
Contents viii
List of Figures xiii
List of Tables xv
Abbreviations xvii
Physical Constants xix
Symbols xxi
1 Introduction 1
1.1 Background . . . 1
1.2 Mission goal . . . 3
1.2.1 Objectives. . . 3
1.2.2 Requirements . . . 3
1.2.3 Spacecraft description . . . 4
1.2.4 Orbital information . . . 4
1.3 Motivation . . . 5
1.3.1 Attitude determination . . . 5
1.3.2 Orbit determination . . . 6
1.4 Scope and road map . . . 7
2 Method 9 2.1 Kinematics . . . 9
2.1.1 Reference systems . . . 9
2.1.2 Rotations . . . 10
2.2 The Attitude and Orbit Determination System . . . 11
2.2.1 The need for attitude determination . . . 11
2.2.2 The need for orbit determination . . . 11 ix
Contents x
2.3 Unscented Kalman Filter theory . . . 12
2.3.1 Filter initialization . . . 13
2.3.2 Prediction phase . . . 14
2.3.3 Update phase . . . 15
3 AODS 17 3.1 Attitude determination. . . 17
3.1.1 Approach . . . 17
3.1.2 Available sensors . . . 17
3.1.2.1 Sun sensors . . . 17
3.1.2.2 Magnetometer . . . 18
3.1.2.3 Star trackers . . . 19
3.1.3 Prefilter . . . 19
3.1.4 Dynamic model. . . 20
3.1.5 Measurement model . . . 22
3.2 Orbit determination . . . 22
3.2.1 Approach . . . 22
3.2.2 Dual-Frequency GNSS Receiver . . . 23
3.2.3 Dynamic model. . . 23
3.2.4 Measurement model . . . 25
3.2.5 Force model . . . 25
3.2.5.1 Geo-potential. . . 26
3.2.5.2 Sun and Moon . . . 27
3.2.5.3 Atmospheric drag . . . 27
3.2.5.4 Solar radiation . . . 27
4 Simulation 29 4.1 Attitude determination system . . . 29
4.1.1 Simulator description . . . 29
4.1.2 Performance verification . . . 30
4.1.2.1 Initial state vector . . . 30
4.1.2.2 Combinations of sensors. . . 30
4.1.2.3 Sampling of the star tracker . . . 31
4.2 Orbit determination system . . . 31
4.2.1 Simulator description . . . 31
4.2.2 Performance verification . . . 31
4.2.2.1 Nominal operation . . . 31
4.2.2.2 Measurement noise implications . . . 32
4.2.2.3 GPS duty cycle implications . . . 32
5 Results 33 5.1 Attitude determination. . . 33
5.1.1 Initial state vector . . . 33
5.1.2 Combinations of sensors . . . 35
5.1.3 Sampling of the star tracker . . . 36
5.2 Orbit determination . . . 39
5.2.1 Nominal operation . . . 39
Contents xi
5.2.2 Measurement noise implications. . . 39 5.2.3 GPS duty cycle implications. . . 43
6 Conclusion 45
References 47
List of Figures
1.1 The SEAM CubeSat with deployed solar panels and transmission anten-
nas (Zimmerhakl, 2014) . . . 5
3.1 Sensor distribution for the SEAM CubeSat (ss - sun sensor; mt - magne- torquer; +X refers to positive direction of axis in Figure 1.1) . . . 18
3.2 Magnetometers used by the SEAM CubeSat (GomSpace, 2014c) . . . 19
3.3 Current design of the star tracker used by the attitude determination system (Shterev, 2014) . . . 20
3.4 OEM615 Dual-Frequency GNSS Receiver (NovAtel, 2014) . . . 23
4.1 Attitude UKF simulator . . . 29
4.2 Orbit UKF simulator. . . 32
5.1 Filter state initialization comparison . . . 34
5.2 Estimated measurements from sigma points and actual filtered measure- ments . . . 37
5.3 Different combinations scenarios of the attitude determination sensors . . 38
5.4 Qualitative depiction of the orbit UKF . . . 40
5.5 Absolute error in position and velocity . . . 41
5.6 Residuals and RMDS for 3 measurement noise cases. . . 42
xiii
List of Tables
5.1 Orientation error of the satellite for a particular combination of attitude sensors. . . 35 5.2 Position error including the filter convergence for 2 different geo-potential
models and different sampling times for the GPS measurements. . . 43
xv
Abbreviations
AU Astronomical Unit ACS Attitude Control System
AOCS Attitude and Orbit Control System AODS Attitude and Orbit Determination System CMOS Complementary Metal-Oxide-Semiconductor ECI Earth-Centered Inertial
EKF Extended Kalman Filter ELF Extremely Low Frequency
GNSS Global Navigation Satellite System GPS Global Positioning System
KF Kalman Filter
KTH Kungliga Tekniska H¨ogskolan LEO Low Earth Orbit
MMSE Minimum Mean Square Error UKF Unscented Kalman Filter UT Unscented Transform
SEAM Small Explorer for Advanced Missions SSO Sun-Synchronous Orbit
TLE Two Line Element VLF Very Low Frequency
xvii
Physical Constants
Astronomical Unit a = 1.496 × 108 km
Earth Standard Gravitational Parameter µ⊕ = 398 600.4418 km3s-2 Gravitational Constant G = 6.67 × 10−11Nm2kg-2 Moon Standard Gravitational Parameter µm = 4 902.8000 km3s-2
Solar Radiation Constant Ps = 1358/c Wm-2
Speed of Light c = 2.997 924 58 × 108 ms−s
Sun Standard Gravitational Parameter µs = 132 712 440 018 km3s-2
xix
Symbols
aa perturbational acceleration vector due to atmospheric drag ad total perturbational acceleration vector
ag perturbational acceleration vector due to Earth’s gravity an perturbational acceleration vector due to Sun and Moon ap perturbational acceleration vector due to solar pressure bmag magnetometer bias vector
ˆ
e Euler principal axis
q rotation quaternion
δq error quaternion
r satellite position vector
v satellite velocity vector
vmg inertial magnetometer vector vss inertial sun sensor vector
x state vector
z transformed measurements vector
A attitude matrix
Isat inertia matrix of the satellite
K Kalman gain
L number of elements in the state vector
Lc torque vector
P covariance matrix
Pxkzk state-measurement cross-covariance matrix Pzkzk measurement covariance matrix
Q process noise covariance matrix
R measurement noise covariance matrix
Z predicted measurements vector
xxi
Symbols xxii
f dynamic system model
h sensor model
m satellite’s mass
A satellite’s area of reference
Cd drag coefficient
Csr solar radiation parameter W0(m), W0(c), Wi(m) UKF weights
α, β, κ UKF constants
λ scaling parameter
ρ air density
χ sigma point vector
ω angular velocity vector
Φ Euler principal angle
Φ(r, φ, λ) Earth’s geo-potential function of geocentric distance, latitude and longitude
Ω skew-symmetric function
Chapter 1
Introduction
1.1 Background
Microsatellites (10-100 kg) and nanosatellites (1-10 kg) have become a present reality of space research with the industry growing rapidly in recent years. Their reduced cost and the use of cheaper launch vehicles (often launched ”piggyback”) make them more attractive than small satellites (100-500 kg) (Buchen & DePasquale, 2014). One type of miniaturized satellite for space research is the CubeSat, a 10 cm by 10 cm cube (1U) that has a volume of exactly one litre and a mass that does not exceed 1.33 kg, although different variants exist: 1.5U, 2U, 3U (Mehrparvar,2014).
