OPTIMAL ORBIT-RAISING AND ATTITUDE CONTROL OF ALL-ELECTRIC SATELLITES
A Dissertation by Suwat Sreesawet Master of Science, Wichita State University, 2014 Bachelor of Engineering, Kasetsart University, 2009
Submitted to the Department of Aerospace Engineering and the faculty of the Graduate School of Wichita State University in the partial fulfillment of the requirements for the degree of Doctor of Philosophy
December 2018 c Copyright 2018 by Suwat Sreesawet
All Rights Reserved OPTIMAL ORBIT-RAISING AND ATTITUDE CONTROL OF ALL-ELECTRIC SATELLITES
The following faculty members have examined the final copy of this dissertation for form and content, and recommend that it be accepted in partial fulfillment of the requirement for the degree of Doctor of Philosophy, with a major in Aerospace Engineering.
Atri Dutta, Committee Chair
James E. Steck, Committee Member
Roy Myose, Committee Member
Animesh Chakravarthy, Committee Member
John Watkins, Committee Member
Accepted for the College of Engineering
Steven Skinner, Interim Dean
Accepted for the Graduate School
Dennis Livesay, Dean
iii DEDICATION
To my family, my mother, father, and sister who have always supported me and provided me the warmth to my heart for entire life, even though I am on the other side of world. To my future wife who always cares for me, and to my friends with whom I have spent many great times together.
iv ACKNOWLEDGMENTS
First of all, I have many thanks for my family, my dad, mon and sister. They always keep supporting and encouraging me since the day I started breathing with my own nose. I always perceive their love and support even I have been in the opposite side of the world for many years. I would like to thank my adviser, Dr. Atri Dutta, for his guidance and support all along this dissertation. He has kindly and sincerely helped since the first day i met him. I have learned a variety of knowledge from him. I would also like to thank the committee members Dr. James E. Steck, Dr. John Watkins, Dr. Animesh Chakravarthy and Dr. Roy Myose for their advice and suggestions for this research. I would like to tan I would also like to thank Ministry of Science and Technology, Thailand for funding my graduate program. I would Thanks also to my colleagues and friends who helped with their suggestions during the course of this research work.
v ABSTRACT
Electric propulsion is gaining popularity among satellite operators due to its fuel efficiency. However, electric propulsion has the limitation of producing a small magnitude of thrust, meaning that the transfer time to geostationary orbit is of the order of several months. The long transfer time adds more complexity to the mission design process due to the long exposure to hazardous radiation belts. Obviously, thrusters require electric power that is generated from sunlight by satellite solar panels. Therefore, the earth’s shadow significantly impacts the orbit-raising maneuver. This study proposes a novel, robust, and fast numerical methodology for generating low-thrust trajectories to the geosynchronous orbit. This methodology utilizes a new set of state variables that has a physical interpretation and exhibits slow variation under a small magnitude of thrust. The absence of mathematical singularities in the equatorial plane adds to the benefits. The new set of state variables, along with a closed-loop guidance scheme and direct optimization methodology, is used to optimize the satellite trajectory. An unconstrained optimization scheme is able to robustly and rapidly generate low-thrust orbit-raising trajectories for a variety of mission scenarios, various initial orbits, application of electric battery to allow thrusting during eclipses, and orbital perturbations due to the earth’s oblateness or a third body. The proposed methodology can be seamlessly integrated into receding horizon control scheme, which recomputes the minimum time trajectory at regular intervals referred to as planning horizon. The attitude of spacecraft must also be maintained in a desired direction which can be time-varying. Regular satellites perform orbit-raising using stowed solar array. In contrast, all-electric satellites perform orbit-raising using deployed solar arrays. Therefore, simple inverse controllers for attitude control with a neural network-based observer has been studied and evaluated. We demonstrate that the performance of the inverse controller is drastically improved with the proposed observer.
vi TABLE OF CONTENTS
Chapter Page
1. INTRODUCTION ...... 1
1.1 Motivation ...... 1 1.2 Research Objective ...... 3 1.3 Literature Review ...... 4 1.3.1 Low-Thrust Orbit-Raising ...... 4 1.3.2 Attitude Control ...... 6 1.4 Contributions ...... 8 1.5 Organization of Dissertation ...... 9
2. SPACECRAFT TRANSLATIONAL DYNAMICS ...... 11
2.1 Two-Body Problem ...... 11 2.2 Perturbation ...... 14 2.2.1 Engine Thrust ...... 14 2.2.2 Third Body ...... 15 2.2.3 Earth Oblateness ...... 16 2.3 Set of Orbital Parameters ...... 17 2.3.1 Classical Orbital Elements ...... 17 2.3.2 Modified Equinoctial Orbital Element ...... 19 2.3.3 Spherical Coordinates ...... 21 2.4 Proposed Set of Orbital State Variables ...... 22 2.4.1 Reference Frames ...... 23 2.4.2 State Variables and Transformation ...... 27 2.4.3 Variation of Proposed State Variables ...... 30
3. SPACECRAFT ROTATIONAL DYNAMICS ...... 38
3.1 Reference Frames ...... 38 3.2 Attitude Representation ...... 39 3.2.1 Euler’s Angles & Direction Cosine Matrix ...... 39 3.2.2 Single Rotation and Axis of Rotation ...... 41 3.2.3 Quaternion or Euler’s Parameters ...... 42 3.3 Attitude Kinematics ...... 44 3.4 Attitude Motion ...... 49
4. NUMERICAL TRAJECTORY OPTIMIZATION ...... 54
4.1 Unconstrained Optimization ...... 54
vii TABLE OF CONTENTS (continued)
Chapter Page
4.1.1 Orbit Segmentation and Eclipse Consideration ...... 55 4.1.2 Optimization Sub-Problem ...... 57 4.1.3 Terminal Conditions ...... 62 4.1.4 Special Case of Planar Transfer ...... 62 4.2 Receding Horizon Control ...... 64 4.2.1 Constrained Non-Linear Optimization Problem ...... 65 4.2.2 Initial Guess Generation ...... 67
5. ATTITUDE CONTROLLER DESIGN ...... 69
5.1 Inverse Controller ...... 69 5.2 Uncertainty and Compensation ...... 70 5.2.1 The Design of Modified State Observer ...... 72
6. SIMULATION RESULTS ...... 79
6.1 Results of Low-Thrust Orbit-Raising ...... 79 6.1.1 Planar Transfers ...... 80 6.1.2 Non-Planar Transfers ...... 86 6.1.3 Sub-Optimality of Computed Solutions ...... 91 6.2 Receding Horizon Control ...... 92 6.3 Attitude Controller ...... 95
7. ATTITUDE CONTROL TESTBED DESIGN ...... 105
7.1 Background ...... 105 7.1.1 State of Spherical Air Bearing Testbed ...... 106 7.2 Testbed Design ...... 107 7.2.1 Mechanical Design ...... 108 7.2.2 Electrical Design ...... 113
8. CONCLUSIONS ...... 126
8.1 Thesis Summary ...... 126 8.2 Future Work ...... 128 8.2.1 Low-Thrust Trajectory Generation ...... 128 8.2.2 Attitude Controller ...... 129 8.2.3 Attitude Control Testbed ...... 130
REFERENCES ...... 131
viii TABLE OF CONTENTS (continued)
Chapter Page
Appendices ...... 141
ix LIST OF TABLES
Table Page
3.1 List of possible rotational sequences and singularities...... 39
−1 3.2 List of function S (θ2, θ3) for all Euler’s angle sequence...... 46
6.1 Summary of orbit-raising results...... 91
6.2 Optimality gap of solutions computed using proposed methodology...... 92
