MASTER'S THESIS
Model Development and Attitude Control for pico-satellite UWE-4
Siddharth Dadhich 2015
Master of Science (120 credits)
Space Engineering - Space Master
Master Thesis
Model Development and Attitude Control for pico-satellite UWE-4
written by
Siddharth Dadhich
November 3, 2014
Julius-Maximilians-University Würzburg
Department of Computer Science, Robotics and Telematics
Examiner I
Prof. Dr. rer. nat Klaus Schilling Professor and Chair
Informatics VII : Robotics and Telematics Informatics Julius-Maximilians-University Würzburg ,Germany
Examiner II
Prof. Dr. Johnny Ejemalm Senior Lecturer
Department of Computer Science, Electrical and Space Engineering Luleå University of Technology, Sweden
Supervisor
MSc. Philip Bangert Research Assistant
Informatics VII : Robotics and Telematics Informatics Julius-Maximilians Universität Würzburg, Germany
Date of the submission
03.11.2014
Declaration
I hereby declare that this thesis is entirely the result of my own work except where otherwise indicated. I have only used the resources given in the list of references.
Würzburg, 03.11.2014 (Siddharth Dadhich)
Contents
Abstract 1
1 Introduction 3
1.1 UWE . . . . 4
1.1.1 UWE - 3 . . . . 4
1.1.2 UWE - 4 . . . . 4
1.2 Vacuum arc thrusters . . . . 6
1.3 State of art . . . . 7
1.4 Orbit control vision . . . . 8
1.5 Summary . . . . 8
2 System Models 9 2.1 Coordinate frames and attitude definition . . . . 9
2.1.1 ECI and ECEF . . . 10
2.1.2 Orbit coordinate frame (OCF) . . . 10
2.1.3 Body coordinate frame (BCF) . . . 10
2.1.4 Control coordinate frame (CCF) . . . 11
2.1.5 Attitude definition . . . 12
2.2 Satellite dynamics . . . 13
2.3 Sensor models . . . 15
2.3.1 Gyroscope . . . 16
2.3.2 Magnetometers . . . 16
2.4 Actuator models . . . 17
2.4.1 Thrusters . . . 17
2.4.2 Magnetic torquers . . . 19
2.4.3 Reaction wheel . . . 19
2.5 Disturbance models (environment) . . . 20
2.5.1 Aerodynamic drag . . . 20
2.5.2 Gravity gradient . . . 21
2.5.3 Residual magnetic field . . . 21
2.6 Summary . . . 21
3 Orbit Propagator 23 3.1 Modified SGP- 4 . . . 23
3.2 Simplified orbit propagator (SOP) with low thrust . . . 24
3.3 Summary . . . 27
Contents Contents
4 Attitude Control 29
4.1 Target attitude . . . 29
4.1.1 In-track target attitude . . . 30
4.1.2 Anti-in-track target attitude . . . 31
4.2 Control Laws . . . 32
4.2.1 PD control . . . 32
4.2.2 Sliding mode control . . . 33
4.3 Magnetic control . . . 34
4.3.1 B-dot control . . . 34
4.3.2 Two axis pointing control . . . 34
4.4 Thruster control . . . 35
4.4.1 One quadrant based two-axis control . . . 35
4.4.2 Two quadrant based two-axis control . . . 36
4.5 Combination of torquers and thrusters . . . 36
4.6 Stability for large angle maneuvers . . . 37
4.7 Summary . . . 37
5 Simulation Results 39 5.1 B-dot simulation . . . 41
5.2 Full magnetic control . . . 42
5.2.1 PD control law . . . 42
5.2.2 SMC law . . . 43
5.3 Full thruster control . . . 45
5.3.1 PD control law . . . 45
5.3.2 SMC law . . . 47
5.4 Combination of thrusters and torquers . . . 48
5.4.1 PD law with one quad thruster . . . 49
5.4.2 PD law with two quad thruster . . . 51
5.4.3 SMC law with one quad thruster . . . 52
5.4.4 SMC law with two quad thruster . . . 54
5.5 Summary . . . 58
6 Implementation 59 6.1 Hardware architecture . . . 59
6.2 Low level code . . . 59
6.3 Summary . . . 61
7 Conclusion and Future Work 63 7.1 Conclusions . . . 63
7.2 Future work . . . 64
7.2.1 Equal distribution of thruster usage . . . 65
7.2.2 Controller stability analysis . . . 65
7.2.3 Efficient use of torquers . . . 65
7.2.4 Orbit Propagator . . . 65
Contents
Acknowledgments 67
Bibliography 69
Nomenclature 73
List of Tables
1.1 Classification of satellites according to size . . . . 4
5.1 Simulation Initialization Parameters . . . 41
5.2 Comparison of Different Simulated Scenarios . . . 58
List of Figures
1.1 Structural overview of UWE-4 . . . . 5
1.2 Design model of UWE-4 showing thruster locations . . . . 6
1.3 Trailing formation of two satellites . . . . 7
2.1 Simulation software block diagram . . . . 9
2.2 Body coordinate frame . . . 11
2.3 Inertial coordinate frames: ECI and ECEF . . . 12
2.4 Orbit coordinate frame and Control coordinate frame . . . 13
2.5 Simulated gyroscopic data (left) vs UWE-3 gyro measurements (right) 15 2.6 Simulated Magnetometer Data . . . 17
