official policy or position of the United States Air Force, Department of Defense, or

Transcription

1

2 The views expressed in this thesis are those of the author and do not reflect the official policy or position of the United States Air Force, Department of Defense, or the U.S. Government.

3 Acknowledgements I would like to thank Dr. Pol Spanos and Rice University for the opportunity to obtain a masters degree from Rice, and especially Dr. Spanos for his academic guidance and tutorship. Special than thanks to Dr. Nazareth Bedrossian for the opportunity to be a DLF. I enjoyed all of our conversations. I am comforted in knowing that if this engineering thing doesn t work out I may have a future in politics! I am extremely grateful for Dr. Renato Zanetti s guidance and assistance in completing this thesis. Every time I needed help, which was a lot, Renato found time to assist me. I m certain Renato is happy that this is the last time he will have to read one of my drafts! I am lucky to have a great family. I am not sure where I would be without their encouragement, support, and the occasional berating. I am very thankful for my almost wife Kristen. Her love, support, and all the healthy things she makes me eat have been essential. I would also like to say thanks to all my Air Force amigos who shared this experience. I am lucky to know you all. Lastly, thanks to the United States Air Force for allowing me to go to graduate school for the first to years of my career.

4 Contents Acknowledgements ii Abstract 1 1 Introduction Inter Planetary Superhighway Spacecraft Trajectory Optimization Contributions Agenda Permanent Solution Equations of Motion for the Circular Restricted Three-Body Problem Libration Points State Transition Matrix Linearization at the Libration Points Stability of the Libration Points Numerical Study of Linear Stability of the Libration Points in the Planar System Zero Velocity Curves in the Earth-Moon CRTBP Invariant Manifolds

5 v Linear and Nonlinear Flow Near Unstable and Stable Fixed Points Computing Manifolds Calculating The Minimum Change in Velocity Fuel Requirement V and Fuel Mass Requirements V and Thrusting Time Realized Trajectory Atmospheric Drag Jacchia Model Results Temporary Solution Equations of Motion Problem Constraints Sequential Quadratic Programming The Karush-Kuhn-Tucker Theorem Quadratic Functions Quadratic Programming (QP) Subproblem Active Set Method Runge-Kutta Method Upper and Lower Bounds Results FMINCON Settings Flight Time and Constraint Considerations Single and Double ATV Solutions

6 vi Gravity and Drag Perturbations Final Performance Results Propellant Mass vs Lifetime Future Work Conclusion 102 Bibliography 104

7 List of Figures 1.1 Earth Moon L 1 Manifolds The Restricted Three Body Problem Lagrange Points Zero Velocity Curve for C=1.1C Zero Velocity Curve for C=C Zero Velocity Curve for C=C Zero Velocity Curve for C=C Linear and Nonlinear Flow Near L Linear and Nonlinear Flow Near L Stable Manifold of a L 1 Lyapunov Orbit About Moon Unstable Manifold of a L 1 Lyapunov Orbit About Moon Stable Manifold of a L 1 Lyapunov Orbit About Earth Zero Velocity Curve for C = (m/s) Zero Velocity Curve for C = (m/s) Zero Velocity Curve for C = (m/s) Transfer from LEO to Lunar Orbit using Invariant Manifolds Transfer from LEO to Lunar Orbit using Invariant Manifolds Density vs. Altitude vii

8 viii 3.2 Hohmann Transfer Decay Time vs. Initial Altitude Area Consideration Maximum Radius Transfer The Active Set Method for a Simple QP Problem Feasible Region of Circle-to-Circle Transfers Minute Test Transfer with 2350kg of Propellant Control Histories for 100 Minute Test Transfer with 2350kg of Propellant Minute Test Transfer with 2350kg of Propellant Control Histories for 200 Minute Test Transfer with 2350kg of Propellant Control Histories for Single ATV 200 Minute Transfer Control Histories for Single ATV 250 Minute Transfer ATV Trajectory Inverse Square Gravity Model ATV Control Histories Inverse Square Gravity Model ATV Trajectory with Gravity and Drag Perturbations Velocity of Free Flying ISS Propellant Mass vs Lifetime

9 List of Tables 1.1 ISS Configuration Fuel Mass Results I SP = 400s Fuel Mass Results I SP = 10, 000s Thrusting Time Results I SP = 10, 000s Theoretical vs Realized L 1 Lunar Transfer Orbit Using VASIMR Fun Tol Comparison ATV Information Control Refinement Validation Mass vs Altitude Trade Off Performance for 2 ATV s

10 Abstract Trade Study of Decommissioning Strategies for the International Space Station by Eric H. Herbort This thesis evaluates decommissioning strategies for the International Space Station ISS. A permanent solution is attempted by employing energy efficient invariant manifolds that arise in the circular restricted three body problem CRTBP to transport the ISS from its low Earth orbit LEO to a lunar orbit. Although the invariant manifolds provide efficient transport, getting the the ISS onto the manifolds proves quite expensive, and the trajectories take too long to complete. Therefore a more practical, although temporary, solution consisting of an optimal re-boost maneuver with the European Space Agency s automated transfer vehicle ATV is proposed. The optimal re-boost trajectory is found using control parameterization and the sequential quadratic programming SQP algorithm. The model used for optimization takes into account the affects of atmospheric drag and gravity perturbations. The optimal reboost maneuver produces a satellite lifetime of approximately ninety-five years using a two ATV strategy.

11 Chapter 1 Introduction The International Space Station (ISS) represents a long-term cooperative effort between NASA, Canadian Space Agency, European Space Agency, Russian Space Agency, and Space Transportation Agency, and provides a long term orbiting platform for high-value scientific research in a micro-gravity environment for unprecedented scientific investigations. Research aboard the ISS benefits a wide variety of technologies on Earth from biotechnology to better methods for drilling oil wells. ISS also allows for the development of a human presence in space, and is a crucial experiment for future international cooperative space ventures [1]. The basic configuration of the ISS is described in Table 1.1. Note that the ISS has a mass of approximately 420 metric tons. This makes a decommissioning strategy for the ISS necessary and very difficult. However, an acceptable decommissioning strategy for ISS does not exist. NASA has examined several options for decommissioning the ISS. The options examined include disassembly and return to Earth using the Space Shuttle (itself now decommissioned), natural orbit decay with random reentry, boosting it to a higher altitude, and a controlled targeted deorbit to a remote ocean area. The cost to disassemble the ISS and return it to Earth is at least 27 Shuttle flights, to return 3

12 4 System Requirements Configuration Design Life 10+years Orbit Inclination 51.6 degrees Average Orbital Altitude 408 km Crew Size 6 Total Mass 420,000 kg First Element Launch November 1997 Total Flights Required Through Assembly 44 Permanent Human Presence Capability May 1998 Complete Assembly June 2011 Resupply Flights U.S. and International Partners Table 1.1: ISS Configuration only those components originally launched by the United States. Obviously, this strategy is no longer an option. Decommissioning through natural orbital decay and random reentry does not guarantee that debris surviving reentry would land in unpopulated areas. Decommissioning by boosting the ISS to a higher orbit was considered using two variations. The first variation considered use of only the assets aboard the ISS at the time it is decommissioned. This strategy results in only a slightly increased orbit altitude which translates to a negligible increase in satellite lifetime. The second variation employed the use of propulsion methods to escape the Earth s gravitational pull, but these methods impose large additional costs burdens. The currently proposed decommissioning approach is a controlled, targeted deorbit into a remote ocean area. The technical feasibility of this option has been discovered to be within the capability of systems already aboard the ISS. Using a debris model NASA estimates between 6 to 19 percent of the ISS (or 24, 260kg to 78, 570kg) would survive reentry. Some of these pieces could have enough kinetic energy to cause

13 5 damage to people and structures upon falling to Earth. The purpose of this thesis is to investigate decommissioning strategies and propose a solution that does not involve re-entry or the risks associated with re-entry. 1.1 Inter Planetary Superhighway For every three body system comprised of two large celestial bodies and a smaller body of negligible mass (e.g. satellite) there is a vast system of interconnected and winding tunnels and channels in space. This system is the interplanetary superhighway (IPS). Jules-Henri Poincaré is credited as the first person to notice the IPS. While calculating trajectories to and from the equilibrium points of the circular restricted three body problem CRTBP, Poincaré noticed that trajectories would always settle for a brief period of time on an orbit around the points [2]. For the CRTBP, the two larger bodies, referred to as the primaries, maintain a constant separation in circular orbits about their barycenter. The equilibrium points are commonly referred as Lagrange points after the French mathematician Joseph Louis Lagrange who calculated their locations in the eighteenth century [3]. The CRTBP has five stationary points, or Lagrange points, where the gravitational forces and rotational forces within the three body system are balanced. Three of the Lagrange points lie on the same axis as the two primaries, and they are referred to as the collinear Lagrange points. These points give rise to unstable halo orbits. These orbits are referred to as halo orbits, because, since Lagrange points are massless points in space, the trajectories appear to be orbiting nothing and cast a halo over the primaries. There are an infinite number of trajectories that asymptotically wind onto and off of the halo orbits of the collinear Lagrange points without requiring maneuvers. These trajectories form a tube-like surface, or manifolds, as seen in Figure 1.1, which displays the stable

14 6 and unstable manifolds of a two-dimensional Lyapunov orbit around Earth-Moon L 1 point. A Lyapunov orbit is a Lissajous orbit that is entirely in the plane of rotation of the two primaries. A Lissajous orbit a quasi-periodic orbital trajectory around a Lagrange point [4]. Figure 1.1: Earth Moon L 1 Manifolds The blue surface in Figure 1.1 is the stable manifold and it contains the trajectories that wind onto the Lyapunov orbit, and the red surface is the unstable manifold which contains the trajectories that wind off of the Lyapunov orbit. These manifolds are contained in the Hill s region. The Hill s region is the region where trajectories at a certain energy are contained, and the shape of Hill s region varies with the Jacobi constant C of the CRTBP [5]. The halo orbit serves as a natural corridor for passage between the neighborhood of one celestial region to another; thus the lunar L 1 point may act as a versatile hub for space transportation systems in the future [6]. These

