COUPLED ORBITAL AND ATTITUDE CONTROL SIMULATION

Transcription

1 COUPLED ORBITAL AND ATTITUDE CONTROL SIMULATION Scott E. Lennox AOE 5984: Advanced Attitude Spacecraft Dynamics and Control December 12, 23 INTRODUCTION In the last few years the space industry has started to change its focus of single large satellite missions to the use of many smaller satellites flying in formation. Formation flying presents many interesting and difficult problems that have not been dealt with in the past. One of these topics is the idea of coupling the spacecraft s attitude and orbital control systems. Coupling the attitude and orbital control systems is required for some of these spacecraft because of the physical constraints that are enforced on the spacecraft as well as the operational constraints to keep the formation together. An example of a physical constraint is the reduced amount of space and mass allotted for the orbital propulsion system. It might not be possible for smaller spacecraft to have complete controllability that a larger spacecraft might have been able to poses. In many cases there might be one or two thrusters that need to be reoriented in the correct direction before an orbital maneuver can be executed. Many of these small spacecraft are using low thrust propulsion systems which require almost continuous thrust. In situations like these, we do not want the propulsion system firing in the wrong direction and sending the spacecraft in a direction that was not intended. Firing the thrusters in the wrong direction is a major reason to couple the attitude and orbital control system. The operations of a formation may also require the spacecraft to have a 1

2 high relative position constraint. This relative position constraint requires a low thrust propulsion system to be able to change the spacecraft s position with a high amount of accuracy on the thrust direction. This high accuracy is coupled with the attitude control system accuracy, thus the need for a coupling of the attitude and control systems. The coupling of the attitude and orbital control systems is a relatively new concept and there are only a few published papers. The first paper published on coupled attitude and orbital control was written by Wang and Hadaegh in They derive and implement the attitude and orbital control laws in a simulation of microspacecraft flying in formations. The simulations are concerned with formation-keeping and relative attitude alignment. The relative attitude alignment of the microsatellites is derived by finding the relative attitude of each microsatellite in the inertial frame. 1 This paper did not discuss if the coupled control system could accomplish formation-maneuvering. This paper is the first step in coupling the attitude and orbital control systems. Naasz et al. discussed and performed simulations of an orbital feedback control law and a magnetic torque coil attitude control system. 2 They applied the control laws to the Virginia Tech Ionospheric Scintillation Measurement Mission, a.k.a. HokieSat, which is part of the Ionospheric Observation Nanosatellite Formation (ION-F) project. 2 The ION-F project will perform formation flying demonstrations while collecting scientific measurements. The orbital feedback control law is proven to be globally asymptotically stable in Ref. 3 and Ref. 4. The attitude control system is described in Ref. 5 and Ref. 6. Further research needs to be completed before a coupled attitude and orbital control system can be used for formation flying missions. For most space missions, orbital maneuvering is controlled by the ground station. The following two examples of orbital controllers could be used for autonomous orbital control. Ilgen develops a Lyapunov-optimal feedback control law for orbital maneuvers. 7 This control law uses Gauss s form of lagrange s planetary equations in classical and equinoctial orbital-element forms. Naasz and Hall 3,4 develop a nonlinear Lyapunov-based control law with mean-motion control to perform autonomous 2

3 orbital maneuvers for formation flying. This control law can be used for formation-keeping and formation-maneuvering. The dynamics of a spacecraft are nonlinear so it reasonable to assume that a nonlinear controller would be a more effective solution to control a spacecraft s attitude motion than a linear controller. Tsiotras presents eight different nonlinear feedback control laws using Lyapunov functions with a quadratic and logarithmic terms. These control laws use Euler Parameters, Cayley-Rodrigues Parameters and Direction Cosines. These controllers are expanded to include Modified Rofrigues Parameters in Ref. 8. Hall el al. uses a Modified Rodrigues Parameter Lyapunov function to derive three attitude tracking controllers using thrusters and momentum wheels. 8 The three controllers were proven to be globally asymptotically stable using LaSalle s Theorem. Schaub et al. uses a linear closed loop control law to model a spacecraft s nonlinear dynamics. 9 The linear closed loop control law is found using an open loop nonlinear control law. An adaptive control law is developed to enforce the closed loop dynamics with large knowledge errors in the moments of inertia and external disturbances. Xing and Parvez derive a nonlinear Lyapunov controller and a robust sliding controller for the tracking control problem. 1 They convert the tracking control problem into a regulator problem using relative attitude state equations. Transforming the tracking problem into a regulator problem simplifies the design procedure of the controllers. 1 In this paper we present an attitude estimator (orbital controller), two attitude controllers, and an orbital estimator (orbit propagator). We then couple these controllers together to preform coupled attitude and orbital maneuvers simulations. SYSTEM We consider the system of two spacecraft, a passive target spacecraft and a controllable rendezvous spacecraft. The target spacecraft is in a parking orbit that is known to the rendezvous 3