2.7 Thruster Plume variation . . . 19
2.8 Simulated Disturbance Torques . . . 22
3.1 Modified SGP-4 Propagator . . . 24
3.2 Simplified orbit propagator . . . 25
3.3 Error in Simplified propagator from SGP-4 output . . . 26
3.4 Demonstration of Thruster effect seen as the separation of two satel- lites generated by Simplified orbit propagator . . . 26
4.1 Schematic of the Controller . . . 30
4.2 Thruster control quadrants . . . 35
5.1 Graphical User Interface (GUI) for simulation experiments . . . 39
5.2 B-dot result from simulation (left) vs UWE-3 de-tumbling experiment result (right) . . . 42
5.3 Full magnetic control with PD law - Attitude and Angular Velocity . 43 5.4 Full Magnetic Control with PD control law - Magnetic Moment Out- put of Torquers . . . 43
5.5 Full magnetic control with SMC law - Attitude and Angular Velocity 44 5.6 Full Magnetic Control with SMC control law - Magnetic Moment Output of Torquers . . . 44
5.7 Full thruster control with PD law - Attitude and Angular Velocity . . 45
5.8 Full thruster control with PD law - Settling Time . . . 46
5.9 Full thruster control with PD law - Settling Accuracy . . . 46
5.10 Full thruster control with PD law - Thruster firings per minute and
Cumulative firings . . . 46
5.11 Full thruster control with SMC law - Attitude and Angular Velocity . 47
List of Figures List of Figures
5.12 Full thruster control with SMC law - Settling Time . . . 47 5.13 Full thruster control with SMC law - Settling Accuracy . . . 48 5.14 Full thruster control with SMC law - Thruster firings per minute and
Cumulative firings . . . 48 5.15 PD law with One Quad Thrusters - Attitude and Angular Velocity . 49 5.16 PD law with One Quad Thruster - Settling Accuracy and Settling Time 50 5.17 PD law with One Quad Thrusters - Thruster firings per minute and
Cumulative firings . . . 50 5.18 PD law with One Quad Thrusters - Commanded Torque Load sharing
between Thruster and Torquers in X and Y axis . . . 51 5.19 PD law with One Quad Thrusters - Commanded Torque Load sharing
in Z-axis and Magnetic Moment Output of Torquers . . . 51 5.20 PD law with Two Quad Thrusters - Attitude and Angular Velocity . 52 5.21 PD law with Two Quad Thruster - Settling Accuracy and Settling Time 52 5.22 PD law with Two Quad Thrusters - Thruster firings per minute and
Cumulative firings . . . 53 5.23 PD law with Two Quad Thrusters - Commanded Torque Load sharing
between Thruster and Torquers in X and Y axis . . . 53 5.24 PD law with Two Quad Thrusters - Commanded Torque Load sharing
in Z-axis and Magnetic Moment Output of Torquers . . . 54 5.25 SMC law with One Quad Thrusters - Thruster firings per minute and
Cumulative firings . . . 54 5.26 SMC law with One Quad Thrusters - Commanded Torque Load shar-
ing between Thruster and Torquers in X and Y axis . . . 55 5.27 SMC law with One Quad Thrusters - Commanded Torque Load shar-
ing in Z-axis and Magnetic Moment Output of Torquers . . . 55
5.28 SMC law with One Quad Thrusters - Attitude and Angular Velocity . 55
5.29 SMC law with One Quad Thrusters - Settling Time . . . 56
5.30 SMC law with One Quad Thrusters - Settling Accuracy . . . 56
5.31 SMC law with Two Quad Thrusters - Attitude and Angular Velocity 57
5.32 SMC law with Two Quad Thrusters - Settling Time . . . 57
5.33 SMC law with Two Quad Thrusters - Settling Accuracy . . . 57
6.1 Hardware architecture of UWE-4[1] . . . 60
Abstract
UWE-4 is the fourth pico-satellite satellite in UWE series. After the successful launch and operation of UWE-3 in 2013, the mission of UWE-4 is to demonstrate electric propulsion technology for formation flying missions at the pico-satellite level of miniaturization. The challenge in this project is to demonstrate precise orbit and attitude control given the extremely limited resources available to the satellite. Four vacuum arc thrusters and one PPU constitute the electric propulsion system.
In this master thesis project, attitude control of a cubesat is studied and a simu- lation platform for attitude and orbit control is developed. More than one control philosophies are developed, compared and implemented to fly on-board UWE-4. Dif- ficulties with using SGP-4 orbit propagator with low thrust action has motivated the development of a low fidelity orbit propagator which should be improved further.