The program began as a collaboration between the California Polytechnic State University and Stanford University’s Space Systems Development Laboratory in 1999.
Currently the project spans over 100 universities worldwide as well as various high schools and private firms. The main purpose was to create a standard for designing nanosatellites as to reduce development cost and time, increase accesibillity to space and sustain frequent launches. To date a considerable number of CubeSats have been launched. Their missions however focused manly on new technology demonstrations (Buchen & DePasquale, 2014). The case for Earth observation and remote sensing missions rises challenges that have just yet started to be addressed manly due to space limitation and sensor capabilities.
The Small Explorer for Advanced Missions (SEAM) project proposes to design and build a 3U CubeSat that serves as a platform for advanced scientific missions. The platform will be then made available to universities, private companies or any third party for a relatively low cost. The goal of the mission is to ”develop and demonstrate in
1
Chapter 1. Introduction 2
flight for the first time a concept of an electromagnetically clean nanosatellite with preci- sion attitude determination, flexible autonomous data acquisition system, high-bandwidth telemetry and an integrated solution for ground control and data handling” (Ivchenko, 2014b).
The present study aims at designing the Attitude and Orbit Determination Systems (AODS) required by the SEAM project to properly fulfil its objectives. As a starting point the simulation environment and the actual flight code of a previous CubeSat mission are used. The GOMX-1 CubeSat is a 2U spacecraft developed by GomSpace, DSE Airport Solutions and Aalborg University and launched in November 2013. The satellite carries a payload capable of tracking trans-oceanic flights by receiving ADS-B signals from the aircraft. The secondary payload, a color camera, is used for Earth observations. Following the launch the satellite’s control system managed to detumble the spacecraft and all deployables performed successfully (GomSpace,2014a).
The simulation environment provided by GomSpace consists of a Simulink model emulating the environment, the ephemeris, the spacecraft dynamics and the on-board code. The sensor emulation block is included in the spacecraft dynamics one. The output of the spacecraft dynamics block consists of 2 type of sensors the satellite is equipped with: magnetometers and sun sensors while the input amounts to the torques received from the magnetorquers. Disturbances are also modelled into these blocks.
The information from the sensors is passed on to the on-board software block which is written in C and wrapped into Matlab. This block contains the attitude determination and control software outputting the magnetic torques which are then fed to the spacecraft dynamics block.
The attitude determination software is based on the theory of the Unscented Kalman Filter (UKF) which is used to estimate the satellite’s attitude. Then based on actual measurements from the sensors the predicted attitude is corrected. The filter in its current form estimates 16 states: a rotation quaternion, angular velocity, gyro- scope and magnetometer bias and quasi-static torque (GomSpace,2011). Furthermore it uses measurement information from 3 different sources: magnetometers, gyroscopes and sun sensors. The SEAM CubeSat however is not equipped with gyroscopes but has two star trackers which on their own output an estimated quaternion. The star trackers are a necessity in order to achieve the precision required for attitude determination. The implementation of these star trackers into the estimation algorithm is part of the current thesis.
Chapter 1. Introduction 3
The current GomSpace model uses the simplified perturbations model (SGP4) to keep track of the satellite in its orbit. The SGP4 orbit propagation is based on the two-line element (TLE) set data format used to convey sets of orbital elements that describe the orbits of Earth-orbit satellites. The format is specified by NORAD. The orbital elements are also determined by NORAD and are freely distributed (Hoots &
Roehrich,1980). The SEAM CubeSat has a more stringent requirement for the known position which can not be achieved by using orbital dynamics and the TLE. Thus a global positioning system (GPS) receiver is used to accurately determine the satellite’s position in orbit. A second UKF filter is designed for this purpose.
1.2 Mission goal
The goal of the SEAM project is to develop and build a set of subsystems that fit the en- velope of a 3U CubeSat with the purpose of creating an affordable platform for advanced scientific experiments. In order to achieve this goal several companies and research orga- nizations join their expertise in various fields including advanced instrumentation, data processing and telemetry systems (Ivchenko,2014a).
1.2.1 Objectives
The objectives of the SEAM project include the development of an electromagnetically clean CubeSat with a flexible data acquisition system, high-band telemetry and sup- ported by ground control and data handling (Ivchenko,2014a).
The satellite itself has two objectives. The first is to provide high resolution mea- surements of Earth’s DC and AC magnetic fields by characterizing the auroral current systems, monitoring the natural very low frequency (VLF) and extremely low frequency (ELF) waves and observing the anthropogenic VLF and ELF waves. The second objec- tive aims at proving the flight capability of novel subsystems developed by the companies involved (Ivchenko,2014b).