x LIST OF FIGURES
Figure Page
1.1 Van Allen radiation belts [1]...... 2
2.1 Radius vectors in two-body problem...... 12
2.2 Perturbation from third body...... 16
2.3 3-1-3 sequential rotation of classical orbital parameter...... 18
2.4 Spherical coordinates...... 22
2.5 First rotation about the basis vector Jˆ of frame I...... 24
2.6 Second rotation about the basis vector Iˆ0 of frame I0...... 25
2.7 Third rotation about the basis vector kˆ of frame O...... 26
3.1 Reference frames to describe actual and desired spacecraft orientations. . . . . 38
3.2 Rotation of a reference frame about an axis...... 41
3.3 Structure of spacecraft with masses...... 50
3.4 Multiple-spin body...... 51
4.1 Spiral shape of low-thrust trajectory (not to scale)...... 55
4.2 Segmentation of the trajectory over a revolution...... 56
4.3 Model of the Earth’s shadow...... 57
4.4 Algorithm flowchart...... 64
4.5 Spacecraft path planning...... 65
4.6 Flowchart of receding horizon scheme...... 68
5.1 Block diagram of inverse feedback control and observer...... 73
5.2 Structure of neural network...... 74
6.1 Result of the circular LEO to GEO transfer...... 81
xi LIST OF FIGURES (continued)
Figure Page 6.2 Result of GTO to GEO planar transfer...... 82
6.3 Result of sub-GTO to GEO planar transfer...... 83
6.4 Result of planar super-GTO to GEO planar transfer...... 84
6.5 Result of circular LEO to GEO Transfer with energy storage...... 85
6.6 Result of non-planar circular LEO to GEO transfer...... 87
6.7 Result of the non-planar GTO to GEO Transfer...... 89
6.8 Result of the non-planar LEO to GEO case with the energy storage...... 90
6.9 Result of the receding horizon control in earth orbit...... 94
6.10 Result of the receding horizon control in Enceladus’s orbit...... 95
6.11 Comparison of performance between with and without MSO...... 97
6.12 Control input torque with MSO...... 97
6.13 Performance of MSO ...... 98
6.14 Comparison of performance between the controllers with and without MSO. . 100
6.15 Control input and its transient period...... 101
6.16 Performance of MSO prediction ...... 101
6.17 Problem configuration...... 102
6.18 Comparison of performance between the controllers with and without MSO. . 103
6.19 Control input...... 104
6.20 Performance of MSO prediction ...... 104
7.1 Shapes of spherical air bearing...... 107
7.2 Examples of spherical attitude control testbeds...... 108
xii LIST OF FIGURES (continued)
Figure Page 7.3 Air bearing structure...... 109
7.4 Designed air bearing structure...... 109
7.5 Spinning directions of momentum wheels...... 110
7.6 Limiting mechanism for roll and pitch angles...... 113
7.7 Li-Po battery and charging equipments...... 114
7.8 DC buck converter...... 115
7.9 DC buck converter...... 116
7.10 Simple Linear model of DC motor...... 117
7.11 Brush DC motor...... 118
7.12 Concept of incremental encoder...... 119
7.13 Arduino Uno...... 121
7.14 Inertia Measurement unit...... 122
7.15 Real-time clock module...... 123
7.16 Arduino Mega2560...... 124
7.17 Raspberry Pi 3 model B...... 125
xiii NOMENCLATURE
ADCS Attitude Determination and Control System
AEHF Advanced Extremely High Frequency
CM Center of Mass
COMSAT Commercial Communication Satellite
GEO Geosynchronous Equatorial Orbit
GTO Geosynchronous Transfer Orbit
I2C Inter-Intergrated Circuit
LEO Low Earth Orbit
Li-Po Lithium-Polymer
MRAC Model Reference Adaptive Control
MSO Modified State Observer
PD Proportional-Derivative
PWM Pulse Width Modulation
RTC Real-Time Clock
SPI Serial Peripheral Interface
TPBVP Two-Point Boundary Value Problem
USB Universal Serial Bus
xiv CHAPTER 1
INTRODUCTION
1.1 Motivation One of the most valuable orbits for satellites around the earth is the geosynchronous equatorial orbit (GEO) because spacecraft in this orbit are stationary with respect to a ground station on the surface of the earth. The advantages of being stationary are usually ex- ploited by commercial communication satellite (COMSAT) operators. In general, COMSATs have relatively high-power generation due to the requirement of their payload (transponders). Moreover, power-to-mass ratios of COMSATs show an increasing trend [2], opening more possibilities of including the power-hungry electric propulsion systems on COMSATs during transfer of the satellite from low altitudes to the GEO. Historically, electric propulsion systems have grown from a small area of high-impulse or deep-space missions [3] into precision control due to their fuel efficiency and capability of continuous operation [4]. However, these systems have gained popularity among satel- lite operators because of their well-known capability of saving fuel compared to chemical thrusters [5]. Recently, two satellites launched in 2015 were designed using the Boeing 702SP electric architecture in which orbital maneuvers rely solely on electric propulsion [6]. In 2010, a U.S. Air Force Advanced Extremely High Frequency (AEHF) satellite was launched into space and encountered a lethal failure of a liquid-propellant (chemical) engine while steer- ing into the operational GEO [7]. Eventually, this satellite was recovered innovatively by using on-board electric propulsion that had been initially designed for station-keeping only, with an inevitable delay of eight months due to the small magnitude of generated thrust [7]. Earlier in 2001, the European Space Agency performed a rescue mission for their COMSAT named Artemis after the launcher failure injected the satellite lower than its intended injec- tion orbit [8]. The on-board electric propulsion, which originally was intended to actuate only north-south station keeping in the GEO, was redesigned to perform tangential thrust-
1 ing for orbit-raising maneuvers from altitude of 30,000 to 35,786 km of the GEO [8]. While, these satellites used electric thrusters to salvage the mission, the Boeing 702-SP architecture allows the use of an electric thruster for all propulsive tasks. The fuel-saving capability of electric thrusters over traditional chemical thrusters allows for the design of lighter and smaller satellites that can be stacked together within a launch vehicle, reducing launch cost. However, in order to incorporate electric propulsion into all types of GEO satellites (beyond a small platform), mission designers have to address several challenges, the crucial one being the long transfer time of several months when maneuvers begin from either a low earth orbit (LEO) or a geosynchronous transfer orbit (GTO). The long transfer time not only complicates the trajectory planning but also degrades the on-board electrical components and solar panels due to prolonged exposure to the hazardous environment of Van Allen radiation belts [9], as depicted in Figure 1.1 [1]. Furthermore, the lack of power generation within the earth’s shadow also extends the transfer time because electric thrusters cannot be operated unless energy storage is provided. Therefore, the transfer time is an important factor that must be minimized. Minimizing the transfer time for low-thrust trajectories is a challenging problem, because it requires the solution of a long-time-scale nonlinear multi-phase optimal control problem.
Rotational Outer Axis Radiation Belt Inner Radiation Belt
Inner Radiation Belt
Magnetic Outer Axis Radiation Belt
Figure 1.1: Van Allen radiation belts [1].