15 7 tubes provide very efficient natural trajectories and much utility is found in them. Manifolds in the CRTBP have been used to explain the erratic trajectories of comets near Jupiter [7]. A remarkable implementation of manifold theory was guiding the International Cometary Explorer from a halo orbit to fly by the Giacobini-Zinner comet using very little fuel for orbital correction [8]. The IPS enables many mission possibilities. NASA is considering placing a Lunar Gateway Habitat in an orbit around the Earth-Moon L 1 Lagrange point. Due to the efficiency of travel by the IPS, the craft can be transfered between the L 1 and L 2 points with only slight maneuvering [9]. NASA believes this environment is an ideal next step for extended human presence in space beyond LEO. This thesis examines the possibility of using the IPS to take the ISS to the Lunar neighborhood via the lunar L 1 gateway as a decommissioning solution. 1.2 Spacecraft Trajectory Optimization The problem of spacecraft trajectory optimization can be stated simply as the determination of a spacecraft trajectory that satisfies some specified initial and terminal conditions required by the mission, while minimizing some important quantity. The quantities to be minimized are usually fuel, time, or both. Complications in finding optimum trajectories arise when there are discontinuities in the state variables, the terminal, initial, or both conditions are not known explicitly, there are time-dependent forces, or the basic structure of the optimal trajectory is subject to optimization, to name a few. The techniques for solving optimization problems fall into two broad categories: analytic and numeric procedures. Analytic techniques are the original approach for spacecraft trajectory optimization. The oldest and most useful analytical solution

16 8 is the Hohmann transfer trajectory, which is the fuel optimal way to go from one circular orbit to another coplanar circular orbit [10]. This thesis deals mostly with numeric procedures; however, the Hohmann transfer method is used as a measure of performance, and its optimality is shown analytically. Direct solutions of continuous optimal control problems transcribe a continuous problem into a parameter optimization problem. For problems structured in this manner, the equations of motion EOM that define the dynamical system are numerically integrated, and discretized control parameters that optimize the trajectory and satisfy the constraints are calculated [11]. The optimal control problem is thus converted to a nonlinear optimization programming problem where the control variables are the parameters that must satisfy the nonlinear constraints. Many recent optimal control methods reduce the problem to parameter optimization problems that are solved by a nonlinear programming NLP problem solver. One parameter optimization method, the method that yields the smallest number of parameters, parameterizes only the control variables and numerically integrate the system equations. Reducing the number of parameters reduces the number of variables that the NLP solver must solve for, which decreases the computational expense. The main advantage of implementing a direct method is that it does not require satisfying the necessary conditions of the Hamiltonian function or using adjoint variables [12]. The method used in this study for solving the NLP problem is the sequential quadratic programming SQP algorithm. The SQP method is perhaps one of the most effective methods for nonlinear constrained optimization. The algorithm uses an iterative approach to find a solution that minimizes the nonlinear problem via a Newton step [13]. An important characteristic of the SQP method is its strength when solving problems with significant nonlinearities in the constraints. One SQP method used to address the inequality constrained optimization problem is the active-

17 9 set method. The active-set method uses Lagrange multipliers and construction of an active set which allows the inequality constrained problem to be modeled as an equality constrained problem, which is much easier to solve. 1.3 Contributions This thesis analyzes the feasibility of using the dynamics in the CRTBP for the Earth- Moon system to transport the ISS from its low Earth orbit LEO to a permanent parking orbit. The minimum necessary changes in velocity to reach each Lagrange point are calculated, and a transfer using the L 1 point is shown to be the most efficient transfer route to a decommissioning destination. High thrust, low specific impulse, and low thrust, high specific impulse propulsion systems are considered for the transfer, and the low thrust, high specific impulse, system is shown to be the best method for the transfer. A trajectory taking the ISS from LEO to the lunar region using L 1 manifolds is calculated. The propellant and time of flight demands necessary to achieve the trajectory prove to be unrealizable, and a more reasonable, albeit temporary, solution is sought. Analysis of the LEO environment shows that satellite lifetime increases exponentially with increasing altitude. Therefore, a trajectory is developed to maximize the altitude of a terminal coplanar circular orbit. This thesis uses control parameterization to solve for the optimal control histories that realize this trajectory. The SQP algorithm with the active set method is the selected NLP solver for the algorithm developed. The European Space Agency s automated transfer vehicle ATV is the platform chosen for the re-boost maneuver. Contraints on the propulsion system are consistent with ATV specifications. The ATV is designed specifically for automated re-boost of the ISS which makes it an ideal selection.

18 10 Due to the difficulty of optimizing the control histories for an engine that is either burning at maximum thrust or completely off, the propulsion system is modeled as throttle-able in the optimization algorithm. A pulse width modulation scheme is then implemented to convert the throttle-able control histories into control histories that are implementable for the ATV. 1.4 Agenda A complete derivation of the equations of motion for the CRTBP is included. The associated Jacobian matrix and state transition matrix are also derived from the equations of motion. The equilibrium points of the equations of motion are found and their stability is analyzed. The Jacobi constant for the equations of motion is derived and used to generate zero velocity curves at various energy levels of the third body of negligible mass. The curves are analyzed and possible trajectories associated with the curves are discussed. A method for calculating the minimum v necessary to take the ISS out Earth s gravitational sphere of influence is derived and implemented to analyze the cost of different end of life trajectories for the ISS. A method for computing the global stable and unstable manifolds of the CRTBP libration points is derived and implemented. A trajectory taking the ISS from LEO to a lunar orbit via the stable manifold of the Earth-Moon L 1 point is computed, and the feasibility of implementing the trajectory is discussed. A discussion of the LEO environment and factors that contribute to limited satellite lifetime in the LEO environment are discussed, and estimated lifetimes for the ISS at initial circular orbits with varying altitudes are computed using STK software. The results of the lifetime calculations are combined with the optimization results to create a lifetime vs. fuel mass approximation.

19 11 Background information on the SQP algorithm and the active set method is provided, and the method is implemented to derive the control histories that maximize the final orbit of the ISS, with the specification that the ATV s engines are providing the thrust. Various test optimization calculations are accomplished in order to determine the best parameter settings and constraint structure for the algorithm. The resulting control histories are then refined so that the engine is restricted to being either on at its maximum thrust or off. The resulting trajectory is documented as well as its adherence to the terminal constraints inflicted on the system.

20 Chapter 2 Permanent Solution The equations of motion for the circular CRTBP describe the motion of a particle P 3 of mass m 3 under the gravitational influence of two massive primaries P 1 and P 2 with masses m 1 and m 2, respectively. It is assumed that P 3 has negligible mass in relation to the two primaries and does not have a gravitational impact on the primaries. Motion in the CRTBP is considered from a rotating frame of reference. The two primaries are assumed to be in circular orbits about the mass barycenter and thus maintain a constant separation. It is convenient to put the system in dimensionless coordinates, or normalize the system. The the distance between the primaries is chosen to be the length unit. The mass of the two primaries is the mass unit, and the time unit is selected such that the orbital period of the primaries about their barycenter is 2π time units. The CRTBP also assumes the universal gravitational constant G is unity, making the mean motion n of the primaries one. The system is thus described by a single parameter, µ, which is defined µ = m 2 m 1 + m 2. (2.1) 12

21 13 By convention, m 1 m 2, making the normalized masses of the primaries m 1 = (1 µ) and m 2 = µ and the radii P 1 = µ and P 2 = 1 µ length units. Figure 2.1 displays the restricted three body problem. Figure 2.1: The Restricted Three Body Problem 2.1 Equations of Motion for the Circular Restricted Three-Body Problem The formulation of the equations of motion for the CRTBP are rooted in the Newtonian approach. The non-dimensional position of the third body of negligible mass, P 3, is expressed r 3 = xˆx + yŷ + zẑ. (2.2) The angular velocity in the synodic rotating frame is written in vector as I R ω = θẑ, where R and I represent the rotating and inertial frames of reference respectively. In the CRTBP the motion of the primaries is assumed to be circular and planar,

22 14 therefore θ is constant and equal to one in non-dimensional units. This leads to the vector kinematic expression for the inertial velocity of P 3, Iṙ = R ṙ + I R ω R r 3 (2.3) = (ẋ θy)ˆx + (ẏ + θx)ŷ + żẑ, (2.4) where the dots represent the time derivatives of the respective variable. Time is dimensionless in the CRTBP. Proceeding with the derivation, the kinematic inertial acceleration of P 3 is expressed I r 3 = R r 3 + I R ω R ṙ 3 + I R ω I R ω R r 3 (2.5) = (ẍ 2 θẏ θ 2 x)ˆx + (ÿ + 2 θẋ θ 2 y)ŷ + zẑ. (2.6) The gravitational field of the two primaries in the CRTBP provides two forces on P 3. In dimensionless units, the gravitational force acting on P 3 in the model is given by, F = 1 µ r r µ r r (2.7) Combining equations (2.6) and (2.7) results in the equations of motion for the CRTBP. That is ẍ = 2 θẏ + θ 2 (1 µ)(x + µ) µ(x 1 + µ) x, (2.8) r13 3 r23 ( 3 ÿ = 2 θẋ + θ 1 µ 2 y µ ) y, (2.9) r13 3 r23 3 and ( 1 µ z = µ ) z. (2.10) r13 3 r23 3

23 15 The terms containing θ are the coriolis accelerations, and the terms containing θ 2 are the centripetal accelerations [14]. The pseudo-potential function, U, is extracted from the right side of the equations of motion, and it is made use of in this thesis. It is defined as U = 1 2 θ 2 (x 2 + y 2 ) + 1 µ r 13 + µ r 23. (2.11) Using this definition and unity θ, the equations of motion are re-written as ẍ = U x ÿ = U x + 2ẏ, (2.12) + 2ẋ, (2.13) and z = U z. (2.14) Poincaré proved that only one integral of motion exists for the restricted threebody problem, the Jacobi Constant, C [15]. The Jacobi constant is named after Carl Gustav Jacob Jacobi, and it is the only conserved quantity for the CRTBP; the expression for the Jacobi Constant is derived from equations (2.8)-(2.10) [16]. Specifically, C = 2U (ẋ 2 + ẏ 2 + ż 2 ). (2.15) The Jacobian defines the Hill s region, or the region of configuration space where P 3 is allowed to travel at a certain energy level. The magnitude of the velocity at any point is uniquely determined once C is fixed. The boundary of the Hill s region is plotted by setting the velocity terms in (2.15) to zero and selecting a value for C. The contours that form the boundary of the Hill s region are known as the zero-velocity contours, because the velocity of trajectories fixed at C are zero when they reach the

24 16 surface and cannot pass beyond the contours. The zero velocity curves are applied later in this thesis. 2.2 Libration Points Since the equations of motion EOM of the CRTBP are autonomous, there exists the possibility of equilibrium points. It is shown that the CRTBP contains five equilibrium points, three collinear and two equilateral points [17]. Using equations (2.8)-(2.10), the stationary points (x, y, z ) are defined as x = (1 µ)(x + µ) r 3 1 y (1 µ)y = r1 3 µ(x 1 + µ), (2.16) r2 3 µy, (2.17) r2 3 and (1 µ)z 0 = r1 3 µz. (2.18) r2 3 Equation (2.18) shows that all equilibrium points lie in the x y plane, because z = 0 is the only solution to the equation. Equation (2.17) shows that y = 0 is a solution, and there are equillibrium points that lie on the x axis with the primaries. These are referred to as the collinear libration points. Since the primaries reside at µ and 1 µ and y = z = 0, the collinear libration points are found using equation (2.16) to obtain x (1 µ)(x + µ) µ(x 1 + µ) x + µ 3 x (1 µ) = 0 (2.19) 3 There are three cases to consider for the collinear libration points: x < µ, µ < x < 1 µ, and 1 µ < x whose solutions are called the Lagrange L 3, L 1, and L 2