1.2.2 Requirements
The project and spacecraft as a whole have numerous requirements but only the ones pertaining to the objective of this thesis are presented here while the remaining require- ments are available in the cited documentation.
Chapter 1. Introduction 4
From the point of view of the orbit determination the requirement states that the position of the spacecraft shall be known with an error less than 1 km. This requirement is closely linked to the science objective of the mission since all the data collected by the satellite need to be correlated with ground observations. Another requirement arising from the scientific objective is the inclination of the spacecraft’s orbit which needs to be 75◦ or more in order to maximize the auroral oval crossings.
The accuracy of the DC magnetic field measurement is the primary driver for the requirement in the attitude determination. Thus the attitude knowledge of the flux gate sensors needs to be within a 10 margin. The requirement for data transmission is not as strict as the satellite’s antennas only have to approximately point towards nadir when the satellite passes in the range of the ground stations. Furthermore the spacecraft does not use active control to reorient itself and the antennas for data transmission.
1.2.3 Spacecraft description
The design of the SEAM satellite follows the 3U CubeSat design specifications. Apart from the standard 10 by 10 by 30 cm frame housing the hardware, the spacecraft is also equipped with 2 deployable booms and solar panels. The sensors responsible for taking scientific measurements include fluxgate and search-coil magnetometers for DC and AC magnetic field measurements respectively, an S-band transceiver for satellite to ground communications and a GPS receiver for positioning and timing. Sun sensors and star trackers are also available and used in determining the satellite’s attitude.
Figure1.1shows the SEAM CubeSat with deployed booms and solar panels and the satellite reference frame. The satellite has two separate electronics stacks: the GomSpace stack containing the batteries and vital hardware (-Z side) and the science stack with the star tracker, S-band transceiver and antennas (+Z side) (GomSpace,2014c). In between the two stacks there is the boom assembly. The boom deploys along the +X and -X axes. The science instruments as well as a star tracker are mounted around the boom tips. This is a design requirement as measurements of Earth’s magnetic field should not be affected by magnetic interference from the satellite.
1.2.4 Orbital information
The SEAM CubeSat will be placed in a Sun-synchronous orbit (SSO) with an ascending node of 2100 at an altitude of 600 km and an inclination of 97◦. Based on the availability
Chapter 1. Introduction 5
Figure 1.1: The SEAM CubeSat with deployed solar panels and transmission antennas (Zimmerhakl,2014)
and position of the solar panels a mean power of 7.2 W is to be expected. Using an integrator model for the orbital dynamics the decay time is found to be around 20 years well above the designed mission lifetime of 1 year. Contact will made at 2 ground stations, one in Kiruna, Sweden and one in Yatharaaga, Australia, giving a mean access time of 48 and 12 minutes respectively assuming it can be achieved at an elevation of 10◦ and above (GomSpace,2014c).
1.3 Motivation
1.3.1 Attitude determination
The attitude of a spacecraft represents its orientation in relation to a given reference frame. The rotational motion about the center of mass of the spacecraft is called atti- tude motion. Knowledge about the attitude and its motion is of great value especially for the attitude control system (ACS) which deals with the reorientation of the space- craft. According to the mission specifications, for example, the spacecraft might need to perform different functions like pointing the solar panels towards the Sun or an antenna towards a station on the ground.
Attitude determination is achieved by taking measurements from the satellite’s sensors. These measurements are then compared with known observations like ephemeris
Chapter 1. Introduction 6
models, magnetic field models or in the case of star trackers with star maps. The requirement for attitude determination plays a vital role in the success of the overall mission. The concept is fundamentally linked to the science objectives the spacecraft needs to achieve. While attitude determination of CubeSats has become common in recent years the precision requirement of 10 sets some challenges. Most common attitude sensors are sun sensors (30%), magnetometers (30%) and gyroscopes (10%) while star trackers are the least common (5%) according to Bouwmeester and Guo (2010) based on 94 pico- and nanosatellites from 1957 to 2009.
One of the key concepts of the SEAM project is to merge data from various high- performance attitude sensors using a principle called sensor fusion. Information deriving from various sources combines in such a way that the result is more accurate than the individual data (Elmenreich,2002). Thus information from sun sensors, magnetometers and star trackers is merged in order to achieve the attitude accuracy.
1.3.2 Orbit determination
The orbit of a spacecraft is connected to its attitude. They are interdependent quanti- ties each affecting the other one. In low Earth orbit (LEO) their relation is extremely important because of the strength of the magnetic field and the density of the atmo- sphere. For example the drag forces a spacecraft experiences while passing through the atmosphere are different depending on its orientation. At the same time an orbit with a lower altitude passes through denser areas of the atmosphere, with stronger effects on its attitude, than one at a higher altitude. Earth’s magnetic field acts in a similar manner introducing forces in the system for spacecraft with magnetizable components and modifying the attitude.
By providing accurate knowledge of the position and attitude of the spacecraft the AODS ensures that the scientific mission is carried out successfully and also that the high-bandwidth downlink is reliable. Satellites orbiting in LEO can acquire information about their position and velocity by using a GPS receiver. Sampling a GPS receiver too often might be outside the resource budget of the spacecraft thus an equation of motion integrator can be used to propagate the orbit and the GPS sampling can be used for correction.
Chapter 1. Introduction 7
1.4 Scope and road map
The main goal of this thesis is to develop simulation models for both attitude and orbit determination of the SEAM CubeSat in order to demonstrate that the precision requirements can be met within the project. Two separate filters, one for attitude and another one for the orbit, are designed and their performance is asserted through simulations.
Firstly, in Chapter2, attitude and orbit determination concepts are presented. The reference frames used throughout this thesis are outlined as well as some key aspects of rotation formalisms in 3 dimensions. The need for an attitude and orbit determination system is also discussed followed by specifically stating the dynamic models involved.
The theory for the unscented Kalman filter is presented.
Secondly, in Chapter 3, the sensors available on the SEAM CubeSat are discussed and their role in the filters is explained. Furthermore both the dynamic and measurement models for each of the filters and their implications are discussed separately.
Thirdly, Chapter4 contains the set-up for the simulation of the filters. A detailed description and a performance verification is carried out for each of them individually.