Insufficient actuation in one axis has motivated the development of a quaternion
based dynamic target attitude solution to achieve precise 2D attitude control while
also having stabilization in the third axis. Due to the advantages of non-linear con-
trol, a sliding mode controller is evaluated against conventional linear PD controller
with the conclusion that a non-linear controller has some advantages over a linear
controller for this particular problem of attitude control where constant large angle
maneuvers are desired for changing the orientation of the satellite. Two operation
modes for the thrusters named as the one quadrant and the two quadrant method
are formulated and simulated in different scenarios. It is found out that the two
quadrant method results in a quicker response for the change in orientation of the
satellite.
1 Introduction
The beginning of space age was marked by the launch of Sputnik1 on October 4, 1957. This event started a space race between the Soviet Union and the USA which lasted for almost two decades and resulted in an accelerated progress of man towards previously unexplored space.
The advancements in all of the fundamental and interdisciplinary fields of science and engineering has made it possible for humans to make use of spacecrafts for applications such as communication, astronomy, navigation, atmospheric studies and more. The oil and natural gas boom across the developed and developing economies also played a huge role in this conquest of space. So far, there have been more than 6800 large or small satellites launched [2] (see Tab. 1.1 for classification of satellites). The increasing demand of fuel and food for the growing population of the planet is putting brakes on spending too much fuel for space missions. The number of large satellites being put in space has been decreasing in the last few decades and recently, in the last few years there has been an increase in the number of satellites being launched contrary to the previous decades [2]. This is because there has been an sharp increase in the number of small satellites, due to their economic advantage over large satellites. This trend will continue and will require more research and development efforts to engineer small satellites for future.
Some space exploration missions can be better accomplished by a combination of small satellites flying together in a formation [3]. There can be three types of formation flying scenarios: trailing, cluster and constellation. A trailing formation is when two or more satellites fly in the same orbit with a certain lag. This formation is useful for providing continuous observation over an area. A cluster formation is when a dense group of satellites work towards a common mission. The advantage being in the possibility to do observations from different angles at the same time.
The constellation formation when many satellites works in complete harmony for proving extensive ground coverage (for example GPS constellation).
Julius-Maximilians-University Würzburg (JMUW) has initiated a pico-satellite pro-
gram which aims to test the capabilities of pico-satellites for future formation flying
missions. So far, three satellites have been launched in this program and the fourth
satellite UWE-4 is under development. This thesis summarizes the work done for
development of a simulation platform and attitude control software for UWE-4.
Chapter 1 Introduction
Large Small
Large satellites >1000 kgs Micro-satellite 10-100 kgs Medium Size 500-1000 kgs Nano-satellite 1-10 kgs Mini-satellite 100-500 kgs Pico-satellite 0.1-1 kgs
Table 1.1: Classification of satellites according to size
1.1 UWE
UWE (Universität Würzburg Experimentalsatellit), is a satellite program for exper- imentation with pico-satellites. The first two satellites in this program were UWE-1 and UWE-2 which were launched in 2005 and 2009 respectively. All UWE satellites are 1U cubesats. Cubesats are required to follow strict specifications, subject to the latest standards initially written by California Polytechnic Institute [4]. The most common sizes for cubesats are 1U, 2U and 3U which is chosen depending on the size of the payload. It is important to mention that the small size of cubesats and other pico-satellites pose difficult engineering challenges. Like JMUW, many universities have their own cubesat programs. Readers are encouraged to refer to [5, 6, 7, 8] for similar works in this field at other universities.
1.1.1 UWE - 3
UWE-3 was launched in Nov 2013. The mission objective of UWE-3 was to de- mostrate an accurate attitude determination and control system (ADCS) for pico- satellites. For attitude determination, UWE-3 has sun sensors, gyros and magne- tometers which are fused in an Isotropic Kalman filter. For attitude control, UWE-3 has six magnetic torques mounted on each face and also one reaction wheel. For details about UWE-3 ADCS, please refer to [9, 10] and, [11, 12] discusses the results and performance demonstrated by UWE-3.
1.1.2 UWE - 4
UWE-4 program started after the successful operation of UWE-3. The aim for
UWE-4 is to demonstrate orbit maneuvering capabilities together with precise at-
titude control with a use of a propulsion system. UWE-4 is the first in this series
to have an propulsion system. A detailed study of various possible propulsion tech-
nologies applicable to cubesats is available in [13]. The propulsion system chosen for
UWE-4 is the Vacuum arc thruster which will be discussed in sec. 1.2. A structural
overview of UWE-4 is shown in Fig. 1.1 which shows the location of the different
hardware components in UWE-4. The four thrusters and one of the magnetic tor-
quer is also visible in the Fig. 1.1.
1.1 UWE
1 2
3 4 5
8 7 6
9
Figure 1.1: Structural overview of UWE-4
Description of different components is as follows: (1) Communication Board (2) OBDH (3) PPU (4) Power board with batteries (5) Magnetic Torquer (6) Thruster
(7) ADCS (8) Front Access board (9) Panel
Due to the size constraints with cubesats which makes the use of deployable solar panels very difficult, there is always very limited power available (1.5-2.5W) [1].
