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
Luleå University of Technology
Department of Computer Science, Electrical and Space Engineering
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
iii
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)
iv
Contents
Abstract
1
1 Introduction
1.1 UWE . . . . . . . . .
1.1.1 UWE - 3 . . .
1.1.2 UWE - 4 . . .
1.2 Vacuum arc thrusters
1.3 State of art . . . . .
1.4 Orbit control vision .
1.5 Summary . . . . . .
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2 System Models
2.1 Coordinate frames and attitude definition
2.1.1 ECI and ECEF . . . . . . . . . . .
2.1.2 Orbit coordinate frame (OCF) . . .
2.1.3 Body coordinate frame (BCF) . . .
2.1.4 Control coordinate frame (CCF) .
2.1.5 Attitude definition . . . . . . . . .
2.2 Satellite dynamics . . . . . . . . . . . . . .
2.3 Sensor models . . . . . . . . . . . . . . . .
2.3.1 Gyroscope . . . . . . . . . . . . . .
2.3.2 Magnetometers . . . . . . . . . . .
2.4 Actuator models . . . . . . . . . . . . . .
2.4.1 Thrusters . . . . . . . . . . . . . .
2.4.2 Magnetic torquers . . . . . . . . . .
2.4.3 Reaction wheel . . . . . . . . . . .
2.5 Disturbance models (environment) . . . .
2.5.1 Aerodynamic drag . . . . . . . . .
2.5.2 Gravity gradient . . . . . . . . . .
2.5.3 Residual magnetic field . . . . . . .
2.6 Summary . . . . . . . . . . . . . . . . . .
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3 Orbit Propagator
23
3.1 Modified SGP- 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2 Simplified orbit propagator (SOP) with low thrust . . . . . . . . . . . 24
3.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
v
Contents
Contents
4 Attitude Control
4.1 Target attitude . . . . . . . . . . . . . . . .
4.1.1 In-track target attitude . . . . . . . .
4.1.2 Anti-in-track target attitude . . . . .
4.2 Control Laws . . . . . . . . . . . . . . . . .
4.2.1 PD control . . . . . . . . . . . . . .
4.2.2 Sliding mode control . . . . . . . . .
4.3 Magnetic control . . . . . . . . . . . . . . .
4.3.1 B-dot control . . . . . . . . . . . . .
4.3.2 Two axis pointing control . . . . . .
4.4 Thruster control . . . . . . . . . . . . . . . .
4.4.1 One quadrant based two-axis control
4.4.2 Two quadrant based two-axis control
4.5 Combination of torquers and thrusters . . .
4.6 Stability for large angle maneuvers . . . . .
4.7 Summary . . . . . . . . . . . . . . . . . . .
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5 Simulation Results
5.1 B-dot simulation . . . . . . . . . . . . .
5.2 Full magnetic control . . . . . . . . . . .
5.2.1 PD control law . . . . . . . . . .
5.2.2 SMC law . . . . . . . . . . . . . .
5.3 Full thruster control . . . . . . . . . . .
5.3.1 PD control law . . . . . . . . . .
5.3.2 SMC law . . . . . . . . . . . . . .
5.4 Combination of thrusters and torquers .
5.4.1 PD law with one quad thruster .
5.4.2 PD law with two quad thruster .
5.4.3 SMC law with one quad thruster
5.4.4 SMC law with two quad thruster
5.5 Summary . . . . . . . . . . . . . . . . .
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6 Implementation
59
6.1 Hardware architecture . . . . . . . . . . . . . . . . . . . . . . . . . . 59
6.2 Low level code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
6.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
7 Conclusion and Future Work
7.1 Conclusions . . . . . . . . . . . . . . . . .
7.2 Future work . . . . . . . . . . . . . . . . .
7.2.1 Equal distribution of thruster usage
7.2.2 Controller stability analysis . . . .
7.2.3 Efficient use of torquers . . . . . .
7.2.4 Orbit Propagator . . . . . . . . . .
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Contents
Acknowledgments
67
Bibliography
69
Nomenclature
73
vii
List of Tables
1.1
Classification of satellites according to size . . . . . . . . . . . . . . .
4
5.1
5.2
Simulation Initialization Parameters . . . . . . . . . . . . . . . . . . 41
Comparison of Different Simulated Scenarios . . . . . . . . . . . . . . 58
ix
List of Figures
1.1
1.2
1.3
Structural overview of UWE-4 . . . . . . . . . . . . . . . . . . . . . .
Design model of UWE-4 showing thruster locations . . . . . . . . . .
Trailing formation of two satellites . . . . . . . . . . . . . . . . . . .
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
Simulation software block diagram . . . . . . . . . . . . . . .
Body coordinate frame . . . . . . . . . . . . . . . . . . . . . .
Inertial coordinate frames: ECI and ECEF . . . . . . . . . . .
Orbit coordinate frame and Control coordinate frame . . . . .
Simulated gyroscopic data (left) vs UWE-3 gyro measurements
Simulated Magnetometer Data . . . . . . . . . . . . . . . . .
Thruster Plume variation . . . . . . . . . . . . . . . . . . . .
Simulated Disturbance Torques . . . . . . . . . . . . . . . . .
3.1
3.2
3.3
3.4
Modified SGP-4 Propagator . . . . . . . . . . . . . . . . . . . . . .
Simplified orbit propagator . . . . . . . . . . . . . . . . . . . . . . .
Error in Simplified propagator from SGP-4 output . . . . . . . . . .
Demonstration of Thruster effect seen as the separation of two satellites generated by Simplified orbit propagator . . . . . . . . . . . .
4.1
4.2
Schematic of the Controller . . . . . . . . . . . . . . . . . . . . . . . 30
Thruster control quadrants . . . . . . . . . . . . . . . . . . . . . . . . 35
5.1
5.2
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(right)
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. 26
Graphical User Interface (GUI) for simulation experiments . . . . . .
B-dot result from simulation (left) vs UWE-3 de-tumbling experiment
result (right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Full magnetic control with PD law - Attitude and Angular Velocity .
5.4 Full Magnetic Control with PD control law - Magnetic Moment Output of Torquers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5 Full magnetic control with SMC law - Attitude and Angular Velocity
5.6 Full Magnetic Control with SMC control law - Magnetic Moment
Output of Torquers . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.7 Full thruster control with PD law - Attitude and Angular Velocity . .
5.8 Full thruster control with PD law - Settling Time . . . . . . . . . . .
5.9 Full thruster control with PD law - Settling Accuracy . . . . . . . . .
5.10 Full thruster control with PD law - Thruster firings per minute and
Cumulative firings . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.11 Full thruster control with SMC law - Attitude and Angular Velocity .
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xi
List of Figures
List of Figures
5.12 Full thruster control with SMC law - Settling Time . . . . . . . . . .
5.13 Full thruster control with SMC law - Settling Accuracy . . . . . . . .
5.14 Full thruster control with SMC law - Thruster firings per minute and
Cumulative firings . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.15 PD law with One Quad Thrusters - Attitude and Angular Velocity .
5.16 PD law with One Quad Thruster - Settling Accuracy and Settling Time
5.17 PD law with One Quad Thrusters - Thruster firings per minute and
Cumulative firings . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.18 PD law with One Quad Thrusters - Commanded Torque Load sharing
between Thruster and Torquers in X and Y axis . . . . . . . . . . . .
5.19 PD law with One Quad Thrusters - Commanded Torque Load sharing
in Z-axis and Magnetic Moment Output of Torquers . . . . . . . . . .
5.20 PD law with Two Quad Thrusters - Attitude and Angular Velocity .
5.21 PD law with Two Quad Thruster - Settling Accuracy and Settling Time
5.22 PD law with Two Quad Thrusters - Thruster firings per minute and
Cumulative firings . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.23 PD law with Two Quad Thrusters - Commanded Torque Load sharing
between Thruster and Torquers in X and Y axis . . . . . . . . . . . .
5.24 PD law with Two Quad Thrusters - Commanded Torque Load sharing
in Z-axis and Magnetic Moment Output of Torquers . . . . . . . . . .
5.25 SMC law with One Quad Thrusters - Thruster firings per minute and
Cumulative firings . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.26 SMC law with One Quad Thrusters - Commanded Torque Load sharing between Thruster and Torquers in X and Y axis . . . . . . . . . .
5.27 SMC law with One Quad Thrusters - Commanded Torque Load sharing in Z-axis and Magnetic Moment Output of Torquers . . . . . . .
5.28 SMC law with One Quad Thrusters - Attitude and Angular Velocity .
5.29 SMC law with One Quad Thrusters - Settling Time . . . . . . . . . .
5.30 SMC law with One Quad Thrusters - Settling Accuracy . . . . . . .
5.31 SMC law with Two Quad Thrusters - Attitude and Angular Velocity
5.32 SMC law with Two Quad Thrusters - Settling Time . . . . . . . . . .
5.33 SMC law with Two Quad Thrusters - Settling Accuracy . . . . . . . .
6.1
xii
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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 simulation platform for attitude and orbit control is developed. More than one control
philosophies are developed, compared and implemented to fly on-board UWE-4. Difficulties 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 control, 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
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 program 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.
3
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 experimentation 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 demostrate an accurate attitude determination and control system (ADCS) for picosatellites. For attitude determination, UWE-3 has sun sensors, gyros and magnetometers 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 attitude 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 technologies 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 torquer is also visible in the Fig. 1.1.
4
1.1 UWE
1
9
2
3
4
5
6
8
7
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.
5
Chapter 1
Introduction
1.2 Vacuum arc thrusters
Vacuum arc thruster (VAT) is the technology chosen for micro-propulsion for UWE4. 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 developed 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.
Antenna
Thruster 2
Thruster 3
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 performed 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.
6
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 UWE4. The UWE-4 project is aiming to push miniaturization to its extreme limit to
demonstrate what could be the future of space missions.
Target
Chaser
Figure 1.3: Trailing formation of two satellites
7
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
106 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.
8
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 simulation 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 basic 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 understanding of the concepts presented in this thesis.
Orbit
in-Track
anti-in-Track
Controller
Disturbances
Actuators
Satellite
Dynamics
Sensors
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.
9
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 geometric 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
h
ωooi = 0 0 −ω0
where ω0 =
2π
T
iT
(2.1)
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.
10
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. ωbbi . Therefore, in order to compute angular velocity of the satellite in the control coordinate
frame we need the the transformation matrix (Tcb ) from BCF to CCF which is
0 0 1


