Control of satellites using environmental forces : aerodynamic drag / solar radiation pressure

The dynamics of a satellite involves orbit and attitude motion. Both orbit and attitude motion are required to be controlled for achieving any mission objectives. This thesis examines both of these motion with a focus on formation flying and attitude stabilization using aerodynamic drag or solar radiation pressure. The concept of satellite formation flying involves the distribution of the functionality of a single satellite among several smaller, cooperative satellites. Autonomous multiple satellite formation flying space missions offer many promising possibilities for space exploration. A large quantity of fuel is typically required onboard any conventional satellite to carry out its attitude and orbital positioning, and satellite formation flying has a higher fuel requirement. As on-board fuel is a scarce commodity, it is important to have control methods requiring little or no fuel. Keeping this in mind, this thesis presents the results of using aerodynamic drag or solar radiation pressure for satellite formation flying. This methodology has potential to have significant commercial advantage as it pertains to almost negligible fuel requirements. A leader/follower formation architecture is considered for the formation flying system. The control algorithms are derived based on adaptive sliding mode control technique. Aerodynamic drag is used to accomplish multiple satellite formation flying in low Earth orbit whereas solar radiation pressure is used in geostationary orbit. Due to the nature of the aerodynamic drag, in-plane control can only be accomplished by the suitable maneuvering of the drag plates mounted on the satellites. Solar radiation pressure is able to accomplish in-plane and out-of-plane control by maneuvering of the solar flaps. Efficacy of the control methodology in performing various scenarios of formation flying including formation reconfiguration is validated by numerical simulation in both the cases. The numerical results demonstrate the effectiveness of the proposed control techniques for satellite formation flying using aerodynamic drag or solar radiation pressure. Formation flying accuracies of less than 5 m are achieved in both the cases. Next, this thesis investigates the use of aerodynamic drag or solar radiation pressure for satellite attitude control. In the case of the aerodynamic drag stabilized satellite, the drag plates are considered to be attached to the satellite while the solar flaps are fixed to the satellite in the case of the solar radiation pressure stabilized satellite. In both the cases, the control algorithm is developed based on adaptive sliding mode control technique. The effectiveness of this methodology is numerically evaluated under various scenarios including the pressure of failure/fault in the solar flaps or drag plates attached to the satellite. The satellite attitude is stabilized within reasonable limits in all the cases thereby demonstrating robust performance in the presence of external disturbances.


Introduction
A utonomous formation flying is currently one of the most important areas of research in satellite dynamics and control. Use of a cluster of smaller, closely flying satellites can distribute the functionality of a single, complex satellite. Space agencies, in particular NASA, ESA and U.S. Air Force have taken a keen interest in developing and deploying reliable autonomous multiple satellite systems. This approach has several advantages including the ability to enhance and/or enable missions through increased resolution of scientific observations, improved flexibility and redundancy. Moreover capability for real-time reconfiguration, adaptable to highly dynamic demands, and lower life cycle cost also drive this methodology. Key challenges involved in these missions include autonomous control of the satellites which are influenced by the different disturbing forces, and to achieve it with minimum fuel consumption. These forces include environmental forces caused by gravity gradient, solar radiation pressure, magnetic, aerodynamic drag, and Earth's oblateness effect [Larson 1999], as well as third body perturbations. It is the Attitude and Orbit Control System (AOCS), that orients and stabilizes the satellite to the desired position in the presence of these external forces. AOCS consists of sensors to measure states of the satellites, actuators to apply the torques needed to re-orient the satellite to the desired position, and also control laws, i.e., a set of computations that determine how the actuators should operate, based on the sensor measurements. However, some of these disturbing forces, if properly utilized may be used to stabilize the satellite orientation and maintain the formation. Several methods of satellite attitude and orbit control have been developed over the last four decades. These methods may be broadly classified as active and passive/semi-passive methods. Active stabilization methods require expenditure of propellant or energy, leading to an increase in weight and space requirements in the satellite, whereas passive/semi-passive methods depend on natural/environmental forces and 2 Chapter 1. Introduction they make use of spin stabilization, dual spin, gravity gradient, solar radiation pressure (SRP), aerodynamic drag, and Earth's magnetic field to achieve the desired performance.
These methods are thus less costly and their advancement may provide a viable solution to the development of low cost AOCS for satellites. However, their development poses several challenges including lack of methodologies to utilize these forces effectively, lower achievable accuracies compared to a conventional system, and unavailability of these forces throughout the mission. This thesis investigates the efficacy of "propellant-less" method based on aerodynamic drag or solar radiation pressure for multiple satellite formation flying or attitude control, and develops innovative control algorithms with the goal of achieving a low cost AOCS which is able to exhibit superior performance.

Research Motivation
A large quantity of fuel is typically required onboard any conventional satellite to perform its attitude and orbital control. Satellite formation flying has a much higher fuel requirement in order to maintain the tight formation (∼10m). As on-board fuel is a scarce commodity, it is important to have satellite control techniques which would require little or no fuel. Hence it would be important to propose a novel method for replacement of on-board fuel. The proposed method must be efficient and guarantee robust system performance. It is here that the method based on environmental forces such as solar radiation pressure or aerodynamic drag find potential usage. Here we propose a technique to harness solar radiation pressure or aerodynamic drag for satellite control. The proposed technique has the potential to have significant cost savings as it pertains to almost negligible fuel requirement in comparison to formation control using thrusters. In other words, this technology will increase the life span of a satellite, and also decrease the overall mass of the satellite thus resulting in significant cost savings for space missions and also enable formation architectures and orbits previously thought to be too fuel-demanding. Furthermore, this approach avoids the problem of thruster plume impingement when satellites are very close in the formation.
Another aspect is the control algorithm involved in the satellite AOCS. The development 1.1. Research Motivation 3 of practically viable autonomous satellite control algorithms has not kept pace with new capabilities enabled by novel hardware designs and increased computational power and typically tends to employ linear control algorithms for its attitude and orbital control.
Here a more robust control algorithm based on nonlinear Sliding Mode Control (SMC) is proposed. Aerodynamic drag has a substantial influence on the motion of near-Earth satellites.
Even though the atmosphere in the low Earth orbits (LEO) is extremely thin, it can have a considerable impact on a satellite's orbit by retarding its orbital motion and leading to eventual decay. The atmosphere at LEO consists of diffuse molecules that have very little, if any, interaction with one another. This region, known as the free molecular flow regime, consists of molecules that no longer behave like a fluid, rather they behave as individual particles [Regan 1984]. At orbital speeds, the individual molecular impacts on the satellite can greatly affect the motion of the satellite especially over an extended period of time. When the molecules impact and bounce off the satellite surface, linear momentum is transferred through the inelastic collisions. This is proportional to the velocity difference between the molecules and the satellite [Wiesel 2003]. Since force is the rate of change of linear momentum, these individual impacts impart an acceleration to the satellite in the direction opposite to its motion. The drag experienced by the satellite is proportional to its drag coefficient, the exposed frontal area, the square of the satellite relative velocity and the density of the local atmosphere. Fig. 1.1 shows a simplified representation of the force due to aerodynamic drag experienced by a flat plate surface of the satellite as it moves in the direction of V r . ζ is the angle between the relative velocity, V R , and the outward unit normal vector n of the flat plate. Existing literature on the favorable applications of aerodynamic drag for AOCS is rather small. This may be due to the complex nature of the free molecular forces, rotation of the atmosphere, and strong dependence of density on height, season, sun-position, and local atmospheric variations. The lack of a complete understanding of the atmospheric model also adds to the limited effort in their utilization.  This is the first instance of a successful mission utilizing solar photons for propulsion. From these recent advancements, we can infer that SRP propulsion is an advancing technology that has the potential to provide cost effective, "propellantless" propulsion that enables 6 Chapter 1. Introduction longer mission lifetimes, and critical mass saving.

Rationale Behind the Use of Solar Radiation Pressure
However, use of aerodynamic drag or solar radiation pressure for autonomous control of a satellite under a wide variety of operating conditions is a challenging task. But with the combination of improved manufacturing techniques and design of a robust control algorithm, use of aerodynamic drag and solar radiation pressure for satellite formation flying or attitude control is highly promising.

Non-Affine Control Inputs
Further development of aerodynamic drag and SRP technology will require a robust control system that is able to properly direct the resultant thrust vectors. The control inputs due to SRP or aerodynamic drag are typically the orientation angles. The relationship between these angles and the resultant force is nonlinear. Review of the existing literature on the use of aerodynamic drag or SRP as a means of propulsion shows that majority of the control strategies used to date have been based on linear techniques. This is due to the manner in which the control formulation for aerodynamic drag or SRP enters the system dynamics. Consider the following system that is nonlinear in the state vector x and linear in the control input uẋ = f (x, t) + B(x, t)u(x, t) (1.1) where x ∈ R n is the state vector, f (x, t) ∈ R n represents the nonlinearity of the system, B(x, t) ∈ R n×m is the input matrix, and u ∈ R m is the control vector. This dynamic system is classified as affine with respect to the control input u, since u appears linearly in the system. A satellite controlled by a conventional propulsion system falls under this category. Now, consider the following system that is nonlinear in the state vector x and nonlinear in the control input uẋ = f (x, t) + B(x, t)g [u(x, t)] (1.2) where x ∈ R n is the state vector, f (x, t) ∈ R n represents the nonlinearity of the system, B(x, t) ∈ R n×m is the input matrix, and u ∈ R m is the control vector, and g [u(x, t)] is the nonlinear control input function. Such systems are classified as nonaffine with respect to the control input u, since the resultant input to the system is a nonlinear function of u. The proposed method of using SRP and aerodynamic drag for satellite attitude and orbit control falls into the category of nonaffine, underactuated system. To address these type of systems, nonlinear control techniques are considered in this thesis.

Nonlinear Control
The design of AOCS for a satellite is a complex task because most satellites exhibit inherently coupled nonlinear dynamics. For a nonlinear system undergoing large dynamic changes, a linear controller is likely to have poor performance, when compensating for nonlinearities in the system. Therefore, nonlinear control techniques based on global representation of satellite dynamics are needed to ensure that the control objectives are met over a large range of operating conditions. Satellites commonly operate in the presence of various disturbances (inherent nonlinearities), including environmental and non-environmental torques. The problem of disturbance rejection is important for satellites because their dynamics is substantially affected by most of these preceding nonlinear disturbances. Therefore, nonlinear control techniques must be developed to improve the performance of space systems in the presence of inherent nonlinearities. High-performance tracking is another motivation to justify the use of nonlinear control methods. Simple linear control cannot handle the dynamic demands of efficiently desired trajectories. Specifically, achieving global stabilization using nonlinear control techniques [Slotine & Li 1991] indicates that it is possible to ensure that the satellite tends to a desired equilibrium or trajectory irrespective of the initial conditions. Assuming that the parameters of the satellite are reasonably well known, linear control techniques only provide local stability because many control problems involve uncertainties in the satellite parameters, which may be constant with respect to time, or may slowly vary with respect to the changing environment in which the satellite 8 Chapter 1. Introduction is operating. A linear controller based on imprecise values of satellite parameters may degrade the performance of the system significantly by causing the satellite dynamics to deviate largely from its nominal regime. In general, it is very important to systematically explore the use of nonlinear control methods for high precision requirements. A robust adaptive control algorithm should be designed, that is rather independent of system parameters or unknown disturbances , where the uncertainties on constant or slowly-varying parameters are reduced by parameter adaptation and other sources of uncertainties and disturbances are handled by inherent robustness of the controller.

Application to Satellite Formation Flying or Attitude Control
Satellite formation flying (SFF) has been identified as an enabling technology for many future space missions [Neeck et al. 2005]. Satellite formation flying is the concept that multiple satellites can work together in a group to accomplish the objective of one larger satellite, and is expected to have several advantages over single satellite like simpler design of small satellite, faster build times, cheaper replacement creating higher redundancy and the ability to enhance and/or enable missions through increased resolution of scientific observations, and the ability to view research targets from multiple angles or at multiple times [Bauer et al. 2001]. Coordinated autonomous control of satellite formations imposes several stringent requirements on the design of AOCS subsystem because the formation objectives can be achieved efficiently only when all individual satellites are tightly controlled to respond rapidly and accurately to formation coordination commands. The development of a robust controller for SFF poses the following challenges, namely: a. Acquisition of a satellite formation: For large formations, typically all satellites are integrated as one single unit during launch and transfer to orbit. When the carrier spacecraft has reached the target location in orbit, the individual satellites sequentially leave the carrier to reach the specified formation pattern. Therefore, to produce a cohesive formation, the control system must ensure proper relative motion of multiple satellites to maintain the desired formation geometry.
b. Reconfiguration of the formation geometry: The ability of a formation to reconfigure 1.2. Literature Review 9 its relative geometry is of primary importance for many missions in order to meet changes in operational needs, or allow for multi-mission capability.
c. Formation keeping of the satellite: This is one of the most important control requirements as each satellite in the formation is required to precisely track desired formation trajectories where the external disturbances cause a drift of both the relative positions of the satellite and the formation center.
Traditional feedback controllers consider the aerodynamic drag or SRP as perturbations. Utilizing these environmental forces for control of satellites is a challenging problem as these environmental forces are unpredictable due to several factors. These factors include variations in the solar irradiance, inaccurate coefficients of the optical properties of the satellite surface materials, and variations in aerodynamic coefficients, atmospheric conditions, and satellite mass. This research focuses on the application of aerodynamic drag or SRP for satellite formation control. A system comprising of satellites attached with flaps to utilize these environmental forces is investigated and a robust control technique is developed for suitably rotating these control flaps to achieve the desired satellite performance (position or pointing) in the presence of various external disturbances and uncertainties in the system parameters.

Literature Review
Satellite attitude and orbit control systems can be classified as active control systems and passive or semi-passive control systems.

Aerodynamic Drag
We start by reviewing the existing literature on the use of aerodynamic drag for satellite attitude and orbit control. First, application of aerodynamic drag for satellite formation flying is reviewed. Existing literature on satellite attitude control using aerodynamic drag is reviewed next followed by the list of missions that employed or proposed to employ aerodynamic drag for satellite control.