Lastly, Chapters 5 and 6 show the results of the simulations and the conclusion respectively.
Chapter 2
Method
2.1 Kinematics
2.1.1 Reference systems
The problem of attitude determination is closely related to reference systems. Firstly a Cartesian coordinate system needs to be defined before the orientation of a satellite can be discussed. Throughout this thesis 3 reference systems are be used: an inertial reference frame, a satellite body frame and controller frame.
The inertial reference frame has its origin in the center of Earth. The x-axis is directed along the vernal equinox while the plane formed by the x- and y-axis is parallel to the equatorial plane. Lastly the z-axis is defined by the cross product between the x- and y-axis and points towards the North Pole. This particular reference system is also called Earth-centered inertial (ECI) (Hall,2003).
The AODS hardware as well as the sensors measurements are expressed in the satellite body frame. This particular reference frame has its origin in one of the corners of the SEAM CubeSat. By aligning the axis of the frame with the structure of the satellite the physical placement of the various sensors and components can be easily described. Figure1.1in Chapter 1shows how the frame is orientated in the case of the SEAM satellite.
The third reference frame called the controller frame originates in the center of mass of the satellite. Its x-axis is orientated along the minor axis of inertia of the satellite while the z-axis is parallel to the major axis of inertia. The y-axis is the result of the cross product between the z- and x-axis and points along the intermediate axis of
9
Chapter 2. Method 10
inertia. This reference frame is especially useful when dealing with spacecraft dynamics computations since the off-diagonal elements of the inertia matrix are null (Wie,1998).
The orientation of the body reference frame and the orientation of the spacecraft are indistinguishable. This makes the problem of discussing and describing the satellite’s attitude dynamics easier to understand and to work with.
2.1.2 Rotations
Rotations are used to express the difference in orientation between two reference frames.
This difference is represented by the transformation matrix which is also called the direction cosine matrix. If the two reference frames are defined as sets of orthonormal vectors then the transformation matrix is populated by the cosine of the angles between their axis. Thus the matrix has 9 entries but only 3 independent parameters with the rest originating from the orthogonality condition (Schaub & Junkins,2009).
The direction cosine matrix is extremely useful when trying to visualize 3 dimen- sional rotations but it is computationally demanding because of the trigonometric func- tions. A different way of describing rotations is by using the Euler parameters often called quaternions. Quaternions are hyper complex numbers composed of a scalar part q1 and a vector part q2:4. They are based on Euler’s principal rotation theorem which states that a rigid body can undergo a rotation from any initial orientation to any desired final orientation only by a single rotation through a principal angle Φ and a principal axis ˆe (Schaub & Junkins,2009). The quaternion q is thus defined as:
q1 = cos(Φ
2) (2.1a)
q2 = e1sin(Φ
2) (2.1b)
q3 = e2sin(Φ
2) (2.1c)
q4 = e3sin(Φ
2) (2.1d)
The quaternion representation of rotations is used extensively in the attitude dy- namics model in Chapter3.
Chapter 2. Method 11
2.2 The Attitude and Orbit Determination System
The AODS combines the functions of attitude determination and orbit determination providing information about the orientation of the spacecraft and about its position and velocity with respect to a given reference frame. The AODS is responsible for acquiring, storing and sending data to the satellite’s other systems. The tasks are performed by 2 separate filters one for the attitude and the other for the position and velocity thus treating the AOD problem as decoupled.
2.2.1 The need for attitude determination
In essence all satellites have one ore more systems that are designed to interact with or observe other objects, phenomena, etc. These systems are generally called payloads. The satellite’s attitude and orbit control system (AOCS) system ensures that the payload is performing according to its design specifications. Be it the case of solar panels being reoriented towards the Sun or an antenna pointing towards a ground base the control system of the satellite makes use of information received from the attitude determination system. This information is a way to express the orientation of the satellite in a given reference frame. Knowing the current and desired attitude the control system using the thrusters, magnetorquers or any system that introduces a moment about the spacecraft’s center of mass can reorient it accordingly.
In the case of the SEAM CubeSat the payload could be considered as the mag- netometers mounted at the tip of the deployable booms. Alongside the magnetometers one of the booms also contains a star tracker. The purpose of the star trackers is to take pictures of the night sky and match the stars in the images with star catalogues saved on-board. By knowing the orientation of the star tracker relative to the CubeSat’s frame the magnetic measurements taken by the magnetometers can be precisely stamped and located. These measurements can be compared with other observations or even ground based measurements of the magnetic field if the attitude at the time of the sampling is accurate enough.
2.2.2 The need for orbit determination
The orbit of the spacecraft is one of the basic elements of a space mission. The SEAM satellite will be placed in a circular, Sun-synchronous orbit at an altitude of 600 km.
Its current orbit and the prediction of future orbits needs to be known both to the
Chapter 2. Method 12
on-board system and to the ground based operators. Being an autonomous satellite tasked with taking science grade measurements of Earth’s magnetic field the on-board information about the orbit is used for mapping and recording these measurements. This is usually achieved with a GPS receiver mounted on the spacecraft. From the ground a satellite’s orbit can be computed by using the TLE data as initial conditions in an orbit integrator. TLEs are provided every couple of days and have an estimated accuracy at epoch of approximately 1 km. The most common orbit integrator model is the simplified perturbations model (SGP4). The model however acquires 1 km in position error each day or around 16 orbits for a satellite in LEO (Dong & Chang-yin,2010). By knowing the exact characteristics of the spacecraft’s orbit ground support operators can schedule communication sessions or plan downlink/uplink transmissions.
2.3 Unscented Kalman Filter theory
The Kalman Filter (KF) is one of the most widely used methods for system estimation due to its optimality and robustness (Julier & Uhlmann, 1997). Considering linear systems, that is when both the dynamics and observation models are linear, a KF can be implemented to compute the minimum mean square error (MMSE). Difficulties arise however when the filter is used to estimate nonlinear systems. A modified version, the Extended Kalman Filter (EKF) can be used by linearising all the nonlinear models in order for the basic KF to be applied. An even better filter for estimating nonlinear systems is the Unscented Kalman Filter (UKF) which uses a set of discretely sampled points to parametrise the mean and covariance.