2 1.2 Research Objective Since the electric energy for operating thrusters needs to be harvested by solar panels, the earth’s shadow constraints and possible inclusion of energy storage must be taken into consideration. This study aims to develop control and optimization tools for analyzing future high-power all-electric satellites. When the satellite is in an eclipse, thrusters need to be idle, due to the lack of power generation, unless the on-board power storage is adequately employed to provide the requisite power. One can envision spacecraft that possibly employ only a partial energy storage required for operation during eclipses. Consequently, thrusters could generate a partial amount of full thrust for sustaining the operation of the propulsion system during the transfer. While there is no solar energy, the orientation of solar panel could be relaxed in order for the attitude control system to be in an idle state. On the other hand, when the spacecraft is in sight of the sun, solar panels are able to generate enough power for supporting full operation of its electric propulsion system. In the case of energy storage support, batteries must be charged to their full capacity of storage during this time. Therefore, an attitude control subsystem is required to keep the solar panel pointing toward the sun. Additionally, the direction of the spacecraft to the sun also changes over time due to the earth revolving around it during several months of the orbit-raising maneuver. The objectives of this research are threefold. The first objective is to determine the low-thrust orbit-raising trajectories from arbitrary initial orbits to the GEO in robust man- ner, considering the constraint of eclipses and potential inclusion of energy storage support. Rapid computation of such trajectories allows the creation of a robust tool for mission de- signers for rapidly evaluating a wide range of technology alternatives (thrusters, battery, solar array, control mechanism). The second objective is to develop a receding horizon con- trol scheme for orbit-raising in order to incorporate the orbital uncertainty of third-body gravity and earth oblateness. The final objective is to demonstrate the suitability of an
3 inverse attitude controller in order to orient a spacecraft towards the desired attitude in the presence of modeling error or external disturbance. 1.3 Literature Review 1.3.1 Low-Thrust Orbit-Raising The problem of low-thrust trajectory optimization has been studied for several decades in a variety of scenarios [10–13]. Originally, the determination of optimal trajectories is based on applications of the calculus of variations [12]. This method, referred to as indirect op- timization, requires the solution of a two-point boundary value problem (TPBVP) defined by a set of ordinary differential equations and constraints on initial or final states (and/or) co-states. Many methods, including the popular shooting method, have been developed for solving the TPBVP. [14, 15]. Usually, the well-know is challenge that an accurate initial guess is required for numerical convergence to an optimal solution [16]. When convergence is achieved, the satisfaction of necessary conditions of optimality ensures that the optimality of solutions is guaranteed. In order to alleviate the difficulty of providing an initial guess, Thorne and Hall developed a method for determining an approximate initial guess for solving the TPBVP [17]. In the following year, they published another method related to their pre- vious work by utilizing the Kustaanheimo-Stiefel transformation on the state variables [18]. In 2000, Marasch and Hall used an indirect method for computing low-thrust trajectories with the deployment of on-board energy storage by dividing a long trajectory into a se- ries of circular arcs [19]. Later in 2010, Liu and Tongue demonstrated an indirect method application for a non-singular set of orbital elements, referred to as equinoctial elements [16]. Since the convergence characteristics of numeric methods is very sensitive to the initial guess, the domain of convergence is usually small, and a small deviation can result in divergence. In order to circumvent the challenge of indirect optimization, the collocation technique is used to convert ordinary differential equations into algebraic equations using quadrature rules. This results in a nonlinear programming problem that may be solved by commercial optimization software such as Sparse Nonlinear OPTimizer (SNOPT) or
4 Interior Point OPTimizer (IPOPT) [20, 21]. The set of equinoctial orbital elements along with the collocation method by Betts was used for analyzing the low-thrust problem [22]. This technique usually involves large-scale parametric optimization due to the long transfer time. The direct method is known to have a better tolerance to arbitrary initial guesses (domain of convergence is greater). However, this method still requires reasonable guesses, and automated convergence is not guaranteed [23]. One factor that affects the performance of this method is the set of state variables. Since these variables are integrated stepwise by a numerical integration technique (quadrature rule), a high number of collocated nodes are required to capture the dynamics of rapid changing in the state variables, leading to an overly complex computational problem. Due to the requirements of initial guesses, some researchers have developed a shape- based technique for generating approximate trajectories that could be used as initial guesses for the direct or indirect methods. The shape-based technique was developed by initially assuming that a trajectory has a certain geometry. Then, the control input is determined by following the constraints of equations of motion along the assumed trajectory. The process of control input being determined from the given trajectory is similar to that of the inverse control problem in aeronautics [24]. In 2004, Petropoulos and Longuski proposed an exponential sinusoidal geometry for spherical coordinates [25]. However, Wall and Conway’s study suggested that the shape of the exponential sinusoidal is far from being optimal, and they proposed a new shape in the form of an inverse polynomial inspired by known optimal trajectories [26]. The shape-based method for different sets of state variables was studied by De Pascale and Vasile in 2006 [27] and Vasile et al. in 2007 [28] by using pseudo- equinoctial elements. Later in 2011, Novak and Vasile introduced a shape-based trajectory on the spherical coordinates and equinoctial orbital elements with a method for improving optimality of the trajectory [29]. In 2012, the finite Fourier series was used to approximate the trajectory shapes [30].
5 Following a different approach, some researchers have developed a methodology sim- ilar to a closed-loop guidance-like scheme. The solution here is obtained by the integration of the trajectory from the initial point to the final destination using some guidance schemes. In 1998, a method referred as a simple guidance scheme was introduced by Kluever [31]. This method determines the optimal thrust direction from the maximum rate of change of the desired orbital elements, which are semi-major axis, eccentricity, and inclination. It tries to drive the current states into the target as fast as possible. Later in 2003, Petropoulos demonstrated the concept of the “proximity quotient” to measure the proximity of the cur- rent orbit and the destination orbit [32, 33]. The solution is determined by optimizing the rate of change of the proximity quotient including all orbital parameters. 1.3.2 Attitude Control During the orbit-raising maneuvering, the attitude of the spacecraft must be con- sidered. Attitude control is used in a variety of applications involving, spacecraft, aircraft, robotic vehicles, underwater vehicles, rockets and missiles. In attitude control, the classic technique of the proportional-derivative (PD) feedback law, has been established by many studies [34–36]. Due to the ease of concept and low risk of implementation, this class of controllers still dominates practical spacecraft operations. In January 1944, a spacecraft, named Clementine, was launched for lunar mapping operation. Its control algorithm was based on a proportional-integral-derivative (PID) controller with an assist from an attitude determination subsystem that uses a simple Kalman filter for sensing the angular rate from two solid optical gyros and two wide field-of-view star trackers. As a result, this control con- figuration yielded a great performance in lunar mapping mission [37]. However, in the design of the attitude controller, there is a mathematical complication, referred to as the unwinding phenomenon, which is due to the nature of the three-dimensional rotation space [38]. The complication here is that continuous controllers unnecessarily drive the orientation through a large angle before stopping at the desired point. Therefore, several studies proposed tech- niques to overcome this problem by introducing a discontinuous control law [39–41]. Some
6 researchers studied the problem that full-state feedback is not available. In 1991, the nonlin- ear observer for estimating angular velocity was applied to the attitude control for the first time [42]. Later, control methodologies that require only the measurement of orientation were published [43–45]. In the field of attitude control, the problem of uncertain system modeling is also taken into an account because of the difficulty in accurately sensing the mass property or the moment of inertia, especially during operation. Therefore, researchers circumvented the problem by designing inertia-free control algorithms which the computation of the control input does not require the information of moment of inertia [46, 47]. Later in 2014, an adaptive controller that is capable of predicting the moment of inertia was proposed [48]. However, moment of inertia is not the only source of uncertainty. External disturbance torque
also degrades performance of the attitude controller. The robust control technique, called H∞ control, was applied to guarantee controller performance in the presence of unknown external
torque [49–51]. Stabilizing a spacecraft by the H∞ control under additional uncertainty induced by sloshing onboard liquid has been demonstrated [52]. Control techniques that consider the combination of different uncertainties have also been proposed [53,54]. Neural network-based system observer, referred to as modified state observer (MSO) for airplane have been used in some studies [55–57]. These observer, along with an inverse controller have been successfully applied in the field of general aviation control [55, 58–60]. Later, the capability of MSO observing uncertain nonlinear states was demonstrated with atmospheric re-entry and earth’s oblateness perturbation problem [56, 57]. In the control problem, the MSO is also designed for addressing the problem of high frequency-oscillation in model reference adaptive control (MRAC) with a fast learning rate. Studies have also addressed the problem of MRAC by cascading a low-pass filter into the system [61–63], usually referred to as L1 adaptive control. Others solved the problem of high oscillations by adding an extra integral term in the neural network updating rule [64,65].
7 1.4 Contributions The contributions of this dissertation are as follows:
• A novel mathematical formulation for describing the translational dynamics of a space- craft has been developed. The state variables are meaningful (have physical interpre- tation), five of them are constant under Keplerian motion, and they vary slowly due to the perturbation or thrust force. The sixth variable locates the spacecraft in orbit and changes quickly with time. Unlike classical orbital elements that have well-known singularity for circular orbits on the equatorial plane, the novel mathematical formu- lation is non-singular on equatorial plane and for circular orbits. However, the model is singular for two cases of polar orbits not encountered for the problem under the study. Related works have been published in proceeding of AAS/AIAA Space Flight Mechanics Meeting [66] and in Journal of Guidance, Control, and Dynamics [67].