4 spacecraft. The rendezvous spacecraft has one variable thrust thruster and a three-axis momentum wheel system. We also assume an ideal space environment without disturbance forces or torques. The coupled attitude and orbital control simulation is comprised of two major functions and two minor functions. The two major functions are the attitude estimator or orbital controller and the nonlinear attitude controller. The two minor functions are the bang-bang linear attitude controller and orbital estimator. We will discuss each of these functions in the following sections. ATTITUDE ESTIMATOR The attitude estimator that we use in this study was developed by Naasz and Hall in Ref. 3 and Ref. 4. Naasz and Hall develop a nonlinear Lyapunov-based control law and a mean motion control strategy. The derivation of the control law begins with the equations of motion for a point-mass satellite 11 r = µ r 3 r + a p (1) where r is the position vector from the mass center of the primary body to the satellite, µ is the gravitational parameter, and a p is the perturbation accelerations. If we set a p = and write the equations of motion in terms of orbital elements we have Gauss form of the Lagrange s planetary equations 12 da dt de dt di dt dω dt dω dt dm dt = 2a2 h ( e sin νu r + p ) r u θ = 1 h (p sin νu r + [(p + r) cos ν + re] u θ ) (3) = r cos θ u h h (4) = r sin θ h sin i u h (5) = 1 he [ p cos νu r + (p + r) sin νu θ ] r sin θ cos i h sin i = n + b ahe [(p cos ν 2re) u r (p + r) sin νu θ ] (7) (2) (6) 4

5 where a is the semi-major axis, e is the eccentricity, i is the inclination, Ω is the right ascension of the ascending node, ω is the argument of periapse, M is the mean anomaly, ν is the true anomaly, θ is the argument of latitude, n is the mean motion, p is the semi-latus rectum, h is the angular momentum, b is the semi-minor axis, u r, u θ, and u h are the radial, transverse, and the orbit normal control. These equations can also be written as œ = f (œ) + G (œ) u (8) where œ is the vector of orbital elements, [a e i Ω ω M] T, G (œ) is the input matrix found using equations (2-7), and u is the vector of controls [u r u θ u h ] T. The equations of motion of the first five orbital elements are η = Gu (9) where η = a a e e i i Ω Ω ω ω = δa δe δi δω δω (1) where ( ) is the target element and G = 2a 2 e sin ν h p sin ν h p cos ν he 2a 2 p hr (p+r) cos ν+re h (p+r) sin ν he r cos (ω+ν) h r sin (ω+ν) h sin i r sin (ω+ν) cos i h sin i u r u θ u h (11) 5

6 Naasz and Hall find a control using a Lyapunov function. 4 The proof of globally asymptotic stability for the control is performed in Ref. 4. The control law is u = G T Kη = 2a 2 e sin ν h p sin ν h p cos ν he 2a 2 p hr (p+r) cos ν+re h (p+r) sin ν he r cos (ω+ν) h r sin (ω+ν) h sin i r sin (ω+ν) cos i h sin i T K a δa K e δe K i δi K Ω δω Kωδω (12) where K a, K e, K i, K Ω, and K ω are positive gains. The angle errors δω and δω are defined between π and π. The mean motion control is accomplished by defining a new target semi-major axis, a, which forces the mean anomaly error to zero a = ( K n δm + 1 ) 2/3 (13) a 3/2 where K n is a positive gain and δm is defined between π and π. We replace δm with δθ in application, so that the mean motion control properly positions the spacecraft within the orbital plane. 3 The Lyapunov-based control law, equation (12), and the mean motion control, equation (13), allows for feedback control for spacecraft orbital maneuvers. 4 Naasz and Hall develop a gain selection method in Ref. 3 and Ref. 4. The gains for the attitude estimator are found using K a = h 2 4a 2 (1 + e) 2 1 t t (14) K e = h2 1 4p 2 (15) t t [ ] 2 h + eh cos (ω + arcsin e sin ω) K i = p ( 1 + e 2 sin 2 ω ) 1 (16) t t K Ω = K ω = e2 h 2 [ h sin i ( 1 + e sin (ω + arcsin e cos ω)) p 2 (1 e2 4 ] 2 1 t t (17) p (1 e 2 cos 2 ω) ) 1 (18) t t where t t is the length of the thruster firing. K n is chosen depending on how aggressively we want to correct the argument of latitude error. 4 6