The algorithm used by the KF takes noisy measurements from sensors and produces estimations of state variables which are more precise than if each of those measurements were used alone. The model and the measurements involved are both estimates because of model approximations or noise respectively; external factors not taken into account in the model also influence the accuracy of the whole filter. In this respect the filter averages a prediction of the system’s state with new measurements using a weighted average (Rojas, 2003). These weights decide which measurements should be trusted more and are computed from the covariance. The covariance is the estimated uncertainty of the state prediction. This process repeats every time step and thus the filter runs recursively using only information from the previous time step.
Chapter 2. Method 13
The EKF is used for nonlinear systems and provides minimum variance estimate of the state based on statistical information about the dynamic model and observations being composed of the time update and measurement update cycles (Gelb, Joseph F.
Kasper, Raymond A. Nash, Price, & Arthur A. Sutherland, 1974). The state vector and the covariance matrix are updated using the dynamical equations but this can not be done directly. A matrix of partial derivatives (the Jacobian) needs to be computed beforehand by linearising the nonlinear equations of the models. The predicted state and the state transition matrix are then calculated by integrating the equations of motion.
The UKF addresses the approximation problems of the EKF by using set of care- fully chosen sample points called sigma points. These points capture the true mean and covariance of the states. The filter then propagates these points through the nonlinear system dynamics capturing the posteriori mean and covariance accurately to the 3rd order for any nonlinearity (Wan & Merwe,2000). The UKF represents a derivative-free approach because the derivation of the Jacobian is not required by the algorithm.
2.3.1 Filter initialization
In the initialization phase of the filter the initial state vector xk|k is defined, the weights W0(m), W0(c), Wi(m)are calculated and the initial covariance matrix Pk|k, the process noise covariance matrix Qk and the measurement noise covariance matrix Rk are defined.
The weights used to calculate the a priori state estimate and error covariance matrix have the following expressions (Haykin,2001):
W0(m)= λ
L + λ (2.2a)
W0(c) = λ L + λ+
1 − α2+ β
(2.2b) Wi(m)= Wi(c) = 1
2(L + λ) i = 1, . . . , 2L (2.2c) where α is a constant describing the spread of the sigma points, β incorporates previous knowledge of the distribution of the state vector, L is the number of elements in the state vector and λ = α2(L + κ) − L is a scaling parameter. According toHaykin(2001) the optimal value for β is 2 and κ = 3 − L if the distribution of the state is Gaussian. For different distributions other values for the constants β and κ might be more appropriate.
Chapter 2. Method 14
2.3.2 Prediction phase
The UKF is based on a sigma point sampling method called the Unscented Transform (UT). The sample size of the UT is 2L + 1 (Grewal & Andrews, 2008). In the case of the orbit UKF the state vector has 6 states and in the case of the attitude UKF it has 9. The sigma points are a structured set of sample points that give adequate coverage of the input and output probability distribution (Vinther, Jensen, Larsen, & Wisniewski, 2011). They are computed from the error covariance matrix Pk−1|k−1 as described in Haykin(2001):
χk−1|k−1
0 = xk−1|k−1 (2.3)
χk−1|k−1
i = xk−1|k−1+
q
(L + λ)Pk−1|k−1
i
i = 1, . . . , L (2.4)
χk−1|k−1
i = xk−1|k−1−
q
(L + λ)Pk−1|k−1
i−L
i = L + 1, . . . , 2L (2.5)
where xk−1|k−1 is the state vector at time k − 1. A total number of 13 sigma points are used for this particular UKF. The parameter α is constant with 0 ≤ α ≤ 1 (Haykin, 2001).
The method described above for computing the sigma points is used in the orbit UKF algorithm. Instead, the attitude UKF is using spherical simplex sigma points.
This method of computing the sigma points reduces their number to only L + 2 instead of the 2L + 1 sigma points needed in the standard UKF algorithm. The points are now in a radius proportional to√
L for an L-dimensional space and their respective weights are proportional to 1/L (Lozano, Carrillo, Dzul, & Lozano,2008).
For the spherical method the sigma points have the following expression:
χk−1|k−1
i= xk−1|k−1+ q
Pk−1|k−1Zi i = 0, . . . L + 1 (2.6)
where Zi is the i-th column of the spherical simplex sigma point matrix computed as per Lozano et al.(2008).
The spherical method of computing the sigma points is preferred when building the attitude UKF because it reduces the number of points that need to be propagated through the dynamic system. In doing so the filter becomes applicable also to systems with restricted computational power.
Chapter 2. Method 15
The next step is propagating the sigma points through the nonlinear system model:
χk|k−1
i = f ((χk−1|k−1)i) i = 0, . . . , 2L (2.7) The a priori state estimate xk|k−1 and the a priori error covariance matrix Pk|k−1 are computed using the propagated sigma points and the weights from Eq. (2.2):
xk|k−1=
2L
X
i=0
Wi(m)
χk|k−1
i (2.8)
Pk|k−1=
2L
X
i=0
Wi(c)
χk|k−1
i− xk|k−1
χk|k−1
i− xk|k−1
|
+ Qk (2.9)
where the matrix Qk is the process noise covariance.
2.3.3 Update phase
The update part of the filter consists in evaluating the sigma points
χk|k−1
i using the sensor model to obtain the predicted measurements
Zk|k−1
i:
Zk|k−1
i= h
χk|k−1
i
i = 0, . . . , 2L (2.10)
and using the weights from Eq. (2.2a) the transformed vector measurement zk|k−1 can be calculated:
zk|k−1=
2L
X
i=0
Wi(m)
Zk|k−1
i (2.11)
Before the a posteriori state estimate xkcan be computed the measurement covari- ance matrix Pzkzk and the state-measurement cross-covariance matrix Pxkzk need to be calculated.