Tcb =  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
c
ωbi
= Tcb ωbbi
(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
11
Chapter 2
System Models
angular velocity of the body w.r.t. OCF (ωcbo ) must be known. To compute that,
the Eq. 2.4 from [24] is used.
(2.4)
ωcbi = ωcbo + ωcoi
Angular velocity of OCF w.r.t ECI in CCF (ωcoi )
ωcoi = Tco ωooi
(2.5)
Using Eq. 2.1 and the transformation matrix from OCF to CCF (Tco ), the angular
velocity of body w.r.t. OCF (ωcbo ) is given as
h
ωcbo = ωcbi − Tco 0 0 −ω0
iT
(2.6)
ZECI ,ZECEF
XECEF
Pointing to 0° Latitude
YECI
YECEF
XECI
Fixed in inertial space
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 hthe two coordinate
frames
i
coincides, it is represented by the attitude quaternion as 0 0 0 1 .
12
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)
13
Chapter 2
System Models
as given in [24] is
c
Is ω̇cbi + ωcbi × Is ωcbi = Ttotal
(2.7)
The total external torque (Ttotal ) is the sum of all individual torques acting on the
satellite.
Ttotal = Ttorquers + Tthrusters + Tdisturbances
(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]
c
Iˆω̇cbi + ωcbi × (Is ωcbi + Aw Iw Ω) = Ttotal
(2.9)
where Aw , Iw , Ω are the layout matrix, inertia tensor and angular velocity of
reaction wheels. and Iˆ is given by
Iˆ = Is − Aw Iw ATw
(2.10)
In presence of wheel a new term, Twheel 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
h
iT
of body w.r.t orbit in OCF is expressed as ωcbo = ωx ωy ωz , the kinematic
equation as given in [24] is
0
ωz −ωy ωx

1  -ω
0
ωx ωy 

q
q̇ =  z

ω
−ω
0
ω
2
y
x
z
−ωx −ωy -ωz 0


(2.11)
The conversion between ωcbo and ωibi has been discussed in sec. 2.1.3 and concluded
in Eq. 2.6. The different values of qinitial and ωinitial can be used as initial conditions
in the simulation for attitude and angular velocity.
14
2.3 Sensor models
0.5
UWE−3 ω ( deg/s)
0
0
x
Simulated ωx (deg/s)
0.5
−0.5
−1
0
20
40
60
Time (min)
80
−0.5
−1
0
100
UWE−3 ω ( deg/s)
100
0.5
y
Simulated ωy (deg/s)
0.5
0
−0.5
0
−0.5
−1
−1
20
40
60
Time (min)
80
100
−1.5
0
20
40
60
Time (min)
80
100
1.5
1.5
1
UWE−3 ωz ( deg/s)
1
Simulated ωz (deg/s)
80
1
1
0.5
0
−0.5
0.5
0
−0.5
−1
−1
−1.5
0
40
60
Time (min)
1.5
1.5
−1.5
0
20
20
40
60
Time (min)
80
100
−1.5
0
20
40
60
Time (min)
80
100
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
15
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 (NW GN ) and gyroscopic drift (Dgyro ) as presented in [26]. The gyroscope measurements are thus given as
ωgyro = ωbbi + NW GN + Dgyro
(2.12)
Ḋgyro = −kf Dgyro + NW GN D
(2.13)
where kf is drift constant and NW 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 kf 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
Bb (~r) = TbECEF BECEF (~r) + NW GM + Bof f set
16
(2.14)
2.4 Actuator models
The estimated value of the magnetic field offset obtained from UWE-3 measureh
iT
ments is −6.1527 5.3040 3.6295 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 Bresidual 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.
-5
Simulated and Ideal (IGRF) Magnetic field ( T)
3
Bsimulated-x
x 10
B
simulated-y
Bsimulated-z
BIGRF-x
2
BIGRF-y
BIGRF-z
1
0
-1
-2
-3
0
20
40
60
80
Time (min)
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
17
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
106 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
Given that the
h the model.
i
ideal firing directions for UWE-4 thrusters in CCF is 1 0 0 . The realistic firing
direction is given as:

Fdirection = 

1
s
1+
δplume plume )
scos(N
δplume 1+
π
π

plume )
ssin(N

δplume 
1+
(2.15)
π
where Nplume is a uniform random number between -π and π. The variance of the
plume angle in degrees (δplume ) 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 δplume = 1 · 10−4
and Thruster 2 (right) δplume = 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 (Fcommand ) from the
controller
h
Fcommand = n1 n2 n3 n4
i
(2.16)
where ni is the number of times ith thruster is to be fired
The thrust and torque produced by such an action is given by
Fthrusters =
n
X
− ni fi pi T0
(2.17)
− ni (pi × fi T0 )
(2.18)
i=1
Tthrusters =
n
X
i=1
pi and fi are the position vector and firing direction of ith thruster respectively. T0
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.
18
2.4 Actuator models
0.1
0.03
0.08
3° cone
0.06
0.04
1° cone
0.02
0.01
0.02
0
0
−0.02
−0.01
−0.04
−0.02
−0.06
−0.03
−0.08
−0.1
−0.1
−0.05
0
0.05
0.1
−0.04
0.15
−0.03
−0.02
−0.01
0
0.01
0.02
0.03
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:
~
Ttorquers = m
~ ×B
(2.19)
~ is the earth’s
m
~ is the magnetic moment produced by torquers in CCF and B
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 ±0.03Am2
and therefore the maximum magnetic moment available on each axis is ±0.06Am2 .
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 atleast 1W of continuous power for the thrusters and the cubesat only has ~1.5W,
19
0.04
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:
b
Twheel = Iw (Ω̇ + Aw ω̇bo
)
(2.20)
2.5 Disturbance models (environment)
A disturbance is an external unwanted influence on the system. The different disturbance 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 (103 -104 ) 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:
1
τaero = Cd ρν 2 AL
2
(2.21)
where Cd 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
h
i of the cubesat and for worst
case, the center of pressure is assumed at 1 1 1 cm.
20
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:
τgg = 3
µe
(c3 × Is c3 )
a3
(2.22)
where µe is the earth’s standard gravitational parameter, a is the semi-major-axis
of orbit, c3 is the third column in rotation matrix Toc and Is 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
~
τresidual = m
~ residual × B
(2.23)
~ is the local earth’s
where m
~ residual is the residual magnetic-moment in CCF and B
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
h
iT
UWE-3 in the worst case situation is −0.045 0.0126 0.006 Am2 .
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. Attitude 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.
21
Chapter 2
-9
−10
1
gg
x 10
Gravity Gradient Disturbance Torque ( τ ) (N−m)
Aerodynamic Disturbance Torque ( aero) (N-m)
1.5
System Models
aero-x
aero-y
0.5
aero-z
0
-0.5
-1
-1.5
0
20
40
60
Time (min)
80
100
3
x 10
1
0
−1
−2
−3
0
τ
1.5
residual−z
1
) (N−m)
τresidual−y
disturbance
τresidual−x
2
2.5
Total Disturbance Torque (T
Residual field Disturbance Torque (τresidual) (N−m)
τgg−z
20
40
60
Time (min)
80
100
−6
x 10
0.5
0
−0.5
−1
−1.5
−2
−2.5
0
τgg−y
2
−6
2.5
τgg−x
20
40
60
Time (min)
80
100
x 10
Tdisturbance−x
2
Tdisturbance−y
Tdisturbance−z
1.5
1
0.5
0
−0.5
−1
−1.5
−2
−2.5
0
20
40
60
Time (min)
80
100
Figure 2.8: Simulated Disturbance Torques
Top-left: Aerodynamic Disturbance torque Top-right: Gravity Gradient Torque
Bottom-left Residual magnetic field Disturbance Torque Bottom-right: Total
Disturbance Torque
22
3 Orbit Propagator
The on-board orbit propagator is crucial for formation flying missions. UWE-3
uses a standard SGP-4 implementation for orbit propagation. Due to the inclusion
of propulsion system, a new orbit propagator is required for UWE-4, which can
incorporate the effect of the low thrust produced by micro-thrusters. An idea for
using SGP-4 for low thrust scenario has been discussed in sec. 3.1. However this
method proved to be unsuccessful. The sec. 3.2 discusses an simple orbit propagator
implementation which takes into account the effect of gravitation, low thrust by
propulsion system and earth’s oblateness (J2 and J3 terms). Orbit propagator with
low thrust input is an active area of research and readers are refereed to [33, 34, 35]
for other studies.
3.1 Modified SGP- 4
SGP stands for Simplified general perturbation. It is the most widely used orbit
propagator used in space industry. According to [36], SGP-4 has an accuracy of
1 km at epoch time and the accuracy decays with the rate of 1-3 km each day.
Therefore for the modified orbit propagator the accuracy should be better than 3
km. For UWE-3, a new TLE is generated every day by NORAD thus orbit error is
limited to 1 km only. In the worst care scenario, one TLE can be used for 3 days
maximum, which means that an orbit error of 10 km may be acceptable for some
missions.
Since SGP-4 does not consider a thrust input, an attempt was made to build an
orbit propagator based on SGP-4 to incorporate continuous thrust from propulsion
system.
The idea behind this attempt of modifying SGP-4 for low thrust came from the
fact that a thrust force effects the instantaneous position and velocity vector in the
inertial frame. A block diagram representation of the modified SPG-4 is shown in
~ vector and velocity vector (~v ) outputs
Fig. 3.1. In this modification, the position (R)
from SGP-4 is modified by the thrust input according to Newton’s second law of
~ 0 ) vector
motion. Then the block “RV to COE” converts the modified position (R
and velocity vector (v~0 ) in ECI frame, to classical orbital elements. The classical
elements are then fed back to SPG-4 as a fresh initial conditions. Keeping the
input update time of SGP4 propagator equal to zero makes sure that the orbit is
23
Chapter 3
Orbit Propagator
propagated one point at one time capturing the effect of low thrust produced by the
propulsion system for one time step.
The result from this modification of SGP-4 came out to be fruitless. The outputs
−
→−
(R,→
v ) were oscillatory and unfortunately, all attempts to stabilize this propagator
failed, therefore any results from this modified SGP-4 are not shown here. It was
concluded that this feedback method is unstable due to improper handling of orbital element vectors due to the nature of the method itself. After this attempt, a
simplified orbit propagator was implemented which is discussed in the next section.
Thrusters
∆r, ∆v
TLE
SGP-4
Propagator
+
RV to
OEV
Figure 3.1: Modified SGP-4 Propagator
3.2 Simplified orbit propagator (SOP) with low thrust
A simplified orbit propagator is developed for realizing the effect of thrusters in
the simulation. It only takes into account the effect of earth’s gravitation, thruster
impulse force and earth’s oblateness. These three terms are summed up to compute
the instantaneous acceleration which is then used in the newton’s second law of
motion. All calculation are done in the ECI frame assuming it as an absolute
inertial frame.
The acceleration due to gravitation pull is the standard result of two body problem
from [24] which is
r̈ = −
µ
r
|r|3
(3.1)
The effect of thruster impulse force is captured directly by Newton’s second. The
24
3.2 Simplified orbit propagator (SOP) with low thrust
acceleration due to oblateness effect in ECI coordinate frame as derived in [37] are
r̈x = J2
rx
3
rx rz
15
(6rz2 − (rx2 + ry2 )) + J3 9 (10rz2 − (rx2 + ry2 ))
7
|r|
2
|r|
2
(3.2)
r̈y = J2
ry
ry rz
3
15
(6rz2 − (rx2 + ry2 )) + J3 9 (10rz2 − (rx2 + ry2 ))
7
|r|
2
|r|
2
(3.3)
r̈z = J2
3
1
rz
9
(3rz2 − (rx2 + ry2 )) + J3 9 (4rz2 (rz2 − 3(rx2 + ry2 )) + (rx2 + ry2 )2 ) (3.4)
7
|r|
2
|r|
2
where, J2 = 1.7555 × 1010 km5 s−2 and J3 = −2.619 × 1011 km6 s−2 .
A block diagram showing the implementation of simplified orbit propagator is shown
in Fig. 3.2. The results from this propagator are not as accurate as SGP-4. Fig. 3.3
shows the relative distance between the output from simplified orbit propagator and
SGP-4 propagator generated by STK by the exact same TLE and initial conditions.
It can be seen that the simplified orbit propagator output deviates from ideal (SPG4) by 10 km only in 16 orbits (~1 day). This accuracy might not be sufficient for
on-board implementation but it could be good enough to simulate the orbit control
scenario as discussed in sec. 1.4.
Gravitation
+
Thrusters
+
a
∫
v
∫
r
J2 J3
Term
Figure 3.2: Simplified orbit propagator
25
Chapter 3
Orbit Propagator
12
Error in SOP from SGP−4 [km]
10
8
6
4
2
0
0
2
4
6
8
10
12
14
16
18
Orbits
Figure 3.3: Error in Simplified propagator from SGP-4 output
0.7
in−Track Firing
anti−in−Track Firing
Relative distance [km]
0.6
0.5
0.4
0.3
0.2
0.1
0
0
2
4
6
8
10
12
14
16
Orbits
Figure 3.4: Demonstration of Thruster effect seen as the separation of two satellites
generated by Simplified orbit propagator
26
3.3 Summary
Fig. 3.4 demonstrate the scenario simulated by simplified orbit propagator when
thrusters are fired (at a rate of 600 firings per minute) on one satellite in formation
with another satellite initially present at the same location in the orbit. In this orbit
simulation, the thrusters are being fired by an on-off controller switching between the
in-track direction (i.e. the propelling mode) and the anti-in-track direction which is
the braking mode. It can be seen that the effect of thrusters is clearly captured by
the propagator as the two satellites initially at the same location exhibit a formation
flying scenario with their relative distance bounded within 0.6 km. However, with
an error of 10 km in one day, it cannot be directly deployed in the cubesat, but if
both the satellites (the chaser and the target) have simplified orbit propagator as in
the case of simulations, the error from the ideal orbit is heavily negated.
3.3 Summary
In this chapter, the need of an on-board propagator is motivated and addressed. An
attempt is made to modify the standard SGP-4 propagator for low thrust scenario
without success. However a simplified orbit propagator is developed as a first step
towards a more accurate on-board propagator. In future, it is desired to append
more terms in the simplified orbit propagator with low thrust to get closer to the
accuracy of SGP-4 propagator.
27
4 Attitude Control
The requirement of high precision attitude control for UWE-4 comes from the need
that the thrust vector should be pointing in-track direction (X-axis in CCF opposite
to the X-axis in OCF) or anti-in-track direction (X-axis in CCF in the direction
of the X-axis in OCF) for obtaining maximum efficiency. Although, a two-axis
control is sufficient for achieving the desired attitude control, it is highly desired to
have some control on the third axis (X-axis in OCF). Thus, three axis control laws
are formulated and implemented together with an interesting solution of dynamic
target quaternion (discussed in sec. 4.1). Two such control laws, PD (proportional
derivative) and SMC (sliding mode control) are discussed in sec. 4.2. In sec. 4.3
and sec. 4.4, implementation of full magnetic control and full thruster control are
discussed. Finally, the use of both magnetic control (via torquers) and thrusters is
discussed in sec. 4.5. The idea behind the combination is to use torquers to their
maximum capacity wherever possible in order to prolong the life time of thrusters.
In sec. 4.6, stability concerns with large angle rotation are brought to notice. Fig. 4.1
shows the skeleton of the control software which will be explained in more details in
this chapter.
4.1 Target attitude
The attitude control problem defined for UWE-4 is a two-axis control (Y and Z in
OCF) and stabilization in the third axis (X in OCF). In order to compute the target
quaternions dynamically, a solution is developed which is explained below.
iT
h
h
iT
If qT = qT 1 qT 2 qT 3 qT 4 is the target attitude and q = q1 q2 q3 q4 is the
current attitude, where the first three elements in q and qT represents the vector
part and last elements (q4 and qT 4 ) represents the scalar part, then the error in
attitude in terms of quaternions qE given as in [38] is
qE1
qT 4
qT 3 −qT 2 −qT 1 q1
q 
−q
 