7 ATTITUDE CONTROLLERS This section is devoted to the derivation of the bang-bang linear attitude controller and the Lyapunov nonlinear attitude controller that is used by the rendezvous spacecraft. To be able to derive these controllers, we need to define the rotational dynamics and kinematics of the system. Dynamics The rotational dynamics of the rendezvous spacecraft is defined in Ref. 8. The dynamics of the system are described as ḣ B = h B J 1 (h B Ah a ) + g e (19) ḣ a = g e (2) h B = Iω B + AI s ω s (21) where h B is the system angular momentum vector, I is the 3 3 moment of inertia matrix of the entire spacecraft, I s is the 3 3 axial moment of inertia matrix for the momentum wheels, A is the 3 3 matrix containing the axial unit vectors of the momentum wheels, h a is the 3 1 matrix of the axial angular momenta of the wheels, g e is the 3 1 matrix of the external torques applied to the spacecraft, g a is the 3 1 matrix of the internal torques applied to momentum wheels, ω B is the 3 1 angular velocity matrix of the body frame expressed in the inertial frame, ω s is the 3 1 axial angular velocity matrix of the momentum wheels with respect to the body, and J is the positive definite inertia-like matrix defined as J = I AI s A T (22) We can now define ω B and h a using equations (21) and (22) ω B = J 1 (h B Ah a ) (23) h a = I s A T ω B + I s ω s (24) 7

8 Kinematics The kinematics of the rendezvous spacecraft is also defined by Hall et al. in Ref. 8. We will use Modified Rodrigues Parameters (MRPs) 13 to describe the kinematics of the rendezvous spacecraft. Modified Rodrigues Parameters are defined as σ = ê tan ( ) Φ 4 where ê is the unit vector along the Euler axis and Φ is the Euler angle. 13 (25) Using MRPs, the differential equations for the kinematics are σ = G (σ) ω (26) where G (σ) = 1 2 ( I 3 + σ + σσ T 1 + ) σt σ I 3 2 (27) and I 3 is the 3 3 identity matrix. Hall et al. show that differential equation for the error kinematics is δ σ = G (δσ) δω (28) where δσ is the rotational error between the desired reference frame and the current attitude of the spacecraft, and δω is the difference in the angular velocities of the desired attitude and the current attitude. 8 Bang-Bang Linear Controller The bang-bang attitude controller is used to obtain an estimate of the amount of time needed to complete the attitude maneuver using the nonlinear attitude controller. The bang-bang controller is derived using Euler s Law 14 ḣ = g (29) 8

9 where h is the angular momentum about the mass center of the system and g is the net applied moment about the mass center. We constrain the problem to be a planar problem which leads to ḣ = g I max θ = gmax (3) where I max is the maximum moment of inertia of the system, θ is the angular acceleration about the moment of inertia axis, and g max is the maximum applied torque that momentum wheels can produce on the system. By integrating equation (3) twice we obtain θ θ o I max θ t I max θdθ = g max dt θ o t o ( ) I max θ θo = g max (t t o ) ( ) dθ t dt θ o dθ = g max (t t o ) dt t o I max (θ θ o ) I max θo (t t o ) = 1 2 g max (t t o ) 2 g max t o (t t o ) (31) We define t o = and we can rearrange equation (31) into the following form I max (θ θ o ) = 1 2 g max t 2 + I max θo t (32) We can assume that θ o =, θ =, and θ is defined as a ramping function. At t 2 there is a discontinuity in θ. This discontinuity leads us to just examine the first half of the maneuver, where ( ) θo I max 2 θ o = 1 2 g max ( ) 2 t (33) 2 Solving equation (33) for t produces t = 2 2I max g max ( ) θo 2 θ o (34) Equation (34) provides an estimate of the time required to complete an attitude maneuver using the nonlinear controller. Lyapunov Nonlinear Controller We use Lyapunov s method to find a nonlinear attitude controller. Lyapunov stated that the solution x of the system ẋ = f (x), f () =, is asymptotically stable if there exists a positive- 9