25 17 points, respectively. In the first case, since L < µ and L < 1 µ, it follows that L + µ = (L + µ)and L (1 µ) = (L (1 µ)). (2.20) Therefore, equation 2.19 becomes L (1 µ)(l + µ) ( 1)(L + µ) µ(l 1 + µ) = 0, (2.21) 3 ( 1)(L (1 µ)) 3 which can also be written as L(L + µ) 2 (L (1 µ)) 2 + (1 µ)(l (1 µ)) 2 + µ(l + µ) 2 (L + µ) 2 (L (1 µ)) 2 = 0. (2.22) Equilibrium is achieved when the numerator of the left side of the equation equals zero. Expanding the numerator results in the following, L 5 + al 4 + bl 3 + cl 2 + dl + e = 0, (2.23) where a = 2(2µ 1), (2.24) b = (1 µ) 2 4µ(1 µ) + µ 2, (2.25) c = 2µ(1 µ)(1 2µ) + 1, (2.26) d = µ 2 (1 µ) 2 + 2(µ 2 (1 µ) 2 ), (2.27) and e = (1 µ) 3 + µ 3. (2.28)

26 18 The equation is solved numerically since there is no analytic solution. The only real root is L 3 = ˆx. (2.29) Using the same method, L 1 satisfies L 5 + al 4 + bl 3 + cl 2 + dl + e = 0, (2.30) where a = 2(2µ 1), (2.31) b = (1 µ) 2 4µ(1 µ) + µ 2, (2.32) c = 2µ(1 µ)(1 2µ) + 1, (2.33) d = µ 2 (1 µ) 2 + 2(µ 2 (1 µ) 2 ), (2.34) and e = (1 µ) 3 µ 3. (2.35) Similarly, L 2 satisfies the same equation with coefficients a = 2(2µ 1), (2.36) b = (1 µ) 2 4µ(1 µ) + µ 2, (2.37) c = 2µ(1 µ)(1 2µ) 1, (2.38) d = µ 2 (1 µ) 2 + 2(µ 2 (1 µ) 2 ), (2.39) and e = (1 µ) 3 µ 3. (2.40)

27 19 Numeric solutions for these cases yield L 1 = ˆx (2.41) L 2 = ˆx. (2.42) Theoretically, a particle placed at these points with zero initial velocity will remain there. However, later it is shown that nonlinearities in the system cause particles placed at the collinear libration points to drift away from the points. Let r 1 = r 2 = 1, and equations (2.16) and (2.17) are satisfied; hence, there is equilibrium at the two locations in the x y plane unit distance from both primaries. These are the equilateral Lagrange points L 4 and L 5, where L 4 = ( 1 2 µ, 3 2 ) and L 5 = ( 1 µ, 3). Since we are concerned with the µ value for the Earth-Moon 2 2 system, the points are located at L 4 = ( , , 0) (2.43) and L 5 = ( , , 0). (2.44) Figure 2.2 shows the location of the five libration points and their relation to the two primaries, in this case the Earth and Moon. The Earth is represented by the blue dot near the center of the figure, and the Moon is represented by the black dot near x = 1. Later it is shown that the three collinear points (marked by blue stars) are unstable, and the two equilateral points (marked by red stars) are marginally stable.

28 20 Figure 2.2: Libration Points 2.3 State Transition Matrix The state transition matrix STM of the CRTBP is an important tool implemented [ ] T for analysis of the dynamical system. Let x = x(t) y(t) z(t) ẋ(t) ẏ(t) ż(t) be the time history of a known solution of the nonlinear CRTBP, the variation y = [ δx(t) δy(t) δz(t) δ x(t) δ y(t) δ z(t)] T, with respect to x, is approximated using a Taylor series expansion about x. If only the first order terms are considered, the approximation becomes ẏ(t) = A(t)y(t), (2.45)

29 21 where A(t) is the 6 6 Jacobian matrix. The Jacobian matrix is written in the following form, A(t) = 0 3 I 3, (2.46) U ij 2Ω 3 where the submatrices I 3 and 0 3 are the 3 3 identity and zero matrices, respectively. The submatrix U ij takes the following form, U xx U xy U xz U ij = U yx U yy U yz, (2.47) U zx U xy U zz where U ij represents the second partial derivative of the pseudo-potential function with respect to the states (x, y, z). The submatrix Ω 3 results in Ω 3 = (2.48) The well known solution to equation (2.45) is expressed in the form y(t) = Φ(t, t 0 )y 0, (2.49) where Φ(t, t 0 ) is the 6 6 state transition matrix. The vector y 0 = y(t 0 ) represents the initial variation relative to x. It is fairly intuitive that when t = t 0, Φ(t, t 0 ) = I 6.

30 Linearization at the Libration Points The Jacobian matrix seen in equation (2.46) is used to linearize about any point in the CRTBP. In the CRTBP coordinate system, r 1 = (x + µ) 2 + y 2 + z 2 (2.50) and r 2 = (x (1 µ)) 2 + y 2 + z 2. (2.51) The calculation of the Jacobian matrix begins by taking the partial of r 1 with respect to x. Specifically, x r 1 = (x + µ)2 + y x + z 2 (2.52) = x + µ. r 1 (2.53) The next phase of the computation involves the inverse of r 1. This yields 1 = x r 1 x r 1 1 (2.54) = x + µ. (2.55) r1 3 The next step in the derivation is 1 x r 3 1 = x r 3 1 (2.56) = 3 x + µ. (2.57) r1 5

31 23 Now that these derivatives have been established, the partial derivatives of the Jacobian can be computed. The second order partials of the pseudo potential function are [ 2 1 U = 1 (1 µ) x2 r1 3 3 ] [ (x + µ)2 1 µ r1 5 r2 3 3 ] (x (1 µ))2. (2.58) r2 5 Similarly [ 2 1 U = 1 (1 µ) y2 r1 3 ] [ 3 y2 1 µ r1 5 r2 3 ] 3 y2 r2 5 (2.59) and [ 2 1 U = 1 (1 µ) z2 r1 3 ] [ 3 z2 1 µ r1 5 r2 3 ] 3 z2. (2.60) r2 5 The mixed second order partials are given by the equations 2 x y U = x y U (2.61) = 3(1 µ)y x + µ r µy x (1 µ), (2.62) r2 5 2 x + µ x (1 µ) U = 3(1 µ)z + 3µz, (2.63) x z r1 5 r2 5 and 2 y z U = 3(1 µ)y z r µy z. (2.64) r2 5 At the collinear libration points, y = z = 0. Therefore, r 1 = (L i + µ) 2 = L i + µ (2.65) and r 2 = (L i (1 µ)) 2 = L i (1 µ). (2.66)

32 24 for i = 1, 2, 3. It follows that 2 x U 2 L i = µ L i + µ + µ 3 L i (1 µ), 3 (2.67) 2 y U 2 L i = 1 1 µ L i + µ + µ 3 L i (1 µ), 3 (2.68) and 2 x U 2 L i = 1 µ L i + µ + µ 3 L i (1 µ). (2.69) 3 All the mixed partials are zero since y = z = 0. If α = 1 µ µ L i + +µ 3 L i, then the (1 µ) 3 complete Jacobian at the collinear libration points is A(t) =. (2.70) 1 + 2α α α For the L 4 and L 5 libration points only z = 0, while x = µ + 1/2 and y = ± 3/2. Therefore, 2 x U 2 L i = 3/4, (2.71) 2 y U 2 L i = 9/4, (2.72) (2.73)

33 25 and Also, 2 x y U L i = 2 z 2 U L i = 1. (2.74) 2 y x U L i = (1 2µ) (2.75) while the rest of the partials equal zero. Therefore, the Jacobi matrix at the equilateral triangle libration points is A(t) = /4. (2.76) 3(1 2µ) (1 2µ) 9/ The Jacobian, thus defined, is used to analyze the stability of the libration points. 2.5 Stability of the Libration Points In order to determine the linear stability of the system near the libration points one must first calculate the eigenvalues and eigenvectors of the associated Jacobi matrix. An eigenvalue λ of a matrix A satisfies det(a(t) λi) = 0, (2.77)

34 26 this results in the following equation for the collinear points λ 6 + 2λ 4 + (1 + 3α 3α 2 )λ 2 + α(1 + α 2α 2 ) = 0, (2.78) where α has the same definition as previously stated. Defining s = λ 2 reduces the previous equation to s 3 + 2s 2 + (1 + 3α 3α 2 )s + α(1 + α 2α 2 ) = 0. (2.79) This third order polynomial has the solutions s 1 = α, (2.80) s 2 = α2 + 2(9α 2 8α), (2.81) and s 3 = α2 2(9α 2 8α). (2.82) The six eigenvalues of A are λ k1,2 = ± s k. If α < 0, then λ 1,2 are real eigenvalues with opposite signs and equal magnitude, and the corresponding eigenmodes have an exponential growing and an exponential decaying term. The dynamics near the libration points is discussed in further detail in the next section Numerical Study of Linear Stability of the Libration Points in the Planar System In this thesis, the planar, two degree-of-freedom CRTBP is analyzed. The focus of this study is the energy required to achieve desired trajectories, and the planar problem is sufficient for this purpose. The third dimension becomes important for geometric

35 27 reasons such as constant access to the sun for solar power or constant access to the Earth for communications. It is also sometimes required to approach a planet or moon out of plane, and to control the latitude and longitude of a spacecraft s escape or approach to or from an orbit require three-dimensional capabilities [18]. However, many of the essential dynamics can be captured well with the planar model [19]. Using the previous results and the coordinates of the Libration points, the Jacobi matrix for the collinear points for the planar dynamics is A(t) =. (2.83) 1 + 2α α 2 0 The planar dynamics for the equilateral triangle points result in the following Jacobian matrix. A(t) = /4. (2.84) 3(1 2µ) (1 2µ) 9/4 2 0 The four eigenvalues for the L 1 libration point to be λ 1 = (2.85) λ 2 = (2.86) λ 3 = i (2.87) λ 4 = i. (2.88) These results suggest the L 1 libration point exhibits center-saddle behavior. The λ 1