Formation Flying using Aerodynamic Drag
Formation keeping using aerodynamic forces has been studied by different researchers. Leonard et al. [Leonard et al. 2004] considered differential aerodynamic drag while Matthews and Leszkiewicz [Matthews & Leszkiewicz 1988], and Steffy et al. [Steffy et al. 1988] varied ballistic coefficients of the satellite for maintaining the formation. Leonard et al. proposed the use of differential drag for orbital control, by simplifying the formation keeping problem to the simultaneous solution of a double integrator and a harmonic oscillator without accounting for J 2 effect. Aorpimai et al. [Aorpimai et al. 1999] assumed satellite equipped with aerodynamic wings and varied the angle of attack to establish the formation. The New Millennium Program Earth Observation-1 (NMP EO-1) is an example of successful application of aerodynamic forces for formation flying [Folta & Hawkins 1996].
Another research carried out by Utah State University studied the use of differential aero- dynamic drag as means of orbital control for formation flying in the Ionospheric Observation Nanosatellite Formation (ION-F) experiment [Swenson et al. 2002]. Here the ballistic coefficient was altered for orbital control and controlled permanent magnet torquing method was used for attitude control with GPS information to keep the formation. In ] a modified Clohessy-Wiltshire equation [Clohessy & Wiltshire 1960] which includes quadratic drag is studied. This modified Clohessy-Wiltshire equation can find application in station keeping and formation flying. In [Kumar et al. 2007b] use of differential drag as a means for nanosatellite formation control is studied and a simple PI control law is derived to adjust the cross sectional area of the satellites to maintain an along track formation. Orbital maneuvering using aerodynamic forces was first proposed by London [London 1962] in 1962. Since then, many researchers have worked on this subject; an excellent survey of the past work was presented by Miele [Miele 1996]. Rao [Rao et al. 2002] considered optimal aero-assisted orbit transfers from low Earth orbit to geostationary orbit with a large plane change. Lohar et al. [Lohar et al. 1998] examined orbit transfer between coplanar elliptical orbits using aero-cruise. The use of aerodynamic force for orbit control of a single satellite has been studied by Palmerini [Palmerini et al. 2005] and Humi . Formation keeping and rendezvous of multiple satellites have also been investigated , Humi & Carter 2001. In more recent studies, Bevilacqua et al. [Bevilacqua & Romano 2008, Bevilacqua et al. 2009] examine multiple satellite rendezvous using differential aerodynamic drag by considering the linearized differential equations by Schweighart and Sedwick [Schweighart & Sedwick 2002].  [Ravindran & Hughes 1972] considered performance optimization of an aerodynamic controller by minimizing a performance index weighted equally with respect to the attitude errors and control surface move-

12
Chapter 1. Introduction ments. Successful application of aerodynamic forces for pitch control of COSMOS-149 was reported by Sarychev [Sarychev 1968]. An optimal aerodynamic stabilization of near-Earth satellites was investigated by Pande and Venkatachalam [Pande & Venkatachalam 1979].
Recently Chen Y. H et al. [Chen et al. 2000] has proposed the combined use of aerodynamic and gravity gradient torque for three axis stabilization of a microsatellite using gravity gradient boom and vertical tail stabilizer. Propellant-less methods have their own limitations and to overcome these limitations, methods using combinations of various environmental forces have been envisaged, such as aerodynamic and gravity (aero-gravity assist), aerodynamic and magnetic forces. Kumar et al. [Kumar et al. 1995] examined passive aerodynamic stabilization and passive magnetic hysteresis damping of attitude rates while Chen et al. [Chen et al. 2000] considered aerodynamic torque for yaw stabilization and the gravity gradient torque to stabilize pitch and roll attitudes. Magnetic torquers with a gravity-gradient boom have been applied by Wisniewski [Wisniewski 1997]. Psiaki [Psiaki 2004] considered passive aerodynamic drag torques for stabilization of pitch and yaw motions, and roll motion was controlled by active magnetic torquing of a shuttle-cock type satellite. -149 [NSSDC 1967], COSMOS-320 [NSSDC 1970], JC2Sat [Kumar et al. 2007a, Mierlo 2009] and New Millennium Program Earth Observation-1 (NMP EO-1) [Folta et al. 2002], are some of the examples where control based on aerodynamic drag has been applied for satellite attitude control and/or formation flying. In the JC2Sat mission, along track formation control is proposed and the relative states will be controlled by varying the pitch attitude of the satellites in the formation. Details of some of the missions that make use of aerodynamic drag as a means of passive/semi-passive control is provided in

Solar Radiation Pressure
In this section, we review at the existing literature on the use of solar radiation pressure for satellite formation flying, attitude control and also list some of the missions that used or proposed to use solar radiation pressure for satellite control.

Formation Flying using Solar Radiation Pressure
Application of SRP for formation maintenance has been investigated by several researchers.
Wang [Wang 2001] and Williams and Wang [Williams & Wang 2002] considered a satellite with a solar wing and it was shown that a solar wing of the correct area can prevent the secular out-of-plane growth in a low-Earth orbit formation that is caused by differential nodal drift. Kumar et al. [Kumar et al. 2004a] and Fourcade [Fourcade 2005] found that SRP is successful in maintaining the desired formation. However, these studies assumed a simple SRP model and the satellite attitude motion was not considered. Another recent research studied SRP based tetrahedron satellite formation. This research studied the use of SRP to deploy and stabilize a three-dimensional satellite formation for a Heliocentric as well as High Earth Orbit [Smirnov et al. 2007], with the diameter of the formation and the stabilization time dependent on the properties of the solar sail. Application of SRP for station-keeping of a geostationary satellite was studied by Black et al. [Black et al. 1968] and Circi [Circi 2005]. Black et al. [Black et al. 1968] considered east-west station-keeping problem where the use of a very small solar sail and a simple thrust strategy were pro-posed, while Circi [Circi 2005] examined both east-west and north-south station-keeping problems. The study concluded that use of solar sail was more beneficial when the total satellite mass was reduced, and it was found that for a 300 kg satellite, the required side for a square solar sail is 44 m. Several kinds of missions with solar sail as a propulsion system for orbit transfer have also been proposed: Salvail and Stuiver [Salvail & Stuiver 1995] investigated the problem of transfer to the moon from a geosynchronous orbit, Leipold et al. [Leipold et al. 1996] [Kumar et al. 2004b] carried out the preliminary study wherein the satellite equipped with two sets of wings -aerodynamic and SRP wingswere successfully able to control in-plane relative position errors using aerodynamic wings while the out-of-plane relative position errors were controlled by SRP wings. Solar Blade Heliogyro Nanosatellite [Blomquist 1999], ODISSEE [Leipold et al. 1999 [Modi 1995] . The attitude control of the satellite has been accomplished by translatory motions of single or several control surfaces relative to the satellite body [Kumar 1988] or by rotating the control surfaces  ]. Some missions have also been flown to verify these concepts. The Mariner IV mission employed solar vanes for achieving passive sun pointing attitude [Scull 1969]. The European Space Agency conducted experiments in which the attitude of the geostationary communication satellite OTS-2 was controlled by rotating solar panels [Renner 1979]. The SRP control torque thus can be utilized to stabilize librational dynamics of a satellite with a desired degree of accuracy. Circi [Circi 2007] recently proposed a satellite attitude controller that combines passive gravity-gradient torque with solar pressure torque for three-axis stabilization of a micro-satellite. A dumbbell satellite configuration was chosen to increase the gravity-gradient torque and three-axis stabilization was performed with the help of small solar sails. Another recent publication was on the use of SRP for satellite pitch attitude control based on a non-linear sliding mode control [Patel et al. 2008, Patel et al. 2009]. Lievre in [Lievre 1986] proposed an attitude control system for geostationary satellites making use of solar sails and demonstrated its advantage as compared to a thruster based or magnetic torquer based control. In [Sidi 1997], Sidi makes reference to Roll-Yaw attitude control with solar torques. The geometry consists of solar flaps attached to the sides of the solar panels in order to achieve a solar torque about a line in the orbit plane and perpendicular to the sun direction. A control mechanism for satellite attitude control is also derived. A comparative study of two solar sail attitude control implementations is carried out in [Bladt & Lawrence 2005]. One implementation employs four articulated reflective vanes located at the periphery of the sail assembly to generate control torque about all three axes. The second attitude control configuration uses mass on a gimballed boom to alter the center of mass location relative to the center of pressure producing roll and pitch torque. It also uses a pair of articulated control vanes for yaw control. The combination of SRP and aerodynamic forces was mainly focused on attitude control problem [Modi & Pande 1974b, Modi & Pande 1974a, Kumar et al. 2004b]. Modi and Pande [Modi & Pande 1974a] have stated an altitude of 800 km as the switch-over point from control based on aerodynamic forces to control using SRP.

Problem Statement
The previous section listed the existing research scenario on the application of SRP and aerodynamic drag for satellite formation flying and attitude control. It is clearly evident that research has largely been confined to the use of either SRP or aerodynamic drag for satellite attitude control or formation flying with simpler linear control schemes. Also the problem of multiple satellite formation flying using either SRP or aerodynamic drag is yet to be studied. Hence this thesis focuses on the development of a robust nonlinear control technique of application of SRP or aerodynamic drag for satellite formation and attitude control. Utilizing these environmental forces for desired control of satellites is a challenging problem as these disturbing forces are unpredictable in nature. Hence the control scheme to be used should be robust and also adaptive in nature. The problems to be addressed can be classified as: [PROB1] Nonlinear System Equations: The inherent nonlinearity in the dynamics of satellite systems is one of the challenges to be overcome for successfully developing a robust system. It is not always the case that a linear model can be built up that sufficiently and accurately describes the dynamic behavior of the system for a wide range of operating conditions.
[PROB2] Non-Affine Control Input: An additional complexity involved in the nonlinear equation of motion used to describe these systems is the means by which the control input enters the equation of motion. A system where the control inputs do not appear linearly in the plant falls in the category of non-affine control. For SRP and aerodynamic drag based propulsion, the control inputs are the angles that define the orientation of the wings/flaps of the satellite. The control input enters the equations [PROB3] Model uncertainties and external disturbances: In space applications, controllers designed based on nonlinear spacecraft models can also be imprecise due to unknown values of some physical parameters, disturbance models, etc. The resulting mismatch between the model and the real system is referred to as model uncertainty.
The problem of disturbance rejection is particularly pronounced for spacecraft that operate in the altitude ranges where their dynamics is affected by various environmental and non-environmental disturbances.
[PROB4] Orbit selection and formation design. It is well known that SRP is the dominant environmental force in the high Earth orbits where as atmospheric drag is dominant in the low earth orbits. Hence orbit selection and formation trajectory generation becomes critical so that both the environmental forces can be utilized to its fullest possible manner.

Research Objectives
To address the problems presented, this dissertation concentrates on the development of nonlinear adaptive control laws for multiple satellite formation flying and attitude control.
From practical viewpoint, the main objective is to design control algorithms that are globally stable and are able to achieve high precision pointing, fast slewing, and large maneuvers in the presence of external disturbances, and model uncertainties. The main objectives are identified as: [OBJ1] Control System Design: In the proposed system, precise system modeling is not always possible due to the lack of exact knowledge of all the parameters associated with solar radiation pressure and aerodynamic drag. Hence a robust closed loop feedback control strategy based on adaptive control theory is developed ( Fig.1.4). The adaptive control law reconfigures the control algorithm using the adaptive estimates of the unknown parameters. The objective of this control

Research Contributions
Acknowledging the advantages of using a passive AOCS, the core of this dissertation con- The feasibility of using SRP for formation control in a geocentric orbit is investigated in this section. Here a nonlinear adaptive control algorithm based on higher order sliding mode control technique is developed. The system is essentially underactuated, hence an area control input is introduced. This additional input transforms the system to a fully actuated system. The performance of the proposed system in the presence of external disturbance, and parameter uncertainties is examined. Feasibility of multiple satellite formation flying using solar radiation pressure is also validated using numerical simulation.
3. Satellite attitude control using aerodynamic drag (Chapter 5)) Satellite attitude stabilization in the local vertical local horizontal (LVLH) frame using aerodynamic drag is examined in this chapter. Existing literature proposes only pitch and yaw control using aerodynamic drag. Roll stabilization is usually achieved by a separate actuator. By developing the control algorithm based on nonlinear theory, here we are able to achieve complete three axis attitude stabilization using aerodynamic drag. The proposed attitude control system demonstrated robust performance in the presence of uncertainties and disturbances.
4. Satellite attitude control using solar radiation pressure (Chapter 6) Satellite attitude stabilization using solar radiation pressure is examined. An adaptive control algorithm is developed that is able to achieve complete three axis attitude stabilization. To date, variety of control methods using SRP has been proposed for attitude control, but it has been confined to the the control of satellite pitch motion.
Here the efficacy of using SRP for three axis attitude control in the presence of uncertainties and perturbations is verified.

SFF System Model
We start the mathematical formulation of the SFF system by assuming that the leader and follower satellites are point masses under the Earth's central gravitational force of attraction 24 Chapter 2. Satellite Formation Dynamics and Control Methodology ( Fig. 2.1). The leader satellite is in an elliptical planar trajectory with the Earth's center at one of its foci and a follower satellite is moving in a desired relative trajectory about the leader satellite. The motion of the follower satellite is described relative to the leader satellite using a relative local vertical local horizontal (LVLH) frame S − xyz fixed at the center of the leader satellite with the x-axis pointing along the local vertical, the z-axis taken along normal to the orbital plane, and the y-axis representing the third axis of the

Equations of Motion
The position vector of the leader satellite in the S − xyz is given as and the position vector of the follower satellite in S − xyz frame is given as Now let us define ρ = (xî + yĵ + zk) and r f = (r l + x)î + yĵ + zk. The velocity of the leader satellite is given as˙ The velocity of the follower satellite is given aṡ r f =˙ r l +˙ ρ =ṙ lî + r lθĵ + ẋî +ẏĵ +żk +θk × xî + yĵ + zk (2.4) or˙ r f = ṙ l +ẋ −θy î + r lθ +ẏ +θx ĵ +żk (2.5) Now we use the Lagrange method of deriving the equations of motions, The kinetic energy of the leader satellite is given as, and the potential energy is given as Now taking the partial derivatives of L, as follows, These form the equations of motion of the leader satellite. Next we derive the relative equations of motion of the follower satellite in similar manner. First the kinetic energy is 26 Chapter 2. Satellite Formation Dynamics and Control Methodology calculated and is given as and the potential energy is given as Now taking the partial derivatives of L with respect to x, y, z,ẋ,ẏ, andż, and performing some manipulations we get the relative equations of the follower satellite. The complete nonlinear relative equations of motion of the leader and follower satellites can now be written along with the force and disturbance terms as follows, where r f = [(r l + x) 2 + y 2 + z 2 ] 1/2 is the position of the follower satellite, f dtj is the net relative perturbations acting on the SFF system, and f j are the components of the control input vector, for j = x, y, z. This mathematical model for SFF is also referred to as the "true model" [Vaddi 2003].

Hill-Clohessy-Wiltshire Equations
Satellite formation flying is principally concerned with the relative motion of the follower satellite with respect to the leader, most commonly expressed in the rotating Hill reference frame. Although the actual relative dynamics of the follower satellite are nonlinear as well as nonperiodic, a set of linearized ordinary differential equations which approximate the full dynamics and have periodic solutions can be derived. These are commonly referred 2.2. Desired Formation Geometry 27 to as the Hill-Clohessy-Wiltshire (HCW) equations. The homogeneous form of the HCW equations is given as [Hill 1878, Clohessy & Wiltshire 1960]: The simplifying assumptions made during the derivation affects the accuracy of the HCW equations. The assumptions include no external perturbations, the leader is assumed to be in a circular orbit around the Earth, and also require that the relative distance between the leader and the follower is much smaller than that of the orbital radius of the leader.
However, the HCW equations are useful in control algorithm design and their periodic solutions help in deriving basic reference trajectories for the controller to track during near circular formation flying.  [Sabol et al. 2001].

Desired Formation Geometry
The mathematical formulation of the these two desired trajectories are as follows: 1. Circular Formation: In this formation, the leader and the follower satellite maintain a constant separation from each other in three-dimensional space and the formation is mathematically defined as x 2 + y 2 + z 2 = r 2 d . The relative motion in the radial/alongtrack (x/y) plane is fixed in eccentricity [Sabol et al. 2001]. The equations of desired circular trajectory are given as follows, where r d is the formation size for circular and projected circular formations, φ is the in-plane phase angle between the leader and the follower satellites, andθ is the mean angular velocity and equals to µ e /a 3 c (µ e is the gravitational parameter of the Earth; a c is the semi-major axis of the leader satellite) [Yan et al. 2009].