10 definite function V (x) such that ( V x ) T f (x) is negative-definite. 15 Hall et al. present the following candidate Lyapunov function in Ref. 8 V = 1 2 δωt Kδω + 2k 1 ln(1 + δσ T δσ) (35) where K = K T, and is a positive-definite matrix, and k 1 >. This is a positive-definite and unbounded function in terms of the errors δω and δσ. 8 The derivative of V calculation yields V = δω T Kδ ω + 4k 1 δ σ T δσ 1 + δσ T δσ (36) Using equation (28) we can rewrite V as V = δω T Kδ σ + 4k 1 δσ T G (δσ) δω 1 + δσ T δσ (37) Using equation (27) we can derive the following identity 4k 1 δσ T G(δσ)δω 1 + δσ T δσ = k 1 δω T δσ (38) Plugging this identity, equation (38), into equation (37) we obtain V = δω T Kδ ω + k 1 δω T δσ (39) = δω T (Kδ ω + k 1 δσ) (4) Choosing K = J and using equations (19), (2), and (23), we obtain V ( ) = δω T ḣ B Aḣa + k 1 δσ = δω T ( h B J 1 (h B Ah a ) + g e Ag a + k 1 δσ ) (41) We need to pick g e and g a for V to be negative-definite. For this study we use g e = g a = A 1 ( h B J 1 (h B Ah a ) + g e + k 1 δσ + k 2 δω ) (42) When we plug equation (42) into equation (41), we obtain V = k 2 δω T δω (43) 1

11 with k 2 >. Using this g a, equation (42), V is negative semi-definite and bounded. To prove that V is negative-definite we need to use LaSalle s Theorem. 16 Equation (43) also yields lim δω = (44) t Since lim t δω =, we can conclude that lim t δ ω =. By examining equations (23), and (44), we can also conclude that lim t ḣ B =. This produces = J 1 Ah a = J 1 Ag a = J 1 k 1 δσ (45) From equation (45) we can see that lim δσ = (46) t We can now conclude that the dynamic and kinematic errors with the feedback control law,equation (42), are globally asymptotically stable. ORBITAL ESTIMATOR The orbital estimator is used to propagate the orbits of both the target spacecraft and the rendezvous spacecraft. The orbital estimator uses the f and g expressions in terms of eccentric anomaly to propagate the spacecraft s orbit. 11 The f and g method requires the input of the initial position (r o ) and velocity (v o ) of the spacecraft, and the propagation time ( t). The f and g equations are 11 f = 1 a (1 cos E) (47) r o a g = t 3 ( E sin E) (48) µ f µa sin E = (49) rr o ġ = 1 a (1 cos E) (5) r 11

12 where a is the semi-major axis of the orbit, E is the change in eccentric anomaly, r o is the magnitude of the initial position vector, and r is the magnitude of the finial position vector. The final position (r) and velocity (v) vectors are calculated using 11 r = fr o + gv o (51) v = fr o + ġv o (52) PROGRAM The coupled orbit and attitude maneuver is simulated using Matlab c. The overall simulation architecture is shown in Figure 1. The simulation begins by defining a position and velocity of the target and rendezvous spacecraft along with the attitude of the rendezvous spacecraft. These initial conditions are used as the first set of inputs into the control loop. The attitude estimator uses the position and velocity data to produce an optimal thrust magnitude and direction in the inertial frame. This thrust direction is converted into a desired attitude of the spacecraft. The bang-bang attitude controller uses the current attitude of the spacecraft and the desired attitude to calculate an approximate time to complete the attitude maneuver. This time, t 1, is used by the orbital estimator to determine the position and velocity of spacecraft at t 1. The new position and velocity data are used by the attitude estimator to determine the optimal thrust magnitude and direction at t 1. This thrust direction is used to determine the desired attitude at t 1. The Lyapunov attitude controller uses the attitude of the spacecraft at t o and the desired attitude at t 1 and runs until the time, t 1 is reached. The current attitude at t 1 is compared to the desired attitude at t 1 by the attitude check function. This function determines if the current attitude is within a set of error limits that the user specifies. If the current attitude is not within the error limits, then the current attitude and the position and velocity of the target and rendezvous spacecraft are used as the new inputs of the control loop. If the current attitude is within the error limits, then the thruster fires according to the current attitude. The thruster is considered an ideal thruster, so 12