36 28 value describes a stable direction, or manifold, and λ 2 indicates the unstable manifold. The imaginary λ 3 and λ 4 values imply a two dimensional center manifold about L 1. The eigenvalues λ 5 and λ 6 are not shown here, because the planar system is considered. However, it is important to note that the eigenvectors at L 1 for the three dimensional system have no out of plane or z component. Therefore, trajectories that begin in the plane with no out of plane velocity will remain in the plane. A halo orbit is an orbit that is out of pane near the collinear libration points, but halo orbits do not exist in the linearization about the collinear points. Halo orbits are a purely nonlinear phenomenon [20]. This analysis shows solutions that emanate to and from L 1 as well as solutions that periodically orbit L 1 in the plane. Continuing this analysis for L 2 and L 3 the eigenvalues are found to be λ 1 = , (2.89) λ 2 = , (2.90) λ 3 = i, (2.91) λ 4 = i, (2.92) and λ 1 = , (2.93) λ 2 = , (2.94) λ 3 = i, (2.95) λ 4 = i, (2.96) respectively. Note the L 2 and L 3 points exhibit the same linear behavior as the L 1

37 29 point. One will also notice the different expansion and contraction rates of the three points. The L 1 libration point has the greatest expansion and contraction rates, and the L 3 point has the smallest. This tells us that a L 1 orbit will have a higher energy than a L 2 or L 3 orbit. This point will be revisited later on in this thesis. The L 4 and L 5 points have the following eigenvalues. λ 1 = i, (2.97) λ 2 = i, (2.98) λ 3 = i, (2.99) and λ 4 = i. (2.100) These eigenvalues signify center-center behavior. There are only two linear frequencies, and, unless sufficient energy is added to the system, a particle placed near one of these points with zero initial velocity will remain indefinitely. The linear dynamics provide a rough idea about what happens near these points. Later, the nonlinear dynamics are analyzed. 2.6 Zero Velocity Curves in the Earth-Moon CRTBP In the first section of this chapter the Jacobi constant, C, is discussed. The projection of the Jacobian into configuration space is called the Hill s region [21]. By selecting a certain energy level described by the Jacobian and setting the velocity terms to zero, the solution to the Jacobi Integral forms a curve in the CRTBP that contains the

38 30 Hill s region. This curve is known as the zero velocity curve. The curve gives great insight into the allowable movement of the third body of negligible mass, given that the energy level described by C remains constant. When the third body reaches the contour of the zero velocity curve its velocity is zero, and it is not able to pass beyond the curve. The inaccessible regions beyond the curve are called the forbidden regions. The libration points are fixed points, meaning a trajectory there has zero velocity, and they are on the zero velocity curve. The Jacobi constants at the libration points are calculated to prove this. The values in equations (2.101)-(2.104) are computed by plugging the coordinates of the libration points into equation (2.15). At this point it is important to note that C is the pseudo energy of the system, and larger values of C actually mean less energy. Inspection of equation (2.15) makes this point clear. The Jacobi constants at the libration points are C 1 = , (2.101) C 2 = , (2.102) C 3 = , (2.103) and C 4 = C 5 = (2.104) The zero velocity curve of the energy level just above C 1 is presented in Figure 2.3. The blue dot at the center of the figure is the Earth, and the black to the right of the plot is the Moon. The plot displays three Hill s Regions: one surrounding the Earth, one surrounding the Moon, and the region outside the outermost curve. Figure 2.3 suggests that

39 31 Figure 2.3: Zero Velocity Curve for C=1.1C1 a trajectory starting near the Earth would remain bounded to the Earth s neighborhood, and a trajectory starting near the Moon would stay bounded to the Moon s neighborhood. Transfer between the primaries is not possible at this energy level. The curve resulting from C = C 1 is seen in Figure 2.4 Figure 2.4: Zero Velocity Curve for C = C 1

40 32 Note the pinching that occurs at L 1. This occurs because, as stated earlier, the libration points lie on the zero velocity curves at their associated Jacobi energy levels. The barrier between the two primaries is just starting to open or tear, hence the name L 1. The two primaries are still disconnected, but if the energy is decreased slightly an opening will occur. Figure 2.5 shows the zero velocity curve for C = C 2. Figure 2.5: Zero Velocity Curve for C=C2 2.5 Here the pinching at L 2 is identifiable. Passage between the two primaries is allowed, and the second tear in the Jacobi curve is noticeable. Hence the name L 2. If the Jacobi constant is decreased slightly there will be a neck region at L 2, and passage out of the Earth-Moon region will be allowed. The zero velocity curve at C = C 3, is seen in Figure 2.6. As expected, the pinching is visible at L 3. If the Jacobi constant is decreased further, the forbidden region continues to shrink. If the energy is decreased all the way to C = C 4 = C 5, the forbidden region completely disappears. Thus, the Jacobi constant defines the permissible trajectories in the CRTBP.

41 33 Figure 2.6: Zero Velocity Curve for C=C3 Later, a method is devised for computing the minimum change in velocity for generating decommissioning trajectories. 2.7 Invariant Manifolds Knowledge of the phase space allows for decomposition of the flow of the states into subspaces that characterize the behavior of the system. Utility is found in exploiting knowledge of the flow in the vicinity of reference solutions. It is shown that the local flow in the vicinity of periodic solutions in this problem can be globalized to identify naturally occurring trajectories to be exploited for low-energy transfer orbits [22]. These trajectories comprise the system s invariant manifolds. Invariant manifolds are a very central element of dynamical systems analysis. Stable, unstable, and center manifolds are of particular importance, and they are analyzed in this thesis. For a

42 34 broad description, consider the nonlinear dynamical system ẋ = f(x), (2.105) where x R n. If S R n is some set, S is said to be invariant in the vector field ẋ = f(x) if for any x 0 S, x(t, 0, x 0 ) S for all t R. A invariant set S R n is a C r,(r 1) invariant manifold if S is a C r differentiable manifold. The term C r means the r th derivative of the function it describes is continuous. Therefore, the r th derivative of the function describing S needs to be continuous in order for it to be a C r invariant manifold. Manifolds are often described as an m-dimensional surface embedded in R n Linear and Nonlinear Flow Near Unstable and Stable Fixed Points The orbit structure near the CRTBP co-linear fixed points produce useful invariant manifolds. As stated earlier, the libration points have different types of orbits or modes, and the system is brought into its Jordan Canonical form in order to analyze these modes. Consider the linear system of the vector field ẏ = Ay. (2.106) where y R n is the deviation from the state x, and A is the Jacobian of f(x). When x R is a fixed point, the Jacobi matrix, A, is a constant n n matrix. The set R n can be represented as the sum of the three subspaces E s, E u, and E c.

43 35 Each are defined as follows: E s = span(e 1,..., e s ), (2.107) E u = span(e s+1,...e s+u ), (2.108) E c = span(e s+u+1,..., e s+u+c ), (2.109) and s + u + c = n. The terms (e 1,..., e s ) are the generalized eigenvectors of A corresponding to eigenvalues with negative real part, (e s+1,...e s+u ) are the generalized eigenvectors of A corresponding to the eigenvalues with positive real part, and (e s+u+1,..., e s+u+c ) are the generalized eigenvectors of A corresponding to eigenvalues with zero real part. It is seen here that R n contains three subspaces. E s, E u, and E c are known as the stable, unstable, and center subspaces, respectively. They are examples of invariant subspaces, or manifolds, because flows with initial conditions contained in one of the subspaces will remain in the respective subspace for all time. Furthermore, flows starting in E s will asymptotically approach y = 0 as t, flows starting in E u will asymptotically approach y = 0 as t, and flows initiated in E c will neither grow nor decay relative to y = 0 in time [23]. The Jacobi matrix is also written as, A = T DT 1, (2.110) where T is a matrix whose elements are the eigenvectors of A. The matrix T is generaly written as.... T = e 1 e 2 e 3 e 4, (2.111)....

44 36 and D is the diagnalized matrix of eigenvalues, λ λ D =. (2.112) 0 0 λ λ 4 The solution of equation (2.106) through the point y 0 R n at t = 0 is written y(t) = e At y 0 = e T DT 1t y 0. (2.113) Equation is also written y(t) = T e Dt T 1 y 0 (2.114) e λ 1t e λ 2t 0 0 = T T 1 y 0 (2.115) 0 0 e λ 3t e λ 4t.... = e 1 e λ 1t e 2 e λ 2t e 3 e λ 3t e 4 e λ 4t T 1 y 0. (2.116).... In previous linear analysis of the stationary points it is shown that the collinear libration points have stable, unstable, and center manifolds. In the planar system, one eigenvalue is negative and real, one is positive and real, and the remaining two eigenvalues are purely imaginary. Therefore, the collinear libration points have the

45 37 linear subspaces E s = span(e 1 ) (2.117) E u = span(e 2 ) (2.118) E c = span(e 3, e 4 ). (2.119) If the point y 0 is a point in R 4, T 1 is the constant transformation matrix that changes the the coordinates of y 0 with respect to the standard basis in R 4 into coordinates with respect to the basis e 1, e 2, e 3, and e 4. For y 0s E s, T 1 y 0 has the form ỹ 1 T 1 0 y 0s =. (2.120) 0 0 Similarly, for y 0u E u, T 1 y 0 has the form 0 T 1 ỹ 2 y 0u =, (2.121) 0 0 and, for y 0c E c, T 1 y 0 has the form 0 T 1 0 y 0c =. (2.122) ỹ 3 ỹ 4

46 38 The flow of each set of initial conditions, y 0s, y 0u, and y 0c, stay on the stable, unstable, and center invariant manifolds, respectively. However, it is important to remember that these are linear approximations, and the nonlinear solutions does not act exactly as the linear approximation predicts. The smaller the neighborhood around the fixed point the better the linear approximation holds up. Consider, for example, Figure 2.7. The dashed line trajectory is the solution to Figure 2.7: Linear and Nonlinear Flow Near L1 the linear system, the solid trajectory is the solution to the nonlinear system, and the star at the center of the plot is the L 1 Lagrange point. The state transition matrix is used to calculate the linear approximation and the nonlinear equations of motion are used to calculate the nonlinear trajectory. Figure 2.7 suggests that the trajectories are in a small enough neighborhood that the short time agreement between the linear approximation and the nonlinear solution is good. Given the same initial conditions in E c, the nonlinear trajectory follows the linear orbit for almost a full period before heading towards Earth. Consider the linear vector field of the collinear fixed points, the system has an s-