With the inclusion of the propulsion system which itself consumes 1 W of power, there is not enough power left for the reaction wheel. Also that the reaction wheel is only a momentum exchange device and that it cannot provide any thrust for orbit control makes it far less competitive against the thrusters. Nevertheless, the incorporation of a reaction wheel in UWE-4 is under discussion.
UWE-4 will inherit the attitude determination system from UWE-3 which means
that all the sensors are likely to remain same with some minor upgrades. UWE-4
will have the magnetic torquers and the propulsion system as actuators.
Chapter 1 Introduction
1.2 Vacuum arc thrusters
Vacuum arc thruster (VAT) is the technology chosen for micro-propulsion for UWE- 4. It is promising due to the desirable small impulse bit and low power consumption [1]. VAT has also been used previously for cubesat ION (Illinois observing satellite) [14]. Currently, a micro-vacuum arc thruster based propulsion system is being de- veloped at the University of Federal Armed Forces in Munich (UniBwM) which is collaborating on this project with JMUW.
The propulsion system has two components: Vacuum arc thrusters (VAT) and one power processing unit (PPU). The four micro-VAT will be located at the head of the four rails all facing the same direction. Thus all the thrusters will be able to produce thrust in one direction and torque in a two dimensional plane. A design model of UWE-4 pointing the location of the thrusters is shown in Fig. 1.2.
Thruster 2
Thruster 3 Antenna
Figure 1.2: Design model of UWE-4 showing thruster locations
For working principles of vacuum arc thruster readers are referred to [15]. Since micro-VAT for UWE-4 is in development phase, an extensive literature survey is per- formed to obtain reasonable estimates of thrusters characteristics. These estimates are based on similar developments of VAT for cubesats as in [15, 16, 17]. From all available literature on VAT including initial publications by UniBwM [1, 18], and from ongoing discussions with the developers, it is concluded that impulse bit of mico-VAT for UWE-4 could be in the range of 0.5-2 μNs with specific impulse of
~1000s.
1.3 State of art
1.3 State of art
Vacuum art thrusters falls in the category of electric propulsion which is on the horizon of technology for propulsion in small spacecrafts. Extensive study of state of art technologies for small satellites has recently been published by NASA in [19].
Precise formation flying mission is one of the driving force for the development of new chemical and electric propulsion technologies. So far, only one cubesat program CanX (Canadian Advanced Nanospace experiments) has successfully demonstrated propulsion technology for cubesats [20]. CanX like UWE, is also a program which has a vision for developing formation flying technology. Earlier, they have demonstrated chemical propulsion technology in CanX-2 which flew in 2008 and now CanX-4 and CanX-5 [21], which are their current satellites were launched in June 2014 to demonstrate precision formation flying. According to [22], within one month of their launch, tightly controlled formation of less than 5 kms has been demonstrated.
UWE-4 will be the first mission to demonstrate electric propulsion technology for cubesats and existing missions like CanX-4 and CanX-5 can be used as bench-marks for performance. A previous cubesat mission to use electric propulsion technology was ION, which was lost due to launch failure [20]. JPL (Jet propulsion laboratory), NASA is also developing a VAT with a thrust of 125μN, Isp of 1500 s, 10 kg mass and 10W power [19] which is much bigger than the one being developed for UWE- 4. The UWE-4 project is aiming to push miniaturization to its extreme limit to demonstrate what could be the future of space missions.
Chaser Target
Figure 1.3: Trailing formation of two satellites
Chapter 1 Introduction
1.4 Orbit control vision
Without orbit control only passive formations are possible. It has been observed that even the satellites launched from the same launcher can deviate to thousands of kilometers if initial orbit corrections are not provided to them [23]. It is shown in [23] that a simple MPC control scheme with an active thrust of Δv = 2.5 m/s per month is sufficient to keep a simple trailing formation mission of two satellites within a range of 1500 km. Life expectancy of UWE-4 thrusters has been evaluated to be 10 6 firings per thruster head, which implies that such a mission can be sustained for 3-6 months. A visualization of a trailing formation is shown in Fig. 1.3.
The limited life of thrusters mandates the most effective use of them. They should be used only when the thrust vector is aligned to the velocity direction (in-track) or opposite to velocity direction (anti in-track) which invokes a requirement of high accuracy attitude control. It is also important to note that the thrusters can them self provide attitude control.
1.5 Summary
This chapter provides the background and motivation for the work behind this thesis
project. UWE-4 will inherit all hardware from UWE-3 except the newly introduced
propulsion system. The propulsion system needs a sophisticated control software due
to its limited life. Successful demonstration of the use of this propulsion technology
for attitude and orbit control for this cubesat will be a milestone towards future
formation flying missions.
2 System Models
This chapter discusses all the building blocks of the simulation software. A block diagram view of the attitude control simulation can be seen in Fig. 2.1. The simula- tion should accurately model the different blocks in order to design a good control system. A lot of parameters while building these models are assumed to be the same from UWE-3. The organisation of this chapter is as follows, sec. 2.1 defines the ba- sic coordinate frames of interest. The four blocks that define, compute and affect the satellite’s attitude in its environment are the satellite dynamics block (sec. 2.2), the sensors (sec. 2.3), the actuators (sec. 2.4) and the disturbance block (sec. 2.5).