qT 4
qT 1 −qT 2 



 q2 
qE =  E2  =  T 3
 
qE3 
 qT 2
−qT 1 qT 4 −qT 3  q3 
qE4
qT 1
qT 2
qT 3
qT 4
q4



 
(4.1)
29
Chapter 4
Attitude Control
Direction
1.
2.
In-Track
Anti-In-Track
Control Law
1.
2.
PD
SMC
Actuators
1.
2.
3.
Torquers
Thrusters
Both
Outputs
1.
2.
Magnetic Moment
Firing Command
1.
2.
One Quad
Two Quad
Figure 4.1: Schematic of the Controller
Since a full 3 axes control is not the goal due to the absence of full controllability
in all axes, qE1 is disregarded and assumed to be zero.
qE1 = qT 4 q1 + qT 3 q2 − qT 2 q3 − qT 1 q4 = 0
(4.2)
By the definition of quaternion, we have
qT2 1 + qT2 2 + qT2 3 + qT2 4 = 1
(4.3)
4.1.1 In-track target attitude
When the thrust vector is in the velocity direction, the thrusters
are facing
the
h
i
trailing orbit and the target attitude takes the form qT = 0 qT 2 qT 3 0 . This
target attitude formulation represents the attitude in in-track direction with any
given rotation in the X-axis. In this form of the target attitude, it is evident that
qT 1 = qT 4 = 0
30
(4.4)
4.1 Target attitude
Solving the Eqs. 4.2, 4.3 and 4.4 gives two sets of solution
qT 2 = q
qT 2 = q
q2
qT 3 = q
q22 + q32
−q2
qT 3 = q
q22 + q32
q3
q22 + q32
−q3
q22 + q32
(4.5)
(4.6)
These two solutions for the target attitude represents the same physical orientation
of the satellite. The choice of selecting one solution over the other will affect the
direction of rotation of the satellite towards its target attitude. The optimal control
should use both the solutions but guaranteeing global stability while using both the
solutions is regarded as a difficult problem [39]. Many research works that provides
the solution to this problem while addressing stability issues have also been pointed
out in [39]. However, in this thesis, the solution from Eq. 4.5 is used which when
used alone is proven as globally stable during simulations.
Thus, the in-track target attitude is given as
q2
qT = 0 √q2 +q2
2
3
√ q23
q2 +q32
0
(4.7)
4.1.2 Anti-in-track target attitude
When the thrust vector is opposite to the velocity direction, the thrusters
are facing
h
i
the forward orbit and the target attitude takes the form qT = qT 1 0 0 qT 4 .
This target attitude formulation represents the attitude in anti-in-track direction
with any given rotation in the X-axis. In this form of the target attitude, it is
evident that
qT 2 = qT 3 = 0
(4.8)
Similar to the sec. 4.1.1, solving the Eqs. 4.2, 4.3 and 4.8 gives two sets of solution
qT 1 = q
qT 1 = q
q1
q12 + q42
−q1
q12
+
q42
qT 4 = q
qT 4 = q
q4
q12 + q42
−q4
q12 + q42
(4.9)
(4.10)
31
Chapter 4
Attitude Control
Following the discussion in the sec. 4.1.1, the solution of Eq. 4.9 is used in the
simulations. Thus, the anti-in-track target attitude is given as
q1
q4
qT = √q2 +q2 0 0 √q2 +q2
1
4
1
4
(4.11)
4.2 Control Laws
Attitude control problem for satellites has been studied for many decades. The
attitude control law determines the required torque to meet the control problem.
The two pointing directions of interest (in-track and anti-in-track) are translated into
target attitudes in sec. 4.1. A proportional derivative (PD) control law discussed in
sec. 4.2.1 is the most widely used technique for attitude control. The other control
techniques include the simple bang bang control [40], LQR with pulse width pulse
frequency [41, 42] and model predictive control [43, 44]. Sliding model control which
is a non-linear control technique has also been fairly well discussed and implemented
for attitude control as in [45, 46, 26]. Given the advantages of non-linear control for
this problem, it is decided to implement a sliding mode control for UWE-4, which
is discussed in sec. 4.2.2.
4.2.1 PD control
PD control law outputs the torque required for in-track or anti-in-track direction.
The PD control law for attitude as given in [38] is
Tcx = −T̂0 (Kx qE1 + Kdx ωx )
(4.12)
Tcy = −T̂0 (Ky qE2 + Kdy ωy )
(4.13)
Tcz = −T̂0 (Kz qE3 + Kdz ωz )
(4.14)
where Kx , Ky , Kz are proportional gains and Kdx , Kdy , Kdz are the derivative gains.
The angular velocities ωx , ωy , ωz are in CCF and qE1 , qE2 , qE3 are computed from
Eq. 4.1.
It is to be noted that in Eq. 4.12, first term is zero because of Eq. 4.2, which
makes Tcx = −T̂0 Kdx ωx . T̂0 is the unit of torque produced by the thrusters and it
32
4.2 Control Laws
is assumed that T̂0 = T̂f actor 10−7 where T̂f actor is a tunable parameter. It is to be
noted that T̂0 is a close relative of T0 from sec. 2.4.1. If there is an effective way to
measure T0 on board, then T̂0 can be replaced by T0 .
qE1 , qE2 , qE3 in the above equations are calculated with the help of Eqs. 4.1 and 4.7
or 4.11.
4.2.2 Sliding mode control
Sliding mode control is a discontinuous type of non-linear control. The control action
produced by sliding mode control forces the system to slide along a designed sliding
surface which is a subspace of all possible values of the sliding variable. This sliding
mode control presented here for UWE-4 is inspired by [46]. The sliding variable (Sv )
is defined at first as
h
c
Sv = ωbo
+ Λs qE1 qE2 qE3
iT
(4.15)
where Λs is a positive definite diagonal matrix assumed as
K11
0
0