47 39 dimensional invariant stable manifold, a u-dimensional unstable invariant manifold, and a c-dimensional invariant center manifold all intersecting at the fixed point. Theorem explains how the dynamical structure is altered when the nonlinear vector field is considered. Theorem (Local, Stable, Unstable, and Center Manifolds of Fixed Points) If a nonlinear vector field is C r, r 2, the fixed point x eq of the nonlinear vector field possesses a C r s-dimensional local, stable manifold, W s loc (x eq), a C r local, unstable manifold, W u loc (x eq), and a C r W c loc (x eq), all intersecting at x eq. u-dimensional c-dimensional local, center manifold, These manifolds are tangent to their respective invariant manifold in the linear vector field at the fixed point [23]. Globalizing the stable and unstable invariant manifolds near a fixed point along an unstable periodic orbit produces naturally occurring trajectories that converge to and away from the periodic solution, like the one seen in Figure 2.7, to yield low energy orbital transfers [24]. If o(x 0) = o h denotes a halo orbit, the set W s (o h ) = [x U : φ(x, t) o h as t ] (2.123) is the stable manifold of o h, and the set W u (o h ) = [x U : φ(x, t) o h as t ] (2.124) is the unstable manifold of o h. W s (o h ) and W u (o h ) are global objects obtained by propagating the local stable and unstable manifolds. Linear analysis coupled with an understanding of the stable and unstable manifolds as the sets of points that flow to o h in forward or backward time provides a method by which the global manifolds are

48 40 numerically calculated. 2.8 Computing Manifolds The computation of the Lyapunov orbit s manifolds is accomplished by dividing the the halo orbit, o h, into N points x k, 1 k N. There is a corresponding time, τ k, such that x k = x (τ k ), where x (t) is the Lyapunov orbit with some initial condition x 0. o h = [x (t) : 0 t < T ] (2.125) The initial conditions for three dimensional halo orbits are found using numerical methods [25]. However, the initial conditions for two dimensional Lyapunov orbits are easily obtained using the analytical linear analysis previously described. Computing the global manifold trajectories requires integration of initial conditions in the local stable and unstable subspaces. Each initial condition is approximated using the state-space information corresponding to a point on the periodic orbit. Any state on the periodic orbit, when integrated using the linearized model, will stay on the periodic orbit; therefore, a perturbation is added to shift the state onto the local stable or unstable subspace and integrating the new initial condition to yield the global solution. The local eigenvector directions are indicated in Figure 2.8. The perturbed state is calculated by defining the shift in the direction of the stable or unstable eigenvectors by a relatively small distance d M. Let the terms ξ u0 and ξ s0 be the unstable and stable eigenvectors, respectively, of the monodromy matrix of a periodic orbit with period T. The monodromy matrix, M, is the solution matrix of a particular initial condition evaluated over the period of the orbit. It is such that δx(t ) = Mδx(0). (2.126)

49 41 Figure 2.8: Linear and Nonlinear Flow Near L1 The eigenvectors of the monodromy matrix and the state transition matrix are used to calculate the perturbation vectors ξ sk and ξ uk at a point x (τ k ) on the orbit using the relationships ξ sk = Φ(0, τ k )ξ s0 (2.127) and ξ uk = Φ(0, τ k )ξ u0. (2.128) These vectors are tangent to the unstable and stable manifolds at x (τ k ). If the eigenvectors associated with the stable and unstable eigenvalues are defined as [ ] T [ ] T ξ u (t k ) = x u y u z u ẋ u ẏ u ż u, and ξ s (t k ) = x s y s z s ẋ s ẏ s ż s, then normalization yields V Ws (t k ) = ξ sk (t k ) x 2 s + y 2 s + z 2 s + ẋ s + ẏ s + ż s (2.129)

50 42 and V Wu (t k ) = ξ uk (t k ) x 2 u + y 2 u + z 2 u + ẋ u + ẏ u + ż u. (2.130) A tolerance d M is set, and if d M is sufficiently small, x sk (t k ) = x k ± d M V Ws (t k ) (2.131) and x uk (t k ) = x k ± d M V Wu (t k ) (2.132) are points in the stable and unstable manifolds, respectively. Typically, for the Earth- Moon system, a value of 50km is sufficient for d M (converted into non-dimensional units) [24]. The alternating sign on the displacement from x k is used because a trajectory can be perturbed towards either primary along the stable or unstable subspace. These initial conditions are integrated over some time interval [0, T f ] to obtain the orbits x sk manifolds respectively. and x uk, which are orbit fibers of the stable and unstable Propagating the states x sk (0) in negative time results in the globalization of the stable manifold. Similarly, propagating the states x uk (0) in positive time allows for the globalization of the unstable manifold. Globalization of the stable and unstable manifolds near L 1 in the direction of the moon is displayed in Figures 2.9 and 2.10, respectively. The star at the center of Figures 2.9 and 2.10 represents the moon, and the star at the right represents L 2. The zero velocity curve associated with the energy of the system is the red contour in the figures. Note how the manifolds reside strictly in the Hill s region and do not traverse the zero velocity curve. Globalization of the stable manifold in the direction of the Earth is displayed in figure The star in the center of figure 2.11 represents the Earth. This manifold is

51 43 Figure 2.9: Stable Manifold of a L 1 Lyapunov Orbit About Moon Figure 2.10: Unstable Manifold of a L 1 Lyapunov Orbit About Moon

52 44 Figure 2.11: Stable Manifold of a L 1 Lyapunov Orbit About Earth useful for finding a trajectory from the vicinity of the Earth to the Moon. If a particle is injected onto this manifold, it will eventually be brought to the Lyapunov orbit and drift off to the vicinity of the Moon according to the unstable manifold seen in figure Thus, there are many natural trajectories to the lunar neighborhood from the terrestrial neighborhood via the stable and unstable L 1 manifolds. The manifold going towards Earth is achieved by using a negative d M term, and the manifolds in the direction of the Moon are achieved by using a positive d M term. Once the manifolds have been established, it is necessary to find out how much energy must be added to the system to get a craft from its current state in the CRTBP onto the desired manifolds.

53 2.9 Calculating The Minimum Change in Velocity 45 It is shown in the background section of this thesis how the Jacobi Constant of the Hamiltonian system is a useful tool for analyzing the possible trajectories of a solution. In this section, a method is developed and utilized to determine the required perturbation, in this case a change in velocity, that must be added to the system to alter the energy of the system to a desired level. The desired energy level is attained by understanding where a solution in the system to should be allowed to traverse. The process begins with the potential function of the CRTBP. In the preceding analysis dimensionless coordinates were used, but, here the objective is to obtain results in SI units. Therefore, some parameters and variables must be defined. The distance between the Earth and the Moon is l = km. The mass of the Earth and Moon is m 1 = kg and m 2 = kg, respectively. The mass ratio in the Earth-Moon system is thus µ = The Moon s period around the Earth is T = s. The positions, in meters, of the Earth and Moon in the CRTBP are p 1 = µlˆx (2.133) and p 2 = (1 µ)lˆx, (2.134) respectively. The distances of the third body from the two primaries become r 1 = (x p 1 ) 2 + y 2 + z 2 (2.135) and r 2 = (x p 2 ) 2 + y 2 + z 2 (2.136)

54 46 where r 1 is the distance from the Earth to the third body, and r 2 is the distance from the Moon to the third body. The Jacobi constant is re-written as C = n 2 (x 2 + y 2 ) + 2 Gm 1 r Gm 2 r 2 (ẋ 2 + ẏ 2 + ż 2 ), (2.137) where G = m3 kg s 2 in terms of the Potential function as and n = 2π. The Jacobi constant can also be rewritten T C = 2U V 2 (2.138) where the potential function U is defined U = 1 2 n2 (x 2 + y 2 ) + Gm 1 r 1 + Gm 2 r 2. (2.139) It is useful to know how much V must be added to the system to get to some desired pseudo energy, C T arget. If our initial condition is the current state of ISS, then the equation for C T arget interms of V is C T arget = 2U ISS (V ISS + V ) 2. (2.140) Solving for V results in the following: V = 2U ISS C T arget V ISS. (2.141) Values of C T arget must be determined to give the desired results. Recall that a Jacobi constant, C, slightly less than C 1 results in a zero velocity curve with a opening or neck region around L 1 where trajectories are permitted to pass from the

55 47 neighborhood of one primary to the neighborhood of the other. Figure 2.12 displays the zero velocity curve associated with this pseudo energy. The Jacobi constant Figure 2.12: Zero Velocity Curve for C SI = (m/s) 2 associated with this pseudo energy is the C T arget value needed to find the minimum V necessary to put the ISS on a manifold that would take it from the terrestrial neighborhood to the lunar neighborhood. A trajectory to transport the ISS past the Moon and past L 2, requires a C T arget slightly less than C 2. The associated zero velocity curve of this energy level is seen in figure The ISS can also be transported out of Earth s neighborhood via the L 3 neck. This would require a C T arget slightly less than C 3. The associated zero velocity curve of this pseudo energy level is seen in figure Note that a second V is required for each case to close the neck regions and deny the third body passage back to the Earth s neighborhood. For the pseudo energy levels displayed in figures 2.12 and 2.13, the V need only be such to bring the system s pseudo energy to C 1 and C 2 respectively. However, for the pseudo energy level displayed in figure 2.14 the V must be such that it brings the system s pseudo energy back to C 2 to deny passage back to Earth s neighborhood.

56 48 Figure 2.13: Zero Velocity Curve for C SI = (m/s) 2 Figure 2.14: Zero Velocity Curve for C SI = (m/s) 2

57 Fuel Requirement Once the required V is calculated, the amount of fuel required to achieve it can be calculated via the Tsiolkovsky rocket equation, or ideal rocket equation, derived by Konstantin Tsiolkovsky. The equation relates V with the effective exhaust velocity and the initial and final mass of the rocket [26]. The equation is V = v e ln ( M0 M 1 ), (2.142) where v e is the effective exhaust velocity, M 0 is the initial mass including propellant, and M 1 is the final mass. The effective exhaust velocity is defined as v e = I sp g 0, where I sp is the specific impulse and g 0 is the acceleration of Earth s gravity at sea level (9.81m/s). With this information, an expression for the mass ratio is defined M 0 M 1 = e V Ispg 0. (2.143) Equation (2.143) is further manipulated to give an expression for amount of propellant mass m p needed to achieve the required V. m p = M 0 (e V Ispg 0 1) (2.144) 2.11 V and Fuel Mass Requirements A Lyapunov orbit with an amplitude of km is chosen for the purpose of this analysis. The linear analysis previously detailed is employed to find some initial

58 50 conditions for such a Lyapunov orbit about L 1. The initial condition ỹ 1 = 0, (2.145) ỹ 2 = 0, (2.146) ỹ 3 = ( i) 10 4, (2.147) and ỹ 4 = ( i) 10 4 (2.148) is chosen to isolate the center eigenmode. These conditions are transformed back to the physical coordinates using the transformation matrix, y 0 = T ỹ to obtain x 0 = (2.149) y 0 = (2.150) ẋ 0 = (2.151) ẏ 0 = (2.152) These initial conditions are in the normalized coordinate system, relative to the L 1 point. Figure 2.7 shows the initial conditions integrated in both the linear and nonlinear models. These same initial conditions are used in equation (2.141) to calculate C T arget. The ISS is currently in an approximately circular orbit about Earth with a semi major axis of about a = 6, 772km. Therefore the ISS has the following conditions in