The orbit block and the controller block in Fig. 2.1 are discussed in chapter 3 and chapter 4 respectively. For the notations used in this chapter, please refer to the nomenclature at the end of thesis. The readers are also referred to the textbooks on attitude control by Sidi [24] and Spacecraft system design [25] for detailed un- derstanding of the concepts presented in this thesis.
Disturbances
Controller Actuators Satellite
Dynamics
Sensors in-Track
anti-in-Track
Orbit
Figure 2.1: Simulation software block diagram
2.1 Coordinate frames and attitude definition
This section covers the five coordinate frames which are used in this work. The sec. 2.1.1 discusses the two earth centered reference frames, also shown in Fig. 2.3.
The orbit coordinate frame, the body coordinate frame and the control coordinate
frame are discussed in sec. 2.1.2, sec. 2.1.3 and sec. 2.1 respectively.
Chapter 2 System Models
2.1.1 ECI and ECEF
The ECI (Earth centered inertial) frame is assumed as the absolute inertial reference frame and most of the calculations in this work are done in the ECI frame. The ECEF (Earth centered earth fixed) coordinate frame is used because the Earth’s magnetic model is easily available in ECEF since this reference frame rotates as the Earth rotates with its magnetic field.
1. ECI - There are difference variations of ECI (for example J2000, TEME), but the center of ECI is always fixed at the earth’s center. According to the most common J2000 ECI reference frame, the X-axis and the Y-axis points to the mean equinox direction and to the earth’s mean equator direction on 1st January 2000 respectively. The Z-axis points at the celestial north pole.
2. ECEF - This coordinate frame rotates as earth rotates on its axis. The center of ECEF is fixed at the center of earth. The X-axis and the Z-axis points to the prime meridian and to the earth’s north pole respectively. The Y-axis is in the equatorial plane and perpendicular to both the X-axis and the Z-axis.
2.1.2 Orbit coordinate frame (OCF)
The orbit coordinate frame moves along the orbit with its center fixed at the geomet- ric center of the satellite. The X-axis is along the velocity direction, Y-axis points to nadir towards earth and the Z-axis completes the triad being perpendicular to both X and Y axis. It is important to note that OCF rotates w.r.t. any inertial frame (say ECI) by a constant angular velocity given by
ω o oi = h 0 0 −ω 0
i T
(2.1)
where ω 0 = 2π T is the orbital angular velocity with orbital period T .
2.1.3 Body coordinate frame (BCF)
The body coordinate frame is the legacy coordinate frame for all UWE satellites
which has been defined from existing standards for Cubesats. The body coordinate
frame is centered at the geometric center of satellite and is fixed along the dimensions
of satellite. The Y-axis of BCF is fixed in the direction opposite to the front panel
and the Z-axis points to the antennas. The X-axis completes the triad with right
hand rule. Refer to Fig. 2.2 for visualization of BCF for UWE-4.
2.1 Coordinate frames and attitude definition
Figure 2.2: Body coordinate frame
2.1.4 Control coordinate frame (CCF)
A new coordinate system called CCF has been defined, which is more suitable for the requirements of the control software. The control coordinate frame like BCF is centered at the geometric center of the satellite and is fixed along the dimensions of the satellite.The X-axis points to the face, which has thrusters mounted on it.
Therefore the thrust vector is always opposite to the X-axis in control coordinate frame. The Y-axis is in the opposite direction of the front panel and Z-axis completes the triad. Refer to Fig. 2.4 for visualization.
The sun sensors and the magnetometer measures in BCF and, also the gyroscope measures the angular velocity of the body w.r.t inertial frame in BCF i.e. ω b bi . There- fore, in order to compute angular velocity of the satellite in the control coordinate frame we need the the transformation matrix (T c b ) from BCF to CCF which is
T c b =
0 0 1 0 1 0
− 1 0 0
(2.2)
Using this transformation matrix, angular velocity of the body w.r.t inertial frame in CCF is
ω bi c = T c b ω b bi (2.3)
From now on, in this entire work, only CCF is used to express the angular velocities,
thrust forces, torquers and all other sensor data. In order to compute attitude, the
Chapter 2 System Models
angular velocity of the body w.r.t. OCF (ω c bo ) must be known. To compute that, the Eq. 2.4 from [24] is used.
ω c bi = ω c bo + ω c oi (2.4)
Angular velocity of OCF w.r.t ECI in CCF (ω c oi )
ω c oi = T c o ω o oi (2.5)
Using Eq. 2.1 and the transformation matrix from OCF to CCF (T c o ), the angular velocity of body w.r.t. OCF (ω c bo ) is given as
ω c bo = ω c bi − T c o h 0 0 −ω 0
i T
(2.6)
Z ECI ,Z ECEF
X ECI X ECEF Y ECI
Y ECEF
Fixed in inertial space
Pointing to 0° Latitude
Figure 2.3: Inertial coordinate frames: ECI and ECEF
2.1.5 Attitude definition
In this work, the attitude of the satellite is defined as the angle between the orbit
coordinate frame and the control coordinate frame. When the two coordinate frames
coincides, it is represented by the attitude quaternion as h 0 0 0 1 i .