Chapter 2. Method 16
Using the weights from Eq. (2.2b), the transformed sigma points and the trans- formed measurement vector the covariance matrices are described as (Haykin,2001):
Pzkzk =
2L
X
i=0
Wi(c)
Zk|k−1
i− zk|k−1
Zk|k−1
i− zk|k−1
|
+ Rk (2.12)
Pxkzk =
2L
X
i=0
Wi(c)
χk|k−1
i
− xk|k−1
χk|k−1
i
− xk|k−1
|
(2.13)
where the matrix Rk is the measurement noise covariance matrix.
The Kalman gain is calculated as:
Kk= PxkzkP−1zkzk (2.14)
and the posteriori state estimate xk can now be computed using the zk measurement vector:
xk|k = xk|k−1+ Kk
zk− zk|k−1
(2.15) The final step before the cycle repeats is the computation of the a posteriori co- variance matrix Pk|k:
Pk|k= Pk|k−1− KkPzkzkK|k (2.16) Since the filter is an estimator of the minimum mean-square error the difference between the actual state and the estimated state is the error in the a posteriori state estimation. The goal is then to minimize the expected value of its squared magnitude.
This translates into minimizing the trace of the a posteriori covariance matrix Pk|k. This trace is minimized since the Kalman gain defined by2.14 is optimal.
The UKF procedure described above can be modified in order to better asses the process and measurement noises by combining the state variables together with the noise components. By doing so information about the odd-order moment is improved.
However this requires that the sigma points computed in the prediction phase to be propagated through the update phase as well. Better results are obtained with the augmented approach when the noise is not additive (Wu, Hu, Wu, & Hu,2005). In the present thesis the noise is assumed to be additive and the non-augmented version of the UKF is used. The sigma points are thus be recalculated once the prediction phase ends.
Chapter 3
AODS
3.1 Attitude determination
3.1.1 Approach
The attitude determination is performed using an unscented Kalman filter. The filter integrates a dynamic model based on Euler’s equations for rigid body rotations. Mea- surements from the satellite’s sun sensors, magnetometers, and a star tracker are used to estimate the attitude, the angular velocity and the magnetometer bias. A Rung-Kutta 7(8)th-order numerical integration method is used to propagate the dynamic model.
The ephemeris models include a sun propagator, an eclipse predictor and a 13th or- der IGRFv11 magnetic field model. A prefilter is also used to manipulate the sensor measurement before they are passed on to the UKF.
3.1.2 Available sensors 3.1.2.1 Sun sensors
A number of 9 coarse sun sensors (photo-diodes) are used by the SEAM satellite.
These are integrated on the NanoPower P110 solar panel boards provided by GomSpace (GomSpace, 2014b) and are operated in short-circuit mode (Ivchenko, 2014a). Their distribution on the spacecraft as well as the position of the other sensors on-board the SEAM CubeSat is shown in Figure 3.1. There are 34 solar panels distributed across 9 panels. Out of these 6 panels have magnetorquers included on them.
17
Chapter 3. AODS 18
Figure 3.1: Sensor distribution for the SEAM CubeSat (ss - sun sensor; mt - magne- torquer; +X refers to positive direction of axis in Figure1.1)
3.1.2.2 Magnetometer
Two types of magnetometers are used on-board the SEAM satellite both for taking scientific measurements of Earth’s magnetic field and to aid with the attitude determi- nation.
Search-coil magnetometers are coils wound around a high magnetic permeability core. The magnetic field lines along with their fluctuations are concentrated inside the coils by the magnetic core. Currents and voltage drops are induced by the fluctuations inside the core and are picked up by the instrument’s electronics (NASA, 2007). The search-coil magnetometer used by the SEAM project is depicted in Figure 3.2a.
Fluxgate magnetometers, shown in Figure3.2b, are also used by the SEAM satel- lite. They consist of a magnetically susceptible core wrapped in two coils of wire. An alternating current is passed through one of the coils magnetizing it which in turn pro- duces a magnetic field. Because of the alternating current the magnetic field changes direction and thus induces an electric field in the second coil. The output signal is mea- sured on the second coil by a detector. By detecting the amplitude of the pick-up signal the magnetic field at the core can be measured.
The measuring range for the fluxgate magnetometer as specified in GomSpace (2014c) is ±64000 nT.
Chapter 3. AODS 19
(a) Search-coil magnetometer (b) Fluxgate magnetometer Figure 3.2: Magnetometers used by the SEAM CubeSat (GomSpace,2014c)
3.1.2.3 Star trackers
A star tracker solution specifically designed for the SEAM CubeSat is in development at KTH, Stockholm. The star tracker is shown in Figure 3.3. Two star trackers are included in the design of the satellite. One is mounted on the CubeSat’s frame and the other is integrated at the top of one of the deployable booms. They are based on a 5-megapixel CMOS monochrome image sensor operated in multiple frames per second.
Real-time image readout is possible using a register-level firmware detecting the stars in the captured images (GomSpace,2014c). The images are compared with star catalogues to determine the direction and orientation of the instruments.
The on-board star tracker consists of one single unit featuring larger aperture optics as compared to the boom star tracker. Even though their functionality is identical the main electronic components of the boom star tracker are located inside the CubeSat frame with only the optical head and the pre-processing hardware being located at the boom tip. As a result the boom star tracker has a slower update rate and a lower tolerance to angular rotations (GomSpace,2014c).
3.1.3 Prefilter
The estimated sensor outputs from the measurement models are compared with the actual measurements from the sensors. For this reason the real measurements are passed through a prefilter before they are used in the UKF.
The sun sensors values are pairwise compared using the Max Current algorithm described in Bhanderi (2005). The algorithm also compensates for Earth’s albedo by choosing the sun sensor with the highest current value for future computations. The resulting sun pointing vector is thereafter normalized and thus comparable with the model out of the UKF.
Chapter 3. AODS 20
Figure 3.3: Current design of the star tracker used by the attitude determination system (Shterev,2014)
The magnetometer vector measurement is also normalized in order to make it comparable with the UKF model.
3.1.4 Dynamic model
In the present thesis the attitude is defined by the quaternion described in Eq. (2.1).