Projected Circular Formation:
In this formation, the leader and the follower satellite maintains a fixed relative distance when the formation is projected onto the alongtrack/cross-track (y − z) plane. This configuration finds applications for ground observing Synthetic Aperture Radar (SAR) missions. Since most SAR applications target objects on the surface of the Earth,it is desirable to achieve a formation plane 2.2. Desired Formation Geometry 29 perpendicular to the radial vector in order to have a field of view that has its target on the Earth. Fig. 2.3 represents the projected circular formation scheme. Mathematically it is defined as y 2 + z 2 = r 2 d .

External Perturbations
The disturbances terms in Eqs. (2.15)-(3.5),f dtj for j = x, y, z constitute time-varying quantities attributed to gravitational field, solar radiation pressure, aerodynamic drag and third body perturbations. For a satellite in low Earth orbit, the second zonal harmonic (J 2 ) of the oblate Earth's potential distribution is by far the most dominant perturbation force followed by aerodynamic drag and solar radiation pressure. The disturbance accelerations due to J 2 are at least an order of magnitude larger than the other perturbations such as third-body gravitational effects, drag due to residual atmosphere, and solar radiation pressure [Sabol et al. 2001]. The effects of the Earth's oblateness on the formation geometry are: 1) precession of the orbital plane leading to differential changes in the right ascension of the ascending node, and 2) secular changes in the argument of perigee and mean anomaly due to rotation of the line of apsides in the orbital plane. Next, we derive the relative acceleration components due to J 2 in the LVLH frame. The leader satellite ( r l ) and follower satellite ( r f ) positions in the LVLH frame are given by r l = r l 0 0 T and r f = r l + x y z T (2.21) and the transformation matrix, T IB required to convert these positions from the relative frame to the Earth-centered inertial frame of reference is obtained by using the following rotation sequence Using the transformation matrix, T IB , along with Eq. (2.21), we obtain the positions of the leader satellite ( R l = [X l , Y l , Z l ] T ) and the follower satellite in the ECI frame. The perturbations due to J 2 in the ECI frame for 2.3. External Perturbations 31 the leader and follower satellites are given by [Schaub & Junkins 2003] where µ e is the Earth's gravitational parameter, R e is the radius of the Earth, and J 2 is second zonal gravitational coefficient, J 2 = 1.08263 × 10 −3 . The full effects of differential J 2 can be added to the nonlinear relative mathematical model by transforming the disturbance forces from the Earth-centered inertial frame to the relative reference frame. Therefore, the external disturbance forces in Eqs. (2.15)-(3.5) is given by

Aerodynamic Drag Model
Next we study the aerodynamic drag models. Aerodynamic drag has substantial influence on the motion of the near-Earth satellites. There are two extremes when discussing how   Reality falls somewhere in between specular and diffuse reflection. Two of the commonly used aerodynamic models are the free-molecular aerodynamic force model and a more simplified aerodynamic force model

Simplified Aerodynamic Force Model
First we consider the simplified aerodynamic force model. It is expressed in terms of the drag, i.e. the force parallel to the satellite velocity direction. The vector equation for computing the drag force is given as ], where C D is the drag coefficient, ρ is the atmospheric density, and A is the projected area of the satellite normal to V rel . V rel is the relative velocity of the satellite with respect to the atmosphere. The relative velocity can be represented by where ω E is the Earth's rotation rate,V rel = V rel V rel is the velocity vector of the surface element with respect to the atmosphere, and i is the orbit inclination. The drag force, atmosphere ineffective, so the density distribution is governed by the relation: where y is an altitude above a reference surface (Earth), y 0 is the altitude at geocentric distance r 0 which has density ρ 0 , and H is the scale height given by where T is the local atmospheric temperature, M is the molecular mass, and R 0 is the universal gas constant. The density model given by Eq. (2.28) is also known as the exponential density model. The density scale height H governs the rate of decrease of density and has a firm temperature dependence. Most uncertain parameter in the aerodynamic force equation is the atmospheric density. Various density models are currently available, but the two of the most widely used density models are the exponential density model (Eq. 2.28) and the NRLMSISE-00(United States Naval Research Laboratory Mass Spectrometer and Incoherent Scatter Radar Exosphere) model [Picone et al. 2000]. NRLMSISE-00 model is one of the widely used empirically derived aerodynamic drag force models and considered to be more accurate especially at higher altitudes. The model utilizes atmospheric composition data from instrumented satellites and temperatures from ground-based radars as well as extensive set of parametric equations, of which the parameters have been determined using large historical data sets and also depend on latitude, local time, day of the year, solar activity index (F 10.7 ) and geomagnetic activity index (A p ) [Picone et al. 2000]. Figure ( 2.6) shows the variations in the density profiles of the exponential and NRLMSISE-00 model.

Free Molecular Aerodynamic Force Model
The second aerodynamic model is the free molecular aerodynamic force model. Based on the free-molecular force model, the force on a flat surface of area A is given by the vector equation ], where p is the total pressure, τ is the shearing stress,V rel = V rel V rel is the velocity vector of the surface element with respect to the atmosphere andn is the outward pointing unit normal vector. The total pressure p and the shearing stress τ are given by [Gombosi 1994] where σ n and σ t are the normal and tangential accommodation coefficients, T s is the absolute temperature of the satellite surface, T a is the atmospheric temperature, q ∞ is the dynamic pressure given by q ∞ = 1 2 ρV 2 rel , and erf(x) is the error function defined by erf(x) = 2 √ π x 0 e −y 2 dy and s is the air speed nondimensionalized by the mean molecular speed of the atmosphere.
As mentioned earlier, the satellite orbit contracts under the influence of aerodynamic 36 Chapter 2. Satellite Formation Dynamics and Control Methodology drag. The effect on the satellite altitude is less in the higher orbits, but the orbit is appreciably affected by drag at lower orbits. Figure (2.7) illustrates the orbital decay of a 10 kg satellite at an initial orbit of 500 km and with an exposed surface are of 1 m 2 . The Solar Radio Flux (F 10.7 ) index is set to 120 and the Geomagnetic Index (A p ) has a value of 3. It can be observed that initially there is a linear decay rate, but at lower orbits the decay rate demonstrates exponential increase. Figure   In this dissertation, we propose the use of differential force to maintain the SFF system.
In the case of aerodynamic drag, this differential force is created by maneuvering the drag plates attached to the leader and follower satellites in the formation. The differential force is given as ( F Df − F Dl ), where F Df and F Dl represents the force due to aerodynamic drag on the follower and leader satellites respectively. Figure  In addition to the aerodynamic drag, which acts in a direction opposite to the satellite's motion,there is also an aerodynamic lift force acting in a direction perpendicular to the satellite's motion. The value of the lift force depends on the orientation of the satellite, geometry and angle of attack. According to Karr [Karr & Cleland 1975], for an energy accommodation factor of 0.9, the value of the lift to drag ratio for a flat plate is less than 0.1 and hence the effects of lift is neglected.

Solar Radiation Pressure Model
The solar radiation pressure force is the largest perturbation acting on high Earth orbit satellites next to the gravitational perturbations from the Earth, Sun, and Moon, and the total force exerted due to SRP can be written as where f r is the force due to reflection, f a is the force due to absorption, and f e is the force due to emission due to re-radiation. From Fig. (2.10) it can be observed that, the SRP force acting on a flat surface in terms of normal and transverse components is given by  is the angle between the force vector and the flap normal and is given by Two material property parameters which totally account for the effects of absorption, specular and diffuse reflection and thermal emission on thrust and moments are, γ = ρ r ρ s , the fraction of the incident flux that is specular reflected and β given by Chapter 2. Satellite Formation Dynamics and Control Methodology which is the fraction of the incident flux that contributes to thrust from diffuse reflection and thermal re-emission. Based on JPL calculations for comet Halley rendezvous study [Wright 1992], the optical parameters for an ideal and non-ideal solar flap are shown in Table 2.1. Assuming that the surface is ideal, the force due to SRP acting on an ideal Ideal 1 1 0 0 2/3 2/3 Non-Ideal 0.88 0.94 0.05 0.55 0.79 0.55 reflecting surface simplifies to a force acting normal to the flap given as, (2.11) shows the error in the relative position of the follower satellite under the influence of the differential SRP in a projected circular formation of size 10 km.

Control Methodology
It is evident from Figs. (2.9) and (2.11), that the formation system is adversely affected by differential aerodynamic drag and SRP. Hence, a suitable control technique is required to make use of these differential forces to control the relative states of the formation. In this thesis, we consider sliding mode control (SMC) technique. SMC is a subset of the variable structure control (VSC) technique. Here a control algorithm is derived such that it is deliberately altered during the control process based on predefined rules which depend on the state of the system. For the purpose of illustration consider the following linear time-invariant system in state-space form ( [Edwards & Spurgeon 1998b]): where X ∈ R 2 = [X 1 , X 2 ] T is the state vector, and U is a scalar control input. Next, define a linear sliding surface given by where p is a positive design scalar. Consider the VSC control law given by where η is a positive design scalar that has a direct effect on the rate at which the sliding surface can be reached. The expression given by Eq. −p. This line divides the phase plane into four regions given by I : X 1 > 0, S > 0 and II : X 1 > 0, S < 0 III : X 1 < 0, S < 0 and IV : For a given initial state, the control input U drives the system trajectory towards the line Consequently, when p |X 2 | < η the system trajectories on either side of the line S = 0 point towards the line. This is demonstrated in Fig. 2.12 which shows the phase portraits intercepting the line S = 0 from different initial conditions. The condition given by SṠ < 0 is referred to as the reachability condition. The trajectory of the system confined to the sliding surface, S = 0, satisfies the differential equation obtained from rearranging the This represents a first-order decay and the trajectories of the system will 'slide' along the line S = 0 to the origin. Such dynamical behavior is described as an ideal sliding mode and the line S = X 2 + p X 1 is termed the sliding surface. During sliding motion, lower-order dynamics dominate the behavior of the system independent of the control. Therefore, the control action only ensures that the sliding surface is reached and the conditions given by Eq. (2.43) are satisfied. The dynamic performance of the system is governed by the choice of the parameter p in the sliding surface. The finite-time (t r ) convergence of the system trajectories to the sliding surface (reaching time) can be shown by recalling that Dividing by |S| and integrating both sides of Eq. (2.45) between 0 and t r gives Since |S(t r )| = 0, the reaching time is given by formation was carried out. The main challenge in such SFF mission involves controlling the relative positions of the satellites in the presence of external disturbances i.e., gravitational perturbation (including the Earth's oblateness (J 2 -effect)), aerodynamic drag, solar radiation pressure, and third body perturbations. The effect of these external disturbances were analyzed in Chapter 2. In this chapter, a methodology to achieve coordinated multiple satellite formation flying using differential aerodynamic drag is proposed. The satellite's relative orbit is controlled by rotating the drag plates attached to the satellites and thereby varying the aerodynamic drag experienced by each satellite. This in turn generates relative differential accelerations among the satellites in the formation. A control algorithm based on sliding mode control (SMC) technique is used to develop the steering law for the drag plates. The proposed method can only be used in low Earth orbit (LEO) missions.
The chapter is organized as follows: Section 3.1 introduces the nonlinear mathematical model of the SFF system. Control laws along with the observer design are described in

46
Chapter 3. Satellite Formation Control Using Aerodynamic Drag

SFF Model and Systems Equations of Motion
The investigation is initiated by formulating the nonlinear equations of motion of the SFF system.

Coordinate Frames and Equations of Motion
The SFF system comprises a leader satellite and a follower satellite as shown in Fig.2.1.
To initiate the study, we recall the assumptions from Chapter 2, i.e., the leader satellite is in a reference orbit that is assumed to be planer and defined by a radial distance from the center of the Earth and a true anomaly, and the follower satellite is moving in a relative trajectory about the leader satellite. The orientation of the relative frame that is fixed at the leader satellite is such that the x -axis points along the local vertical, the z -axis is taken along normal to the orbital plane, and the y-axis represents the third axis of this right-handed frame. The equations of motion of the follower satellite reproduced here. The orbital equations of motion for the leader satellite is given by Eqs.
where r l is the radial distance from the center of the Earth, and r = (r l + x) 2 + y 2 + z 2 1 2 .
f j and f dtj denote the control and disturbance forces along the j -axis, j= x,y,z, respectively. However, in this study, control force is available only in the y-axis. This force, f y , is created by the drag plates on the satellites while the disturbance force, f dtj , is attributed 3.1. SFF Model and Systems Equations of Motion 47 to gravitational perturbation, solar radiation pressure, aerodynamic drag, third body perturbations and the Earth's oblateness effect. Since aerodynamic drag is considered as the control input, only disturbing effect from that will be the variation in atmospheric density.

Aerodynamic Drag Model
Follower Satellite Leader Satelite where C D , is the drag coefficient, ρ, is the atmospheric density, and A is the projected area of the satellite normal to V rel . V rel , is the relative velocity of the satellite with respect to the atmosphere andV rel is a unit vector in the direction of the satellite's velocity. The drag force, F D , thus acts in the direction opposite to the relative velocity, V rel . The coefficient C D is dependent on the geometry of the satellite and varies with altitude, and the angle of attack. The differential drag created by the drag plates attached to the satellites in the formation can be formulated using Eq. (3.6), and the force (f y ) in Eq.(3.4) acting on the follower satellite is given by where ρ j , V jrel , A j , α j , for j = f, l correspond to the local atmospheric density, the relative velocity of the satellite with the local atmosphere, the area of the drag plate for the follower and leader satellites, respectively. The drag plates are assumed to be of negligible thickness. A pictorial representation of the system is shown in Fig.3

Design of Control Law
The control objective here is to determine the orientation of the drag plates (α f ,α l ) which will drive the follower satellite to the desired formation trajectory with respect to the leader satellite in the presence of external disturbances and initial deployment errors. We develop a control law based on SMC technique. Considering the mathematical formulation of the leader follower SFF model and aerodynamic drag model presented in the previous section, we define the relative state vector and the desired relative trajectory as, respectively. The performance measure is defined as the tracking error, e (t) ∈ R 4 , The objective is to develop a control algorithm for Eqs.(3.3) and (3.4) in the SFF system such that the control force f y drives the relative states of the system to the desired relative trajectories as t → ∞, and thereby the tracking errors, Eq.(3.8), converges to zero.
In order to simplify the control law design, we consider α f = α 0 + δα/2 and α l = α 0 − δα/2, where α 0 is the initial orientation of the drag plates. Assuming the first order approximation for the differential drag plate's orientation δα, and considering the parameters ρ j , V jrel , A j , for j = f, l are same for both the leader and follower satellites (ρ j = ρ, , we have

Design of Control Law 49
Please note that these assumptions on the aerodynamic drag system parameters are applied only during the control law formulation but not considered during the numerical simulations.
Next we present the design procedure to implement SMC for changing δα to achieve satellite formation control. The overall design can be divided into two main steps. (i) A stable sliding surface/manifold is designed such that this surface will ensure that the follower satellite dynamics precisely tracks the desired trajectory once the system is restricted to that manifold. (ii) A feedback control algorithm that can drive the relative states to the sliding manifold in finite time and maintain it on the manifold is derived. The sliding manifold should be designed such that not only the closed loop system stability is guaranteed in the presence of unmatched perturbations, but also the desired dynamic behaviour should be exhibited when the dynamics of the system is on the sliding manifold.