K22
0 
Λs =  0

0
0
K33


(4.16)
The control torque by sliding mode law has two parts. First part (Tsm ), incorporates
the non-linear dynamics of the system and keeps the system on the sliding manifold
i.e. it makes sure that time derivative of sliding variable is zero (Ṡv = 0). The
second part (Tsv ) brings the sliding variable to zero in a finite time. From [46], the
values of the two control torques are:
Tsv = −λs sign(Sv )
(4.17)
where λs is a tunable parameter like Λs .
1
c
c
c
c
c
q4 +ωbo
×qvec ) (4.18)
Tsm = ωbi
×Is ωbi
−3ω02 (koc ×Is koc )+ω0 Is (ico ×ωbo
)− Is Λs (ωbo
2
where ico and koc are the first and the third column in the transformation matrix
from OCF to CCFh (i.e. Tco ) irespectively. The qvec is the vector part of the current
attitude given by q1 q2 q3 .
The demanded torque by sliding mode control law is then
Tc = Tsm + Tsv
(4.19)
33
Chapter 4
Attitude Control
4.3 Magnetic control
Magnetic control infers to the use of magnetic torquers for attitude control. Due
to project limitations (e.g. budget, technology), many cubesats only use magnetic
control for attitude control. But even with magnetic torquers only, many different
experiments can be performed.
The B-dot control (sec. 4.3.1) is used to detumble the satellite from high angular
velocities that can happen during the release from launch vehicles. The two axis
pointing control (sec. 4.3.2) formulates an actuator law to produce the necessary
torques ordered by the control law. As discussed earlier, the torquers can only
produce torques perpendicular to the earth’s local magnetic field, thus a precise
attitude control is not possible.
4.3.1 B-dot control
The B-dot algorithm for torquers [8, 10, 46] is used to stabilize a fast spinning satellite. It computes the rate of change in earth’s local magnetic field and accordingly
takes some actions. The B-dot law is given as
→
−
m = −K Ḃ
(4.20)
Several variation of B-dot control have been presented in [10] and applied on UWE3. A comparison of B-dot simulation results with actual B-dot experiment data from
UWE-3 is shown in sec. 5.1.
4.3.2 Two axis pointing control
Two axis pointing control computes the magnetic moment to be generated to produce
the required control torque. Therefore, the purpose is to find m,
~ in the Eq. 4.21
→
−
−
Tc = →
m×B
(4.21)
→
−
Cross multiplying B on both sides of Eq. 4.21 from left and assuming that m
~ is
→
−
almost perpendicular to B , gives the following solution
→
−
B × Tc
→
−
m = →
− 2
B 34
(4.22)
4.4 Thruster control
4.4 Thruster control
The arrangement of thrusters can produce torques in Y-axis and Z-axis in CCF. But
here a precise two axis control is guaranteed. Two different methods of actuation of
thrusters has been developed and discussed in this section. One quadrant method
(sec. 4.4.1) is based on using one thruster to produce the required control torque
while two quadrant method (sec. 4.4.2) uses two thrusters to produce required control
torque demanded by the control law. It is intuitive to say that two quadrant method
is computationally expensive because it needs to solve an integer problem to compute
how many times the two selected thrusters should fire in every update cycle to PPU.
Since thrusters can only produce torques in two axis, they only act on Tcy , Tcz from
iT
h
a required control torque command Tc = Tcx Tcy Tcz . The Fig. 4.2 shows the
thruster control quadrants which tells which thruster number should to be fired
according to the sign of Tcy and Tcz . For example if Tcy > 0 and Tcz < 0, thruster 4
should be fired.
Tcy
Thruster 4
Thruster 1
Tcy + , Tcz -
Tcy + , Tcz +
Tcz
Thruster 3
Thruster 2
Tcy - , Tcz -
Tcy - , Tcz +
Figure 4.2: Thruster control quadrants
4.4.1 One quadrant based two-axis control
One quadrant based control method commands one thruster to provide the required
control torque. It computes the thruster action in two steps:
1. Compute the thruster number from Fig. 4.2 based on the sign of Tcy and Tcz .
2. Compute the number of firings of the selected thruster according to the magnitude of Tcy and Tcz with Eq. 4.23.
Nf iring =
min(Tcy , Tcz )
T0
(4.23)
35
Chapter 4
Attitude Control
A saturating number on Nf iring is placed based on the stability of the closed loop
control.
4.4.2 Two quadrant based two-axis control
Two quadrant based control method commands two thrusters to provide the required
control torque. It computes the thruster action in these steps
1. Compute the primary thruster number (Qf p ) according to the sign of Tcy and
Tcz exactly as in sec. 4.4.1. The quadrant of primary thruster direction is given
as
h
Qf p = sign Tcy −Tcz
iT
2. Compute the secondary thruster number and secondary thruster quadrant
h
iT
(Qf s ) by the sign of Tcy −Tcz if max(Tcy , Tcz ) = Tcy or by the sign of
h
−Tcy Tcz
iT
if max(Tcy , Tcz ) = Tcz .
3. Solve for primary thruster firing number Nf p and secondary thruster firing
number Nf s by solving the following equation
Nf p Qf p + Nf s Qf s =
1
Tc2q
T0
(4.24)
where Qf p and Qf s are quadrants of primary and secondary thrusters, both a
subset of all possible quadrants.
Qf p , Qf s ∈
h
h
1 1
and Tc2q = Tcy Tcz
iT
iT
h
iT
1 −1
h
−1 1
iT
h
iT −1 −1
.
4. A saturation check is done on total number of firings. If Nf p + Nf s exceeds
Nlinit , both Nf p and Nf s are reduced keeping their relative ratio same and
making Nf p + Nf s = Nlimit
Two quadrant method makes optimal use of more than one thruster to produce the
control torque.
4.5 Combination of torquers and thrusters
To improve the life expectancy of attitude control hardware and to reduce the average power consumption, it is desired to use magnetic torquers as much as possible.
Therefore, combining the actions from magnetic torquers and thrusters is an advantageous solution. It is done by dividing the demanded control torque Tc between
36
4.6 Stability for large angle maneuvers
torquers and thrusters. A simple strategy to split the control torque is used which
introduces a new variable Rsplit in the software. The Eqs. 4.25 and 4.26 are used to
compute control torque demanded by thrusters and torquers respectively.
h
Tc−thrusters = Rsplit 0 Tcy Tcz
h
Tc−torquers = Tcx 0 0
iT
iT
(4.25)
h
+ (1 − Rsplit ) 0 Tcy Tcz
iT
(4.26)
In this work, Rsplit is kept constant at 0.5 but a better method to compute it onboard is a part of future work. The combination of torquers and thrusters improves
the control in all three axes. The control in X-axis of CCF is still dependent on the
earth’s magnetic field, so a full control in this axis is not possible.
4.6 Stability for large angle maneuvers
Quaternion based attitude control problem is not globally stable [38] especially for
large angle rotations. However they are still preferred over Euler angles because
they provide singularity free analysis. The strongest tool to establish the stability
of a control system is the Lyapunov stability criteria. As discussed in [38, 47], for
large angle errors, the quaternion based control laws can result in infinite control
action. This needs to be avoided in all cases by limiting the angular rates to small
values which can then guarantee global stability of the attitude control.
In the performed simulations in this work, this limit on angular rate is put on by
the variable Nlimit , used for thrusters. It is also noticed that if Nlimit is set very high
for example at 50, instability is observed in some cases, possibly where Liapunov
stability criteria is not satisfied. A theoretical study of the stable zones for the two
controllers is not performed in this work. The attitude control stability strongly
depends on the thruster characteristics (i.e. impulse bit, operation frequency and
plume) therefore this will be addressed later while doing system level tests when the
thruster characteristics are fully known.
4.7 Summary
This chapter describes the attitude control software developed for UWE-4. Given
there is no requirement for control and presence of weak actuation in X-axis in CCF,
a new concept of dynamic target attitude is developed to avoid taking excessive
action in this direction. A conventional control law - PD and an advanced control
law - sliding mode have been formulated and presented. The torque demanded by
control law can be realized with a pure magnetic control using magnetic torquers or
by newly incorporated vacuum arc thrusters or by a combination of both.
37
5 Simulation Results
The simulation is developed in Matlab and Simulink environment without using any
additional libraries other than the base package. Fig. 5.1 is the the graphical user
interface developed for the simulation experiments. Based on the discussions in the
previous chapter, a variety of controllers can be formed by choosing a set of options
from Fig. 4.1. In this chapter, the results from all such possible controllers are
presented and compared against each other.
Figure 5.1: Graphical User Interface (GUI) for simulation experiments
39
Chapter 5
Simulation Results
All the control modes, the two direction and different controller parameters in
Fig. 5.1 have been discussed other than a few thruster parameters. The explanation
of the thruster parameters not introduced previously are as follow:
• Tcl: It is the lower cutoff of the demanded control torque i.e. if Tc < T cl,
the control demanded torque Tc is set to zero. This is done to not send small
currents to the magnetic torques and to avoid small fractional calculations for
thrusters, which will only burden the PPU microprocessor.
• Firing limit: It is the maximum number of allowed firings in 100 ms. This
means that 10·F iringLimit is the maximum operating frequency of all thrusters
combined. Although this number depends on the PPU design and the time
constants involved in thruster operation, the main limit on this variable is
coming from control stability aspect which is discussed in sec. 4.6. It is observed empirically that keeping Firing limit high (~50), i.e. 500Hz, introduces
instability in attitude control.
• T̂f actor : In the simulations, the base torque unit T̂0 is assumed as T̂0 =
T̂f actor 10−7 . However, it is desired that this number should be calculated onboard. The T̂f actor leaves the space for this requirement in the simulation.
An initialization script is always run before the simulation which imports the assumed parameters for each simulation run. Table 5.1 shows the list of some parameters and their values. The controller parameters are found out empirically and are
kept constant for all simulations for the sake of clean comparison. For better visualization, all the simulation results shown in this chapter are done for one orbit. The
simulations for more orbits have also been done but no new observations were found
in them. The sec. 5.1 compares the B-dot control implementation in the simulation
with the B-dot de-tumbling experiment done on UWE-3. A full magnetic and a full
thruster based control are simulated in sec. 5.2 and sec. 5.3. In these sections the
results with both the PD control and the SMC are presented. The sec. 5.4 discusses
four different scenarios with variation in the control law and the type of thruster
operation, when using a combination of thruster and torquers for attitude control.
Finally, sec. 5.5 summarizes the findings from simulation experiments.
40
5.1 B-dot simulation
Parameter
ms
Description
Mass of cubesat
Is
Tensor Matrix
stSimulation
stADS
initialEuler
Simulation Time Step
Sampling period of ADS
Initial Attitude
h
wInitial
Kbdot
Kx , Ky , Kz
Kxd , Kyd , Kzd
K11 , K22 , K33 , λ
Mresidual
ρ
Cd
Aaero
rCM
rCP
Value
1 kg