13 Figure 1: The overall simulation architecture the magnitude of the thrust is variable and is determined by the attitude estimator. The thrust is applied to the rendezvous spacecraft in the form of a V over a user specified time interval. The orbital estimator uses the thrust time along with the current position and velocity of the spacecraft to determine the position and velocity of the spacecraft at t 2. The position and velocity at t 2 of both spacecraft and the attitude of the rendezvous spacecraft are used as the new inputs to the control loop. The user defines the number of orbits to simulate and the control loop keeps track of the total time of the simulation. RESULTS Before we can test the coupled orbital and attitude control simulation we need to test each controller separately. In the following sections we discuss the attitude estimator, the attitude controller, and the coupled orbital and attitude control technique simulations. 13

14 Attitude Estimator To test the orbital estimator, we need to assume that the spacecraft is able to thrust in any direction that is requested. The initial conditions for the simulation are a = 6823 (km) e =.1 i = 28 Ω = 135 ω = 9 ν = ( ) 2 da = de = di = dω = dω = dν = 6823 t t = 1 (sec) m = 1 (kg) K n =.5 where ( ) are the target spacecraft orbital elements, d( ) are the change in orbital elements of the rendezvous spacecraft compared to the target spacecraft, t t is the duration of time that the thruster will fire, m is the mass of the spacecraft, and K n is the mean motion control gain. Figure 2 shows the position error between the two spacecraft in the orbital frame. The position error between the two spacecraft becomes zero after approximately 14 orbits. Figure 3 shows the change in semimajor axis of the rendezvous spacecraft throughout the simulation. Figure 4 is the magnitude of the thrust applied to the rendezvous spacecraft. Figure 5 shows the thrust direction in the orbital frame. The thrust direction changes pretty rapidly over the course of the simulation. These figures are representative of similar simulations that we performed using different initial conditions. We conclude that the attitude estimator is effective in formation-keeping and formation-maneuvers. Attitude Controller We need to test the nonlinear attitude controller to insure globally asymptotic stability. The following initial conditions are used I = ( kgm 2 ) I s = ( kgm 2 ) A =

15 Figure 2: The position error in the orbital frame as seen by the rendezvous spacecraft. Figure 3: The change in semi-major axis of the rendezvous spacecraft(a) compared to the target spacecraft(a ). 15

16 Figure 4: The magnitude of thrust applied to the rendezvous spacecraft. Figure 5: The thrust direction of the rendezvous spacecraft in the orbital frame. 16

17 Figure 6: The change in the Modified Rodrigues Parameters of the spacecraft. ω i = ( ) rad s σ i = ω = ( ) rad s σ = k 1 = 1 k 2 = 2 where I is the moment of inertia matrix of the system, I s is the moment of inertia matrix of the momentum wheels, A is the axial unit vector matrix of the momentum wheels, ω i is the initial attitude, σ i is the initial angular velocity of the system, σ is the desired attitude, ω is the desired angular velocity, k 1 is the attitude gain, and k 2 is the angular velocity gain. Figure 6 shows how the attitude varies throughout the simulation. Figure 7 is the variation of the body angular velocity. Figure 8 is the applied torque provided by the momentum wheels throughout the simulation. This simulation is representative of the other attitude simulations. We can conclude that the nonlinear attitude controller is effective in controlling nonlinear attitude dynamics. 17

18 Figure 7: The change in the angular velocity of the spacecraft. Figure 8: The applied torque on the spacecraft. 18

19 Coupled Attitude and Orbital Control We have now proven analytically and numerically that the attitude estimator and nonlinear attitude controller are globally asymptotically stable. We can now couple these controllers. The initial conditions for the simulation are a = 6823 (km) e =.1 i = 28 Ω = 135 ω = 9 ν = ( ) 2 da = de = di = dω = dω = dν = 6823 t t = 1 (sec) m = 1 (kg) K n =.5 I = ( kgm 2 ) I s = ( kgm 2 ) A = ω i = ( ) rad s σ i = k 1 = 1 k 2 = 2 T = 1 where T is the orientation of the thruster in the body frame of the rendezvous spacecraft. Figure 9 shows the relative error between the two spacecraft in the orbital frame. It is very hard to see but the coupled controller takes a little longer to reach the target spacecraft than the attitude estimator controller (Figure 2). This is expected, but it is good that the change in time is small (approximately a half of an orbit). Figure 1 shows the variation of the semi-major axis over the simulation time. There is not much of a difference between this figure and Figure 3. The thrust magnitude can be seen in Figure 11. The values for the thrust are low, which is good for propulsion systems on formation flying spacecraft. Figure 12 shows the attitude error between the desired attitude and the current attitude that was found after the attitude maneuver was completed and the acceptable attitude error (5 in all directions for this simulation). The thruster did not fire when the attitude 19