Design of Sliding Manifold
The sliding manifold is designed using the linear representation of the nonlinear system equations of motion Eqs.(3.3) and (3.4),(i.e., Hill's Equations [Hill 1878] or the Clohessy-Wiltshire equations (HCW) [Clohessy & Wiltshire 1960]), that can be represented in terms of new coordinates p 1 ∈ R 3×1 and p 2 ∈ R as follows: y,ẋ], is the state vector associated with the states that are not under explicit control authority called the unactuated states, p 2 =ẏ, is the system state under explicit control authority called the actuated state, U = δα, and Based on Eqs.(3.11) and (3.12), the following error dynamics can be formulated: By utilizing the coupling between the actuated and the unactuated states, the sliding surface, S can be defined as a function of the tracking errors and the desired states as follows: S = e 1 ∈ R 3×1 , e 2 ∈ R : e 2 + Ke 1 = 0 (3.14) where K ∈ R 1×3 is a weighting matrix. The sliding motion starts as soon as the system reaches the sliding surface, S = 0∀t > t r , where t r is the reaching time and the system dynamics is governed by the following relation

Control Formulation
Here an adaptive control algorithm that will manoeuvre the system trajectories to the sliding manifold is designed. In the SFF system controlled by aerodynamic drag, parameter uncertainties such as aerodynamic density, relative velocity and drag coefficient can adversely affect the performance of the system if not appropriately addressed. In the adaptive control methodology the unknown or slowly time-varying parameters are estimated online and these estimated quantities are used in place of the uncertain parameters in the feedback control law. In order to design control algorithm, the equations of motion that are of interest in this study Eqs.(3.3) and (3.4) are rewritten in the form: Here the nonlinear terms in the equations of motion have been lumped into E (X), A represents the linear part of the system, C is the control matrix, where B is as shown in Eq.(3.11), U is the control input, and F dt the differential perturbation forces. The control law is derived such that, the system response during the transient phase is improved and the closed loop system is maintained on the sliding surface at all time. Next we make use 52 Chapter 3. Satellite Formation Control Using Aerodynamic Drag of the estimates on the upper-bound of the uncertainties and perturbation on the system to design the control algorithm. It is assumed that the nonlinear component, E (X), in the equations of motion are bounded and the choice of the desired reference trajectories, X d , impose the following constraint given by, where d 1 > 0 and d 2 is the Lipschitz constant of the nonlinear vector field associated with E (X). This information is found by varying the formation disc size from 1 km to 150 km and choosing the maximum value, i.e., max ( E (X) / X ) [Vaddi 2003]. All the external perturbations such as J 2 , solar perturbations, and magnetic forces, are assumed to be bounded as well and included in the following chosen uncertainty bound (with scalar Next the adaptive control algorithm is derived. Here the parameter B given in Eq.(3.11), is assumed as an unknown quantity and is hence estimated online and then used in the control algorithm. The sliding manifold derived in Eq.(3.14) is rewritten as Now consider the Lyapunov candidate function defined as follows, where V > 0 and γ is a positive constant. Taking the derivative of V along its trajectories Taking the parameter estimation error asB =B −B (estimate -true value), and expanding Eq.(3.25), we havė Rearranging the preceding Eq.(3.26) yields, Now the bounds on the uncertainties and nonlinear components, given by Eqs.(3.18) and (3.19), can be combined together and written as Now in order to makeV negative, choose the adaptive law aṡ Chapter 3. Satellite Formation Control Using Aerodynamic Drag Also let where η is a positive constant and η » sup L . Substituting Eqs.(3.31) and (3.30) into Thus,V is negative and hence the reaching condition is satisfied. Using Theorem-3 of LaSalle's principle [LaSalle 1960] it is proved that the system is completely stable and hence the system states converge onto the sliding surface which is the invariant set here.
Next, Eq.(3.31) is rearranged to derive the control law as, It is to be mentioned that the system states are driven to the sliding surface by the discontinuous part in the control law (i.e.,ηsgn (S)). The orientation of the drag plates (α f ,α l ) can now be determined and used in computing the differential drag, Eq.(3.7).This is then used in the numerical simulations, Eq.(3.4). Though sliding mode control is a robust control technique, chattering is a major drawback because of the discontinuous control nature.
This problem, however, can be solved by replacing the nonlinear signum function with a suitable boundary layer of nonlinear saturation function. Using this approach, the phase trajectory oscillates within the sliding surface without switching controller action. Hence, the control law, Eq.(3.33) can be rewritten incorporating the boundary layer function as where τ , the width of the boundary layer is a positive constant.

Dynamic System Stability Analysis
Next we examine the robustness of the system during the sliding regime. It is well known that the system states are insensitive to matched disturbances, i.e., once on the sliding surface, the uncertainties and perturbations do not affect the explicitly actuated states. But  [Shyu et al. 1998] together with the procedure detailed in Spurgeon et.al. [Spurgeon & Davies 1993], [Edwards & Spurgeon 1998a] for the reduced order systems with unmatched disturbances. Consider the error dynamics of the SFF system on the sliding surface referring to Eq.(3.13) in conjunction with uncertainties or disturbances,ė where f um (X) ∈ R 3 and f ma (X) ∈ R are the bounded terms representing unmatched and matched uncertainties or disturbances in the system, respectively. The unmatched uncertainties are assumed to be bounded f um (X) ≤ 1 + d 5 X . The equation representing the error dynamics on the sliding surface, obtained by substituting S = 0 in Eq.(3.35), is given asė To verify the stability of the system in the presence of unmatched uncertainties we propose the following theorem.
Theorem 3.1: For a reduced order system with uncertainty, described bẏ wheref (x 1 ) is the unmatched uncertainty and which satisfies the uniform Lipschitz con- 1 , the uncertain system on the sliding surface S = 0 is asymptotically stable. Here range of k is given by 0 < k < 0.5λ min Q / P with P and Q ∈ R (n−m)×(n×m) .P and Q are symmetric, positive-definite matrices satisfying the Lyapunov equation (A 11 ) T P + P (A 11 ) = −Q.
Proof : Consider a candidate Lyapunov functioṅ Now by using Eq.(3.39) in conjugation with the Rayleigh principle, we have HenceV is negative. This shows that the system is insensitive to the unmatched disturbances and all the states reach the sliding surface and remain bounded.

Observer Design
From previous sections on control algorithm design, it is clear that state variables are required in the feedback control formulation. These can be measured or estimated for the tracking control. A Carrier-phase Differential GPS (CD Global Positioning System) receiver could be used to obtain the relative positions of the satellites. Here it is considered that only the relative position vector is readily available. We either use numerical differentiation to obtain the velocities or use an observer to estimate the states. Here an observer is designed for estimating the velocities to be used in the feedback control law. The observer is based on the modified second order sliding mode super twisting algorithm [Davila et al. 2005], Sirdi et al. 2008]. This particular sliding mode observer formulation is used due to its finite time convergence to the true value of the observed states, robustness with respect to uncertainties and also requires smaller gain values and is more accurate in terms of estimation [Davila et al. 2005]. Considering the state variables x 1 = x, x 2 =ẋ, y 1 = y, y 2 =ẏ and z 1 = z, the nonlinear equations of motion of the system given by Eq.(3.17) can 3.2. Design of Control Law 57 be rewritten without considering any external disturbance terms as follows.
. Assuming that the system states and the control input, f y , are bounded then the existence of constants f + 1 and f + 2 are ensured such that where p is a constant in the range 0 < p < 1. It has been proved in Davila [Davila et al. 2005] that if the observer parameters β j and δ j , j = 1,2 are chosen in accordance with Eq.(3.46) and if the condition in Eqs.(3.45) holds for the set of the system equations of motion, then the variables of the observer will converge in finite time to the actual system states, i.e.,(x 1 ,x 2 ,ŷ 1 ,ŷ 2 ) → (x 1 , x 2 , y 1 , y 2 ). The separation principle is also satisfied, thereby allowing the design of the observer and the control law separately.

Performance Evaluation
In order to examine the effectiveness and performance of the proposed formation control strategy, the detailed response is numerically simulated using the set of governing equations The SFF systems parameters and the orbital parameters used in the numerical simulation are listed in Table 3.1. The controller and observer design parameters are presented in Table 3.2. It should be noted here that in the development of the control law it is assumed that the leader is in an unperturbed reference orbit. However, numerical simulation is conducted on the system model that considers the leader satellite under J 2 -effect, variation in atmospheric density with position, and variation in orientation of the drag plates. The atmospheric density and the relative velocity in Eq.(3.7) are updated depending on the 3.3. Performance Evaluation 59 satellite position during the simulation. The Matlab function ATMOSNRLMSISE00 is used in the calculation of the atmospheric density based on NRLMSISE-00 scheme.
The control law development refers to the exponential atmospheric density model given Here h is the actual altitude above the Earth's surface, ρ 0 is the reference density, h 0 is the reference altitude and H is the scale height. Table 8-4 in [Vallado 2004] is used to obtain the density values. The adaptive control algorithm makes 60 Chapter 3. Satellite Formation Control Using Aerodynamic Drag use only of an initial estimate of the atmospheric density. Here the initial value of ρ 0 is taken as 1.454 ×10 -13 kg/m 3 , which corresponds to the density for an altitude between 600-700 km. The SFF system is in an orbit of 350 km altitude. This bias in the initial estimate is used to examine the efficacy of the control algorithm in a more realistic environment.
With regard to the observer data, it is assumed that errors in the available position measurements are of the order of 5 cm. This is simulated by adding white noise to the position vector such that the variance for the relative position is 5 / 3 2 . This will generate biased position vectors with maximum error of 5 cm [Bevilacqua et al. 2009].

Projected circular formation and Circular formation
First we examine the performance of the proposed SFF system in maintaining a projected circular formation. The phase angle (φ) between the leader and the follower satellites is assumed to be zero degree with a desired formation radius,r d = 0.5 km with the leader satellite in an orbit of inclination, i = 45 0 . In PCF, the leader and the follower satellite maintain a fixed relative distance on the y-z plane. But due to the lack of control on the z-axis, the PCF will deteriorate over time due to the influence of the J 2 perturbation.
Hence we make use of the iterative scheme given in [Damaren 2007] to derive the initial conditions to reduce the effect of J 2 on z -axis. These initial conditions are derived based on the nonlinear equations of motion rather than the HCW equations and yield relative orbits that are nearly periodic in the presence of J 2 perturbation, and hence ascertain the validity of performing projected circular formation with differential aerodynamic drag.
Such special initial conditions are also derived in [Sabatini et al. 2008]. Fig. (3.2), shows the relative error plots for the follower in the proposed SFF system for one day in orbit using the following initial state vector given by,  The adaptive parameter converges to −5.72× 10 −8 . Fig. 3.4 shows the convergence of the observed states (ẋ,ẏ) to its true values (ẋ,ẏ). The convergence is rapid and takes place within 0.1 orbit.

62
Chapter 3. Satellite Formation Control Using Aerodynamic Drag

Along-track formation
The desired relative motion in this type of formation flying is to maintain a constant separation between the leader and follower in the along track direction. Examples of such formation includes the TanDEM-X [Krieger & Zink 2007] and JC2Sat [Mierlo 2009] missions.
The desired separation distance is taken as 0.5 km and the leader satellite is considered to be in a Sun synchronous orbit with inclination of i = 97.44 deg. Fig. 3.7 shows the response of relative position errors in the radial and along track directions, along with the drag plate orientation (α F ) when the SFF system is subjected to J 2 disturbance and has an initial offset error of 50 m. It is observed that the relative states converge to the desired states in less than an orbit with the relative position errors being bounded within ±0.3 m and

Multiple Satellite Formation
Feasibility of multiple satellite formation flying and reconfiguration is investigated in this section. The projected circular formation is examined in the multiple reconfiguration problem. The problem of multiple satellite formation maneuvering for the proposed SFF system is more challenging as the achievable levels of differential drag between each follower and the leader are mutually constrained. A heuristic algorithm is developed to analyze this sce- During each orbit the drag plate orientation required for each individual leader-follower satellite configuration for maintaining the projected circular formation of size, r d = 0.5 km, is calculated based on the proposed control methodology. The followers align their corresponding drag plates with this calculated orientation. In the first orbit, the drag plates on the leader satellite are oriented with the calculated value with respect to the follower with phase angle 0 deg. This yields the required differential drag for the follower (Satellite 1 with phase = 0 deg). In the next orbit the same procedure of calculation of the required wing orientation is performed, but the drag plates on the leader is oriented according to the follower satellite with phase 90 deg (Satellite 2). This method is continued for the 68 Chapter 3. Satellite Formation Control Using Aerodynamic Drag successive orbits with the orientation of the drag plates on the leader satellite switching between the four follower satellites. The differential drag achieved by the other three follower satellites in each orbit may not be the same as required by each individual follower satellite in relation to the leader satellite. But it is found during simulation that with the proposed algorithm the projected circular formation is maintained with reasonable relative errors (Fig. 3.9).The control input response is shown in Fig. 3.10. The switching after each successive orbit can be seen in Fig. 3.10 as well.  Next we consider the case of multiple satellite reconfiguration using the proposed heuristic algorithm. This could be for a variety of reasons; especially in a multiple formation flying 70 Chapter 3. Satellite Formation Control Using Aerodynamic Drag system where in a new follower satellite may need to be inserted into the cluster or a faulty follower may need to be replaced with a new satellite. Here two satellites in a projected circular formation of initial formation size, r d = 0.25 km and r d = 0.5 km with the phase angle of each equal to 0 deg, are reconfigured to final formation sizes of r d = 0.5 km and r d = 0.75 km, respectively (Fig. 3.12). A smooth reconfiguration is achieved with the application of the heuristic algorithm.

Performance Evaluation 71
Finally to analyze the robustness of the proposed heuristic algorithm, Monte Carlo simulation is carried out. The effect of different initial conditions on the proposed methodology is evaluated by 1000 simulation runs. Four follower satellites in a projected circular formation are considered. Different initial conditions are generated by varying the desired formation size, r d , from 0.5 km to 5 km. Each simulation is run for 5 orbits and the mean value of relative error in the x -axis and y-axis is calculated and plotted against the corresponding value of r d (Fig. 3.13). Here '+' indicates the mean value of e x and '.' indicates that of e y .It can be observed that the relative errors in both the axes remain bounded within -8m.  Next, the performance of the proposed SFF system in the case of the leader in a low eccentric orbit is evaluated. Fig. 3.14 and Fig. 3.15 shows the relative errors in the case of   to the leader satellite is examined.

Performance Evaluation 73
To understand the effect of aerodynamic drag and external disturbances on the leader satellite and its subsequent effect on the formation system, the proposed system is evaluated on a high fidelity model. The equations of motion describing the leader satellite is given where r = X 2 L + Y 2 L + Z 2 L and F t j for j = x, y, z represent the aerodynamic drag and external disturbances affecting the system. The relative equations can be written as where F and L represent the leader and follower satellite respectively and U is the control input required to maintain the desired formation. A projected circular formation scenario is examined with i = 45 • and e = 0.001. Fig. 3.16 shows the orbital elements (a, e, i andθ) of the leader satellite over a period of one day. As expected the semimajor axis of the leader satellite is affected by the aerodynamic drag and it decreases over the course of each orbit. Fig. 3.17 shows the 3 − D plot of the leader satellite under the influence of aerodynamic drag. The corresponding performance of the follower satellite, Fig. 3.18, shows a bounded system response.