• An unconstrained optimization framework that generates sub-optimal low-thrust tra- jectories without the need for a user-inputted guess has also been developed. The algorithm transforms a long-time scale orbit-raising optimal control problem into a series of parameter optimization problems that are readily solvable by a non-linear optimization solver such as fminuncon (MATLAB). The works have been published in proceeding of AAS/AIAA Space Flight Mechanics Meeting [66, 68] and Journal of Guidance, Control, and Dynamics [67].
• The proposed unconstrained optimization framework is developed by including the effect of earth’s oblateness and rotating conical shape of earth’s shadow. This devel- opment also adds the capability in analyzing the scenarios of different thruster types. The transfer by different thruster types allow the spacecraft to reduce radiation dose from earth’s hazardous radiation region. The work has been publish in AIAA SciTech Forum [69].
8 • Taking advantage of the robustness of the optimization methodology, this method is implemented within a receding horizon framework suitable for onboard implementa- tion. The framework includes constraints typical of a direct optimization scheme and improves the solution of the unconstrained optimization solution. This framework was tested within a simulation environment that takes into account J2 and third-body perturbation. The results have been published in American Control Conference [70].
• A proportional-derivative inverse controller was designed and improved by appending a modified-state observer for attitude control during the orbit-raising with simulation. This work has been published in AAS/AIAA Space Flight Mechanics Meeting [71]. A testbed for attitude-control experiments was also designed.
1.5 Organization of Dissertation This dissertation is organized as follows:
• Chapter 2 reviews the translational dynamics of the two-body problem, which describes the motion of a spacecraft orbiting around the earth with perturbations. This chapter also discusses the set of state parameters that are used in orbital motion along with their relative advantages and disadvantages. A set of state parameters, with the derivation of its time rate of change due to arbitrary force (thrust, pertubation), is also proposed.
• Chapter 3 reviews the rotational dynamics of a spacecraft by beginning with the atti- tude representation and the transformation from each representation to another with their time rate of change due to an angular velocity. Then, the equations of motion of a rigid body and multiple-spin body are derived. This chapter also discusses the error dynamics in the rotational motion of a rigid body.
• Chapter 4 discusses the algorithm to generate low-thrust orbit-raising trajectories by using an unconstrained optimization framework. Also discussed is the receding horizon control that allows for automated generation of trajectories implementable in a real- time platform.
9 • Chapter 5 discusses the design of an inverse controller with uncertainty compensa- tion and stability analysis. The appended neural network-based observer along with stability analysis are also discussed.
• Chapter 6 demonstrates the numerical results by beginning with the low-thrust trajec- tories and comparing them with the literature. This chapter also presents results from a receding horizon control, comparing the computational time and objective value be- tween unconstrained and constrained optimizations. Simulation results of the attitude controller are also presented.
• Chapter 7 discusses the design of attitude control testbed. It begins with the discussion of the mechanical structure, appearance, conditions, and specifications. The electrical design of the power supply, batteries, electronic components, their specifications and a wiring diagram are also discussed. Finally, the chapter provides a simple example code and explanation for operating each components.
• Chapter 8 summarizes the dissertation and discusses future work for possibly improving the methodology.
10 CHAPTER 2
SPACECRAFT TRANSLATIONAL DYNAMICS
In this dissertation, the motion of spatial bodies is separated into two types; trans- lational and rotational. The analysis is developed based on Newtonian mechanics which always requires the inertial reference frame (I). This frame is defined to be stationary in space. In this chapter, the translational motion of the spatial bodies is discussed, and the rotational motion is discussed in chapter 4. The translational motion of objects in space primarily follows the Newton’s gravitational law: any two bodies exert attracting force each other with the magnitude being proportional to the product of their masses and inversely proportional to the square of the distance between them. With the combination of Newton’s laws of motion, the motions of celestial bodies can be determined. Furthermore, the problem is simplified by assuming that there are only two objects and these two body have the shape of perfect sphere. This problem is now called two-body problem. However, in the reality, the assumptions are not always valid. For examples, there could be the third body exits or the shape of one body is not perfectly spherical. The effects is still able to be incorporated into the problem by considering them as the perturbation to the two-body problem. However, let us discuss the case that the assumptions are valid. 2.1 Two-Body Problem The two-body problem is a special case of n-body which assumes only two objects in
the system. Let the vector r1 and r2 are the vectors represent the position of the two bodies whose masses are m1 and m2 with respect to the origin of the frame I. The radius vector
pointing from mass m1 to mass m2 is r as depicted in Figure 2.1, so we have
r = r2 − r1. (2.1)
11 Figure 2.1: Radius vectors in two-body problem.
By applying the Newton’s gravitational law and motion, the second time derivative of equation (2.1) is
r¨ = r¨2 − r¨1 (2.2) F F = 2 − 1 (2.3) m2 m1 1 Gm1m2 1 Gm1m2 = ( 3 )(−r) − ( 3 )r (2.4) m2 r m1 r G(m + m ) = − 1 2 r (2.5) r3 µ = − r (2.6) r3
−11 where µ = G(m1 +m2) and G is Newton’s universal gravitation constant of G = 6.674×10
2 2 N.m /Kg . However in the case of a spacecraft orbiting around a planet, the mass m1 is
much greater than m2, the value µ can be approximated by µ ≈ Gm1. Next, if there exists an arbitrary force (F ) acting on the mass m2, the equation of motion becomes
µ F ¨ r + 3 r = . (2.7) r m2
In an addition, the equation (2.7) can be rewritten into the system form of x˙ = f(x) as follows
r˙ = v (2.8) µ F ˙ v = − 3 r + , (2.9) r m2
12 where v is the velocity vector, in the other words v = r˙. In the absence of perturbation, the trajectory has the shape of conic section, and there exists constants which are specific angular momentum (h), eccentricity vector (e) and energy (). These constant parameters are define as below,
h = r × v (2.10) v × h r e = − (2.11) µ r v2 µ = − (2.12) 2 r
The specific angular momentum vector is always perpendicular to the orbit so the orbit plane can be obtained from this vector. The eccentricity vector points into the direction of the closet approach of the orbit.
However, in the presence of force acting on m2, these two parameters are not necessary to be constant and their variation due to the perturbation [72] can be determined by,
F h˙ = r × , (2.13) m2 1 e˙ = [v × (r × F ) + (F × h)] . (2.14) µm2
The derivations of equations. (2.13) and (2.14) can be found at Ref. [73]. In the context of this research, the the masses are the Earth and a spacecraft. The m2, representing the mass of the spacecraft, is denoted as m. The origin of the frame I is placed at the center of the Earth, so the r1 is zero vector. The basis vectors of the frame I are Iˆ, Jˆ and Kˆ where Iˆ points toward vernal equinox, Kˆ points toward the north pole and Jˆ follows the proper orthogonal rule. Next, in order to place a spacecraft into the desired orbit, the engines need to produce thrust which is one of the perturbation on the two-body problem.
13 2.2 Perturbation The perturbation of the two-body problem occurs from several reasons for example;
additional forces acting on mass m1 or m2, the imperfectness of spherical shape of masses or the dragging force with each other. The last example could be the drag from earth atmo- sphere when the satellite stays in a low altitude orbit. Here, we consider three perturbations,
engine thrust FT , third body Fthird and body oblateness FJ2. So the total perturbation to the system is their summation as follows
F = FT + Fthird + FJ2 (2.15)
In this dissertation, the components of perturbation are in the directions of the radius vector rˆ, local horizon nˆ and angular momentum vector hˆ. Note that the directions of basis vectors have to follow the proper orthogonal law which is defined as
nˆ = hˆ × rˆ. (2.16)
Therefore, the perturbation is
ˆ F = Frrˆ + Fnnˆ + Fhh (2.17) T FR = Fr Fn Fh (2.18) where the subscript R means that the vector is expressed by the coefficients along basis vectors of the frame R. 2.2.1 Engine Thrust
When the engine is operating, the thrust (FT ) is considered to be the arbitrary force of the two-body system acting on the mass m2. Additionally, fuel expenditure of the engines
14 decrease the spacecraft mass by
F m˙ = − , (2.19) Ispg0
where F is magnitude of the thrust, Isp is specific impulse of the engine and g0 is Earth’s
2 gravitational acceleration at the surface (g0 ≈ 9.81 m/s ). The required power of the engines can be determined as follows
g I F P = 0 sp (2.20) 2λ where λ is efficiency of the engine. 2.2.2 Third Body As mention earlier, there is an assumption that only two masses exist within the system of the two-body problem. Then, in the case of the existence of the third body, this body affects the two original bodies motion as well as is affected by the two original bodies. However, the effect of the two original body on the third one is discussed here. Hence, the position of the third body is assumed to be known. Now consider the Figure 2.2, there are the third body located at the vector r13 from the mass m1 and the gravitational forces acting on the masses m1 and m2 as F13 and F32 respectively. Now, due to the existence of perturbation, the second time derivative of radius r as earlier shown in equation (2.2) now becomes
r¨ = r¨2 − r¨1 (2.21) F F F F = ( 2 + 23 ) − ( 1 + 13 ). (2.22) m2 m2 m1 m1
We can applying the Newton gravitational law to equation (2.22). Then, we have
µ 1 G(m m ) 1 G(m m ) ¨ 2 3 1 3 r = − 3 r + 3 (−r32) − 3 r13 (2.23) r m2 r32 m1 r13
15 µ −r32 r13 = − 3 r + Gm3 3 − 3 (2.24) r r32 r13
From the Figure 2.2, we know that r32 = r − r13. As a result we get
Figure 2.2: Perturbation from third body.