20 Figure 9: The position error in the orbital frame as seen by the rendezvous spacecraft. (coupled control) error was greater than the acceptable attitude error. We conclude that the coupled attitude and orbital control can be used for formation flying missions while not significantly increasing the time to accomplish orbital maneuvers. CONCLUSIONS We have derived and proven asymptotic stability for a Lyapunov-based attitude estimator and a nonlinear Lyapunov attitude controller using momentum wheels. These controllers were used in conjunction with a linear attitude controller and an orbit estimator to create a coupled attitude and orbital control system. Simulations were completed which lead to the conclusion that a coupled control system will work for formation-keeping and formation-maneuvers for formation flying spacecraft missions. 2

21 Figure 1: The change in semi-major axis of the rendezvous spacecraft(a) compared to the target spacecraft(a ). (coupled control) Figure 11: The magnitude of thrust applied to the rendezvous spacecraft. (coupled control) 21

22 Figure 12: The magnitude of the error between the desired thrust direction and the actual thrust direction. REFERENCES [1] P. Wang and F. Hadaegh, Coordination and Control of Multiple Microspacecraft Moving in Formation, The Journal of the Astronautical Sciences, vol. 44, pp , July-September [2] B. J. Naasz, M. M. Berry, H. Y. Kim, and C. D. Hall, Integrated Orbit and Attitude Control for a Nanosatellite with Power Constraints, AAS/AIAA Space Flight Mechanics Conference, Ponce, Puerto Rico, February 9-12, 23. AAS 3-1. [3] B. J. Naasz, Classical Element Feedback Control for Spacecraft Orbital Maneuvers, Master s thesis, Virginia Polytechnic Institute and State University, Blacksburg, VA, May 22. [4] B. J. Naasz, Classical Element Feedbhack Control for Spacecraft Orbital Maneuvers, Journal of Guidance, Control and Dynamics, 24. (to appear). 22

23 [5] K. L. Makovec, A Nonlinear Magnetic Controller for Nanosatellite Applications, Master s thesis, Virginia Polytechnic Institute and State University, Blacksburg, VA, May 21. [6] K. L. Makovec, A. J. Turner, and C. D. Hall, Design and Implementation of a Nanosatellite Attitude Determination and Control System, in Proceedings of the 21 AAS/AIAA Astrodynamics Specialists Conference, Quebec City, Quebec, 21. [7] M. R. Ilgen, Low Thrust OTV Guidance Using Lyapunov Optimal Feedback Control Techniques, Advances in the Astronautical Sciences, vol. 85, no. Part 2, pp , [8] C. D. Hall, P. Tsiotras, and H. Shen, Tracking Rigid Body Motion Using Thrusters and Momentum Wheels, The Journal of the Astronautical Sciences, vol. 5, no. 3, 22. [9] H. Schaub, M. R. Akella, and J. L. Junkins, Adaptive Control of Nonlinear Attitude Motions Realizing Linear Closed Loop Dynamics, Journal of Guidance, Navigation and Control, vol. 24, January-February 21. [1] G. Q. Xing and S. A. Parvez, Nonlinear Attitude State Tracking Control for Spacecraft, Journal of Guidance, Control, and Dynamics, vol. 24, pp , May-June 21. [11] R. R. Bate, D. D. Mueller, and J. E. White, Fundamentals of Astrodynamics. Dover Publications, [12] R. H. Batin, An Introduction to the Mathematics and Methods of Astrodynamics, Revised Edition. American Institute of Aeronautics and Astronautics, [13] H. Schaub and J. L. Junkins, Analytical Mechanics of Space Systems. American Institute of Aeronautics and Astronautics, 23. [14] C. D. Hall, AOE 414 Spacecraft Dynamics and Control Lecture Notes. Available at chall/courses/aoe414/, 23. [15] P. C. Hughes, Spacecraft Attitude Dynamics. John Wiley & Sons, [16] H. K. Khalil, Nonlinear Systems. Macmillan Publishing Company,