Qualitative Analysis
Next a quantitative analysis in terms of the differential aerodynamic drag generated with respect to drag plate area and altitude of the satellite is carried out. This analysis will provide required surface area of the drag plate for achieving desired satellite formation.
Here, the projected circular formation examined earlier is considered for different altitudes ranging from 350 km to 650 km from the Earth's surface. The maximum drag plate area is assumed to be 2m 2 as a realistic assumption. As expected, it can be observed from Fig.   74 Chapter 3. Satellite Formation Control Using Aerodynamic Drag   (3.19) that the drag force increases with increase in projected area as well as decrease in altitude. Formation reconfiguration at lower altitude is achieved with a smaller projected area compared to the required projected area at higher altitudes. It is observed that to maintain a projected circular at an altitude of 650 km, a projected area of 1.78m 2 is required. All the cases examined above suggest that use of differential aerodynamic drag is a viable alternative for multiple satellite formation.

Hardware-In-Loop (HIL) testing
The validity of the control scheme is evaluated through Hardware-In-Loop (HIL) simulation using a prototype model. The HIL arrangement is shown in Fig. (3.20). The satellite prototype is attached with plates as shown in Fig. (3.20). These plates with servo motors are attached to the satellite body. The computer running Matlab/Simulink simulates the leader follower satellite formation flying system. The control algorithm calculates the

Summary
In this chapter we examined the use of differential aerodynamic drag for multiple satellite b) development of an heuristic control methodology for multiple satellite formation flying using aerodynamic drag. c) validation of the proposed system in the presence of parameter uncertainties like variation in the density and altitude and navigation and initial offset errors. Overall, the theoretical and numerical results suggest that the use of differential aerodynamic drag is a viable alternative for multiple satellite formation.

Chapter 4
Satellite Formation Control Using Solar Radiation Pressure S atellite formation flying (SFF) in low Earth orbit (LEO) using differential aerodynamic drag was studied in the previous chapter. Application of LEO formation flying mainly addresses Earth observation missions. Here we examine the feasibility of using solar radiation pressure (SRP) for maintaining and maneuvering satellite formation system in geostationary orbits. Such a formation system has the advantage of continuous regional navigation and also act as a positioning system [Hui et al. 2006]. Here we proposes a method based on differential solar radiation pressure to achieve coordinated control of multiple satellites flying in formation. The relative orbit is controlled by varying the level of solar radiation pressure experienced by the solar flaps mounted on each satellite and thereby generating relative differential accelerations among the satellites in the formation.
Solar radiation pressure provides control authority in both in-plane as well as out-of-plane direction thereby allowing for complete control of the formation system as compared to control using aerodynamic drag. Sliding mode control (SMC) theory is used to develop an adaptive control algorithm for maneuvering the solar flaps to maintain the satellite formation.
This chapter is organized as follows: Section 4.1 revisits the nonlinear mathematical model of the SFF system. Development of the adaptive control algorithm is described in Section 4.2. For a detailed assessment of the system performance under the proposed control strategies, the results of numerical simulations incorporating different formation scenarios are presented in Section 4.3. The conclusions of the study are summarized in Section 4.4.

SFF System Model and Equations of Motion
The leader follower system shown in Fig.2.1 is used to describe the SFF system. The assumptions stated in Chapter 3, Section 3.1.1 holds here as well and is not repeated for brevity. The mathematical model of the SFF presented in Chapter 2: Section 2.1 is reproduced here. Eqs.(4.1) and (4.2) describe the orbital equations of motion for the leader satellite while Eqs.(4.3) through (4.5) are the relative equations of motion of the follower satellite:r where r f = [(r l + x) 2 + y 2 + z 2 ] 1/2 is the position of the follower spacecraft, f dtj is the net relative perturbations acting on the SFF system, and f j are the components of the control input vector, for j = x, y, z. Here the control forces, f j for j = x, y, z, is created by the rotation of the solar flaps on the satellites.

Solar Radiation Pressure Model
In geostationary orbits, SRP is the most dominant external force acting on a satellite. The Earth's oblateness effect has a very reduced effect at these orbits. The mathematical model of SRP is given as where s represents the Sun vector, n, is the solar flap normal vector, ζ s , is the specular reflection factor, and other coefficients are given by, C srp is the solar flap characteristic acceleration and is given by where the nominal SRP force is p 0 = 1.

Design of Control Law
The control objective is to determine the orientation of the solar flaps (α sj ,φ sj ), for j = F, L where F stands for the follower satellite and L stands for the leader satellite. A control algorithm based on SMC technique is developed to drive the follower satellite to the desired formation trajectory with respect to the leader satellite in the presence of external disturbances and initial deployment errors. Considering the mathematical formulation of the leader follower SFF model and SRP model presented in the previous section, the relative state vector and the desired relative trajectory can be defined as, X (t), and X d (t) ∈ R 6 respectively. The performance measure is defined as the tracking error, e (t) ∈ R 6 , The resultant control forces f j for j = x, y, z in Eqs. (4.3) -(4.5) will drive the relative states of the SFF system to the desired trajectories as t → ∞ and thereby the tracking errors, Eq. (4.12), converge to zero. Control algorithm development is simplified by the following assumptions on the SRP model. The solar flap is assumed to be an ideal highly reflective surface with only specular reflection and no absorption, and the Sun vector is defined as s = [1, 0, 0] T . This yields the SRP force relation to be written as, The transformation of this force equation to the orbital frame from the Sun centered inertial frame is given as F s = C B/I · C I/S · F . Here, C B/I represents the transformation matrix from the geocentric inertial to the orbital frame given by, where ot represents (ω + θ), ε is the angle between the ecliptic and equatorial plane, c () and s () represents cos () and sin () respectively. C I/S represents the transformation from

Design of Control
the Sun centered inertial to the geocentric inertial frame as follows, where ψ is the solar aspect angle. Following these modifications, the differential SRP force can be expressed as, With respect to the solar flaps on the leader and follower satellites in Eq.(4.14), we assume that the leader and follower spacecraft are both equipped with solar sails having the same area to mass ratio, also we assume ζ s = 1 and ε f B f = ε b B b . With these simplifications, the equations used for control law design are written in the following form, The nonlinear terms in the equations of motion have been lumped into E (X), A represents the linear part of the SFF system, g(U ) is the control matrix, and F dt the differential perturbation forces. The control forces f j for j = x, y, z is given by rates. This information is then used in the system dynamics to calculate the force due to solar radiation pressure.

Adaptive Control Formulation and Stability Analysis
First we define the higher order sliding manifold given by where ξ is a positive constant. The lower order sliding plane, S is given by In the adaptive control law, C srp is taken as the unknown quantity due to the presence of uncertainity in the its associated parameters. Consider the candidate Lyapunov function defined as where V > 0 and ζ is a positive constant. Taking the derivative along the trajectory gives, Taking the parameter estimation error asC srp =Ĉ srp − C srp , and expanding Eq. Rearranging the preceding Eq.(4.23) yields, Now in order to makeV negative, choose the adaptive law aṡ where η is a positive constant. Substituting Eqs.(4.26) and (4.25) into Eq.(4.24) we have, Thus,V is negative and hence the reaching condition is satisfied. Using Theorem-3 of LaSalle's principle [LaSalle 1960] it is proved that the system is completely stable and hence the system states converge onto the sliding surface. Next, Eq.(4.26) is rearranged to derive the control law that is capable of formation maneuvering and maintenance is given η sgn (σ) + ΛAẊ (4.28) whereU = δȧα sφs T and U = U dt. The discontinuous sgn(σ) operator in Eq.
(4.28), is replaced by a boundary layer to remove instantaneous switching and it is given

Performance Evaluation
In order to examine the effectiveness of the proposed formation control strategy, the detailed system response is numerically simulated using the set of governing equations of motion

Performance Evaluation 89
Eqs. (4.1) -(4.5) along with Eq. (4.6) in conjunction with the proposed control algorithm given by Eq. (4.29), and the adaptation law, given by Eq. (4.25). The integration is carried out in Matlab using the fourth order Runge-Kutta solver with a fixed step size of 0.1 sec.
The desired states of the system are given by the types of formation, as per Eqs. (2.19) and (2.20). The SFF systems parameters and the orbital parameters used in the numerical simulation are listed in Table 4.1. The controller and parameters are presented in Table 4.2.
It should be noted here that in the development of the control law it is assumed that the leader is in an unperturbed reference orbit. However, numerical simulation is conducted on the system model that considers the leader satellite under J 2 -effect, and variation in orientation of the solar flaps. A satellite in the geostationary orbitis considered here and hence the Earth shadow effect is assumed to be negligible as its effect is experienced for only about 5% of the orbital period ].

Projected circular formation and Circular formation
First we examine the performance of the proposed SFF system in maintaining a projected circular formation. The phase angle φ between the leader and the follower satellites is assumed to be zero degree with a desired formation radius,r d = 10km with the leader satellite in an orbit of inclination, i=45. The relative position errors remain bounded to [1.0 2.5 2.5 4 2.5 4] Initial estimate of Area,Â (m 2 ) 0.5 Initial estimate of p,p (N/m 2 ) 4.563 × 10 −6 Initial Mass,m f (kg) 2.5 ζ 10 −2 less than 1 m in the radial direction and to within 5 m in the along track and cross track direction ( Fig. 4.2). The in-plane (φ) and out-of plane (α) orientations of the solar flaps have a maximum deflection of 7 • (Fig. 4.3) and the maxmium area change is less than 1 m 2 . Next we examine the performance of the proposed SFF system in the presence of an      reconfiguration is another important requirement of the SFF missions, and is studied next.
The projected circular formation of size 10 km is reconfigured into a circular formation of 15 km after 1 orbit. From Fig. (4.6) it can be observed that reconfiguration is possible with the proposed control methodology, and the steady state errors remain bounded. A smooth reconfiguration is established with the solar flaps deflecting to a maximum orientation of 24 • in the in-plane and 10 • in the out-of-plane directions during the reconfiguration phase ( Fig. 4.7).

Along-track Formation
Next, along-track formation using solar radiation pressure is evaluated. The constant separation is desired to be 10 km. Fig. 4.8 shows the relative errors in the radial and along track direction during along-track formation flying with an initial offset error of 100 m.

Multiple Satellite Formation
Feasibility of multiple satellite formation flying and reconfiguration using SRP is investigated in this section. The projected circular formation is examined in the multiple reconfiguration problem. Similar to the multiple satellite formation flying using aerodynamic drag, the constraint on the level of differential SRP achievable between each follower and the leader makes this problem equally challenging. The heuristic algorithm proposed in the previous chapter is rewritten here for clarity, and is tested in a configuration of two followers with the phase angles of the followers at 0 • , and 180 • , respectively. The algorithm is given as: During each orbit the solar flap orientation required for each individual leader-follower satellite configuration for maintaining the projected circular formation of size, r d = 5 km, is calculated based on the proposed control methodology. The followers align their corresponding solar flaps with this calculated orientation. In the first orbit, the solar flap on the leader satellite is oriented with the calculated value with respect to the follower with phase angle 0 • . This yields the required differential solar radiation pressure for the follower (Satellite 1 with phase φ = 0 • ). In the next orbit the same procedure of calculation of the required solar flap orientation is performed, but the flap on the leader is oriented according to the follower satellite with phase 180 • (Satellite 2). This method is continued for the successive orbits with the orientation of the solar flaps on the leader satellite switching between the two follower satellites. Eventhough, the differential force achieved by the follower satellites in each orbit may not be the same as required by each individual follower satellite in relation to the leader satellite, it is found during simulation that with the proposed algorithm, the projected circular formation is maintained with reasonable relative errors ( Fig. 4.9). Next we consider the case of multiple satellite reconfigurations using the    satellite formation is also confirmed in the study. Multiple satellite formation reconfiguration is also examined with excellent results. The theoretical and numerical results suggest that the use of differential SRP is a suitable alternative to multiple satellite formation even in the presence of external disturbances.

Chapter 5
Satellite Attitude Control Using Aerodynamic Drag I n Chapter 3 and Chapter 4, the feasibility of performing satellite formation flying using either aerodynamic drag or solar radiation pressure was studied and evaluated. The satellite attitude dynamics was not considered and attitude was assumed stabilized. The attitude control system provides the satellite with the maneuvering, tracking and pointing capability. For many Earth orbiting satellite three axis attitude stabilization is essential to meet the nadir pointing requirement of the mission. This chapter looks to expand upon the previous efforts on the use of aerodynamic drag for passive satellite attitude control. Existing literature proposes the combined use of aerodynamic and magnetic torquing [Psiaki 2004], control of the two of principal axes using torque due to aerodynamic drag [Pande & Venkatachalam 1979], [Modi & Shrivastava 1973], or using an additional yaw stabilizer along with the drag plates [Chen et al. 2000] for achieving three axis satellite attitude control. The proposed system comprises of a satellite with two pairs of oppositely placed drag plates, and the satellite attitude is controlled by the suitable rotation of these drag plates. This arrangement of the drag plates results is an inherently underactuated system with the number of available control inputs less than the degree of freedom to be controlled.
To compensate for unknown disturbances and parameter uncertainties, an adaptive nonlinear control design based on the theory of sliding mode is proposed as the control technique for satellite attitude control.
The chapter is organized as follows: Section 5.1 introduces the nonlinear equations of motion of an Earth orbiting satellite. The adaptive control algorithm based on sliding mode technique is formulated with the corresponding stability conditions for robustness against uncertainties and disturbances in Section 5.2. The detailed numerical simulation of 100 Chapter 5. Satellite Attitude Control Using Aerodynamic Drag the governing system equations of motion under the proposed control strategy, including the effects of various system parameters on the controller performance is given in Section 5.3. Conclusions are stated in Section 5.4.

Proposed System Model
We start the study by formulating the nonlinear equations of motion of the rigid body satellite. The proposed system consists of a rigid body satellite in an elliptical planar trajectory with the Earth's center at one of its foci. However, Euler angles exhibit singularities whenever the roll angle (φ) has a value of ± π 2 . But singularity does not reflect any physical limitations of the rigid satellite rather it is an inherent property of the Euler angle representation. The orientation of the bodyfixed reference frame, B − x y z, with respect to the LVLH reference frame, L − x 0 y 0 z 0 using the (3 − 2 − 1) rotation sequence is described by the direction cosine matrix where cφ = cos φ and sφ = sin φ, R x (γ) denotes the rotation matrix for γ rotation about the x-axis, R y (φ) denotes the rotation matrix for φ rotation about the y-axis, and R z (α) denotes the rotation matrix for α rotation about the z-axis.
Here β p for p = 1, 2 represents the orientation of the drag plates with its normal initially aligned with the y-axis rotated about the z-axis and β p for p = 3, 4 represents the orientation of the pair of drag plates with its normal initially aligned with the y-axis, and then rotated about the x-axis.