2.2 Satellite dynamics
Figure 2.4: Orbit coordinate frame and Control coordinate frame
2.2 Satellite dynamics
This section will present the basic attitude dynamic equations used in the model.
Since the attitude dynamic equations are non-linear coupled equations therefore, a non-linear model is preferred over a linearized approach.
Non-linear model
The dynamic equation gives us the angular acceleration of satellite which can be integrated to obtain angular velocity of the satellite in ECI frame. Then the attitude (represented as quaternions) of the satellite is computed with kinematic equations.
Dynamics
The attitude dynamics of the satellite is formulated by assuming that the satellite is
a rigid body. The firing of thrusters will result in a change in the inertia tensor of the
satellite but for a small duration the inertia tensor can be assumed as constant. An
On-board estimation of the inertia tensor is already available from UWE-3 and will
be used to update the inertia tensor during operation after certain usage intervals
of thrusters. The attitude dynamic equation (also called Euler’s Moment equation)
Chapter 2 System Models
as given in [24] is
I s ˙ω c bi + ω c bi × I s ω c bi = T total c (2.7)
The total external torque (T total ) is the sum of all individual torques acting on the satellite.
T total = T torquers + T thrusters + T disturbances (2.8)
The disturbance torques will be covered in sec. 2.5.
If a reaction wheel is included in the satellite (UWE-4), the dynamic equation will be given as in [26]
ˆI˙ω c bi + ω c bi × (I s ω c bi + A w I w Ω) = T total c (2.9) where A w , I w , Ω are the layout matrix, inertia tensor and angular velocity of reaction wheels. and ˆI is given by
ˆI = I s − A w I w A T w (2.10)
In presence of wheel a new term, T wheel will be added to the right side of Eq. 2.8.
Kinematics
A satellite’s attitude is easier to represent in quaternions. When the angular velocity of body w.r.t orbit in OCF is expressed as ω c bo = h ω x ω y ω z
i T
, the kinematic equation as given in [24] is
˙q = 1 2
0 ω z −ω y ω x
-ω z 0 ω x ω y
ω y −ω x 0 ω z
−ω x −ω y -ω z 0
q (2.11)
The conversion between ω c bo and ω i bi has been discussed in sec. 2.1.3 and concluded
in Eq. 2.6. The different values of q initial and ω initial can be used as initial conditions
in the simulation for attitude and angular velocity.
2.3 Sensor models
0 20 40 60 80 100
−1
−0.5 0 0.5
Simulated ωx (deg/s)
Time (min)
0 20 40 60 80 100
−1
−0.5 0 0.5
Time (min)
UWE−3 ωx ( deg/s)0 20 40 60 80 100
−1.5
−1
−0.5 0 0.5
1 1.5
Simulated ωy (deg/s)
Time (min)
0 20 40 60 80 100
−1.5
−1
−0.5 0 0.5 1 1.5
Time (min)
UWE−3 ωy ( deg/s)0 20 40 60 80 100
−1.5
−1
−0.5 0 0.5
1 1.5
Simulated ωz (deg/s)
Time (min)
0 20 40 60 80 100
−1.5
−1
−0.5 0 0.5 1 1.5
Time (min)
UWE−3 ωz ( deg/s)Figure 2.5: Simulated gyroscopic data (left) vs UWE-3 gyro measurements (right)
2.3 Sensor models
Accurate simulation of magnetic field and angular velocity is a crucial step in control
design for cubesats. On-board UWE-4, there are three magnetometers and three
gyros mounted on the ADCS board and six magnetometers and six sun sensor, one
Chapter 2 System Models
on each panel. In the sensor model, three magnetometers and three gyro are present which simulate the magnetic field and angular velocity for each axis. The sun sensors are not simulated in this work because there is no direct usage of the sun sensor measurements in the AOCS software.
2.3.1 Gyroscope
The result of the dynamics equation from the simulation is the noise free angular velocity. Now in order to simulate the measurements from gyroscope, we augment these values with white guassian noise (N W GN ) and gyroscopic drift (D gyro ) as pre- sented in [26]. The gyroscope measurements are thus given as
ω gyro = ω b bi + N W GN + D gyro (2.12)
˙D gyro = −k f D gyro + N W GN D (2.13)
where k f is drift constant and N W GGD is white Gaussian noise in gyroscopic drift.
Sensor noise of the gyroscope used in UWE-3 is around 0.15 deg/sec [27]. Thus for simulations, the variance of white gaussian noise is assumed as 6.854 · 10 −6 and the drift constant k f is assumed zero as on-board implementation of isotropic Kalman filter is capable of removing the gyro drift. The implementation of isotropic Kalman filter and the attitude determination software for UWE-3 was developed as a part of a previous master thesis in [28]. In Fig. 2.5, the simulated gyroscopic data (left) is shown against measurements from UWE-3 gyros (right) to establish the correctness of estimation in sensor noise. The UWE-3 measurements and the data from simulated scenario in Fig. 2.5 are both collected at the end of de-tumbling experiments where the tumbling satellite is stabilized with the B-dot algorithm.