−3
1.67 · 10
0
0


0
1.67 · 10−3
0


−3
0
0
1.67 · 10
0.1 sec
0.1 sec i
h
−50 −50 50 deg
i
2 2 2 deg/s except (sec 5.1 and 5.2)
Initial Angular velocity
Controller
Gain of B-dot controller
6000
Proportional gain of PD law
30, 30, 30
Derivative gain of PD law
500, 500, 500
Gain parameters of ASMC
0.1, 0.1, 0.1, 10−6
Magnetic Disturbance
iT
h
−0.045 0.0126 0.006
Residual magnetic moment
Aerodynamic drag
Air density at cubesat altitude
1 · 10−13 kg/m3
Coefficient of drag
2.5
Cross-sectional area
1.732
· 10−2
m2
h
i
0 0 0 m
Position of center of mass
h
i
Am2
10−2 10−2 10−2 m
Position of center of pressure
Table 5.1: Simulation Initialization Parameters
5.1 B-dot simulation
The data from UWE-3 de-tumbling experiment provided a good basis for validating
the simulation. The B-dot simulation
with the same inih experiment was initialized
i
tial velocity from UWE-3 data with −35.282 0 −69.285 deg/sec. Two important
difference which can be observed between the B-dot experiment from simulation and
UWE-3 as seen in Fig. 5.2 can be explained as follows:
1. The results of B-dot experiment in the simulation have few overshoots and it
settles 10 mins later than the UWE-3 detumbling experiment. This can be
explained by the imperfect controller tuning in the simulation experiment.
2. The simulation results are missing the initial oscillations in all three axes.
This is the effect of assuming a diagonal inertial matrix in the simulation
experiment.
Another notable difference between the simulation and UWE-3 is that the torquers
in UWE-3 are digital, producing either ±0.6Am2 , while in the simulation they are
41
Chapter 5
Simulation Results
treated as analog actuators. The torques in UWE-4 are also going to be analog.
The B-dot simulation comparison shows that the simulation correctly captures the
dynamics of attitude in satellite’s environment and is thus suitable for validating
other controllers.
30
10
ωx
ωx
ωY
20
ωz
−10
Angular velocity (deg/s)
Angular velocity (deg/s)
10
0
−10
−20
−30
−40
−20
−30
−40
−50
−60
−50
−70
−60
−70
ωY
0
ωz
0
10
20
30
40
50
60
70
80
90
Time (min)
100
−80
0
10
20
30
40
50
60
70
80
90
Time (min)
Figure 5.2: B-dot result from simulation (left) vs UWE-3 de-tumbling experiment
result (right)
5.2 Full magnetic control
A full magnetic control is possible in UWE-3 but so far it has not been extensively
tested. The simulations of the full magnetic control can reveal the capabilities of
torquers for attitude control. The results with full magnetic control using PD law
and SMC law are shown in sec. 5.2.1 and sec. 5.2.2 respectively. It is suggested that
SMC should be implemented on UWE-3 for comparing the simulation performance
with in-orbit performance, before using it in UWE-4.
The initial angular
velocity
h
i
for full magnetic control simulations is assumed as 0.05 0.05 0.05 deg/s.
5.2.1 PD control law
The attitude and angular velocity of full magnetic control with PD control law is
shown in Fig. 5.3. It can be seen that the attitude could not be controlled in this
scenario but the angular velocities are bound within a range for each axes. Nevertheless, the performance of this controller is even worse than the B-dot control. In
Fig. 5.4, it can be seen that magnetic torquers are commanded to produce magnetic
moment far below their capability, which is due to the failure of demanded control
torque to adapt to the absence of strong actuation force.
42
5.2 Full magnetic control
3
ωx (deg/s)
φ (deg)
200
100
0
−100
−200
0
10
20
30
40
50
60
70
80
90
2
1
0
−1
100
0
10
20
30
40
Time (min)
50
0
−50
0
10
20
30
40
50
60
70
80
90
80
90
100
60
70
80
90
100
60
70
80
90
100
0
−2
100
0
10
20
30
40
50
Time (min)
60
2
ωz (deg/s)
ψ (deg)
70
−1
Time (min)
40
20
0
60
1
ωy (deg/s)
θ (deg)
100
−100
50
Time (min)
0
10
20
30
40
50
60
70
80
90
1
0
−1
100
0
10
20
30
40
Time (min)
50
Time (min)
Figure 5.3: Full magnetic control with PD law - Attitude and Angular Velocity
mx (Am2)
0.04
0.02
0
−0.02
−0.04
0
10
20
30
40
50
60
70
80
90
100
60
70
80
90
100
60
70
80
90
100
Time (min)
my (Am2)
0.04
0.02
0
−0.02
0
10
20
30
40
50
Time (min)
mz (Am2)
0.02
0
−0.02
−0.04
0
10
20
30
40
50
Time (min)
Figure 5.4: Full Magnetic Control with PD control law - Magnetic Moment Output
of Torquers
5.2.2 SMC law
The results from sliding mode control law for full magnetic control are a major
improvement from PD law as can be seen in Fig. 5.5. The attitude in Y-axis and
Z-axis are in full control with the settling time of ~15 mins for small maneuvers
and ~25-30 mins for large angle maneuvers (change of direction from in-track to
anti-in-track). The oscillations observed in Fig. 5.5 and Fig. 5.6 from 35 to 60 mins
can be explained by the presence of strong disturbance torques due to the high
residual magnetic field. The SMC controller spins the satellite to 6 to 7 deg/s but
due to the lack of strong action and the presence of disturbance torque which are
as strong as the control action, the controller fails to slow down the satellite. In
simple words, given such high disturbance torques, the controller of this scenario
is working very close to instability. It is important to notice in Fig. 5.6, that the
43
Chapter 5
Simulation Results
magnetic torquers are used to their capacity when the satellite is far away from
its commanded attitude. But since PD law is simple and robust, more simulation
experiments are performed (sec. 5.3 and sec. 5.4) to conclude on the idea that sliding
mode control can be a better alternative over traditional PD control for UWE-4.
5
ωx (deg/s)
φ (deg)
200
100
0
−100
−200
0
−5
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
ωy (deg/s)
θ (deg)
50
0
−50
80
90
100
60
70
80
90
100
60
70
80
90
100
2
0
−2
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
50
Time (min)
5
ω (deg/s)
200
150
100
0
−5
z
ψ (deg)
70
4
Time (min)
50
0
60
6
100
−100
50
Time (min)
Time (min)
−10
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
50
Time (min)
Time (min)
Figure 5.5: Full magnetic control with SMC law - Attitude and Angular Velocity
mx (Am2)
0.1
0.05
0
−0.05
−0.1
0
10
20
30
40
50
60
70
80
90
100
60
70
80
90
100
60
70
80
90
100
Time (min)
my (Am2)
0.1
0.05
0
−0.05
−0.1
0
10
20
30
40
50
Time (min)
mz (Am2)
0.1
0.05
0
−0.05
−0.1
0
10
20
30
40
50
Time (min)
Figure 5.6: Full Magnetic Control with SMC control law - Magnetic Moment Output of Torquers
44
5.3 Full thruster control
5.3 Full thruster control
A full thruster based control is the next step in this work with an intend to check
the implementation of thruster code and to access the capability of thrusters alone,
to handle the attitude control. As mentioned before, firing limit and T̂f actor are kept
as variables for experimentation and also because the thrusters are still in the design
phase. Since the thrusters can’t produce any significant torque in X-axis, no control
in this axis is realizable in this scenario. The sec. 5.3.1 and sec. 5.3.2 presents the
results with PD and sliding mode control respectively. The thruster operation in
these simulations is based on the one quadrant method as discussed in sec. 4.4.1.
5.3.1 PD control law
The attitude and angular velocity results for this scenario are shown in Fig. 5.7.
As can be seen in this plot, a direction change from in-track to anti-in-track is
performed at 50 min. The settling time for this direction change is found out to be
150 sec, which can also be seen in Fig. 5.8. The accuracy in attitude is measured
in terms of settling interval during steady state. From Fig. 5.9, it can be seen
that settling interval of attitude in Y-axis is ±2.5° and in Z-axis is 2 ± 2°. The
X-axis angular velocity modulates from its initial velocity of 2 deg/s due to the
disturbance torques and minor torques arising due to the thrusters plume. Fig. 5.10
shows the firing frequency of the four thrusters on the left and, total and cumulative
number of firings by individual thrusters on the right. It is important to observe
that thrusters 1 and 3 fire more often than 2 and 4 which is a common observations
in all further simulation results in this chapter. The total number of thruster firings
in this scenario are 46,153.
3
ωx (deg/s)
φ (deg)
200
100
0
−100
−200
0
10
20
30
40
50
60
70
80
90
2
1
0
100
0
10
20
30
40
ωy (deg/s)
θ (deg)
0
−20
−40
0
10
20
30
40
50
60
70
80
90
80
90
100
60
70
80
90
100
60
70
80
90
100
1
0
0
10
20
30
40
50
Time (min)
Time (min)
2
ωz (deg/s)
ψ (deg)
70
2
−1
100
200
150
100
50
0
60
3
20
−60
50
Time (min)
Time (min)
0
10
20
30
40
50
Time (min)
60
70
80
90
100
0
−2
−4
0
10
20
30
40
50
Time (min)
Figure 5.7: Full thruster control with PD law - Attitude and Angular Velocity
45
Chapter 5
Simulation Results
2
200
1
150
0
100
−1
50
ψ (deg)
θ (deg)
−2
−3
0
−4
−50
−5
−100
−6
−150
−7
−8
50
50.5
51
51.5
52
52.5
53
53.5
54
54.5
−200
50
55
Time (min)
50.5
51
51.5
52
52.5
53
53.5
54
54.5
55
Time (min)
Figure 5.8: Full thruster control with PD law - Settling Time
4.5
4
2
3.5
1.5
3
1
2.5
ψ (deg)
θ (deg)
3
2.5
0.5
2
0
1.5
−0.5
1
−1
0.5
−1.5
0
−2
10
15
20
25
30
35
40
Time (min)
−0.5
10
15
20
25
30
35
40
Time (min)
Figure 5.9: Full thruster control with PD law - Settling Accuracy
4
1200
Thruster firings per minute
1000
800
600
400
200
5
x 10
4.5
Thruster firings cumulative
Thruster 1
Thruster 2
Thruster 3
Thruster 4
Total
4
3.5
Thruster 1
Thruster 2
Thruster 3
Thruster 4
Total
3
2.5
2
1.5
1
0.5
0
0
0
10
20
30
40
50
Time (min)
60
70
80
90
100
0
10
20
30
40
50
60
70
80
90
100
Time (min)
Figure 5.10: Full thruster control with PD law - Thruster firings per minute and
Cumulative firings
46
5.3 Full thruster control
5.3.2 SMC law
The results of the full thruster control by sliding mode control law are shown in
Fig. 5.11. For the sake of comparison with the previous scenario, the direction
change is made at 50 min. The sliding mode control provides a more aggressive
action than PD law as it can be seen that the angular velocity in Y and Z-axis goes
close to 10 deg/s compared to 3 deg/sec in Fig. 5.7 for PD. Due to this aggressive
action, the overshoot is more and it settles in 240 seconds as seen in Fig. 5.12. The
attitude accuracy observed from Fig. 5.13 is ±0.3° in Y-axis and 1.2 ± 0.3° in Z-axis,
which is better than the PD scenario. But this comes at the expense of more usage
of thrusters which can be seen in Fig. 5.14. The total number of thruster firings are
292,667 which is more than 6 times the total firings in full thruster control with PD
law.
3
ωx (deg/s)
φ (deg)
200
100
0
−100
−200
2
1
0
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
ω (deg/s)
θ (deg)
0
80
90
100