59 51 the CRTBP x 0 = 6, 772km + p 1 (2.153) y 0 = 0 (2.154) GM 1 ẋ 0 = 6, 772km (6, 772km + p 1)n (2.155) ẏ 0 = 0. (2.156) Evaluating equation (2.141) results in V = km s. This analysis is repeated for the L 2 and L 3 libration points. The propellant mass is calculated using Equation Since it is known that the dry mass of the ISS is 420, 000kg, the required propellant mass needed to achieve the change in velocity can be calculated. All the results are summarized in Table 2.1. Mass ratios and fuel mass assume a specific impulse, I SP, of 400s. Point of Exit V (km/s) Mass Ratio (Unitless) Fuel Mass (tons) L L L Table 2.1: Fuel Mass Results I SP = 400s The values for the mass ratios and propellant masses make this strategy unrealizable, especially since these calculations use the minimum V attainable. According to the rocket equation, an engine with a higher I SP is necessary to drive down the fuel cost. Electric engines, such as the variable specific impulse magneto plasma rocket (VASIMR) are recommended. The fuel requirements of VASIMR s 10,000s I sp engine is given in Table 2.2 [27]. The resulting propellant masses VASIMR requires are reasonable. However, since

60 52 Point of Exit V (km/s) Mass Ratio (Unitless) Fuel Mass (tons) L L L Table 2.2: Fuel Mass Results I SP = 10, 000s VASIMR is a low thrust engine, it will take much longer to reach the necessary change in velocity V and Thrusting Time The VASIMR engine has an efficiency factor of 60%, and the available power from the ISS is P = 110kW [28]. Therefore, the engine will produce 66kW of useable power. This output power is converted to thrust with the following equation: T = P g 0 I SP. (2.157) Hence, VASIMR will generate a thrust of N. Given a constant thrust, T, the total change in velocity after time t f is V = tf 0 T dt, (2.158) M 0 Ṁt where M ISS (t) = M 0 Ṁt. M 0 is the initial mass including the mass of fuel, and Ṁ is the mass rate of the fuel leaving the engine (Ṁ = (2.158) is analytically evaluated to yield: T I SP g 0 ). The integral in Equation V = T t f Ṁ ln(m 0 Ṁt). (2.159) 0

61 53 Solving for t f gives t f = M 0 M 0 e Ṁ T V Ṁ (2.160) The thrusting time results are summarized in Table 2.3. Point of Exit V (km/s) Thrusting Time (years) L L L Table 2.3: Thrusting Time Results I SP = 10, 000s These values are just the thrusting times and do not reflect the entire total transfer time. The transfer time would be longer with the inclusion of coasting times Realized Trajectory The V s used to calculate the values in Tables are theoretical results and are practically unattainable. Therefore, the thrusting times and the fuel mass required to accomplish the transfers will be higher. An actual trajectory is computed using continuous thrusting tangent to the velocity, the most efficient way to increase the orbit s energy, of the third body (ISS in this case) to put the vehicle on the manifold. The objective is to take the ISS from LEO with an altitude of 410km to the lunar neighborhood using stable and unstable manifolds of L 1. Continuous thrusting is used to take the ISS from LEO to the stable manifold seen in Figure Once on the stable manifold, the ISS gets a free ride to the L 1 Lyapunov orbit and is naturally injected onto the unstable manifold seen in Figure The target Lyapunov orbit has a radius of 1, 218km and a period of twelve days. This Lyapunov orbit is chosen because it is relatively small. A smaller

62 54 target Lyapunov orbit translates to a smaller V requirement to get from LEO to the stable manifold. Figure 2.15 displays the trajectory in the rotating frame of reference. The trajectory is put in the inertial frame of reference using a coordinate frame transformation matrix [29]. Figure 2.16 shows the realized trajectory in the inertial frame of reference. Figure 2.15: Transfer from LEO to Lunar Orbit using Invariant Manifolds (rotating frame) The results of the realized transfer are compared to the theoretical results for the same transfer in Table 2.4. The coasting time for the trajectory is only 73 days. The vast majority of the transfer is spent thrusting. The appeal of the inter planetary superhighway IPS is the ultra-low energy transfers it provides. Once a spacecraft is on the manifolds produced by the collinear

63 Figure 2.16: Transfer from LEO to Lunar Orbit using Invariant Manifolds (Earthcentered inertial frame) 55

64 56 libration points, relatively small V s are required to complete trajectories. However, it proves very expensive and time consuming getting a spacecraft from LEO onto one of these manifolds, especially a larger craft like the ISS. V (km/s) Fuel Mass (tons) Thrusting Time (years) Lower Bound Realized Table 2.4: Theoretical vs Realized L 1 Lunar Transfer Orbit Using VASIMR This solution is unacceptable. Thirty tons is an exceedingly large amount of propellant, and a propulsion system will not last for over a hundred years.

65 Chapter 3 Atmospheric Drag The ultimate goal of this thesis is to provide an appropriate end of life solution for the ISS. The previous method proposed to accomplish this goal is infeasible at the present time, given current constraints on propulsion systems. Therefore, a practical solution needs to be devised in order to address the problem. Orbital decay arrises because of atmospheric density ρ. When a spacecraft travels through the atmosphere a drag force acts on the craft opposite its direction of motion. The acceleration resulting from this drag force is given by the equation a D = 1 2 C A D m ρv2, (3.1) where a D is the acceleration of the satellite due to drag, C D is the satellite s coeficient of drag, m is the mass of the satellite, and v is the magnitude of the satellite s relative velocity [30]. If the velocity is expressed as a vector, the equation becomes a D = 1 2 C A D ρ v v. (3.2) m With the exception of atmospheric density, ρ, these values are easilly obtained. The 57

66 58 atmospheric density is a highly variable quantity, making it very difficult to predict. The Jacchia Atmospheric Model is implemented in this study for the purpose of approximating the atmospheric density at the ISS s position. 3.1 Jacchia Model The Jacchia Atmospheric Model provides values for atmospheric density and other properties for altitudes from 90km to 2500km. Although the model was last updated in 1977, it is still the most commonly used model for artificial satellite motion applications. The Jacchia model includes latitudinal, seasonal, geomagnetic, and solar effects, and it assumes that the atmosphere rotates with the Earth as a rigid body. The model must be supplemented for applications at lower altitudes; however, our application deals with altitudes within Jacchia s effective range. The inclusion of solar effects in the Jacchia model is important, because atmospheric density tracks very closely with solar flux. In fact, studies have shown high degrees of correlation (r 0.9) between atmospheric density and solar flux [31]. This means that higher solar flux translates to higher atmospheric density. This seems counter intuitive, as one would expect thermal expansion to cause a decrease in atmospheric density. However, the upper atmosphere follows the opposite trend. The increase in upper atmospheric density is due to an outward expansion of the atmosphere at lower altitudes. Since lower altitudes have higher atmospheric density, the expansion results in increased density in the upper atmosphere. It is also shown that magnetic activity directly impacts air density. When all other factors are held constant, observations show increased air density results from a rise in the magnetic activity index. Latitudinal and seasonal considerations are also important for the model to consider, because these factors have a considerable impact on the atmo-

67 59 spheric density [32]. For example, the June solstice results in the lowest densities, and maximum densities occur during the two equinoxes. During the June minimum, the polar south is in winter and thus experiences even lower atmospheric densities. 3.2 Results The solar F10.7 index is a widely used measure of solar flux, and it is a measure of the solar radio flux per unit frequency at a wavelength of 10.7cm. The altitude vs. atmospheric density plot seen in Fig. 3.1 assumes the nominal solar flux according to the Schatten solar cycle. Figure 3.1: Density vs. Altitude Fig. 3.1 illustrates why it is important to get the ISS to the highest altitude the constraints will allow. The higher the altitude of the ISS, the lower the impact of atmospheric density and the longer the satellite will stay in orbit. After inspecting equation equation (3.1), one would expect a circular orbit, rather than an elliptical orbit, produces the longest lifetime, because in an elliptical orbit

68 60 the satellite is moving fastest closer to Earth where the density is higher. In order to test this hypothesis, some tests are performed using STK. A given amount of fuel can be used to put a satellite in an elliptical orbit or a circular orbit. If the same amount of fuel is used to put two satellites in orbit, one in an elliptical orbit and the other in a circular orbit, the satellite in the elliptical orbit will be higher than the other satellite when it is at apogee and lower than the other satellite when it is at perigee. Given some amount of fuel m fuel and the mass flow rate of the engine ṁ the burn time t burn is given by the equation t burn = m fuel ṁ. (3.3) The burn time is then used to calculate the change in velocity the ATV can provide the ISS using the equation which results in V = tburn 0 T dt, (3.4) m AT V + m ISS + (m fuel ṁt) V = Ṫ m [ln(m AT V + m fuel + m ISS ) ln(m AT V + m fuel + m ISS ṁt burn )]. (3.5) This represent the total change in velocity that the thrusters can attain. The Hohmann transfer consists of two burns, one that takes the satellite out of it s initial circular orbit of radius r 1 into the elliptical transfer orbit and another that re-circularizes the orbit once it has reached its target orbit of radius r 2 as depicted in Figure 3.2.