2.3.2 Magnetometers
Several decades of research has led to accurate model of earth’s magnetic field. The International Geomagnetic Reference Field (IGRF), which is a standard formulation of earth’s magnetic field data is used in this work. IGRF model is based on the early works of [29, 30, 31]. Since the magnetic field of earth rotates with earth, the IGRF takes the position of satellite in ECEF frame as the input and, outputs the magnetic field in ECEF.
Adding the magnetic field offset (arising due to the residual magnetic field) (sec. 2.5.3) and white Gaussian noise, the final output of magnetometer in BCF is estimated to
B b (~r) = T b ECEF B ECEF (~r) + N W GM + B of f set (2.14)
2.4 Actuator models
The estimated value of the magnetic field offset obtained from UWE-3 measure- ments is h − 6.1527 5.3040 3.6295 i T μT. The magnetic field offset can actually be calculated on-board, and thus its effect is compensated by feeding the calibrated magnetic field data to the control system. Therefore in the simulations B residual is assumed as zero for the magnetometers.
The variance of white gaussian noise in Eq. 2.14 is estimated as 7.86 · 10 −13 , which is derived from the calculated standard deviation of 1.5358 · 10 −6 T in magnetic field strength in [32] from UWE-3’s magnetometer data.
An example of the simulated magnetometer data is shown in Fig. 2.6 comparing against the standard IGRF output. The deviation of the simulated data from IGRF data is the result of noise from sensors which can not be eliminated.
0 20 40 60 80
-3 -2 -1 0 1 2 3x 10-5
Time (min)
Simulated and Ideal (IGRF) Magnetic field ( T)
Bsimulated-x Bsimulated-y Bsimulated-z BIGRF-x BIGRF-y BIGRF-z
Figure 2.6: Simulated Magnetometer Data
2.4 Actuator models
2.4.1 Thrusters
Micro-vacuum arc thrusters are in the early development phase in UniBwM. It is still
early to characterize its performance and develop accurate models. One theoretical
model of a vacuum arc thruster has been previously developed in [16]. Developing a
theoretical model for micro-VAT for UWE-4 needed expertise in physics of VAT and
Chapter 2 System Models
direct involvement in the development which was not the main goal of this thesis.
So for simplicity, an empirical model is being used. The current measurements from micro-VAT suggests a thrust performance of 0.5-2μN. Therefore, in simulations a linear decay function is assumed over the life period of the thruster which is 10 6 pulses according to current measurements. This is named as the life model in the simulation. The data in the life model can be changed later when accurate measurement are available towards the end of thruster development.
It is very realistic to assume the thruster discharge will be in the form of a plume.
Therefore a plume deviation angle is also considered in the model. Given that the ideal firing directions for UWE-4 thrusters in CCF is h 1 0 0 i . The realistic firing direction is given as:
F direction =
1
s
1+
δ plume
π
cos(N
plume)
s
1+
δ plume
π
sin(N
plume)
s
1+
δ plume
π
(2.15)
where N plume is a uniform random number between -π and π. The variance of the plume angle in degrees (δ plume ) is a guassian random number. In simulations, the variance of plume angle is assumed as 1 · 10 −4 for all thrusters. The wider the plume will be, more the thrusters performance will degrade. In Fig. 2.7, the effect of variation of the plume angle is shown with Thruster 1 (left) having δ plume = 1 · 10 −4 and Thruster 2 (right) δ plume = 1 · 10 −6 . In Fig. 2.7, 90% of the firings occur in a cone of 3° and 1° in Thruster 1 and Thruster 2 respectively.
The command to the thrusters is in the form of a firing vector (F command ) from the controller
F command = h n 1 n 2 n 3 n 4 i (2.16)
where n i is the number of times i th thruster is to be fired The thrust and torque produced by such an action is given by
F thrusters = X n
i=1
− n i f i p i T 0 (2.17)
T thrusters = X n
i=1
− n i (p i × f i T 0 ) (2.18)
p i and f i are the position vector and firing direction of i th thruster respectively. T 0
is the magnitude of thrust depending on the life model of the thruster.
The main aim of UWE-4 is to demonstrate orbit control with miniature thrusters
and therefore all four thruster are placed in a single plane firing in the same direction
to obtain maximum thrust.
2.4 Actuator models
−0.1 −0.05 0 0.05 0.1 0.15
−0.1
−0.08
−0.06
−0.04
−0.02 0 0.02 0.04 0.06 0.08 0.1
3° cone
−0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04
−0.04
−0.03
−0.02
−0.01 0 0.01 0.02 0.03
1° cone
Figure 2.7: Thruster Plume variation
2.4.2 Magnetic torquers
Magnetic torquers are widely used actuators for cubesats. The main advantage with them is that they consume very little energy and are highly reliable due to the absence of any moving parts. They produce a magnetic-moment which interacts with the earth’s magnetic field and provide a torque to rotate the satellite. The torque produced from the magnetic torquers is:
T torquers = ~m × ~B (2.19)
~
m is the magnetic moment produced by torquers in CCF and ~B is the earth’s magnetic field in CCF.