µ r − r r ¨ 13 13 r = − 3 r + µ3 3 − 3 (2.25) r r32 r13 µ Fthird = − 3 r + , (2.26) r m2
where µ3 = Gm3 and
r13 − r r13 Fthird = mµ3 3 − 3 . (2.27) r32 r13
2.2.3 Earth Oblateness The Earth does not have the perfectly spherical shape. The deviation from the spherical shape is considered as the perturbation of the two-body system. This perturbation is mathematically modeled as a summation of a infinite series. The first term of this series represents the attraction of spherical shape while the other terms represent the disturbing force due to the shape deviation. The major term is the second term, J2, which is in order of 103 times as large as the other terms for the case of Earth. This perturbation is mathematically expressed as
3z 3 15z2 F = −µJ R2 Kˆ + − rˆ , (2.28) J2 2 E r5 2r4 2r6
16 −3 where J2 is a constant coefficient of 1.08263 × 10 , RE = 6378 km is mean equatorial radius of the earth and z is a component of radius vector along Kˆ . 2.3 Set of Orbital Parameters In previous section, the radius and velocity vectors are discussed and shown that they can be used for describing the motion in two-body problem. However, the set of these two vectors is not the only way for describing the motion. There exist several set that can be used with different advantages and disadvantages. In order to demonstrate, some of the set will be discussed here. 2.3.1 Classical Orbital Elements The advantage of classical orbital elements is a clear geometrical shape and orientation of the orbit. This set is also one of most well-known set. The set consists of six parameters, the first two are eccentricity e and semi-major axis a, the other four are angles, right ascension of ascending node Ω, inclination angle i, argument of perigee ω and, the last one, true anomaly angle θ. The eccentricity defines the shape of the orbit. It can be obtained by the norm of eccentricity vector in equation 2.11. On the other hand, the semi-major axis gives the information of the size which can be obtained as follows
h2 a = , (2.29) µ(1 − e2)
where h is magnitude of the angular momentum. The true-anomaly angle is the angle between the eccentricity vector and the radius vector. This angle gives information of position of the spacecraft within the orbit. Therefore these three parameters defines shape, size of the orbit and position of spacecraft in the orbit. The other three parameters gives the information of orientation of the orbit plane and will be further discussed. Now, consider the inertial frame I, as mentioned earlier, its origin is located at the center of the Earth, its first basis unit vector Iˆ points in the direction vernal equinox, its
17 third basis unit vector Kˆ points into the north poles and the second basis unit vector Jˆ points in the direction following proper orthogonal principle. The angles in classical orbital elements are specified by 3-1-3 sequential rotation which is depicted at Figure 2.3. The frame I is rotated by an angle Ω about the Kˆ direction such that the new first axis point into the ascending node. Next, the new frame is rotated about its first axis by angle i in order to align the third axis into the angular momentum. Then, the frame is rotated again about its third axis, in the direction of angular momentum, by an angle ω such that it first axis align with eccentricity vector.
closest approach
line of nodes
Orbit
ascending node
Figure 2.3: 3-1-3 sequential rotation of classical orbital parameter.
For the presence of perturbing force, the variations of parameters are determined by Gauss’s variation form [74]
1 e˙ = {p sin θF + [(p + r) cos θ + re] F } (2.30) mh r n 2a2 p a˙ = e sin θF + F (2.31) mh r r n r sin(θ + ω) Ω˙ = F (2.32) mh sin i h r cos(θ + ω) i˙ = F (2.33) mh h 1 r sin(θ + ω) cos i ω˙ = [−p cos θF + (p + r) sin θF ] − F (2.34) emh r n mh sin i h h 1 θ˙ = + [p cos θF − (p + r) sin θF ] (2.35) r2 emh r n
18 h2 h2 where p is semi-latus rectum and defined as p = µ , and r is computed by r = µ(1+e cos θ) . Although, the set of classical orbital elements has the physical interpretation of the orbit shape, size, and orientation, its main drawback is that there exists the singularity at the circular orbit and at the equatorial plane which are the properties of the GEO. However, this set of parameters has another advantage besides physical interpretation, which is the variations of all parameters except θ are slow in the case that the perturbation is small. The reason is that the multiplication of perturbation components in equations (2.30) to
(2.34), obviously if Fr, Fn and Fh are relatively small value, the rate of change of those five orbital parameters are close to zero. The slow variation yields a major benefit in numerical computation which requires sampling nodes. The trajectory with the slow variation can be properly captured without extensive number of nodes. 2.3.2 Modified Equinoctial Orbital Element The set of modified equinoctial orbital element is developed from the work in Ref. [75]. This set has advantages over the classical orbital elements of that the singularity does not exist at the circular or equatorial orbit. The set consists of 6 parameters which are computed from classical orbital parameters as
2 y1 = p = a(1 − e ) (2.36)
y2 = e cos(ω + Ω) (2.37)
y3 = e sin(ω + Ω) (2.38) i y = tan cos Ω (2.39) 4 2 i y = tan sin Ω (2.40) 5 2
y6 = Ω + ω + θ (2.41)
As shown in the above equations, the singularity at the equatorial plane is eliminated due to the fact that the value of ω + Ω is still be able to be defined at the equatorial, even though ω
or Ω can not be defined separately. Similarly, y4 and y5 are also be defined at the equatorial
19 plane, i.e. y4 = y5 = 0. However, some parameters do not have physical interpretations. This set of parameters is related to radius and velocity vectors expressed in the frame I as [76]
r 2 s2 (cos y6 + s2 cos y6 + 2y4y5 sin y6) 4 r = r 2 , (2.42) I 2 (sin y6 − s2 sin y6 + 2y4y5 cos y6) s4 2r 2 (y4 sin y6 − y5 cos y6) s4
1 q µ 2 2 − s2 y (sin y6 + s2 sin y6 − 2y4y5 cos y6 + y3 − 2y2y4y5 + s2y3) 4 1 q v = 1 µ 2 2 , (2.43) I − s2 y (− cos y6 + s2 cos y6 + 2y4y5 sin y6 − y2 + 2y3y4y5 + s2y2) 4 1 2 q µ 2 (y4 cos y6 + y5 sin y6 + y2y4 + y3y5) s4 y1
where
s1 = 1 + y2 cos y6 + y3 sin y6 (2.44) y r = 1 (2.45) s1 2 2 2 s2 = y4 − y5 (2.46) q 2 2 s3 = y4 + y5 (2.47)
2 2 s4 = 1 + s3 (2.48)
Recall that the subscription means that the components are defined by the basis vectors of the frames. Now, in the existence of perturbation, the dynamics of the parameters are
r 2y1 y1 Fn y˙1 = (2.49) s1 µ m r y1 Fr Fn y3Fh y˙2 = sin y6 + ((s1 + 1) cos y6 + y2) − (y4 sin y6 − y5 cos y6) (2.50) µ m ms1 ms1 r y1 Fr Fn y2Fh y˙3 = − cos y6 + ((s1 + 1) sin y6 + y3) + (y4 sin y6 − y5 cos y6) (2.51) µ m ms1 ms1
20 r 2 y1 s4 cos y6Fh y˙4 = (2.52) µ 2ms1 r 2 y1 s4 sin y6Fh y˙5 = (2.53) µ 2ms1 r 2 y1 (y4 sin y6 − y5 cos y6)Fh √ s1 y˙6 = + µy1 2 (2.54) µ ms1 y1
In the case that the perturbation are small, all parameters in the set except y6 also have the slow variations similar to the classical orbital elements. However, equinoctial elements do not have physically interpretations. 2.3.3 Spherical Coordinates The spherical coordinates were successfully used for analyzing the radiation dose along the transfer [77]. The set of parameters consists of the radius (r), two angles (ψ1, ψ2) and three velocities in radial, tangential of the angles direction (vr, vψ1, vψ2). All the parameters are depicted at Figure 2.4. The frame I is placed at the center of the Earth, so azimuthal and polar angle are ψ1 and ψ2, respectively. The relationships with the radius and velocity vectors are
T rI = r cos ψ1 cos ψ2 sin ψ1 cos ψ2 sin ψ2 (2.55) cos ψ1 cos ψ2 − sin ψ1 − cos ψ1 sin ψ2 vr vI = sin ψ cos ψ cos ψ − sin ψ sin ψ v (2.56) 1 2 1 1 2 ψ1 sin ψ2 0 cos ψ2 −vψ2
This set of state parameters does not have the slow variation under the small pertur- bation. This fact can be clearly seen by observing the its equations of motion as follows