60
70
80
90
100
60
70
80
90
100
0
−10
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
50
Time (min)
Time (min)
10
ωz (deg/s)
200
ψ (deg)
70
−5
y
−50
150
100
50
0
60
5
50
−100
50
Time (min)
Time (min)
5
0
−5
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
50
Time (min)
Time (min)
Figure 5.11: Full thruster control with SMC law - Attitude and Angular Velocity
40
200
150
20
100
0
ψ (deg)
θ (deg)
50
−20
0
−50
−40
−100
−60
−150
−80
50
51
52
53
54
55
Time (min)
56
57
58
59
60
−200
50
51
52
53
54
55
56
57
58
59
60
Time (min)
Figure 5.12: Full thruster control with SMC law - Settling Time
47
Chapter 5
Simulation Results
0.3
1.5
0.2
1.4
1.3
0
ψ (deg)
θ (deg)
0.1
−0.1
1.2
1.1
−0.2
1
−0.3
0.9
−0.4
10
15
20
25
30
35
40
Time (min)
0.8
10
15
20
25
30
35
40
Time (min)
Figure 5.13: Full thruster control with SMC law - Settling Accuracy
5
3000
Thruster firings per minute
2500
2000
1500
1000
500
0
3
Thruster firings cumulative
Thruster 1
Thruster 2
Thruster 3
Thruster 4
Total
2.5
2
10
20
30
40
50
Time (min)
60
70
80
90
100
Thruster 1
Thruster 2
Thruster 3
Thruster 4
Total
1.5
1
0.5
0
0
x 10
0
10
20
30
40
50
60
70
80
90
100
Time (min)
Figure 5.14: Full thruster control with SMC law - Thruster firings per minute and
Cumulative firings
5.4 Combination of thrusters and torquers
Since thrusters have limited fuel and thus a limited life, it is highly desirable to
assist thrusters with any other attitude control method. The magnetic torquers can
be used in parallel with thrusters to supplement them with some torque in all three
axes.
During simulations it was established that angular velocity of more than 2 deg/s
in X-axis will effect the control of Y and Z axes due to cross-couplings. Thus, two
objectives are set up for torquers. With the help of torquers, it is desired to
48
5.4 Combination of thrusters and torquers
1. Keep the angular velocity in X-axis, less than 1 deg/s
2. Produce a portion of the demanded control torque to maintain the desired
direction
The demanded control torque is divided between thrusters and torquers by a Split
variable as discussed in sec. 5.4. For the simulation experiments in this section, the
Split variable is kept constant at 0.5 (also shown in Fig. 5.1). Two theoretical operation strategies for thrusters namely one-quadrant and two-quadrant were discussed
previously in sec. 4.4.1 and sec. 4.4.2. These two operation modes are evaluated with
the PD control law in sec. 5.4.1 and sec. 5.4.2 and with the sliding mode control in
sec. 5.4.3 and sec. 5.4.4.
5.4.1 PD law with one quad thruster
The attitude and angular velocity plots for this scenario are shown in Fig. 5.15. Two
direction changes are made in this simulation, one at 33 min and second at 68 min.
The setting time for attitude in Z-axis is 120 sec and settling accuracy in Y-axis is
±2° and in Z-axis is 2 ± 3.5°. One plot for each settling time and settling interval
is shown in Fig. 5.16. The performance of this scenario is comparable with the full
thruster control with PD law but the total number of firings here are 20,865, which
is less than half from there, which indicates the help from torquers to thrusters.
2
ωx (deg/s)
φ (deg)
200
100
0
−100
−200
1
0
−1
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
ωy (deg/s)
θ (deg)
0
−50
80
90
100
60
70
80
90
100
60
70
80
90
100
0
−4
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
50
Time (min)
4
ωz (deg/s)
200
ψ (deg)
70
−2
Time (min)
150
100
50
0
60
2
50
−100
50
Time (min)
Time (min)
2
0
−2
−4
0
10
20
30
40
50
Time (min)
60
70
80
90
100
0
10
20
30
40
50
Time (min)
Figure 5.15: PD law with One Quad Thrusters - Attitude and Angular Velocity
49
Chapter 5
Simulation Results
2.5
200
2
150
1.5
100
1
ψ (deg)
θ (deg)
50
0.5
0
0
−50
−0.5
−100
−1
−150
−1.5
−2
10
12
14
16
18
20
22
24
26
28
30
−200
30
31
32
Time (min)
33
34
35
36
37
Time (min)
Figure 5.16: PD law with One Quad Thruster - Settling Accuracy and Settling
Time
4
700
Thruster firings per minute
600
500
400
300
200
100
0
2.5
Thruster firings cumulative
Thruster 1
Thruster 2
Thruster 3
Thruster 4
Total
10
20
30
40
50
Time (min)
60
70
80
90
100
Thruster 1
Thruster 2
Thruster 3
Thruster 4
Total
2
1.5
1
0.5
0
0
x 10
0
10
20
30
40
50
60
70
80
90
100
Time (min)
Figure 5.17: PD law with One Quad Thrusters - Thruster firings per minute and
Cumulative firings
It can be noticed from Fig. 5.17 that thrusters need to fire on an average of 200 FPM
(firings per minute) only to keep the attitude but nevertheless the main purpose of
thrusters will be orbit correction, which means that firings per minute should not
be used as an absolute indicator of performance.
Plots in Fig. 5.18 and Fig. 5.19 (left) depicts the magnitude of torque produced by
torquers and thrusters relative to each other and Fig. 5.19 (right) shows the magnetic
moment output of the torquers. In the X-axis, the torque contribution by thrusters
is only due to plume, thus far less than the contribution by torquers. In Y and
Z axes, despite the intend to split the demanded control torque equally between
thrusters and torquers, it can be seen that torquers are incapable of producing the
same amount as thrusters. The reason for this is the failure of Eq. 4.22 to find a
good solution for magnetic moment (m)
~ to produce required torque. Since there
50
5.4 Combination of thrusters and torquers
−7
1.5
−6
x 10
1.6
x 10
Torquers
Thrusters
Thrusters
Torquers
1.4
|Tth−y| & |Ttoq−y| (N−m)
|Tth−x| & |Ttoq−x| (N−m)
1.2
1
0.5
1
0.8
0.6
0.4
0.2
0
10
20
30
40
50
60
70
80
0
90
10
20
30
Time (min)
40
50
60
70
80
90
Time (min)
Figure 5.18: PD law with One Quad Thrusters - Commanded Torque Load sharing
between Thruster and Torquers in X and Y axis
−6
x 10
−3
Thrusters
Torquers
−5
−10
−15
0
10
20
30
40
50
60
70
80
90
100
60
70
80
90
100
60
70
80
90
100
Time (min)
1
0.02
m (Am2)
0.8
y
|Tth−z| & |Ttoq−z| (N−m)
1.2
x 10
0
x
1.4
5
m (Am2)
1.6
0.6
0
−0.02
−0.04
−0.06
0
10
20
30
40
50
Time (min)
0.4
m (Am2)
0.04
z
0.2
0
0.02
0
−0.02
−0.04
10
20
30
40
50
60
70
80
90
0
Time (min)
10
20
30
40
50
Time (min)
Figure 5.19: PD law with One Quad Thrusters - Commanded Torque Load sharing
in Z-axis and Magnetic Moment Output of Torquers
is no guarantee of a solution, this equations attempts to find the closest distance
~ which
between commanded control torque (Tc ) and local magnetic field strength (B)
in the end results in producing only a fraction of expected torque from torquers.
5.4.2 PD law with two quad thruster
This scenario is aimed to test the two-quadrant based operation theory for thrusters.
As discussed previously in sec. 4.4.2, it solves a two integer problem to compute which
two thrusters should be used and the number of times they need to fire. The attitude
and angular velocity plots for this controller scheme with two direction changes at
33 min and 68 min are shown in Fig. 5.20. The attitude accuracy for this case is ±6°
in Y axis and ±8° in Z axis and settling time is 90 seconds (shown in Fig. 5.21). The
51
Chapter 5
Simulation Results
advantage of using two-quadrant based operation is really in quick direction change
compared to one quadrant based operation. The decrease in accuracy is attributed
to small number of thruster firings as can be seen in Fig. 5.22. The total number of
firings are 7,619 which is 2.7 times less than last scenario. The relative contribution
of commanded control torque in the three axes and magnetic moment output by
torquers are shown in Fig. 5.23 and Fig. 5.24.
2
ωx (deg/s)
φ (deg)
200
100
0
−100
−200
0
10
20
30
40
50
60
70
80
90
1
0
−1
100
0
10
20
30
40
Time (min)
0
−50
80
90
100
0
10
20
30
40
50
60
70
80
90
60
70
80
90
100
60
70
80
90
100
0
−2
100
0
10
20
30
40
50
Time (min)
2
ωz (deg/s)
200
ψ (deg)
70
2
Time (min)
150
100
50
0
60
4
ωy (deg/s)
θ (deg)
50
−100
50
Time (min)
0
10
20
30
40
50
60
70
80
90
0
−2
−4
100
0
10
20
30
Time (min)
40
50
Time (min)
Figure 5.20: PD law with Two Quad Thrusters - Attitude and Angular Velocity
2
200
0
150
100
−2
ψ (deg)
θ (deg)
50
−4
−6
0
−50
−8
−100
−10
−150
−12
30
31
32
33
34
Time (min)
35
36
37
−200
30
31
32
33
34
35
36
37
Time (min)
Figure 5.21: PD law with Two Quad Thruster - Settling Accuracy and Settling
Time
5.4.3 SMC law with one quad thruster
In this scenario, the sliding mode control with one quadrant method is simulated.
The direction changes takes place at 51 min and 67 min. The attitude and angular
velocity results are shown in Fig. 5.28. It is important to notice that X-axis is
significantly more stabilized here when compared to the last two scenarios. The
52
5.4 Combination of thrusters and torquers
600
Thruster firings per minute
500
400
300
200
100
8000
Thruster firings cumulative
Thruster 1
Thruster 2
Thruster 3
Thruster 4
Total
7000
6000
5000
4000
3000
2000
1000
0
0
0
10
20
30
40
50
60
70
80
90
Thruster 1
Thruster 2
Thruster 3
Thruster 4
Total
100
0
10
20
30
40
50
60
70
80
90
100
Time (min)
Time (min)
Figure 5.22: PD law with Two Quad Thrusters - Thruster firings per minute and
Cumulative firings
−6
−7
1.5
x 10
1.2
x 10
Thrusters
Torquers
Torquers
Thrusters
|Tth−y| & |Ttoq−y| (N−m)
|Tth−x| & |Ttoq−x| (N−m)
1
1
0.5
0.8
0.6
0.4
0.2
0
0
10
20
30
40
50
Time (min)
60
70
80
90
10
20
30
40
50
60
70
80
90
Time (min)
Figure 5.23: PD law with Two Quad Thrusters - Commanded Torque Load sharing
between Thruster and Torquers in X and Y axis
occasional shoot off of the angular velocity in X-axis is in that phase of orbit where
the earth’s magnetic field is getting close to being parallel to X-axis and thus no
control action can be obtained from magnetic torquers in X-axis. The settling
time for direction change as can be observed in Fig. 5.29 is ~200 sec. The attitude
accuracies from Fig. 5.30 in Y-axis is ±0.25°and in Z-axis is +1.25° ± 0.25°. The
average number of firings used by this controller is close to 3000 per minute, which is
very high and, the total firings in one orbit is 285,051 (Fig. 5.25). As can be seen in
Fig. 5.26 and Fig. 5.27, the torquers provide only a tiny fraction of demanded torque
but they are quite used to their full capacity.
53
Chapter 5
Simulation Results
−6
1.5
x 10
0.01
0
−0.01
−0.02
0
10
20
30
40
50
60
70
80
90
100
60
70
80
90
100
60
70
80
90
100
Time (min)
1
0.1
y
m (Am2)
|Tth−z| & |Ttoq−z| (N−m)
x
m (Am2)
Thrusters
Torquers
0.05
0
−0.05
−0.1
0.5
0
10
20
30
40
50
Time (min)
0.02
0
z
m (Am2)
0.04
−0.02
0
10
20
30
40
50
60
70
80
90
0
10
20
30
40
50
Time (min)
Time (min)
Figure 5.24: PD law with Two Quad Thrusters - Commanded Torque Load sharing
in Z-axis and Magnetic Moment Output of Torquers
5
3000
Thruster firings per minute