Satellite Angular Velocity
Let the unit vectors associated with the reference frames B − x y z and L − x 0 y 0 z 0 , be represented as {î,ĵ,k} and {î 0 ,ĵ 0 ,k 0 } respectively. Using the (3−2−1) rotation sequence, the transformation from the L − x 0 y 0 z 0 orbital frame to the body-fixed frame B − x y z is given by, cφ cα cφ sα −sφ sγ sφ cα − cγ sα sγ sφ sα + cγ cα sγ cφ cγ sφ cα + sγ sα cγ sφ sα − sγ cα cγ cφ For a rigid satellite in a circular orbit around the Earth, the angular velocity of the bodyfixed frame B relative to the ECI frame E is given by where ω B/L is the angular velocity of B relative to the orbital frame L, and ω L/I is the angular velocity of the orbital frame with respect to the ECI frame. From Fig. (5.1), the angular velocity of L relative to E is whereθ is the orbital rate of the satellite. Here,θ = √ µp R 2 , where µ represents the gravitational parameter of the Earth, p is the semi-latus rectum, and R is the distance of the satellite from the center of the Earth. From Eq. (5.2) we have cφ cα sγ sφ cα − cγ sα cγ sφ cα + sγ sα cφ sα sγ sφ sα + cγ cα cγ sφ sα − sγ cα −sφ sγ cφ cγ cφ Substituting fork 0 from Eq. (5.5) into Eq. (5.4) yields, The angular velocity of the satellite with respect to the orbital frame L is given by whereα is about the k 0 axis in the frame L − x 0 y 0 z 0 , followed byφ about the j 1 axis in the intermediate frame S − x 1 y 1 z 1 , and finallyγ about the i axis in the frame B − x y z.
This can also be written as The unit vectors are related to each other based on rotation matrices as follows Chapter 5. Satellite Attitude Control Using Aerodynamic Drag Using Eqs. (5.7)-(5.9), the angular velocity of the satellite with respect to the orbital reference frame can now be expressed as Finally, using Eqs. (5.6) and (5.10) in Eq. (5.3), the angular velocity of the satellite relative to the ECI frame E expressed in the body frame B is

Free Molecular Aerodynamic Drag Model
The free-molecular aerodynamic drag model introduced in Chapter 2 is reproduced here where p is the total pressure and τ is the shearing stress,V rel = V rel V rel is the velocity unit vector of the surface element with respect to the atmosphere andn is the outward-pointing unit normal vector The total pressure p and shearing stress τ are given by [Gombosi 1994] Here σ n and σ t are the normal and tangential accommodation coefficients, T s is the absolute temperature of the surface, T a is the atmospheric temperature, q ∞ is the dynamic pressure given by and s is the air speed, nondimensionalized by the mean molecular speed of the atmosphere and given by where M a is the mean molar mass of the atmosphere and R * is the universal gas constant.
M a and T a vary with altitude and can be obtained from the 1976 U.S. Standard Atmospheric model. The accommodation coefficients σ n and σ t , represent the fact that some impinging air molecules come to thermal equilibrium with the surface and then get re-emitted thermally while others reflect specularly. Specular reflection means that the angle of incidence equals the angle of reflection and that the relative speed remains unchanged. The values σ n = σ t = 0 implies specular reflection of all the impinging air molecules, while σ n = σ t = 1 implies full thermal accommodation of all the impinging air molecules. Eventhough there are no reliable data for these coefficients, typical values of σ n and σ t lie between 0.85 and 1, which means that most of the air molecules get re-emitted thermally. It is to be noted that sin ζ =n TV rel . The outward-pointing unit normal vectors are given as n p = − sin β pî + cos β pĵ p = 1, 2 (5.17) n p = cos β pĵ + sin β pk p = 3, 4 (5.18) Another aspect to be considered is the effect of shading, which refers to the situation where one aerodynamic surface lies directly upstream of another aerodynamic surface causing reincidence of emitted air molecules. This can perturb the incident velocity distributions that The resultant torque on the surface can be calculated using the equations for pressure and shearing stress Eqs. (5.13) in conjunction with the unit vectorsn andV rel . The net torque can then be computed by summing the moments over all the four surfaces, given by where r j is the position vector of the geometric center of the individual drag plates measured from the satellite center of mass. For the proposed system configuration, it can be inferred that T a will have components only along the x and z direction, i.e., T aα and T aγ respectively.

Equations of Motion
The governing nonlinear coupled differential equations of motion of the system are obtained where F α , F φ , and F γ are the nonlinear terms given by The coefficients p α , p φ , and p γ in the preceding Eq. (5.23) are given by, The elements of the matrix N in Eq. (5.22) are given by N 11 = sin 2 γ + k yz cos 2 γ k yz cos 2 φ , N 12 = (1 − k yz ) sin γ cos γ k yz cos φ N 13 = N 11 sin φ, N 21 = N 12 , N 22 = cos 2 γ + k yz sin 2 γ k yz The dimensionless parameters representing the principal moment of inertia ratios of the satellite are defined as Chapter 5. Satellite Attitude Control Using Aerodynamic Drag For a specific orbit, the orbital radius R is given by 1 + e cos θ = µ 1/3 (1 − e 2 ) Ω 2/3 (1 + e cos θ) (5.26) and the orbital rate,θ, is obtained bẏ Since the true anomaly θ(t) is implicitly known from the Kepler's equation, it is convenient to replace t with θ as the independent variable. The corresponding derivatives with respect to θ are represented asq where q is given as q ∈ R 3 = [α, φ, γ] T . After carrying out algebraic manipulation and nondimensionalization, the governing nonlinear equations of motion of the system can be written in the matrix notation as follows where M (q, k xz , k yz ) ∈ R 3×3 and F (q, q , e, k xz , k yz ) ∈ R 3×1 are matrices containing nonlinear functions, andT f ∈ R 3 = [T aα ,T aφ ,T aγ ] T corresponds to the dimensionless control torques. The nondimensionalized torque due to aerodynamic drag is given by, (1 + e cos θ) 3 In this proposed system configuration, the control torquesT aα andT aγ are obtained using aerodynamic drag as given by Eqs.(5.20) and (5.21), but the torque componentT aφ is zero.

Design of Control Law
In this section, a nonlinear control algorithm that makes use of torque due to aerodynamic drag to stabilize the satellite is developed. Taking into account the parameter uncertainty associated with aerodynamic drag, the control law is made to be adaptive in nature. As The force due to aerodynamic drag in this case is represented as follows wheren is the outward normal, andv b is the unit vector in the direction of the translational velocity. The torque due to aerodynamic drag is obtained by computing the moments over the drag plate surface, given by Here again, the torques along the pitch and yaw axes are given as To facilitate the control law design, the state space representation of the system can be formulated as follows:   x 11 Chapter 5. Satellite Attitude Control Using Aerodynamic Drag , d 1 and d 2 represent the external disturbances, andĀ matrix constitutes the nonlinear terms and is given bȳ The control torques, U ∈ R 2×1 = T aα ,T aγ T is in dimensionless form and is given aŝ To simplify the control algorithm development, the following assumptions are made with respect to the orientation of the drag plates. It is assumed that β 2 = −β 1 and β 4 = −β 3 . Eq. (5.35), is rewritten in the following form, F (q, q ) represents the vector containing all the nonlinear terms including the Coriolis and centrifugal contributions and is described in Eq. (5.22), and M is the mass matrix.
Next, we present the design procedure to implement an adaptive sliding mode control for changing the orientation of the drag plates inorder to achieve satellite attitude control.
The design can be divided into two main steps. First, we design a stable sliding surface and this is followed by the development of the feedback control algorithm that drives the system dynamics to this sliding surface in finite time.

Design of Sliding Manifold
The control inputs i.e., the orientation of the drag plates, (β i , for i = 1 − 4), appears in non-affine nature in the system formulation. A nonlinear control algorithm based on 5.2. Design of Control Law 111 higher order sliding mode is developed to calculate the orientation of the drag plates. This increase in the order of the system helps in deriving the control law that can extract the angular rates of the drag plates (β i , i = 1 − 4) from Eq. (5.39). The corresponding angular orientation of the drag plates can then obtained by integration of these angular rates. This information is then used in the system dynamics to calculate the torque due to aerodynamic drag. First a sliding surface with order less than the system dynamics is introduced; it is given as whereq = q − q d , andq = q − q d are the state error variables and K A and K B are positive constants given by Now a higher order sliding manifold is defined combining S and its derivative, i.e, S , it is given as

Adaptive Control Formulation and Stability Analysis
The next stage is the adaptive control law formulation based on Lyapunov stability theorem.
The control law that drives the states to be along the sliding manifold, σ = 0 for all t > t r , can be derived by choosing the Lyapunov candidate function defined as follows: where V > 0 and ς is a positive constant. Taking the derivative along the trajectory gives Chapter 5. Satellite Attitude Control Using Aerodynamic Drag where σ = S + ξS . The control algorithm design is simplified by expressing the system nonlinearities and external disturbances present in system dynamics by using its worst case upper-bound in the controller design. This assumption gives us the following relation: It is to be noted that L contains all the terms including the system nonlinearities, parameter uncertainties and external disturbances, and the upper bound of this lumped disturbances is given by Now, expanding σ in Eq. 5.44 yields Here, q represents the derivative of the equations of motion [Eq. 5.39] along its trajectories given by where U a = [T aα T aγ ]. Substituting for this in Eq. 5.47, we get, Rearranging the preceding Eq. (5.52) yields Now in order to make V negative, choose the adaptive law aŝ where ζ is the adaptation gain. Also let Eq. (5.56) is semi-negative definite, but by using Theorem-3 of LaSalle's principle [LaSalle 1960] it is proved that the system is completely stable and hence the system states converge onto the sliding surface which is the invariant set here. For a system satisfying Eq. where c = η − d. The reaching time (t r ) in this case is derived as where σ(t = t r ) = 0 as per the definition. For the second case of σ(t = 0) > 0, Eq. (5.57) yields σ c (5.60) Hence, the reaching time (t r ) in this case is obtained as Combining both the cases and referring to Eqs. (5.59) and (5.61), the reaching time, (t r ), is given by Next, Eq. (5.55) is rearranged to derive the control law as A boundary layer is introduced in the control input, Eq. (5.63), formulation to remove the instantaneous discontinues switching and it is given by where δ is the width of the boundary layer.

Performance Evaluation
To study the effectiveness and performance of the proposed adaptive controller, the detailed system response is numerically simulated using the set of governing equations of motion, given by Eq.(5.30) along with proposed adaptive control law given by Eq. (5.64). The integration is carried out Matlab using the fourth order Runge-Kutta solver.The simulation parameters used are shown in Table 5

Nominal Performance
The first case to be evaluated is the stabilization of the satellite in a circular orbit from the large initial disturbances about its three axes given by

Variations in Mass Moment of Inertia
In the nominal case the the values of both k 1 and k 2 were positive, signifying the stable gravity gradient configuration. Now, the performance of the proposed adaptive control algorithm in the presence of variations on the satellite moments of inertia is examined.
First, a satellite in an axially symmetric configuration is considered, then the performance of the proposed system as affected by the product of inertia terms (I xy , I xz , I yz ) is evaluated, and finally the efficacy of the adaptive control algorithm for a satellite in an unstable gravity gradient configuration is examined. The inertia matrix for the first and second cases is as  Table 5.2.
The system performance for an axis-symmetric satellite is shown in Fig. (5.4). The attitude response converges asymptotically and remains identical to the previous case. Fig. (5.5) shows the corresponding angular rates and the adaptive parameterB.
We next examine the performance of the proposed control technique on a satellite model containing the product of inertia terms. The attitude of the satellite is stabilized (Fig. 5.6) It can be observed that the product of inertia terms does have an adverse effect on the satellite as the steady state satellite attitude is non zero.
The performance of the control method for a satellite in an unstable gravity gradient

Eccentricity and External Disturbances
First, the effectiveness of the control strategy for satellite in an elliptical orbit is studied.
The orbital eccentricity is increased from e = 0.01 (Fig. 5.8) to e = 0.1 (Fig. 5.9). The state error in the pitch axis is bounded by |α| max = 0.51 • , roll axis by |φ| max = 0.15 • and the roll axis by |γ| max = 0.065 • . The maximum deterioration is observed in the pitch response as compared to the previous numerical results. This is expected as the orbital eccentricity directly affects the pitch motion of the satellite. The drag plates in the pitch axis continuously maneuvers to counter this disturbance due to eccentricity. The maximum deflection of the drag plates is observed in the transient stage and the steady state deflection is nearly 5 • . Overall the satellite attitude remains stabilized within reasonable limits even in elliptic orbits using the proposed methodology.
Finally, the effect of external disturbance on the performance of the system is verified.
The system may experience external disturbance torques due to several factors including 120 Chapter 5. Satellite Attitude Control Using Aerodynamic Drag solar radiation pressure and other environmental forces. The disturbances due to solar radiation pressure is more significant and can be expressed as (in Nm) [Kaplan 1976] Here T s j for j = α, φ, γ represents the disturbance torque due to solar radiation pressure.
It can be observed from Fig. (5.10) that the controller is able to regulate the satellite attitude by the maneuvering of the drag plates, and hence establishes the robustness of the proposed methodology to time varying disturbances external disturbance. The above results bring out the powerful features of the proposed adaptive control algorithm. The performance of the controller does not deteriorate even when the system parameters are changing and yet provides excellent attitude response.

122
Chapter 5. Satellite Attitude Control Using Aerodynamic Drag

Summary
This chapter examines the feasibility of three axis satellite attitude control using aerodynamic drag. The proposed system configuration consists of a satellite with two pairs of oppositely placed drag plates. Attitude maneuvering and stabilization are accomplished by suitable rotation of these drag plates. Such a configuration has not been proposed in the existing literature. This system configuration provides explicit control authority only along its two principal axes, i.e., the pitch and yaw axes. Complete three axis attitude control using aerodynamic drag is achieved by the application of the control law developed based on sliding mode theory. The controller enables the satellite to achieve any arbitrary orientation. The performance of the control strategy in attitude stabilization is examined in detail using high-fidelity nonlinear system model. Unlike earlier results on the use of aerodynamic drag for satellite attitude control, numerical simulation results were obtained under many scenarios including presence of external disturbances, high initial attitude rates and uncertainty in the mass moments of inertia and the the proposed control scheme illustrated excellent performance in successfully regulating the satellite attitude motion in each case. The satellite attitude was observed to be stabilized within 0.2 • in all the cases except for the elliptic orbit (e = 0.1) and stands as a good candidate for use in future space missions.

Chapter 6
Satellite Attitude Control Using Solar Radiation Pressure I n Chapter 5, the feasibility of performing satellite attitude control using aerodynamic drag was studied and evaluated. In this chapter three axis satellite attitude control using solar radiation pressure is analyzed. An adaptive nonlinear control design based on the theory of sliding mode is proposed to control the attitude of a satellite using solar radiation pressure. The system comprises of a satellite with two pairs of oppositely placed solar flaps. The nonlinear analytical model describing the system is used to derive an adaptive control law, based on Lyapunov stability theorem, in the presence of unknown, slow-varying external disturbances. This control law suitably rotates the solar flaps to achieve desired satellite attitude performance. The detailed numerical simulation of the governing nonlinear system equations of motion including the effects of various system parameters on the controller performance establishes the feasibility of the proposed control strategy. The numerical results show the robustness of the proposed adaptive control scheme in controlling the satellite attitude in the presence of uncertainties and external disturbances.
The system model and its equations of motion are presented in Section 6.1. Section 6.2 presents the closed-loop nonlinear control law based on SMC along with the stability analysis of the closed-loop system. In Section 6.3 numerical simulation is carried out for a detailed assessment of the proposed attitude control strategy. The effects of various system parameters on the performance of the controller are examined and the effectiveness of the proposed controller in the presence of parameter uncertainties and external disturbances is studied. Finally, the findings of the present study are summarized in Section 6.4.