r˙ = vr (2.57) ˙ vψ1 ψ1 = (2.58) r sin ψ2 v ψ˙ = ψ2 (2.59) 2 r
21 Figure 2.4: Spherical coordinates.
µ v2 + v2 F v˙ = − + ψ1 ψ2 + r (2.60) r r2 r m v v + v v cot ψ F v˙ = − r ψ1 ψ1 ψ2 2 + n (2.61) ψ1 r m 2 −vrvψ2 + v cot ψ2 F v˙ = ψ1 − h (2.62) ψ2 r m
From the above equations, even in the absence of any perturbation, the variations of all parameters are not necessarily small. Note that, the equations motion written here are different from the literature due to the definition of perturbation direction. Additionally, the tangential direction to the angle ψ2 is opposite to the angular momentum vector for the prograde orbit and in same direction for the retrograde orbit. So they are valid only for the prograde orbit. This set of parameters does not any singularity however the variations of the parameters are fast even in the case of small perturbations. 2.4 Proposed Set of Orbital State Variables The translational motion of the spacecraft can be described by several ways as dis- cussed in the previous section. However, the existing sets can not be perfectly applied into the numerical methodology which will be discussed later in Chapter 4. The proposed set of orbital variables, which can be utilized in the numerical method, is developed and has three main properties:
• The singularities do not exist at the circular or equatorial orbit.
22 • All parameters have obvious physical interpretations.
• The condition of revolutions have to be obviously defined by the parameters.
In order to demonstrate and discuss in detail, first of all we need to introduce other reference frame besides I and R which have already been introduced. 2.4.1 Reference Frames Now, let us remind about the frame inertial frame I and radial frame R which have already been discussed in the previous section. The frame R can be obtained after performing the 3-1-3 sequential rotation by angle of Ω, i and ω + θ, respectively. Mathematically, this sequence of rotation yield the singularity when angle i is zero. The proposed parameters are designed to avoid the singularity at equatorial plane by utilizing the rotational sequence of 2-1-3 instead of the traditional one. However, the proposed sequence still has the singularity which will be discussed later. Initially, the inertial frame I, which comprised of basis vectors Iˆ, Jˆ and Kˆ , is rotated by an angle ζ about the second basis vector Jˆ in order to align the third axis into the projection of angular momentum on the X-Z plane. Let consider Figure 2.5, after the first rotation, a new frame, which is called I0 and comprised of the basis vectors, Iˆ0, Jˆ0 and Kˆ 0, is constructed. The projection of the angular momentum on the X-Z plane is simply the vector which has the same components along Iˆ and Kˆ directions as angular momentum vector and the component along Jˆ direction is zero. From the given definition, the basis vector Kˆ 0 can be expressed on frame I as
T ˆ 0 √ hX √ hZ KI = 2 2 0 2 2 (2.63) hX +hZ hX +hZ
ˆ ˆ where hX and hZ are the components of specific angular momentum along the I and K ˆ ˆ ˆ directions, respectively, or written in to form of h = hX I + hY J + hZ K. Remind that
the component hX , hY and hZ can be obtained from equation (2.10). The relationship in
23 equation (2.63) is used for defining the angle ζ as
sin ζ = √ hX , cos ζ = √ hZ , tan ζ = hX . (2.64) 2 2 2 2 hZ hX +hZ hX +hZ
projection of on X-Z plane X-Y plane(equatorial plane)
X-Z plane
Figure 2.5: First rotation about the basis vector Jˆ of frame I.
Then, the basis vectors Iˆ0 and Jˆ0 can be determined by
√ hZ cos ζ 2 2 0 hX +hZ ˆ0 ˆ0 ˆ I = 0 = 0 , J = JI = 1 (2.65) I I √ hX − sin ζ − 2 2 0 hX +hZ
Secondly, consider the Figure 2.6 the frame I0 is rotated about the direction of its first axis, Iˆ0 by an angle η in order to align the new third axis into the angular momentum. The new three basis vectors generated by this process are ˆi, jˆ and kˆ, respectively. Therefore, the ˆi - jˆ plane is aligned into the orbit plane. So, let us call the newly generated frame as Oˆ. From the definition, the direction kˆ is simply be determined by
h T ˆ I h h h kI = = X Y Z (2.66) |h| h h h
and unit vector ˆi is equal to Iˆ0 which is
T ˆ √ hZ √ hX iI = 2 2 0 − 2 2 (2.67) hX +hZ hX +hZ
24 X-Y plane(equatorial plane)
X-Z plane
Figure 2.6: Second rotation about the basis vector Iˆ0 of frame I0.
So the basis vector now can be computed by the proper orthogonality as
jˆ = kˆ × ˆi (2.68) T h2−h2 jˆ = √−hX hY √ Y √−hY hZ (2.69) I 2 2 2 2 2 2 h h −hY h h −hY h h −hY
π π Next, geometrically, the angle η is in the set of [− 2 , 2 ] and can be obtained by
√ h2 +h2 hY X Z (2.70) sin η = − h , cos η = h
The third rotation is depicted at Figure 2.7. The frame O is rotated about the kˆ by an angle φ in order to align its first axis into the radius vector. The frame which is obtained from the last rotation is the frame R as used earlier at equation (2.17). Geometrically, angle φ can be obtained from the dot product between radius vector and ˆi, or radius vector and ˆi as below
sin φ = rˆ · jˆ (2.71)
cos φ = rˆ · ˆi (2.72)
25 orbit plane
Figure 2.7: Third rotation about the basis vector kˆ of frame O.
As a summary of this section, the 2-1-3 rotation sequence is summarized as follow:
ζ η φ I Iˆ, Jˆ, Kˆ −−−→ I0 Iˆ0, Jˆ0, Kˆ 0 −−→ O ˆi, jˆ, kˆ −−→ R rˆ, nˆ, hˆ . Jˆ=Jˆ0 Iˆ0=ˆi kˆ=hˆ
O R Let us define two direction cosine or rotational matrices, RI and RO . The superscribe O means that basis vector of the frame O and subscribe I means that they are expressed on
O the frame I. Therefore, based on the equation (2.66), (2.67) and (2.69), the matrix RI is written as
O ˆ ˆ ˆ RI = iI jI kI (2.73) √ hZ √−hX hY hX h2−h2 h h2−h2 h Y Y h2−h2 = 0 √ Y hY (2.74) h h2−h2 h Y √−hX √−hY hZ hZ 2 2 2 2 h h −hY h h −hY
R Similarly, the matrix RO can be written by following the given definition with equation (2.16) or applying Euler’s rotational matrix about third axis with equation (2.71) and (2.72) as follows
R ˆ RO = rˆO nˆ O hO (2.75)
26 cos φ − sin φ 0 = sin φ cos φ 0 (2.76) 0 0 1 rˆ · ˆi −rˆ · jˆ 0 = rˆ · jˆ rˆ · ˆi 0 (2.77) 0 0 1
2.4.2 State Variables and Transformation All of the proposed variables have the meaningful interpretation but some of them are required the definition of reference frame which they are expressed on. And, all of the related reference frames were discussed in the previous sections. Now we are able to discuss about the propose state parameter and their conversion between standard Cartesian variable, r and v.