2500
2000
1500
1000
500
0
3
Thruster firings cumulative
Thruster 1
Thruster 2
Thruster 3
Thruster 4
Total
10
20
30
40
50
60
70
80
90
100
Thruster 1
Thruster 2
Thruster 3
Thruster 4
Total
2.5
2
1.5
1
0.5
0
0
x 10
0
10
Time (min)
20
30
40
50
60
70
80
90
100
Time (min)
Figure 5.25: SMC law with One Quad Thrusters - Thruster firings per minute and
Cumulative firings
5.4.4 SMC law with two quad thruster
The two quadrant operation method of thrusters with sliding mode control is simulated with direction changes at 50 min and 67 min and the results of attitude and
angular velocity are shown in Fig. 5.31. As can be seen from Fig. 5.33 that the attitude accuracy of this controller is the same as previous scenario. But as observed
from Fig. 5.32, the settling time after direction change is ~160 sec, which is better
than SMC with one thruster method.
54
5.4 Combination of thrusters and torquers
−5
−6
1.2
x 10
2
x 10
Thrusters
Torquers
Torquers
Thrusters
1.8
1
|Tth−y| & |Ttoq−y| (N−m)
|Tth−x| & |Ttoq−x| (N−m)
1.6
0.8
0.6
0.4
1.4
1.2
1
0.8
0.6
0.4
0.2
0.2
0
0
10
20
30
40
50
60
70
80
10
90
20
30
40
50
60
70
80
90
Time (min)
Time (min)
Figure 5.26: SMC law with One Quad Thrusters - Commanded Torque Load sharing between Thruster and Torquers in X and Y axis
−5
x 10
Thrusters
Torquers
0.1
0.05
0
x
1.8
m (Am2)
2
1.6
−0.05
0
10
20
30
40
50
60
70
80
90
100
60
70
80
90
100
60
70
80
90
100
Time (min)
0.1
1.2
m (Am2)
1
0.05
0
y
|Tth−z| & |Ttoq−z| (N−m)
−0.1
1.4
0.8
−0.05
−0.1
0
10
20
30
40
50
Time (min)
0.6
m (Am2)
0.1
0.4
0
0.05
0
z
0.2
−0.05
−0.1
10
20
30
40
50
60
70
80
90
0
10
20
30
40
50
Time (min)
Time (min)
Figure 5.27: SMC law with One Quad Thrusters - Commanded Torque Load sharing in Z-axis and Magnetic Moment Output of Torquers
2
ωx (deg/s)
φ (deg)
200
100
0
−100
−200
0
10
20
30
40
50
60
70
80
90
1
0
−1
100
0
10
20
30
40
Time (min)
50
0
y
−50
80
90
100
0
10
20
30
40
50
60
70
80
90
60
70
80
90
100
60
70
80
90
100
0
−10
100
0
10
20
30
40
50
Time (min)
5
ω (deg/s)
200
150
100
50
z
ψ (deg)
70
10
Time (min)
0
60
20
ω (deg/s)
θ (deg)
100
−100
50
Time (min)
0
10
20
30
40
50
Time (min)
60
70
80
90
100
0
−5
−10
−15
0
10
20
30
40
50
Time (min)
Figure 5.28: SMC law with One Quad Thrusters - Attitude and Angular Velocity
55
Chapter 5
Simulation Results
30
200
150
20
100
10
ψ (deg)
θ (deg)
50
0
0
−50
−10
−100
−20
−150
−30
50
50.5
51
51.5
52
52.5
53
53.5
54
54.5
55
−200
50
50.5
51
51.5
52
Time (min)
52.5
53
53.5
54
54.5
55
Time (min)
Figure 5.29: SMC law with One Quad Thrusters - Settling Time
0.3
1.5
1.45
0.2
1.4
0.1
ψ (deg)
θ (deg)
1.35
0
−0.1
1.3
1.25
1.2
−0.2
1.15
−0.3
1.1
−0.4
10
15
20
25
Time (min)
30
35
40
1.05
10
15
20
25
30
35
40
Time (min)
Figure 5.30: SMC law with One Quad Thrusters - Settling Accuracy
The use of thrusters in terms of average firings per minute and the cumulative firings
and, their relative use compared to torquers are also very similar to previous case,
therefore most of the plots are not shown for this controller. The total number of
firings used by this controller are 281,334 which is less than the previous scenario. It
is concluded that the only advantage of using a two-quadrant operation of thrusters is
the quicker response during direction change from in-track to anti-in-track direction
or vice versa.
56
5.4 Combination of thrusters and torquers
2
ωx (deg/s)
φ (deg)
200
100
0
−100
−200
1
0
−1
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
ω (deg/s)
θ (deg)
0
80
90
100
60
70
80
90
100
60
70
80
90
100
0
−10
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
50
Time (min)
Time (min)
10
ωz (deg/s)
200
ψ (deg)
70
−5
y
−50
150
100
50
0
60
5
50
−100
50
Time (min)
Time (min)
5
0
−5
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
50
Time (min)
Time (min)
Figure 5.31: SMC law with Two Quad Thrusters - Attitude and Angular Velocity
40
200
30
150
20
100
10
ψ (deg)
θ (deg)
50
0
−10
0
−50
−20
−30
−100
−40
−150
−50
50
51
52
53
54
55
56
57
58
−200
50
59
Time (min)
51
52
53
54
55
56
57
58
59
Time (min)
0.3
1.7
0.2
1.6
0.1
1.5
0
1.4
ψ (deg)
θ (deg)
Figure 5.32: SMC law with Two Quad Thrusters - Settling Time
−0.1
1.3
−0.2
1.2
−0.3
1.1
−0.4
10
15
20
25
Time (min)
30
35
40
1
10
15
20
25
30
35
40
Time (min)
Figure 5.33: SMC law with Two Quad Thrusters - Settling Accuracy
57
Chapter 5
Simulation Results
Control law &
Settling Interval
Settling
Thruster type
Time
Full
PD
uncontrolled
SMC
Y: ±1.2° Z:1.5° ± 1.2°
28 min
Magnetic
Full
PD - One quad
Y: ±2.5° Z:2 ± 2°
150 s
◦
SMC - One quad
Y: ±0.3° Z:1.2 ± 0.3°
240 s
thrusters
◦
Combination PD - One quad
Y: ±2.0° Z:2 ± 3.5°
120 s
of
PD - Two quad
Y: ±6.0° Z:±8.0°
90 s
◦
torquers &
SMC - One quad Y: ±0.25° Z:1.25 ± 0.2°
200 s
thrusters
SMC - Two quad Y: ±0.25° Z:1.25◦ ± 0.2°
160 s
Table 5.2: Comparison of Different Simulated Scenarios
Avg.
FPM
476.2
3017.2
215.1
78.5
2938.4
2900.1
5.5 Summary
Results from many possible variations of the controller implementations are presented in this chapter. The entire simulation environment is developed based on
the theories in chapters 2 and 3. Using the skeleton of the controller in Fig. 4.1
and all concepts in chapter 4, different scenarios were simulated and showcased with
occasional 1-1 comparison. A full comparison of all the simulated scenarios is presented in Tab. 5.2. The next chapter addresses the architecture of UWE-4 hardware
and implementation of the developed control philosophies on the hardware platform.
The discussion about attitude control is continued in chapter 7 where many details
observed in this chapter will be highlighted and discussed.
58
6 Implementation
The control theory developed as a part of this thesis work needs to be implemented
on the ADCS board and it has to be done in the same framework supporting all
other hardware components. In this chapter an overview of the other hardware
components related to orbit and attitude control system with their interactions is
presented in sec. 6.1. The attitude control developed in this work is implemented
in C-language and has been tested on the ADCS board which will be discussed in
sec. 6.2. Since the UWE-4 project is in its beginning phase, a new software framework is also under development. It is likely that introduction of this new framework
which includes a new operating system will change many on-board implementation
aspects. Therefore, implementation and extensive program in loop testing is not in
focus right now.
6.1 Hardware architecture
The general purpose communication bus to be used in UWE-4 is I2C. As per the
current concept (shown in Fig. 6.1) there are two I2C buses. Out of the two I2C
buses, there is one dedicated to orbit and attitude control, on which ADCS communicates with PPU and Panels. The other I2C bus is used for cross talk between
OBDH and ADCS, power system, communication and other boards.
The ADCS already has an attitude determination system in place which has been
demonstrated in UWE-3. The performance aspects of ADS from UWE-3 are presented in [11, 12, 27]. The PPU interface with ADCS is in development as a part of
another Master thesis project and the interface of ADCS with Panels is already in
place in the form of an isotropic Kalman filter. For UWE-4, the isotropic Kalman
filter implementation needs to be integrated in the control software. A new operating system for better communication with the ground segment and among different
boards is also in development.
6.2 Low level code
Most of the control code was developed in Matlab/Simulink as they provide quick
ways to simulate and visualize the results but Matlab/Simulink is not a microcontroller language. The micro-controller used by ADCS and other hardware is
59
Chapter 6
Implementation
Figure 6.1: Hardware architecture of UWE-4[1]
Texas Instruments MSP430. C-language is used as the programming language for
previous UWE missions and it will continue to be so for UWE-4. The performance
of code simulated on two different hardware platforms always differ. Therefore, it
is very important to implement the code on the hardware to be used and perform
validation tests in a hardware in loop environment. As a part of this work, the
attitude control code is translated in C-code and static runs of the C-code were
compared with Matlab/Simulink output to test for any bugs in implementation.
The resulted implementation of the control code in lower level C code has produced
7 programs which are listed below:
1. attitude_control.c - The main control routine which when executed calls other
subroutines to form the controller.
2. pd_attitude_control_law.c - This is the PD control law. It needs current
c
attitude (q) and angular velocity (ωbi
) and produces the control torque demand
(Tc )
3. asmc_attitude_control_law.c - Sliding mode control law is the alternative to
PD control. It requires one more input, semi major axis (a), than PD control
law.
4. load_divide.c - It is to divide the demanded control between torquers and
60
6.3 Summary
thrusters. Currently it contains a simple implementation but with the finding of this work and directions of future work discussed in next chapter,
load_divide.c should be improved further.
5. command_torquers.c - It outputs the magnetic moment (m)
~ to be produced
by the torquers taking Tc−torquers as input. In the next chapter, direction for
improvement in this function is discussed.
6. command_thruster_one_quad.c - This is the one quadrant based operation
code. The input to this function is Tc−thrusters and it outputs a firing command
Nf iring to one thruster.
7. command_thruster_two_quad.c - This is the two quadrant based operation
code, an alternative to one quadrant based operation. The input to this function is Tc−thrusters and it outputs two firing command Nf p and Nf s to the
primary thruster and secondary thruster.
6.3 Summary
In this chapter, the hardware architecture for UWE-4 which is under development is
discussed. The result of this work in terms of implementation are the C codes which
are listed and explained. It is expected that the mode of communication of ADCS
board with PPU, OBDH and Panels is likely to change the interaction of control
parameters like the controller gains with the ground station. Thus, it is suggested
to adapt the control code accordingly in the future.
61
7 Conclusion and Future Work
7.1 Conclusions
In this master thesis work, a simulation environment for a combined attitude and
orbit control system for UWE-4 has been developed. It is established that an accurate simulation is the backbone of any project which deals with dynamics and thus