Proposed System Model and Equations of Motion
The nonlinear equations of motion of the rigid body satellite derived in Section 5.1 is discussed here in brief for clarity. The system model comprises of a satellite with two pairs of oppositely placed light-weight solar flaps along the y and z axes of the satellite as shown in Fig. (6.1). The system center of mass lies on the center of mass of the satellite. The LVLH orbital reference frame, L − x 0 y 0 z 0 , has its origin fixed at the center of mass of the satellite. The x 0 -axis points along the local vertical, the z 0 -axis is normal to the orbital plane, and the y 0 -axis points along the orbit direction. The corresponding principal bodyfixed coordinate frame of the satellite is denoted by O − x y z. The nodal line represents the reference line in orbit for the measurement of the true anomaly (eccentric orbit) or angle θ (circular orbit). For the solar flaps along the y-axis, its normal considered initially aligned with the x-axis is rotated by an angle β about the z-axis (normal to the orbit plane); and for the pair of solar flaps along the z-axis with its normal initially aligned with the x-axis is rotated by an angle ς about the y-axis.

Solar Radiation Pressure Model
The force acting on the solar flap-j, j =1-4 due to SRP is given by where s represents the Sun vector, n, is the solar flap normal vector, ζ s , is the specular reflection factor, and other coefficients are given by, where ot represents (ω + θ), c () and s () represents cos () and sin () respectively. C I/S represents the transformation from the Sun centered inertial to the geocentric inertial frame as follows,  where ψ is the solar aspect angle. The vector normal to the solar flap-j, n j is given by Here β and ς represents the orientation of the solar flaps. Thus, the torque exerted by the solar flap-j on the satellite is obtained as T sj = r j × F sj = r jĵ × F sj , j = 1, 2 (6.8) T sj = r j × F sj = r jk × F sj , j = 3, 4 (6.9) Considering the proposed system configuration along with the Eqs. (6.8) and (6.9), it can be inferred that control torques available only about the pitch (T sα ) and roll (T sφ ) axes.
But we make use of the coupling between the roll and yaw axes to achieve complete three axis satellite attitude control. Next we look at the formulation of the complete equations of motion of the proposed system.

Equations of Motion
The governing nonlinear coupled differential equations of motion of the system are obtained The nonlinear terms in the matrix N and F q , for q = [α, φ, γ], were provided in the previous chapter [Eq. (5.23)] and are not repeated here for brevity. Following a similar approach of nondimensionalization as in the previous chapter, the governing nonlinear equations of motion of the system can be written in the matrix notation as follows where M (q, k xz , k yz ) ∈ R 3×3 and F (q, q , e, k xz , k yz ) ∈ R 3×1 are matrices containing nonlinear functions, andT f ∈ R 3 = [T sα ,T sφ ,T sγ ] T corresponds to the dimensionless control torques. The nondimensionalized torque due to SRP is given by, (1 + e cos θ) 3 In this proposed system configuration, the control torquesT sα andT sφ are obtained using SRP as given by Eqs.(6.8) and (6.9), but the torque componentT sγ equals to zero.

Design of Control Law
In this section, an adaptive control law based on sliding mode technique is designed. Parameter uncertainties are associated with the SRP stabilized system, and these uncertainties can cause numerous problems in the control tasks and lead to inaccuracy and instability of the control system. Hence an adaptive control methodology is required to achieve consistent stable performance. The unknown parameters are estimated online and used in the feedback control law. The control inputs are the orientation of the two sets of solar flaps attached to the satellite. Considering Eqs. (6.1) through (6.9) and (6.10), it is evident that it is difficult to extract the angular rotations of the solar flaps from the coupled equations of motion. But if we choose the angular rates as the control inputs, the corresponding angular orientation of the solar flaps can be obtained by integrating the angular rates.
Hence, to facilitate the control law design, the state space representation of the system can be formulated as follows, A 11 = sin 2 φ sin 2 γ + k yz cos 2 γ k yz cos 2 φ + 1 k xz andĀ 12 = sin φ ( sin 2 γ+k yz cos 2 γ ) To simplify the control algorithm development, the following assumptions are made with respect to the orientation of the solar flaps. It is assumed that β 2 = −β 1 and ς 2 = −ς 1 , also we assume that the cross-sectional area of the solar flaps and the distance between the system center of mass O and the center of pressure for both sets of solar flaps are the same (i.e. A j =A, r j =r for j = 1,2,3,4). Now assuming the solar flaps to be highly reflective surface (i.e., ρ d = 0; no absorption, specular reflection only, ρ t = 0, no transitivity as well), The corresponding torque due to SRP can be expressed as The torque equations are then converted into dimensionless form by making use of C srp = 2ρspAr I z Ω 2 . Now, Eq. (6.12) is rewritten in the following form, F (q, q ) represents the vector containing all the nonlinear terms including the Coriolis and centrifugal contributions, and M is the mass matrix.The basic objectives for the design of the control law are a) To drive the system errors to zero without oscillations or overshoots; b) To compensate external disturbances from the beginning.
First we design the linear sliding manifold which guarantees the desired dynamic behavior for the nominal system in the presence of uncertainties and disturbances, and this is followed by design of the sliding mode control law such that the system states are driven to this sliding plane and the system remains stable.

Design of Sliding Manifold
For the proposed system given by Eq. (6.16), the orientation of the solar flaps, (β i and ς i , for i = 1, 2), are the control inputs and appear in non-affine nature in the system formulation. Hence a nonlinear control algorithm based on higher order sliding mode is developed. Increasing the order of the system helps in extracting the angular rates of the solar flaps (β i , ς i , i = 1, 2) from Eq. 6.16. The corresponding angular orientation of the 132 Chapter 6. Satellite Attitude Control Using Solar Radiation Pressure solar flaps is obtained by integration of these angular rates. These control inputs are then used in the system dynamics to calculate the torque due to SRP. First a sliding surface with order less than the system dynamics is introduced, it is given as Now a higher order sliding manifold is defined combining S and its derivative, i.e, S ; it is given as,

Adaptive Control Formulation and Stability Analysis
In this section the adaptive control algorithm is formulated and its stability is analyzed.
The time varying and uncertain parameter is the the solar parameter C srp , hence it is assumed to be unknown and is estimated online and then used in the control algorithm.
Using Lyapunov stability theorem, the control law that drives the system states to be along the sliding manifold, σ = 0 for all t > t r , is derived. where σ = S + ξS . Next, the system nonlinearities and external disturbances present in system dynamics are expressed using its worst case upper-bound in the controller design, leading to the following relation, Now, expanding σ in Eq. (6.21) yields, where q represents the derivative of the equations of motion, Eq. (6.16) along its trajectories given by where U a = T sαT sγ . Substituting Eq. (6.24) in Eq. (6.23), we get, Now, substituting σ , from the from preceding equation in Eq. (6.21), along with the consideration on the bounds given in Eq. (6.22) we get Now considering the parameter estimation error in B as and expanding Eq. (6.26), we have Rearranging the preceding Eq. (6.28) yields Now in order to make V negative, choose the adaptive law aŝ where ζ is the adaptation gain. Also let where η is a positive constant and η >> sup L . Substituting Eq. (6.31) and Eq. (6.30) into Eq. (6.29), we have Now, we can prove that the system is completely stable using theorem-3 of LaSalle's principle [LaSalle 1960], and hence the system states converge onto the sliding surface. The reaching time, (t r ), for adaptive sliding mode derived in Chapter 5 holds in this case as well. It is given as |σ (t = 0)| /c, where c = (η − d) and d is assumed to be a bounded value.
Next, Eq. (6.31) is rearranged to derive the control law as And the control law with the introduction of the boundary layer in Eq. (6.33), is given by, where δ is the width of the boundary layer.

Performance Evaluation
To study the effectiveness and performance of the proposed adaptive controller, the detailed system response is numerically simulated using the set of governing equations of motion given by Eq. (6.10) along with proposed adaptive control law given by Eq. (6.34). The integration is carried out Matlab using the fourth order Runge-Kutta solver.The simulation parameters used are shown in Table 6.1.  It can be observed that the adaptive controller successfully stabilizes the satellite. The Euler angles asymptotically approach zero and the maximum deflection of the solar flaps |β| max (|ς| max ) is 28 • (9 • ), observed during the transient state. Fig. (6.4) shows the angular rates and the adaptive parameterĈ srp .

Variations in Mass Moment of Inertia
The performance of the proposed adaptive control algorithm in the presence of variations on the satellite mass moment of inertia is examined next. A satellite in an axially symmetric configuration is considered, then the performance of the proposed system as affected by the product of inertia terms (I xy , I xz , I yz ) is evaluated, and finally the efficacy of the adaptive control algorithm for a satellite in an unstable gravity gradient configuration is examined.
The inertia matrix for the first and second cases is chosen same as shown in Table 5.2 in Chapter 5. The system performance for an axisymmetric satellite is shown in Fig. (6.5). The attitude response converges asymptotically. Fig. (6.6) shows the corresponding angular rates and the adaptive parameterĈ srp . Next, the effect of incorporating product of inertia terms in the satellite model is evaluated. The attitude of the satellite is stabilized within ona and half orbits (Fig. 6.7). The steady state error is non-zero,as expected from the addition of the product of inertia terms. The performance of the control method for

Eccentricity and External Disturbances
The effectiveness of the control methodology for a satellite in an elliptical orbit is examined next. The performance of the system for an orbital eccentricity of e = 0.1 is shown in Finally, the effect of external disturbance on the performance of the system is verified.
The system may experience external disturbance torques due to several factors including SRP modeling errors (i.e. ρ d = 1), solar flap misalignment and other environmental forces.
The satellite model is subjected to disturbance of a periodic nature, and the performance of the controller is evaluated. It can be observed from Fig. (6.11)  changing and yet provides excellent attitude response.

Summary
This chapter examines the feasibility of three axis satellite attitude control using solar radiation pressure. A satellite with two sets of oppositely placed light weight solar flaps is proposed. This configuration provides explicit control only on two of the axes, the pitch and roll axis. But, three axis attitude maneuvering and maintenance is accomplished by rotating the solar flaps with accordance to the control law developed based on sliding mode theory.
The performance of the control strategy in attitude stabilization is examined in the detail including its performance in the presence of external disturbances, variation in the solar 6.4. Summary The system model and its equations of motion in the case of pitch attitude control using solar radiation pressure is presented in Section 7.1. In section 7.2, the nonlinear adaptive sliding mode control law is derived. In Section 7.4 numerical simulation is carried out for a detailed assessment of the proposed attitude control strategy, and the effects of different failure scenarios on the controller performance is examined. Next, in section 7.5, the controller performance in the case of three axis attitude control using either aerodynamic drag or solar radiation pressure is evaluated in the presence of faults and failures. Finally, the findings of the present study are summarized in Section 7.6.  Using Euler's method the governing equation of motion of the system is written as where τ d is the disturbance torque, τ g is the gravity gradient torque given by and τ s denotes the torque due to solar radiation pressure and it is derived as follows: Considering a highly reflective surface (i.e., ρ d = 0; no absorption, specular reflection only, ρ t = 0, no transitivity as well), the force acting on the flap-j due to SRP is given by Here s j is the unit vector of the incoming light from the Sun on the solar flap-j and is expressed in the satellite body-fixed reference frame as The vector normal to the solar flap-j, n j is given by n j = [− sin β j ]ĵ + [cos β j ]k, j = 1, 2 (7.5) Thus, the torque exerted by the solar flap-j on the satellite is obtained as Assuming that the cross-sectional area of the flap and the distance between the system center of mass S and the center of pressure for both the solar flaps are the same (i.e. A j = A, r j = r), the components of the total solar torque about the satellite body axes 148 Chapter 7. Fault Tolerant Satellite Attitude Control Using Solar Radiation Pressure / Aerodynamic Drag can be written as τ s = τ s 1 + τ s 2 = 2ρ S pAr [| s 1 · n 1 | ( s 1 · n 1 ) cos β 1 − | s 2 · n 2 | ( s 2 · n 2 ) cos β 2 ] (7.7) Let α be the orientation of the satellite with respect to the inertial axis Y 1 (Fig.7.1). We have the following relation where θ = Ωt is the orbital angle, and Ω = µ/R 3 is the orbital rate. Substituting, α L = α − θ, in τ g and τ s in Eq. (7.2) and Eq. (7.7) , respectively, and expressing the derivatives with respect to θ and using following relations: the resulting governing equation of motion of the system is obtained as follows: Here sgn (f) the denotes signum function. Also the solar aspect angle (ψ) is a function of θ and varies from 0 to 2π radians in one year.

Design of Control Law
In this section, an adaptive sliding mode control law is derived. Adaptive control methodology is employed to deal with the uncertainties due to unknown or slowly time-varying parameters. Here the solar parameters C 1 and , C 2 and the satellite mass distribution parameter K are assumed to be unknown and are estimated online and then used in the control law.

Pitch trajectory tracking
A specific trajectory is assumed for the purpose of attitude tracking. A simple polynomial based trajectory to achieve final attitude orientation is considered.

Hurwitz filter trajectory
The following fourth order polynomial is adopted since the control law requires a third order desired pitch response.
Since we require that the tracking error converge to zero without oscillations, the characteristic polynomial given by the sliding plane (Eq. (7.13)) must be critically damped and it should have two real roots. This would ensure that oscillations or overshoots do not take place neither before nor after the sliding plane is reached. The condition for critically damped second order system is given by λ 2 = 2 λ 1 (7.44) and the parameters λ 1 and λ 2 must be strictly positive constants to make the system stable in the sliding manifold.

Performance Evaluation-Pitch Attitude Control 155
where A d is the adaptive parameter estimation error given by The estimation error A d will converge to zero as the estimated parameters converge to their original value. The preceding Eq. (7.46) is a third order linear differential equation of error dynamics and it does not contain any external disturbance term. This error equation is asymptotically stable as θ → ∞, and the tracking errorα(θ) → 0. Furthermore, the term η sgn (S) forces the error dynamics to stay onto the sliding plane which makes the system insensitive to system parameter uncertainty and disturbances.

Performance Evaluation-Pitch Attitude Control
To study the effectiveness and performance of the proposed adaptive controller, the detailed system response is numerically simulated using the set of governing equations of motion, Eq. (7.10) in conjugation with the proposed control laws given by Eq. (7.34) and (7.35).
The integration is carried out using International Mathematical and Statistical Library (IMSL) routine DDASPG based on the Petzold-Gear BDF method. Here we studied the attitude manoeuvre of the satellite starting from the initial attitude α 0 = 110 0 to the final desired attitude α d = 0 0 with an initial tracking error of 100. The simulation parameters used are shown in deflection angle is 10 • in the transient period and 6 • in the steady state period. The flap deflection in the steady state is to counter the periodic gravity gradient torque T g . Fig.7.2 also shows the estimates of the satellite mass distribution parameter and solar parameterŝ K,Ĉ 1 andĈ 2 respectively. The estimated parameters converge toK = 0.08,Ĉ 1 = 2.49 and C 2 = 2.52. The satellite pitch attitude tracking remains unaffected proving the robustness of the adaptive control scheme. The maximum solar flaps deflection |β j | max remains less than 6 • .

Solar flap failure
Next the performance of the system when one of the solar flaps fails, say solar flap 2 (i.e., C 2 = 0), is studied (Fig.7.3). Failure is simulated to occur at 0.3 orbit. But this failure information is not passed onto the controller. The defective solar flap then fails to generate any useful torque. Referring to Fig.7.3, the operational flap, flap 1, compensates this failure by increasing the angle of rotation to achieve the desired trajectory and to maintain the attitude of the satellite. The satellite pitch attitude tracking remains unaffected. Note in this case the system using sliding mode control without adaptation fails to meet the desired pitch attitude and the solar flap deflection deviates to unreasonable orientations (Fig.7.4).