69 61 Figure 3.2: Hohmann Transfer The required change in velocities of the two burns are described by µ 2r2 v 1 = ( 1) (3.6) r 1 r 1 + r 2 µ 2r1 v 2 = (1 ). (3.7) r 2 r 1 + r 2 Since we know that r 1 = 410, 000m + r earth, r 2 is found with the equation V = v 1 + v 2, (3.8) because r 2 is the only unknown. Therefore, r 2 is the semi major axis of the terminal orbit with zero eccentricity. The satellite could also be placed in an elliptical orbit by devoting all the fuel to

70 62 one burn. To accomplish this equation (4.79 and solved for r 2. This yields r 2 = ( v 1 µ r1 2 ( v 1 µ r1 + 1) 2 r 1 + 1) 2. (3.9) The elliptical orbit has a radius at perigee of r p = 410km + r Earth and a radius at apogee of r a = r 2. Using the either equations r p = a(1 e) (3.10) r a = a(1 + e) (3.11) the eccentricity e is found. The semi major axis is the same. The satellite lifetime feature of STK is then used to compute the lifetime of a satellite in each orbit. For this exercise a fuel mass of 2 ATV payloads is used, and the Jacchia atmospheric density model is chosen to approximate density. Table 3.1 shows the results of the experiment. These results suggest, in terms of fuel vs. lifetime, Orbit Ecentricity Semimajor Axis (km) Lifetime (years) Circular 0 6, Years Elliptical , Years Table 3.1: Fun Tol Comparison putting a satellite in a circular orbit is a more judicious use of resources since the satellite in the circular orbit has about three times the lifetime of the satellite in the elliptical orbit. An initial circular altitude vs satellite lifetime relationship is established so that the optimization results will translate to an associated satellite lifetime. Once again, the lifetime feature in STK is used to establish this relationship. The simulation re-

71 63 sults of some different initial altitudes are documented in Fig The computations assume a coefficient of drag of C D = 2 and a decay altitude of 200km. Figure 3.3: Decay Time vs. Initial Altitude The exponential nature of the relationship provides ample motivation for seeking the optimal trajectory to maximize satellite altitude. Any increase in altitude is exponentially rewarded with more lifetime. Another influential variable in equation (3.1) is the cross-sectional area A. Intuitively, one understands that varying A results in a varying orbit decay rate. A few simulations are completed in STK to gage the impact of varying A. The simulations used to construct Fig. 3.4 use C D = 2, a decay altitude of 200km, and an initial altitude of 600km. The cross-sectional areas range from 700m 2 to 2300m 2 and produce an expected lifetime range of approximately years. According to these predictions, a 78% reduction in area results in a 200% increase in the satellite s lifetime. Consider a satellite with a minimum cross-sectional area of 700m 2 and a maximum cross-sectional area of 2300m 2 but maintains a cross-sectional area around 1000m m 2 throughout

72 64 Figure 3.4: Area Consideration its lifetime. If a control system is successfully implemented to maintain a crosssectional of 700m 2, the satellite s lifetime would increase by approximately 50%. These results lead one to consider deploying a control system to maintain the minimum cross-sectional area. However, one must take into account the fact that most of a satellite s cross-sectional area is attributed to the solar panels used to provide power to the craft, and these panels must be oriented in a manner to extract energy from the sun. Therefore, it may be impossible to satisfy the power needs of the control system while simultaneously maintaining the minimum cross-sectional area. The analysis of the upper atmospheric conditions consistent with LEO lead to the conclusion that a trajectory should be developed that situates the satellite in the highest circular orbit possible. The feasibility of a minimum cross-sectional area control system could be looked into, later, because it would produce considerable improvements to satellite lifetime.

73 Chapter 4 Temporary Solution This chapter documents the development and implementation of an algorithm implementing the control parameterization approach to finding an optimal trajectory to maximize the altitude of the ISS in a coplanar circular orbit. 4.1 Equations of Motion The equations of motion for the optimization problem model the ISS in an Earth centered, planar, radial, and transverse coordinate frame. This frame is chosen to reduce the size of the problem. The planar coordinate system requires only two control variables, one for the thrust magnitude and another for the thrust direction [33]. The Cartesian reference frame requires three control variables, a thrust magnitude for each direction. A planar coordinate frame is sufficient, since it is assumed that a plane change maneuver is not contained in the optimal trajectory. Plane changes are expensive and not useful in this case. The nomenclature is described in Figure

74 66 Figure 4.1: Maximum Radius Transfer The position of the ISS is described by r = rû r (4.1) where r is the magnitude of the distance of the ISS from Earth s center, and û r is the unit vector describing the radial direction. The angular velocity vector of the ISS is perpendicular to the plane of motion. Therefore, the velocity of the ISS is expressed as v = ṙû r + r θû θ (4.2) where dots represent time derivatives, and û θ is the unit vector in the transverse direction. Finally, the acceleration of the ISS is a = ( r r θ 2 )û r + (r θ + 2ṙ θ)û θ (4.3) where the double dot represent second time derivatives. The tangential velocity,v, of a particle is v = r θ, and the tangential acceleration is thus v = ṙ θ + r θ. If we further define the nomenclature,

75 67 u = radial component of the velocity m = mass of the ISS and ATV at time t ṁ = fuel consumption rate φ = thrust direction angle µ = gravitational constant of Earth T = magnitude of the thrust I sp = specific impulse of ATV thruster g 0 = nominal gravity the equations of motion become ṙ = u (4.4) u = v2 r µ r + T sinφ 2 m v = uv r + T cosφ m (4.5) (4.6) θ = v r (4.7) ṁ = T. I sp g 0 (4.8) In order to create a high fidelity model, we take into account that fact that Earth is not a perfect sphere or ellipsoid. The World Geodetic System (WGS) 84 is implemented to address this issue. The WGS 84 datum surface is an oblate ellipsoid with major radius a = 6, 378, 137m at the equator and flattening f = 1/ The nominal sea level of the WGS 84 is defined by a spherical harmonic series of 360 degrees [34]. The equations of motion also take into account the drag effects of a

76 68 rotating atmosphere. A transformation matrix TM is used at each step of the integration to take the states from the planar polar coordinate system to three-dimensional J2000 coordinates with the same orbital elements as the ISS [35]. These coordinates are then used in the models to calculate gravity and atmospheric density. The J2000 coordinates are also used to calculate the magnitude of the velocity of the ISS relative to the rotating atmosphere v norm = v ISS ω r ISS, (4.9) where v ISS and r ISS are the velocity and position of the ISS in J2000 coordinates, respectively. The angular velocity vector has magnitude consistent with the rate at which the earth rotates about its axis and is normal to the plane of rotation. The high fidelity equations of motion become ṙ = u (4.10) u = v2 r g + T sinφ m 1 2 v = uv r + T cosφ ṁ = C d Aρ u m v p v2 norm (4.11) C d Aρ v m v p v2 norm (4.12) m 1 2 T (4.13) I sp g 0 where g is the gravitational acceleration defined by the output of the WGS 84, C d is the drag coeficient of the ISS, A is the cross-sectional area of the ISS, and v p is the velocity in the polar planar system v p = [u, v]. The ρ term describes the atmospheric density at the current position of the ISS, and is based in the Jacchia Atmospheric Model.

77 Problem Constraints The optimization problem assumes that the ISS starts from an initial circular orbit with an altitude of 410km. It is also required that the terminal orbit be circular. It is also required that the thrusters be allowed to fire only while there is fuel available to burn. Thus, the dynamic constraints are r(0) = 410, 000m + r Earth (4.14) u(0) = 0 (4.15) µ v(0) = r(0) (4.16) m(0) = m ISS + m AT V + m fuel (4.17) u(t f ) = 0 (4.18) µ v(t f ) = r(t f ) (4.19) m(t f ) = m ISS + m AT V (4.20) where t f is the final time of transfer, r Earth is the radius of the Earth, and m AT V, m ISS, and m fuel are the dry mass of the ATV, dry mass of the ISS, and the fuel mass, respectively. It is assumed that the initial orbit is circular and required that the final orbit is circular. Table 4.1 lists some key specifications of the ATV [36]. Automated Transfer Vehicle Main Engine 4 x 490N, Aerojet (GenCorp) Model R-4D-1 Thrusters 28 x 220N, Astrium Lampoldshausen Prop. for Re-Boost 4,700 kg Launch Payload 7,085 kg Table 4.1: ATV Information

78 70 There are also constraints associated with the ATV and its thrusters. The maximum thrust achievable for the R-4D-1 engines is T max = 1956N [37]. Although the propellant designated for re-boost is 4,700kg, it is assume that the entire payload for the quasi end of life mission is devoted to re-boost. The thrust generated by the ATV is not throttle-able and is at either T max or zero. Constraining the thrust to reflect this creates difficulties when trying to find the optimal control histories, so the inequality constraints 0 T T max (4.21) are used instead. This method of constraining the the thrust magnitude is acceptable granted some further refinement of the control histories is later applied. The change in velocity created by each burn in the control history is given by V = T dt m(t) (4.22) where dt is the numerical integration step used to propagate the equations of motion. This change in velocity is then used to calculate how long the ATV would need to fire at maximum thrust to achieve the same V. t = m(t) V T max (4.23) If we assign the variables t i and t f to be the start and stop times of dt respectively, i.e. dt = t f t i, (4.24) and t h is halfway between t i and t f, then the times that the actual burns start and

79 71 end are t s = t h t 2 (4.25) and t e = t h + t 2, (4.26) respectively. There is no constraint on the thrust direction, φ. The control problem is solved by discretizing the control time histories into nodes. Each node contains two control variables, the thrust magnitude T and the thrust direction φ. Therefore, if the problem is discretized into 400 nodes, there is a total of 800 control variables that must be identified. Here one witnesses the importance of limiting the number of control variables. Had Cartesian coordinates been implemented there would be a total of 1200 control variables to find. The extra 400 control variables would significantly increase the computational expense of the problem. This method of parameterizing the control variables aptly referred to as control parameterization since only the control variables, and in this case some of the terminal states, are parameterized, while the state equations are numerically integrated. Thus, the selection of the number of nodes, N, and and the integration time step, dt, imply the final time constraint, t c. Therefore, t f is assigned, not free. If N is the chosen number of nodes, and we say that T = x(1 : N) and φ = x(n + 1 : 2N + 1), the control problem becomes min x r f (x) (4.27)

80 72 such that 0 x i T max, i = [1, 2,..., N], (4.28) u(0) = 0 (4.29) µ v(0) = r 0 (4.30) u(t f ) = 0, (4.31) µ v(t f ) =, r f (4.32) m(t f ) = m ISS + m AT V. (4.33) The terms r 0 and r f are the radii of the ISS s initial and terminal orbits, respectively. The final radius is a function of x(1 : 2N +1), since it depends on the control histories. The fmincon function in MATLAB is used to solve this problem, and it implements a sequential quadratic programming (SQP) algorithm to minimize functions with inequality constraints. 4.3 Sequential Quadratic Programming One of the most effective methods for addressing nonlinearly constrained optimization problems is an iterative process that models each iterate x k, k N 0 of the nonlinear problem (NLP) as a Quadratic Programming (QP) subproblem. Hence the method is commonly referred to as Sequential Quadratic Programming (SQP) [38]. The solution to each QP subproblem is used to construct a new iterate x k+1 until the optimal solution is found. The procedure is conducted in such a way that the sequence of iterates (x k ) converges to a local minimum denoted x. The method is also described as applying Newton s method to find the stationary point of the Lagrangian of the

81 73 optimization problem; therefore, SQP is also referred to as the Lagrange-Newton method The Karush-Kuhn-Tucker Theorem The Karush-Kuhn-Tucker (KKT) Theorem provides a set of necessary conditions for the inequality-constrained optimization problem. Consider the NLP min x f(x) (4.34) such that c i (x) = 0, i E, (4.35) c i (x) 0, i I, (4.36) and x = [x 1, x 2,...x n ] T. (4.37) where f and the functions c i are continuously differentiable. The variables E and I represent finite sets of indices. The function f is the objective function to be minimized, and c i, i E are the equality constraints and c i, i I are the inequality constraints. Another important set to identify is the active set, A(x). The active set is defined A(x) = E [i I c i (x) = 0]. (4.38) Intuitively, one sees that the equality constraints i E are always active. For a feasible point x, the inequality constraint i I is active if c i (x) = 0 and inactive if c i (x) < 0 is satisfied.