The proposed set of state variables is x ∈ R6,
T x = h hX hY ex ey φ (2.78)
where h is magnitude of specific angular momentum, hX , hY are the components of specific ˆ ˆ angular momentum along the directions of I and J respectively, ex and ey are the components of eccentricity vector along the directions of ˆi and jˆ respectively and φ is the rotation angle from frame O to frame R. In the absence of perturbation, the all parameters except φ are constant. Moreover, the variable φ can considered to be the fast variable. Additionally, the usage of variable h in stead of hZ as the state variables yields a drawback that they can not differentiate pro-grade orbit and retro-grade orbits. However, this drawback does not affects the problem of low-thrust orbit raising due to that the involved trajectories are never be the retro-grade. Therefore, in order to apply this set of variables to the different problem, this fact is needed to be concerned or avoided by using hZ if necessary. Another point that is
27 needed to be concerned is that this set has the singularity when h = ±hJˆ. This singularity exists due to the 2-1-3 rotational sequence. However, the condition is only for polar orbits which can be considered to be far away from GEO. The transformation of standard Cartesian variables, radius and velocity vector, into the proposed variables can be computed easily by following their given definition. Let us start with the definition of angular momentum in equation (2.10). Assuming the radius and velocity expressed on frame I are known, first three proposed variables can be computed as follows
h = |r × v| (2.79) and
hX h = hI = rI × vI (2.80) Y hZ
Next, the variable ex and ey can be computed by applying the definition of eccentricity at equation (2.11) and direction cosine matrix at equation (2.74) as follow
ex OT vI × hI rI e = eO = R − (2.81) y I µ r 0
Lastly, angle φ is determined from equation (2.71) and (2.72) and summarized as below
cos−1 rˆ · ˆi if rˆ · jˆ ≥ 0 φ = (2.82) 2π − cos−1 rˆ · ˆi otherwise
28 Conversely, we can transform proposed variables back into radius and velocity vectors by starting at the below relationship
h2 h2 = r + r · e =⇒ r = (2.83) µ µ(1 + rˆ · e)
By utilizing direction cosine matrix in equation (2.76) and expressing the unit vector rˆ in terms of components in frame O, we have
T T R rˆ · e = eORO 1 0 0 cos φ − sin φ 0 1 = e e 0 sin φ cos φ 0 0 x y 0 0 1 0
= ex cos φ + ey sin φ (2.84)
Therefore, we can compute the magnitude of the radius vector by substituting equation (2.84) into equation (2.83) as follows
h2 r = (2.85) µ(1 + ex cos φ + ey sin φ)
T At this point the radius vector expressed on the frame R is simply r 0 0 , however for the matter of completeness, the radius vector expressed on the frame I is able to be determined by applying direction cosine matrix from equation (2.74) and (2.76) as follow
T O R rI = RI RO r 0 0 . (2.86)
29 Next, let consider the velocity vector by starting with the velocity along nˆ direction, vn. This speed can be determined by the relationship with angular momentum as follow
h v = (2.87) n r
By substituting equation (2.85), we get
µ v = (1 + e cos φ + e sin φ) (2.88) n h x y
Next, for the component of velocity in the radial direction, it can be shown that
µ v =r ˙ = (e sin φ − e cos φ) (2.89) r h x y
The derivation of the equation (2.89) will be provided later because it needs the discussion of angular velocity of reference frames. Finally the velocity expressed on the frame I is
µ vr (ex sin φ − ey cos φ) h O R O R µ vI = R R v = R R (1 + e cos φ + e sin φ) . (2.90) I O n I O h x y 0 0
2.4.3 Variation of Proposed State Variables In the exist of perturbation, the angular momentum and the eccentricity vectors are no longer to be constant as well as the orientation of the frame O. First of all, let analysis the angular velocity of the frame O due to the perturbation at equation (2.18),
T O ωR = ω1 ω2 ω3 (2.91) where the superscript O stands for the angular velocity of frame O. In order to analyze
O ωR, we set up the relationship with the time rate of change of angular momentum which is
30 defined at equation (2.13),
r Fr 0 1 ˙ rF hR = 0 × F = − h (2.92) m n m rFn 0 Fh m and also
Odh h˙ = R + ωO × h (2.93) R dt R R where the left superscript of O means the differentiation with respect to the rotating frame O. Then, by the definition of the frame O which its third axis always aligns with the angular momentum, this term is equal to h˙ kˆ. Now us let calculate the equation (2.93) and equate with equation (2.92)
0 0 ω1 0 ω2h ˙ rF hR = − h = 0 + ω × 0 = −ω h m 2 1 rFn ˙ ˙ m h ω3 h h 0 ω2h − rFh = −ω h (2.94) m 1 rFn ˙ m h
Now we can specify that
rFh (2.95) ω1 = mh , ω2 = 0
31 In order to compute ω3, let us write the angular velocity in terms of the rates of change of angles as
ω = ζ˙Jˆ +η ˙ˆi (2.96)
And from the direction cosine matrix, we get
T T ˆ ˆ JR = cos η sin φ cos η cos φ − sin η , iR = cos φ − sin φ 0 (2.97)
By substituting equation (2.97) into equation (2.96) and equating with equation (2.91), we have
rFh ˙ ω1 ζ cos η sin φ +η ˙ cos φ mh ωR = ω = 0 = ζ˙ cos η cos φ − η˙ sin φ (2.98) 2 ˙ ω3 ω3 −ζ sin η
From the relationship along second axis, we get
cos η cos φ η˙ = ζ˙ (2.99) sin φ
Substitute in the relationship of the component along the first axis into equation (2.99)
rF cos η cos2 φ cos η h = ζ˙ cos η sin φ + ζ˙ = ζ˙ (sin2 φ + cos2 φ) (2.100) mh sin φ sin φ
rF sin φ ζ˙ = h (2.101) mh cos η
Then, along the third axis, we have
rF sin φ rF sin φ ω = − h sin η = − h tan η. (2.102) 3 mh cos η mh
32 For the range of η ∈ (− π , π ), the relationship of tan η = −√ hY is valid, so 2 2 2 2 h −hY
rFhhY sin φ ω3 = (2.103) p 2 2 mh h − hY
Finally, the angular velocity of the frame O is therefore given by:
T ωO = rFh 0 rFh√hY sin φ (2.104) R mh 2 2 mh h −hY
Next, in order to determine the angular velocity of the frame R. we start with
Rdr v = R + ωR × r (2.105) R dt R R vr r˙ r R v = 0 + ω × 0 (2.106) n R 0 0 0
Now, let consider the term angular rotation rate of frame R. it can be written as
ωR = ωO + φ˙hˆ (2.107) rFh ω1 mh R ω = ω = 0 (2.108) R 2 ˙ 0 ω3 + φ ω3
Now, substituting equation (2.108) into equation (2.106) and considering the component
0 along second axis, we have vn = ω3r. And from equation (2.87), we can determine that
h2 ω0 = (2.109) 3 r
33 and
T R 2 ω = rFh h (2.110) R mh 0 r
Now we are ready to derive the equations of motion for the proposed state variable. Initially parameters A and B for simplicity of the following notations as,
A = ex sin φ − ey cos φ (2.111)
B = 1 + ex cos φ + ey sin φ (2.112)
Let start with the simplest one which is
rF h˙ = n (2.113) m substituting r from equation (2.85) and B from equation (2.112), we get
h2 h˙ = F (2.114) µmB n
Next, the time rate of change of hX and hY are determined from
1 h˙ = RORR (r × F ) (2.115) I m I O R R ˙ h2 hX Fr 0 µB 1 O R O R 2 h˙ = R R 0 × F = R R − h F (2.116) Y m I O n I O mµB h ˙ h2 hZ 0 Fh mµB Fn
We then obtain the following:
2p 2 2 2 ! ˙ hhX h h − hX − hY hhX hY hX = Fn + sin φ + cos φ Fh (2.117) mµB p 2 2 p 2 2 mµB h − hY mµB h − hY