The relation between the attitude matrix and the quaternion as expressed in Lefferts, Markley, and Shuster (1982) is:
A(q1:4) =
|q1|2− |q2:4|2
Isat+ 2q2:4q|2:4+ 2q1Ω(q2:4) (3.1)
where
Ω(q2:4) =
0 q4 −q3
−q4 0 q2 q3 −q2 0
is a skew-symmetric matrix derived from the q2:4 three-vector.
The rate of change of the attitude matrix given the angular velocity ω(t) of the spacecraft is:
d
dtA(t) = Ω(ω(t))A(t) (3.2)
thus the corresponding rate of change of the quaternion is given by:
d
dtq1:4(t) = 1
2Ω(ω(t))q1:4 (3.3)
Chapter 3. AODS 21
where
Ω(ω(t)) =
0 ω3 −ω2 ω1
−ω3 0 ω1 ω2
ω2 −ω1 0 ω3
−ω1 −ω2 −ω3 0
The Euler rotational equations of motion make up the second part of the dynamic model. As shown in Schaub and Junkins(2009) they are:
Isatω = −Ω(ω)I˙ satω + Lc (3.4)
and by choosing a reference frame aligned with the principal body axes and fixed to the satellite:
Isat11ω˙1 = −(Isat33 − Isat22)ω2ω3+ L1 (3.5) Isat22ω˙2 = −(Isat11 − Isat33)ω3ω1+ L2 (3.6) Isat33ω˙3 = −(Isat22 − Isat11)ω1ω2+ L3 (3.7)
where Lcis the torque vector.
The spacecraft dynamic model describing the satellite’s orientation is thus:
˙x =
1
2Ω(ω(t))q1:4 −Isat−1 Ω(ω)Isatω + Lc
0 0 0
|
(3.8)
The state variables are chosen to be:
x =
q ω bmag
|
(3.9)
where bmag is the magnetometer bias.
The advantage of the quaternion representation of attitude is that it is singularity free. However the quaternion has a unity constraint on its parameters. This means that the quaternion only contains 3 pieces of useful information. Also the mean of a set of unit quaternions by a weighted sum can not be calculated. In order to overcome this
Chapter 3. AODS 22
issue an error quaternion δq is introduced as per Vinther et al. (2011):
δ ¯qk|k = qk|k⊗ q−1k|k−1 (3.10)
δqk|k =
δ1q1 δq2 δq3
|
(3.11)
where δqk|k is the update to the predicted quaternion qk|k−1 and giving the estimated qk|k.
Using a multiplicative update step it is possible to retain the unity constraint and expand the three-element error quaternion back into a proper quaternion (Vinther et al.,2011):
δqk|k= Kk
zk− zk|k−1
(3.12) qk|k=
δq|k|k q
1 − δq|k|k· δqk|k
|
⊗ qk|k−1 (3.13)
3.1.5 Measurement model
There are 3 models implemented in the attitude UKF one for each type of measurement.
In order from top to bottom, sun sensor model, magnetic field model and star tracker model:
bvss= q−1vssq (3.14)
bvmag = q−1vmgq +bbmag (3.15)
q = q ⊗ qe w (3.16)
In Eq. (3.16)q is the disturbed quaternion and qe wis the noise quaternion computed as stated inKraft(2003).
3.2 Orbit determination
3.2.1 Approach
In order to achieve the orbit position precision requirement of the mission an unscented Kalman filter is used. Perturbation models due to the gravity of the Sun and Moon, solar radiation pressure, atmospheric drag and the gravity of a non-spherical Earth
Chapter 3. AODS 23
are applied to the dynamic model. A Runge-Kutta 7(8)th-order numerical integration method is used for the orbit propagation. The measurement model is based on navigation solutions, position and velocity vectors, received from the GPS sensor.
The satellite’s orbital motion is modelled with respect the the ECI coordinate system. The output data of the GPS receiver is assumed to be of similar type to the state vector of the UKF.
3.2.2 Dual-Frequency GNSS Receiver
A NovAtel OEM615 Dual-Frequency GNSS Receiver (Figure 3.4) is integrated with the spacecraft on-board computer. The sensor provides navigation solutions which are used as measurement inputs for the filter. The GPS ephemeris data can also be used to calibrate the spacecraft’s system time. However, due to power limitations its duty cycle is around 10% (GomSpace,2014c).
VERSION:1.0 2014-03-31 26/52
the fluxgate sensor. The onboard star tracker will have all the electronics integrated in a single unit, and features larger aperture optics. The boom star tracker, although functionally identical to the onboard one, will only carry the optical head and the essential pre-processing hardware on the boom tip, with the rest of electronics mounted inside the satellite; this design reduces the magnetic disturbances caused by the star tracker. The optical aperture of the boom star tracker will necessarily be smaller than that of the onboard unit. This will result in the boom star tracker having a slower update rate and a lower tolerance to angular rates.
5.5.5 Global Positioning System (GPS) Receiver
A NovAtel OEM615 GPS receiver will be integrated with the NanoMind A712 board. The GPS receiver will mount directly to the NanoMind motherboard. The NanoMind will use the GPS ephemeris data for accurate time stamping of system data. The GPS also supplies a 1PPS signal (after lock) which is used by the DPU for time-keeping purposes. However, due to power limitations, the GPS will run at roughly 10% duty cycle.
Figure 19: OEM615 GPS Receiver
5.6 Structure
The satellite structure is crucial for ensuring the satellite components will survive the vibrational load of launch. The SEAM mission uses a unique deployable boom which enables the science mission.
5.6.1 CubeSat Solid Structure
KTH will provide a custom 3U solid structure for the SEAM satellite. This structure interfaces the physical hardware with the deployment device and ensures that all spacecraft components survive the vibrational load of launch. The solid structure is designed to comply with the CubeSat Design Specification (CDS) which defines the interfaces necessary for integration with the CalPoly Poly Picosat Orbital Deployer (PPOD). The details of the structure will be defined early in the development process to ensure that the interfaces necessary for other components are set.
5.6.2 Mechanical Design of the Deployable Booms Assembly The deployable booms assembly is composed of:
• A central support frame, which mounts to the CubeSat structure with four screws, Figure 20(a).