Actuator stuck fault
Next, we consider a fault scenario in which the actuator gets stuck at a particular position, i.e., the solar flap 2 stops abruptly at 0.3 orbit (Fig.7.5). It is found that β 1 compensates for the fault by overshooting to a maximum value of 40 • as soon as β 2 is jammed. But β 1 eventually settles with a maximum deflection of 17 • after the satellite pitch attitude reaches the desired angle. Thus, even when one of the actuators is stuck in position, the proposed fault-tolerant controller demonstrated excellent performance in providing desired satellite attitude control.
In the case of sliding mode without adaptation, the actuator stuck fault degrades the system performance and is unable to provide the desired response (Fig.7.6). The advantage of using an adaptive sliding mode control is very much evident in the presence of actuator fault and failures.

Actuator loss of effectiveness
In this fault scenario, the actuator has a loss of its control effectiveness (i.e., impaired control surfaces). As a result of this fault, while executing manoeuvres, the torque produced by the defective solar panel is considerably less than its nominal value. Here we have simulated an actuator performance degradation of 90% in solar flap 1 (Fig.7.7). The operational solar flap 2 compensates for this loss in effectiveness of solar flap 1 by increased deflection to maintain the desired satellite attitude. The maximum deflection in solar flap 2, |β 2 | max , is 13.5 • . Furthermore the pitch tracking remains unaffected.

Three Axis Satellite Attitude Control
In this section, the efficacy of the adaptive control methodologies developed in Chapter 5 and 6 is examined in the presence of the faults and failures. The proposed three axis satellite attitude control system is subjected to similar faults and failures as in Section 7.4, and the system response is analyzed.

Satellite Attitude Control Using Solar Radiation Pressure
First, the performance of the proposed method of three axis satellite attitude control using solar radiation pressure in evaluated in the presence of faults or failures. The system equations of motion is modified by replacing the common term used to represent solar parameter (C srp ) with individual solar parameter terms (C srp j ) corresponding to the four solar flaps (j = 1 − 4) associated with the proposed system configuration (Fig. (6.1)). The resulting control law is given as

Actuator stuck fault
Next, the fault caused due to the actuator remaining stuck at a particular position is considered. The solar flap 4 stops abruptly at 0.5 orbit (Fig.7.10). It can be observed that the solar flap 3 (ς 1 ) compensates for this fault as soon as ς 2 is jammed. It can be observed from Fig.7.10, that the orientation of ς 1 increases from 0.5 orbit to a maximum value of 10.5 • . The controller demonstrates excellent performance in stabilizing the satellite even with one of the actuators stuck in position in this underactuated system configuration.

Actuator loss of effectiveness
Systems performance under loss of control effectiveness in the actuator is examined next.
A 90% performance degradation in solar flap 4 (ς 2 ) is simulated (Fig.7.11). The fully operational solar flap 3 compensates by increased deflection to achieve a smooth stabilization of the satellite attitude. The maximum deflection |ς 1 | max is 22.5 • .

Satellite Attitude Control Using Aerodynamic Drag
The fault tolerant system performance of a satellite controlled using aerodynamic drag is evaluated in this section. The control law is developed by modifying the term used to represent uncertainties (B) with individual parameters for each of the four drag plates associated with the proposed system configuration (Fig. (5.1). The resulting control law is given as

Drag plate failure
First, we evaluate the system performance in the case of failure of one of the drag plate.
Drag plate 4 (β 4 ) is assumed to have failed after 0.3 orbit. Referring to Fig.7.12, it can be observed that the operational flap in the corresponding axis, flap 3 (β 3 ), compensates this failure by increasing its orientation to achieve smooth satellite attitude stabilization.

Actuator stuck fault
Next, the fault caused due to the actuator remaining stuck at a particular position is considered. The drag plate 3 (β 3 ) is assumed to have stopped abruptly at 0.5 orbit (Fig.7.13).
It can be observed that the oppositely places drag plate β 4 compensates for this fault as soon as β 3 is jammed. The maximum value of the drag plate orientation is |β 4 | max is 20.5 • . The controller demonstrates excellent performance in stabilizing the satellite even when with one of the actuators stuck in position in this underactuated system configuration.

Actuator loss of effectiveness
Systems performance under loss of control effectiveness in the actuator is examined next. A 90% performance degradation in actuator 4 is simulated (Fig.7.14). The operational drag plate 3 in the corresponding axis compensates by increased deflection to achieve a smooth stabilization of the satellite attitude. The maximum deflection in solar flap 2, |β 2 | max , is 20.5 • .

Summary
Results from chapters 5 and 6 confirmed the robustness of the proposed attitude control strategy in the presence of system disturbances and uncertainties. This chapter examines system performance of the proposed attitude control systems in the presence of some possible failure or fault scenarios with respect to the solar flaps or drag plates respectively.
First, the pitch attitude control of a satellite controlled using solar radiation pressure in the presence of different faults or failure is examined. A comparison to a similar control technique without adaptive control is also carried out. Compared to sliding mode control without adaptation, the proposed adaptive sliding mode control results in high attitude performance especially in actuator fault and failure scenarios. The performance of the proposed three axis satellite attitude control methodologies using aerodynamic drag or solar 7.6. Summary

Chapter 8
Conclusions F ormation flying of multiple satellites is an enabling technology for many future space missions including deep space as well as Earth observation. As mentioned in Chapter 1, achieving the mission objectives with minimum fuel expenditure in the presence of various external disturbances and uncertainties is one of the challenges in coordinated satellite formation flying. In a typical satellite control system, the twin environmental forces of aerodynamic drag and solar radiation pressure constitute two of these disturbance forces. However, in this thesis we analyze the use of aerodynamic drag or SRP for satellite formation and attitude control and use of these forces is examined in detail to achieve multiple satellite formation flying as well as satellite attitude control. Satellite formation and attitude control in low Earth orbit is achieved using aerodynamic drag, while solar radiation pressure is used as the control input in the geostationary orbit. Nonlinear adaptive control strategies that are robust to parameter and system uncertainties and external disturbances are developed to accomplish the proposed satellite control technique. This chapter summaries the major results and contributions of this dissertation, and discusses some recommended future research efforts to extend the current work.

Contributions Outline
Existing literature on the use of the twin environmental forces for satellite control focuses on the development of linear control techniques and is evaluated for a limited operating range.
In this thesis, variable structure based nonlinear control algorithms that are capable of accurate trajectory tracking, fast satellite maneuvering and reconfiguration are developed.
The proposed methodology is evaluated for a wide range of operating conditions in a more realistic space environment. The problem of controlling single as well as multiple follower satellites using aerodynamic drag or solar radiation pressure in a leader-follower configuration is evaluated. The following is a brief summary of the main contributions of this thesis: • Underactuated Satellite Control Use of aerodynamic drag or SRP for satellite formation or attitude control represents an underactuated control problem, and the existing literature proposes hybrid controllers or proposes control methodologies based on linear techniques resulting in restricted range of operation. Here we approach this as an underactuated problem and develop control algorithms that properly address this scenario and provide stability proofs for the resulting closed-loop system.

• Sliding Mode Control
To address the uncertainties associated with the twin environmental forces, a robust control methodology is of paramount importance. Here we employ the sliding mode control algorithm to address these issues. Higher order sliding mode control technique is also implemented to counter the non-affine structure of the control inputs generated by aerodynamic drag or SRP.

• Adaptive Control
Due to the lack of precise knowledge of all the parameters in the environmental models and its time varying nature, adaptive control laws are developed. Here the unknown parameters are estimated online and then used in the feedback control law.
A brief discussion of these contributions are provided in the following sections.

Satellite Formation Control using Aerodynamic Drag
The existing literature on the use of aerodynamic drag for satellite formation flying in LEO is based on simplified system dynamics and environmental models. Also, these proposed control algorithms are based on the linearized plant models, and are validated over a limited range of operation. We start the investigation by considering a single follower satellite in the leader-follower formation system. The satellites in the formation are assumed 8.1. Contributions Outline 171 to be attached with maneuverable drag plates. A nonlinear adaptive control algorithm based on sliding mode control methodology to maintain the satellite formation is developed. Different formation scenarios including different types of formations and formation reconfiguration maneuvers are evaluated. The effect of navigation and initial offset errors on the formation system are also examined. Finally, the concept of using aerodynamic drag for satellite formation flying is extended to the case of multiple follower satellites. A heuristic control technique is proposed to maintain the multiple satellite formation system. The viability of this heuristic control methodology for different formation scenarios is evaluated using numerical simulation including Monte Carlo simulation. All numerical simulations are performed on a high fidelity satellite dynamics and environmental model. The effect of variations in the system parameters such as variation in the density and altitude, on the system performance is also evaluated. Overall, the theoretical and simulation results establish the robustness of the proposed satellite formation control scheme and can be considered in future satellite formation systems including multiple satellite formation flying systems.

Satellite Formation Control using Solar Radiation Pressure
Next, the application of solar radiation pressure for satellite formation control is examined. A formation flying system with the satellites attached with maneuverable solar flaps is proposed. In the case of SRP, in-plane and out-of-plane control is achievable by the appropriate orientation of the solar flaps. From a control perspective, the system with two control angles represents an underactuated control problem. The proposed system is modified into a fully actuated control problem by incorporating an additional input in the form of change in area of the solar flaps. Nonlinear adaptive control algorithm based on sliding mode is developed to accomplish satellite formation control. The system response to different types of desired formation is examined. The performance of the proposed system in the presence of uncertainties including solar aspect angle ψ, is also examined. Initial offset and navigation errors are introduced and the performance of the formation system in this scenario is evaluated. Finally, the concept of using SRP for satellite formation flying is extended to the multiple follower satellite system. A heuristic control methodology is proposed to maintain the multiple satellite formation system, and evaluated for all the desired formations. The numerical simulations are performed on a high fidelity satellite dynamics and environmental model. Overall, the theoretical and simulation results establish the robustness of the proposed control system. Combined with the recent developments in solar sail technology this study will help in the advancement of the proposed methodology of using SRP for satellite formation control.

Satellite Attitude Control using Aerodynamic Drag
Use of aerodynamic drag for satellite attitude control is examined. The proposed system comprises of a satellite with two pairs of oppositely placed drag plates, and the satellite attitude is controlled by the suitable rotation of these drag plates. One set of drag plates is attached along the x-axis and another along the z-axis. Hence there is no explicit control available in the roll axis. This underactuated system configuration along with the proposed nonlinear control algorithm based on the theory of sliding mode generates the required torque to achieve complete three axis attitude control of the satellite. The adaptive nonlinear control algorithm achieves fast system stabilization and also provides robustness to system uncertainties. A high-fidelity nonlinear model is used to carry out the numerical simulation and the system performance is analyzed for a variety of scenarios including large initial conditions and unstable gravity gradient inertia configuration. Results of the numerical simulation shows that the proposed attitude control strategy is robust against disturbances and uncertainties. Thus, the proposed nonlinear adaptive controller can be considered as an ideal candidate for satellite attitude control in future LEO space missions.

Satellite Attitude Control using Solar Radiation Pressure
Feasibility of achieving three axis satellite attitude control using solar radiation pressure is investigated. The proposed system consists of two pairs of oppositely placed lightweight solar flaps. The proposed system configuration provides explicit control authority in the pitch and roll axis of the satellite. Using nonlinear control strategy, the complete three axis attitude control is achieved. The synthesized closed-loop control law based on adaptive sliding mode control rotates the solar flaps suitably to obtain proper SRP torque for desired attitude performance. The rotation angle of solar flaps is continuously adjusted as per the control laws. The efficacy of the control methodology is tested on a nonlinear system model considering various uncertainties in the system parameters including satellite inertia matrix and solar flaps. Results of the numerical simulation shows that the proposed attitude control strategy is robust against disturbances and uncertainties as well as very effective with initial attitude tracking errors. Based on the numerical simulations, the following main conclusions can be drawn: a) The controller exhibits excellent performance in the presence of uncertainties in system parameters including solar aspect angle ψ, and stabilizes the desired satellite attitude rapidly. b) The controller shows robust performance in the presence of external disturbances. Thus, the proposed nonlinear adaptive controller can be used for satellite attitude control in future geostationary space missions.

Fault Tolerant Satellite Attitude Control Using Solar Radiation Pressure / Aerodynamic Drag
The performance of the proposed system in the presence of failure scenarios such as failure of one of the drag plates/solar flaps, one of the drag plates/solar flaps getting blocked or jammed at a particular position is also examined. First, an adaptive control algorithm for pitch attitude control of an SRP based satellite is evaluated for the different fault/failure scenarios. The pitch attitude tracking was not affected by these faults/failures and demonstrated robust performance. Next, The efficacy of the three axis attitude control system for both aerodynamic drag and SRP based satellite is evaluated. The fault/failures did not cause any disturbance on the attitude of the satellite, and the different fault/failures were compensated by the healthy drag plates/solar flaps on the satellite. A comparison between the system controlled by an adaptive control algorithm and a non-adaptive control algorithm is performed to validate the robustness of the prosed control technique. The control law without adaptation was not able to handle the fault/failure scenarios leading to an unstable satellite attitude orientation.

Limitations of the proposed technique
In this section it is also appropriate to mention some of the limitation/disadvantages in using aerodynamic drag or solar radiation pressure for satellite control. With respect to satellite formation flying, the achievable relative positions cannot be controlled to a very high accuracy as compared to a SFF controlled using external thrusters. In the case of aerodynamic drag for SFF, cross track drift is difficult to control due to lack of control authority in that axis. Since aerodynamic drag is a retarding force, it will cause the satellite to loose altitude during the satellite mission, making the need for orbit raising.
In the case of SRP, the solar flaps can undergo optical parameter degradation due to continuous exposure to sunlight resulting in decreased efficiency in the performance of the control system. During formation reconfiguration overshoot is observed in the relative errors during the initial part of the formation stabilization. This is a result of the trade-off between the maximum allowable orientation of the drag plates and the faster convergence of relative errors. Similarly in the case of satellite attitude control highly precise pointing accuracy in the range of less than 0.01 • in all axes is not achievable with the proposed satellite attitude control system. The increased exposed surface due to the drag plates and solar flaps can also result in micro-meteorite collisions, which in turn might affect the control performance.

Future Work
This study proposes techniques to utilize the aerodynamic drag or solar radiation pressure for satellite formation or attitude control. Theoretical results are examined through numerical simulation. There remain several areas yet to be explored. Some of these area for future work are outlined as follows: Here we consider satellite formation control and attitude control as two individual problems. During satellite formation flying, it is assumed that the attitude of the satellite is controlled, and during satellite attitude control, formation maintenance is not considered.
Hence simultaneous satellite formation as well its attitude control will be an interesting topic for future research.

Future Work 175
We propose satellite control using aerodynamic drag in low Earth orbit and solar radiation pressure in the geostationary orbits. The feasibility of using solar radiation pressure in LEO can be examined in detail, and this is especially promising for out-of-plane control during satellite formation in LEO. This can help achieve complete "propellant-less" formation control.
The drag plates and solar flaps considered here are assumed to be perfectly flat. The effect of billowing on the system performance can also be examined in detail. In the case of multiple satellite formation system, the control architecture needs to address the issue of collision avoidance between satellites in the formation. This is very important for a close formation flying scenario.