34 ! hh hph2 − h2 h˙ = Y F + − Y cos φ F (2.118) Y mµB n mµB h
2 ! ˙ hhZ h hX hhY hZ hZ = Fn + − sin φ + cos φ Fh (2.119) mµB p 2 2 p 2 2 mµB h − hY mµB h − hY
After that, we compute the rates of change of ex and ey by
e˙x O deO R O e˙ = = R (e˙R) − ω × eO (2.120) y dt O O e˙z
T O where eO is ex ey 0 , ωO determined by the transformation of equation (2.104) and the direction cosine matrix equation (2.76) as
T ωO = RRωO = rFh cos φ rFh sin φ − rFh√hY sin φ (2.121) O O R mh mh 2 2 mh h −hY
And the term e˙R is computed by equation (2.14), yielding the following:
2h µm Fn 1 h hA e˙R = (2(v · F )rR − (F · r)vR − (v · r)FR) = − F − F (2.122) µm µm r µmB n rvr − mµ Fh
By putting equation (2.122) and all other parameters given earlier into equation (2.120), we get
2h rFh e˙x Fn cos φ ex µm mh R rF e˙ = R − h F − hA F − h sin φ × e y O µm r µmB n mh y rF h sin φ rvr h√Y e˙z − mµ Fh 2 2 0 mh h −hY
35 h sin φ F + 2h cos φ + hA sin φ F + rey√hY sin φ F µm r µm µmB n 2 2 h mh h −hY h cos φ 2h sin φ hA cos φ rexhY sin φ = − Fr + − Fn − √ Fh (2.123) µm µm µmB mh h2−h2 Y 0
Additionally, there is a necessary condition in order to always be the orbit plane of x-y surface. the condition is that ez has to always be zero and it is satisfied by equation (2.123). Then, eliminate the parameter, r, by substituting equation (2.85) into equation (2.123), we get rate of change of other two state variables as
h sin φ 2h cos φ hA sin φ heyhY sin φ e˙x = Fr + + Fn + Fh (2.124) µm µm µmB p 2 2 mµB h − hY h cos φ 2h sin φ hA cos φ hexhY sin φ e˙y = − Fr + − Fn − Fh (2.125) µm µm µmB p 2 2 mµB h − hY
For the last state variable, φ can be computed by equation (2.108) and equation (2.109) as
˙ h hhY sin φ φ = − Fh r2 p 2 2 mµB h − hY 2 2 µ B hhY sin φ = − Fh (2.126) h3 p 2 2 mµB h − hY
As a summary of this section, the equations of motion could be written in a more compact form as
2 ˙ 0 h 0 h µmB √ h2 h2−h2 −h2 ˙ 0 hhX √ X Y sin φ + hh√X hY cos φ hX mµB mµB h2−h2 mµB h2−h2 Fr Y √ Y 2 2 hh h h −hY h˙ = 0 Y − cos φ F Y mµB mµB n he h sin φ e˙ h sin φ 2h cos φ + hA sin φ y√Y F x µm µm µmB mµB h2−h2 h Y h cos φ 2h sin φ hA cos φ he h sin φ e˙y − − − x√Y µm µm µmB 2 2 mµB h −hY (2.127)
36 The variations of proposed state variables are clearly shown in equation (2.127) that once the perturbation vector is small the variations also be slow. This is the condition that is need for the orbit-raising strategy which will be discussed in Chapter 4. As a conclusion, the two-body problem has been used to model the motion of space- craft orbiting around the globe. The other forces except the gravitational attraction of the two main bodies are considered to be perturbation to the system. With context of two-body problem, we develop a set of state variables that has suitable properties for implementing in numerical algorithm. However, only the orbital motion does not completely describe the dynamics of the spacecraft, it also has the rotational motion that will be discussed in next chapter.
37 CHAPTER 3
SPACECRAFT ROTATIONAL DYNAMICS
In last chapter, we discussed about the translational dynamics of spacecraft which relates to the position and velocity of the spacecraft respect to the earth. In this chapter, we consider about the orientation and angular velocity of the spacecraft. This is called spacecraft rotational dynamics or spacecraft attitude dynamics. In order to describe the rotational dynamics, first of all, let us discuss about the reference frames. Any analysis in classical physics requires the inertial reference frame. From the previous chapter, the inertial frame I is defined by locating its origin at the center of the Earth. Additionally, we have defined the frame R which rotates with respect to the radius vector. In this chapter we want to emphasize on rotational dynamics. Therefore, we will introduce new reference frames in order to decouple the translational and rotational motions so that we can primarily focus on the rotational dynamics. 3.1 Reference Frames
Figure 3.1: Reference frames to describe actual and desired spacecraft orientations.
The non-rotating frame I00 is defined to be fixed in space. Next, we define a body frame, B, which is fixed with a rotating body and a desired frame, D is oriented in the desired direction as depicted in Figure 3.1. The principal axes of each frame are label as are labeled as x, y and z subscripted by the frame notation. The origins of all the frames,
38 for now, are assumed to be coincided so all the origins in Figure 3.1 should be at the same point, however, the non-rotating frame is separated out for the matter of aesthetic. 3.2 Attitude Representation Orientation of one frame with respect to another frame can mathematically be pre- sented by several formulations. However, the selected formulation, which is used for the controller design, is Euler’s parameters or unit quaternion. In order to profoundly demon- strate the visualization and reasons of this selection, we need to discuss some of other attitude representations for comparisons. Let start with the representation that used in the previous chapter, Euler’s angles. 3.2.1 Euler’s Angles & Direction Cosine Matrix The Euler’s angles are 3 consecutive rotation about a basis axis of reference frames for aligning one frame to another. In the previous chapter, we used 3-1-3 by angles (Ω, i, ω) for the classical orbital elements and 2-1-3 by angles (ζ, η, φ) for the proposed state variables. There are totally 12 rotational sequences which are possible. Each of them always has some specific orientations that can not be uniquely defined by angles in the sequences which is called singularity. For example, the angle Ω and ω can be defined once orbits are on the equatorial plane or i is zero. We can only specify only the value of Ω + ω. All possible rotational sequences are shown in Table 3.1. It can be identified that the singularity exists when the second rotation aligns the third rotation axis into the first one.
Table 3.1: List of possible rotational sequences and singularities.
Rotational Singularity Rotational Singularity Sequence Condition Sequence Condition 1-2-1 2nd rotation is 0◦ or 180◦ 2-3-2 2nd rotation is 0◦ or 180◦ 1-3-1 2nd rotation is 0◦ or 180◦ 2-3-1 2nd rotation is ±90◦ 1-3-2 2nd rotation is ±90◦ 2-1-3 2nd rotation is ±90◦ 1-2-3 2nd rotation is ±90◦ 3-1-3 2nd rotation is 0◦ or 180◦ 2-1-2 2nd rotation is 0◦ or 180◦ 3-2-3 2nd rotation is 0◦ or 180◦ 3-2-1 2nd rotation is ±90◦ 3-1-2 2nd rotation is ±90◦
39 Next, let consider in term of the transformation of a vector expressed on one frame to another. We require a matrix that contains information of three basis vectors of one frame expressed on another, and this matrix is called direction cosine matrix (DCM). This matrix
O have been used at equation (2.74) and (2.76). Remind that the DCM RI at equation (2.74) contains the basis vectors of frame O by expressing them on frame I. Any vector expressed on frame O multiply with O becomes that vector which expressed on the frame I, so let
O we define this DCM as RI : O → I. The DCM in the reverese direction can be easily determined by