• Two tape spring spools mounted on axles with polymer bearings, Figure 20(b).
• Four 1 m long glass fiber tape springs with radius 7 mm and thickness 0.3 mm coiled on the spools, Figure 20(c).
Figure 3.4: OEM615 Dual-Frequency GNSS Receiver (NovAtel,2014)
3.2.3 Dynamic model
The dynamic model describing the satellite’s motion in orbit is based on Newton’s law of universal gravitation. The law states that the force of gravity between two bodies is proportional to the product of their masses and inversely proportional to the square of the distance between them:
F = Gm1m2
r2 (3.17)
where F is the force between the masses, G is the gravitational constant, m1,2 are the masses and r is the distance between the mass’ centres.
Chapter 3. AODS 24
This force is constantly acting on the satellite and so the velocity and position of the craft can be obtained by integrating the resulting acceleration once and twice respectively. The equation of motion in its simplest form, works well only if special conditions are satisfied. It can be rewritten as:
¨
r = −µ⊕
r3r (3.18)
where ¨r is the second inertial derivative of the position vector r, the force F has been replaced with −gm1 and µ⊕= Gm2 is the standard gravitational parameter of Earth.
The conditions mentioned earlier restrict the problem of determining the orbit to situations where these assumptions apply. In the case of the SEAM project perturbation models need to be added in order to increase the precision of the orbit estimation. The Cowell method of special perturbation is used to refine the orbit model. The method is a direct integration of accelerations acting on the satellite. Eq. (3.18) becomes:
¨r = −µ⊕
r3 r + ad (3.19)
where ad is the perturbational acceleration.
According toSchaub and Junkins(2009) this perturbation method should be used if the acceleration ad is of the same order of magnitude as the main gravitational ac- celeration. This is not the case of satellites orbiting in LEO, and thus SEAM, where perturbational accelerations are 3 or more times smaller than the acceleration due to Earth’s gravity. However Cowell’s method is still to be preferred, on basis of computa- tional reasoning, to the alternative (Encke’s method) even if does not take advantage of how close the perturbed solution is to the analytical one.
The model of the orbit UKF is described by:
˙x =
v ad
|
(3.20)
where v is the velocity vector and ad is the perturbational acceleration defined in Eq.
(3.19).
The state variables are chosen to be:
x =
r v
|
(3.21)
Chapter 3. AODS 25
where r is the position vector in the ECI frame and v is the velocity vector in the same frame.
The equations of motion are integrated with a Runge-Kutta 7(8)th-order method.
3.2.4 Measurement model
The measurement model integrated in the UKF uses GPS navigation solutions as ob- servations. These solutions are generated internally on the GPS receiver board by a navigation algorithm using information about the pseudorange and the pseudorange rate (NovAtel,2014). The relationship between a nonlinear dynamic system (the space- craft dynamics) and a linear system (the GPS solutions) can be expresses as (Yoon, Lee,
& Choi,2000):
˙
x(t) = f (x(t), t) + ω(t) (3.22)
y(t) = x(t) + v(t) (3.23)
where x(t) is the state vector containing the position and velocity vectors at time t, f (x, t) is the equation of motion of the dynamic system, ω is the dynamic model error, y(t) is the measurement vector and v(t) is the measurement noise.
Since the observation vector is obtained from the GPS solution provided by the receiver it is not a function of the GPS receiver clock bias. Therefore partial derivatives of the observations can not be calculated. The UKF does not require partial derivatives in its implementation however if the state vector includes the GPS clock bias as an estimation, the navigation solution measurement model would not be suitable. One advantage of having the measurement vector identical to the state vector is that there is no information loss since no linearisation is taking place.
3.2.5 Force model
Four additional forces are added to the basic model: 2 gravitational and 2 non-gravitational.
These are the geo-potential of Earth, the gravity of the Moon and Sun and forces due to atmospheric drag and solar pressure respectively. The gravity model applied in the present thesis is the EGM96 (Pavlis et al., n.d.) up to the 18th order and degree. The Moon and Sun are assumed to be point masses and their ephemeris computed based on numerical methods explained in Flandern and Pulkkinen(1979). The acceleration due to atmospheric drag is based on the 1976 U.S. Standard Atmosphere model for density,
Chapter 3. AODS 26
the velocity of the satellite along the orbital trajectory and its mass and a drag coeffi- cient corresponding to a flat plate of appropriate dimensions. Finally the solar radiation pressure is a function of the satellite’s mass, its surface area, distance from the Sun and the reflectivity of the material.
The term ad from Eq. (3.19) is thus:
ad(r, v, t) = ag+ aa+ an+ ap
where ag is the acceleration due to gravity, aais the acceleration due to the atmospheric drag, anis the acceleration due to the Sun and the Moon and ap is the acceleration due to the solar pressure.
3.2.5.1 Geo-potential
The spherical harmonic representation of Earth’s geo-potential function is given by (Pavlis et al.,n.d.):
Φ(r, φ, λ) = µ r+µ
r
∞
X
n=1
Cn0 R r
n
Pn0(u)+µ r
∞
X
n=1
∞
X
m=1
R r
n
Pnm(u)[Snmsin mλ+Cnmcos mλ]
(3.24) where φ is the geocentric latitude of the satellite, λ is the geocentric longitude of the satellite and r is the geocentric distance of the satellite. Furthermore S and C are the harmonic coefficients of the geo-potentional and P are the associated Legendre polyno- mials of degree n and order m with the argument u = sin φ (Pavlis et al.,n.d.).
The satellite’s acceleration due to Earth’s gravity field has the following expression:
ag(r, t) = ∇Φ(r, t) (3.25)
The acceleration vector ag is a combination between a general two-body gravity acceleration term and the acceleration derived from the higher order non-spherical terms.
The inertial rectangular Cartesian components of the satellite’s acceleration vector as well as the partial derivatives of the geo-potential with respect to the geocentric latitude, longitude and distance and the Legendre polynomials are computed as per Kuga and Carrara(2013). Also the gravity model coefficients used in Eq. (3.24) are not normalized and should be transformed according to Kuga and Carrara(2013) if an order or degree higher than 18 needs to be added.