a crucial step towards implementation.
The fundamental concepts of modeling the cubesat with its orbit and environment
are discussed in the initial chapters. The data from UWE-3 sensors have been
used extensively to estimate the realistic sensor noises and disturbance parameters
with their corresponding effect on the attitude of the cubesat. A non-linear model
of satellite’s attitude is considered over a linear model. The linear modeling and
control theory is also not suitable due to the requirement of large angle maneuver
from in-track to anti-in-track direction or vice versa.
The advantages of SGP-4 propagator could not be exploited as an attempt to modify
SGP-4 propagator to include low thrust effect was not fruitful. Nevertheless, a
simplified orbit propagator was implemented which deviates from SGP-4 output by
10 kms in one day. Although, a more precise orbit simulation will be necessary
to simulate formation flying scenarios, the current status of orbit propagator is
sufficient enough to simulated relative effect of thrusters for orbit maneuvers. The
simplified orbit propagator is without doubt fit for attitude control simulations.
For the sake of orbit correction or maneuvering, the cubesat should be able to
change between in-track and anti-in-track direction. Limited action present in Xaxis motivated the search for dynamic target quaternions which when used should
not provoke the need of precise attitude control in X-axis. The two dynamic attitude
target equations obtained for each direction, also aim for attitude stabilization in
X-axis which in turn also helps in better attitude control in Y and Z axes.
Two types of control laws are tested for the attitude control problem; a PD control
law which is linear and a sliding mode control law which is non-linear. Since the
satellite’s attitude is non-linear and it is also modeled as a non-linear plant, the
linear control methods are by nature not the best choice for the problem. The
linear control like PD can be used when attitude control is desired without any
requirement of change in the orientation. In this thesis, the main focus of study is
the attitude control for the purpose of orbit control which needs frequent large angle
maneuvers to change the orientation. With the presented tuning of the two control
63
Chapter 7
Conclusion and Future Work
laws, the sliding mode control performs better than the PD control law in terms of
the stability of the X-axis and the attitude accuracy in terms of settling interval.
This can be explained by the fact that the PD law has no knowledge of the plant
dynamics while the sliding mode control law compensates for the dynamics. It can
be clearly observed from Tab. 5.2 that the attitude accuracy is directly proportional
to firings per minute (FPM). Both controller were found to be stable but it is
important to mention that global stability is not guaranteed if the saturation limit
on thruster firing frequency is increased. Detailed study of the unstable zones for
attitude control is pointed out as future work.
From the point of actuators, three modes of control are considered, a full torquers
based action, a full thrusters based action and a combination of thrusters and torquers. The output torque by torquers is only a fraction of the commanded torque
due to the dependency of earth’s magnetic field. In-fact, the torquer equation (Eq.
4.22) rarely finds a solution which results in limited control when torquers are used.
Vacuum arc thrusters which are still in development are capable of producing thrust
of 0.5μN to 20μN. The torque that can be generated with this thrust together with
the fast operating frequency of thrusters can provide precise attitude control (±0.25°
in Y and Z axis with 1.25° bias in Z-axis) which is demonstrated by the simulations.
Two modes of operation by the thrusters are implemented, one-quadrant and twoquadrant based. The two-quadrant based operation gives faster response than the
other.
Based on the B-dot simulation, it is concluded that the simulation captures cubesats
dynamics in its environment very well. Other simulation results suggest that sliding
mode control is better for precise attitude control in Y and Z axes and also for
better stability in X-axis. The simulation of sliding mode law for full magnetic
control motivates to experiment with sliding mode control law already in UWE-3.
The output of the thesis is the Simulink model and control code functions in Matlab/Simulink and in C-language. The control code implementation in the new framework which is currently in development will require some additional modifications of
the functions in C-code. The Simulink model is however ready for further research
on attitude and orbit control of UWE-4.
7.2 Future work
Since UWE-4 is a state of art research project in its beginning phase, it is important
to point out the directions of future work for the continuation of research. Three
important areas of future work in continuation of this work are discussed here in
sec. 7.2.1, sec. 7.2.2 and sec. 7.2.3.
64
7.2 Future work
7.2.1 Equal distribution of thruster usage
It is observed that whenever thrusters are used Thruster 1 and Thruster 3 are used
much more than the other two. It is suspected that this can just be an artifact in
the simulation arising from the way the random numbers are generated in Simulink.
Nevertheless, this is unacceptable for the simulation and for the mission, and thus
needs to be investigated further. It needs to be investigated if unequal use of thruster
is only an artifact effect from simulation or could it be a real issue with the satellite
by the nature of the control implementation itself. It is further suggested that, in
order to completely avoid this affect, a spin stabilization control on X-axis with
angular rate of 1 deg/s could be implemented. This can be done easily with small
modification of the control laws.
7.2.2 Controller stability analysis
As mentioned in sec. 7.1, a comprehensive study of stability of attitude control is
required. The zone of instability needs to be quantified in terms of angular acceleration so that appropriate saturation limits could be used. This study needs to be
controller specific too. If in the future more controllers are implemented, they all
must go under a strict stability check before their implementation.
7.2.3 Efficient use of torquers
It is essential to maximize the use of magnetic torquers because they are robust
and have longer life. In the current implementation, a simple equation to compute
magnetic moment from the demanded torque and local magnetic field is used. For
UWE-4 mission, a precise attitude control is desired only in Y and Z axes. The
equation for magnetic moment however does not take this into account, but it treats
all axes with equal importance. This information together with the conclusion from
the work for sec. 7.2.1 should be combined to come up with a new strategy for
efficient use of torquers.
7.2.4 Orbit Propagator
As mentioned in chapter 3, the simplified orbit propagator needs to be improved
in terms of its accuracy. This should be done by incorporating more terms in the
Newton’s second law and by finding out an strategy to make corrections for the
still unaccounted terms. While doing so, the simplified orbit propagator should be
compared with SGP-4 to quantify the error. The simplified orbit propagator also
needs to be implemented on the hardware which is left out as a future work.
65
Acknowledgments
I want to express my deep gratitude towards everyone who have helped me or supported me during my thesis work. First I would like to thank Prof. Dr. Klaus
Schilling from JMUW for providing me this opportunity to work on a space mission. Challenging work has always motivated me and this definitely has been one of
them. I would also like to thank Prof. Dr. Johnny Ejemalm for acting as a remote
examiner from Luleå Technical University.
Next, I would like to thank my supervisor Philip Bangert who has encouraged me
to take ownership over my work but still has mentored me in the best possible way.
With the guidance of Philip, it was very easy to cross the roadblocks in the project.
Many thanks to Stephan Busch and Ali Kheirkhah for mentoring me during the
team design project which eventually became my master thesis topic. I would also
like to thank Karthik Ravandoor who has helped me when I was struggling with
orbit propagators.
Now, i would like to thank my companions in UWE-4 project, Alex Kramer, Jesko
Bahr and Arunkumar Rathinam. Alex has helped me many times especially with
the hardware. I wish all three of them the very best for completing their master
thesis project and in their further career. I want to thank John Panikulam, Janis
Gailis and Karim Bondoky as my friends. John has been a great friend to me during
the time spent in Würzburg.
Very importantly, I want to thank my parents for providing me the best environment
to grow and learn. I also thank my sisters and family members who have supported
me throughout my life. I also want to thank my girlfriend who has provided me
invaluable emotional support. Lastly I extend my gratefulness to all my friends for
helping and encouraging me to move forward in my life and career.
67
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72
Nomenclature
Tˆ0
Torque unit magnitude
Ω
Angular speed of wheel
ω0
Angular velocity of the orbit
p
ωqr
Angular velocity of ’q’ in w.r.t ’r’ reference frame in ’p’ reference
frame
τ
Disturbance torque in CCF
Aw
Layout matrix of wheel
Bx
Magnetic field vector in x frame (if unspecified, x = CCF)
Is
Moment of Inertia of satellite
Iw
Moment of inertia of wheel
m
magnetic moment in CCF
Nf
Number of firing (primary or seconday thruster)
q
Quaternion
Qf
Quadrant of firing (primary or seconday thruster)
T
Torque
T0
Thruster torque magnitude
Tyx
Transformation Matrix from x to y
ADCS
Attitude determination and control system
ADS
Attitude determination system
BCF
Body coordinate frame
CCF
Control coordinate frame
73
Nomenclature
ECEF
Earth center earth fixed
ECI
Earth center inertial
GPS
Global Positioning Satellite
JMUW
Julius-Maximilians-University Würzburg
MPC
Model predictive control
OCF
Orbit coordinate frame
PD
Propotional Derivative
PPU
Power processing unit
SGP-4
Simplified General Perturbations Model-4
SMC
Sliding Mode Control
UniBwM
University of Federal Armed Forces in Munich
UWE
Universität Würzburg Experimentalsatellit
VAT
Vacuum arc thrusters
w.r.t
with respect to
WGN
White guassian noise
74