Since the torquers can only produce a torque perpendicular to earth’s magnetic field, they have a performance limitation due to this dependency on the strength and direction of earth’s magnetic field. Although they are good for de-spinning a satellite to lower angular rates, they cannot provide a three axis control, easily.
On UWE-4, there will be one magnetic torquer on each panel, thus in total 6 of them. Each magnetic torquer can produce magnetic moment between ±0.03Am 2 and therefore the maximum magnetic moment available on each axis is ±0.06Am 2 .
2.4.3 Reaction wheel
A reaction wheel may be incorporated in UWE-4 in X-axis in CCF. The major
problem with reaction wheel is its power consumption. Since PPU unit needs at-
least 1W of continuous power for the thrusters and the cubesat only has ~1.5W,
Chapter 2 System Models
there is not enough available power for a reaction wheel [23]. Thus the reaction wheel and thrusters cannot be used simultaneously. The moment generated by the wheel as given in [26] is:
T wheel = I w ( ˙Ω + A w ˙ω bo b ) (2.20)
2.5 Disturbance models (environment)
A disturbance is an external unwanted influence on the system. The different dis- turbance sources acting on the attitude of the satellite are
1. Solar radiation pressure
2. Third body perturbation (mainly moon) 3. Aerodynamic drag
4. Gravity gradient torque, and 5. Effect of residual magnetic field
The first two in the list above are not significant for a cubesat in LEO orbit and thus can be easily neglected [26]. The later three are discussed below and their worst case effects are taken into account in the simulation (chapter 5). In Fig. 2.8, the simulated disturbances are shown. The information of the residual magnetic field is taken from UWE-3. It is clear that the disturbance torque due to residual magnetic field is higher by many orders (10 3 -10 4 ) than that of other disturbance torques. But with improvements in design of UWE-4 over UWE-3, the residual magnetic disturbance torque can be kept much lower than that.
2.5.1 Aerodynamic drag
Aerodynamic drag arises from the air resistance force acting against the velocity of satellite. When the center of pressure of the satellite deviates from the center of gravity, the aerodynamic drag creates a moment which as defined in [26], is:
τ aero = 1
2 C d ρν 2 AL (2.21)
where C d is the drag coefficient, ρ is the atmospheric density, ν is the satellite
velocity, A is the cross-sectional area perpendicular to the velocity and L is the
distance between the center of gravity and center of pressure. In simulations, the
center of gravity is assumed at the geometric center of the cubesat and for worst
case, the center of pressure is assumed at h 1 1 1 i cm .
2.6 Summary
2.5.2 Gravity gradient
The gravity gradient disturbance is caused by the uneven mass distribution along the three axis of the satellite and is given as:
τ gg = 3 µ e
a 3 (c 3 × I s c 3 ) (2.22)
where µ e is the earth’s standard gravitational parameter, a is the semi-major-axis of orbit, c 3 is the third column in rotation matrix T o c and I s is the satellite’s inertia tensor.
2.5.3 Residual magnetic field
The presence of electronics and ferromagnetic material can produce an additional unwanted magnetic field by the satellite which is called as the residual magnetic field. When this residual magnetic field interacts with earth’s magnetic field a disturbance torque is produced. The residual magnetic field produces a torque in the same manner as that of the torquers and thus the disturbance torque is given by
τ residual = ~m residual × ~ B (2.23)
where ~m residual is the residual magnetic-moment in CCF and ~B is the local earth’s magnetic field in CCF.
The estimated value of the residual magnetic moment for UWE-3 is quite high but since it is time invariant, it can be accurately estimated and taken into consideration for the control design . The estimated value of residual magnetic moment from UWE-3 in the worst case situation is h − 0.045 0.0126 0.006 i T Am 2 .
2.6 Summary
In this chapter the main components of satellite’s attitude simulation are discussed.
A description of all the useful coordinate systems for UWE-4 is also provided. At-
titude dynamics and kinematics equations are formulated to compute the satellite
attitude in terms of quaternions. All attitude related sensors and actuators present
in UWE-4 are modeled. Disturbance sources are discussed and their worst case
effect is also shown. It is assumed in this chapter, that position and velocity of the
satellite are already known in ECI frame. This is the subject of the next chapter in
which orbit simulation is discussed.
Chapter 2 System Models
0 20 40 60 80 100
-1.5 -1 -0.5 0 0.5 1 1.5x 10-9
Time (min) Aerodynamic Disturbance Torque ( aero) (N-m)
aero-x
aero-y
aero-z
0 20 40 60 80 100
−3
−2
−1 0 1 2 3 x 10
−10Time (min)
Gravity Gradient Disturbance Torque ( τgg) (N−m)τgg−x τgg−y τgg−z
0 20 40 60 80 100
−2.5
−2
−1.5
−1
−0.5 0 0.5 1 1.5 2 2.5 x 10
−6Time (min)
Residual field Disturbance Torque (τresidual) (N−m)τresidual−x τresidual−y τresidual−z