82 74 The Lagrangian for the of the above optimization problem is written L(x, λ) = f(x) + i E I λ i c i (x). (4.39) The necessary conditions for the local solution of the optimization problem are first order conditions, because they refer to properties of the gradients of the objective function and constraint functions. Assuming that the Lagrange multiplier λ i exist, the stationary point x that satisfies min x f(x) = f(x ) (4.40) c i (x ) = 0, i E (4.41) c i (x ) 0, i I (4.42) is found by employing the conditions: x L(x, λ ) = 0, (4.43) λ L(x ) = 0, i E, (4.44) λ L(x ) 0, i I, (4.45) (λ ) 0, i I, (4.46) λ i c i (x ) = 0, i E I. (4.47) These conditions are referred to as the KKT stationary conditions [39]. Condition ( 4.47) is known as complimentary conditions, and it implies that the constraint i is active, or λ i = 0, or both. Thus, the Lagrange multipliers corresponding to the

83 75 inactive constraints are zero, and we can rewrite condition ( 4.43) as x L(x, λ ) = f(x ) + λ i c i (x ). (4.48) i A(x ) Later on in this thesis these conditions are applied to a QP subproblem to form an iterative process that takes us to x from an initial guess x Quadratic Functions Each iteration of the SQP algorithm models the optimization problem as a quadratic function. Therefore, a discussion of quadratic functions and relevant properties of quadratic functions is necessary. A quadratic function on E n is written in vector-matrix notation as f(x) = c + b T x xt Ax (4.49) where b T = [b 1,..., b n ] and A = [a ij ] is an n n matrix. If A is proven to be positive definite, then f(x) is called a positive definite quadratic function [40]. It is apparent that the gradient vector of the quadratic function described by Eq is f(x) = Ax + b. (4.50) Assuming the quadratic function f(x) is twice continuously differentiable, there exists an extremum at one of the stationary points given by the solution of the equation f(x) = 0. By further assuming that f(x) is a positive definite quadratic function, meaning the Hessian of f(x), A, is positive definite, one can conclude that f(x)

84 76 attains a unique minimum at x = A 1 b. (4.51) Equation 4.51 provides a nice analytical solution. However, few NLP s lend themselves to analytical solutions, and the need to solve a NLP numerically brings leads to SQP Quadratic Programming (QP) Subproblem Consider the equality-constrained minimization problem min x f(x) (4.52) subject to c(x) = 0. (4.53) The SQP approach models the optimization problem at the current iterate x k as a QP subproblem and uses the solution to the QP subproblem to define a new iterate x k+1. The quadratic model is based on a Taylor series approximation of the Lagrangian about x k and λ k. The QP subproblem takes on the form min d L(x k, λ k ) + L(x k, λ k ) T d dt H(L)(x k, λ k )d (4.54) where H(L) is the Hessian of the Lagrangian, and d = [d k, d λ ] where d k = x k+1 x k (4.55) d λ = λ k+1 λ k. (4.56)

85 77 Applying the KKT conditions to the QP subproblem, we obtain H(L)(x k, λ k )d = L(x k, λ k ) T, (4.57) or c d k = c 0 2 xxl d λ f k cλ k c k. (4.58) The d k and d λ terms solve the Newton-KKT system, and the Newton step from the current iterate is provided by x k+1 = x k + d k. (4.59) λ k+1 λ k d λ If the Hessian is positive indefinite at some iterate, it is modified on an element by element basis in such a manner to ensure minimum distortion until it is positive definite [41]. This ensures a unique minimum is attained Active Set Method The SQP framework just described is easily applied to the inequality problem. The active-set, A(x), method is commonly used to address the inequality problem, and it is the method used by fmincon. The active set method generates iterates that are feasible with respect to the constraints c i (x) = 0, i E, c i (x) 0, i I, (4.60)

86 78 while decreasing the objective function f(x). The method finds the step from one iterate to the next by solving the quadratic subproblem, where some of the inequality constraints and all of the equality constraints are imposed as equality constraints. In other words, the active set is modeled as equality constraints, and the inequality constraints that do not fall in the active constraint category are ignored. This is accomplished using Lagrange multipliers. The Lagrange multipliers of any constraint measures the sensitivity of the objective function to changes in that constraint. For a given iterate, a positive Lagrange multiplier, for a particular constraint, indicates that the constraint is active. A negative Lagrange multiplier indicates that the constraint is inactive. Intuitively, this makes sense, because a positive Lagrange multiplier would indicate that the constraint is limiting the minimization while a negative Lagrange multiplier assists minimization of the cost function. Consider the problem subject to min x f(x) (4.61) a T i x = b i = 0, i A (4.62) At x k the Lagrange multiplier vector is λ = A 1 f, (4.63) where A is the n n matrix whose columns are a i,i A, and the columns of A T are s 1, s 2,..., s n. At x k Equation 4.63 is evaluated and the vectors s i, i A I are

87 79 the directions of the feasible edges at x k, and λ i are the slopes of f(x) along these edges. If λ i 0, i A I, then no feasible descent direction exists, and x k is optimal. Otherwise, the most negative λ i, i A I, called λ q is selected, and a search is made along x = x k + αs q, (4.64) where α 0. At x k the i th constraint has the value c k i = a T i b i and a value of c i (x) = c k i + αa T i s q at points along the edge. This search is terminated by the first inactive constraint to become active. The constraint function becomes zero when α = b i a T i x k a T i s. (4.65) q An illustration for the active-set method for a simple QP problem of two variables, f(x) = f(x 1, x 2 ), is displayed in Figure 4.2. For this particular QP problem, the Figure 4.2: The Active Set Method for a Simple QP Problem constraints are the bounds x 0 and the general constraint a T x b. At the first iteration x 1 the bounds x 1 0 and x 2 0 are both active. Calculating λ 1 shows that the constraint x 2 0 has a negative λ, and it becomes inactive. Thus, only

88 80 x 1 0 is active. The corresponding equality constrained problem is then solved by x 2, which is feasible. Calculation of λ 2 reveals that the constraint x 1 0 has a negative multiplier, so both constraints have negative multipliers and A is empty. The same is true for λ 3. At x 3 the equality constrained problem is solved by x which is infeasible. Therefore, the search direction s 3 is calculated, and a line search along s 3, using the relationship 4.65, gives x 4 as the best feasible point. The constraint a T x b is active at x 4 and is added to A. The equality constrained problem is solved and yields x 5. Calculation of multipliers at x 5 indicates that it is the optimal solution, and the algorithm is terminated [42]. 4.4 Runge-Kutta Method Variable time step ODE-solvers in MATLAB are used to calculate the trajectories in Section They are used because the thrust is held constant. Implementing the control parameterization method requires that the number and length of the integration time steps be specified, because the control value over each time step of the trajectory is treated as a different variable. The fourth-order Runge-Kutta (RK4) method is used in this optimization algorithm to numerically integrate the equations of motion. The RK4 method is one of the most accurate and popular numerical methods used to obtain solutions to first-order initial value problems [43]. There are Runge- Kutta methods of different orders, but the fourth-order Runge-Kutta method is so widely used that it is often referred to as the classical Runge-Kutta method. The RK4 method is a generalization of the basic Euler formula.

89 81 Given the ordinary differential equation ẏ = f(y, t) where the value y n is the variable y at time t n. The RK4 method approximates y n+1 using y n and t n. The method uses the weighted averages of numerical values of f(y, t) at several points within the interval (t n, t n + h) where h is the time step [44]. The RK4 algorithm is defined y n+1 = y n + h 6 (k 1 + 2(k 2 + k 3 ) + k 4 ) (4.66) where k 1 = f(y n, t n ) k 2 = f(y n hk 1, t n + h 2 ) k 3 = f(y n hk 2, t n + h 2 ) (4.67) k 4 = f(y n + hk 3, t n + h). 4.5 Upper and Lower Bounds In order to test and quantify the effectiveness of the solution, upper and lower performance bounds are identified. The upper performance bound is established using a Hohmann transfer. The Hohmann transfer is the globally optimized solution for a minimum-fuel transfer between coplanar circular orbits. Feasibility of the transfer requires that the orbit intersect both circular orbits of radii r 1 and r 2, where r 2 > r 1. Infeasible orbits are those for which either perigee lies outside the orbit with radius r 1, or the apogee lies inside the terminal orbit of radius

90 82 r 2. This means that the transfer orbit is an elliptical orbit with perigee at or less than r 1 and apogee at or greater than r 2. These conditions are summarized by r p = p 1 + e r 1 (4.68) r a = p 1 e r 2, (4.69) where p is the semi latus rectum and e is the eccentricity of the elliptical transfer orbit. Equations (4.68) and (4.69) can be rewritten in the form p r 1 (1 + e) (4.70) p r 2 (1 e). (4.71) Thus, each conic transfer orbit is characterized by its values of p and e. Equations (4.70) and (4.71) describe the feasible region in the (p, e) plane, as seen in Figure 4.3. Also shown are the regions in the (p, e) plane where elliptical (0 < e < 1), parabolic (e = 1), and hyperbolic (e > 1) orbits occur [45]. According to the law of cosines, the velocity change needed to exit or enter a circular orbit is given by: v 2 = v 2 + v 2 c 2v c vcos(φ) (4.72) where v is the velocity of the orbit coming into or leaving the circular orbit, v c is the velocity of the circular orbit, and φ is the angle between the two trajectories. Since the burns will be tangent to the two circular orbits, φ = 0, the equation is re-written v 2 = v 2 + v 2 c 2v c v. (4.73)

91 83 Figure 4.3: Feasible Region of Circle-to-Circle Transfers From the conservation of energy, the velocity of a particle in an orbit described by the semi major axis a at a point described by the radius r is v 2 = µ( 2 r 1 ). (4.74) a Using the relationship p = a(1 e 2 ), equation (4.75) is re-written v 2 = µ( 2 r e2 1 ). (4.75) p The total change in velocity for the transfer is given by v T = v 1 + v 2 (4.76) where v 1 occurs at r 1 and v 2 occurs at r 2. The v term in equation (4.73) can be replaced by v k, where k = 1, 2. Evaluating the gradient of equation 4.73 with