Optimal Transfers Between Unstable Periodic Orbits Using Invariant Manifolds

Transcription

1 Celestial Mechanics and Dynamical Astronomy manuscript No. (will be inserted by the editor Optimal Transfers Between Unstable Periodic Orbits Using Invariant Manifolds Kathryn E. Davis Rodney L. Anderson Daniel J. Scheeres George H. Born Received: date / Accepted: date Abstract This paper presents a method to construct optimal transfers between unstable periodic orbits of differing energies using invariant manifolds. The transfers constructed in this method asymptotically depart the initial orbit on a trajectory contained within the unstable manifold of the initial orbit and later, asymptotically arrive at the final orbit on a trajectory contained within the stable manifold of the final orbit. Primer vector theory is applied to a transfer to determine the optimal maneuvers required to create the bridging trajectory that connects the unstable and stable manifold trajectories. Transfers are constructed between unstable periodic orbits in the Sun-Earth, Earth-Moon, and Jupiter-Europa three-body systems. Multiple solutions are found between the same initial and final orbits, where certain solutions retrace interior portions of the trajectory. All transfers created satisfy the conditions for optimality. The costs of transfers constructed using manifolds are compared to the costs of transfers constructed without the use of manifolds. In all cases, the total cost of the transfer is significantly lower when invariant manifolds are used in the transfer construction. In many cases, the transfers that employ invariant manifolds are three times more efficient, in terms of fuel expenditure, than the transfer that did not. The decrease in transfer cost is accompanied by an increase in transfer time of flight. Keywords Three-Body Problem Unstable Periodic Orbits Invariant Manifolds Dynamical Systems Theory Transfer Trajectories Primer Vector Theory Introduction In the last three decades, libration point orbits (LPOs, a special class of unstable periodic orbits in the three-body problem, have been utilized in missions that require orbits vastly different than traditional Keplerian orbits to meet scientific goals. There are currently several missions operating in LPOs in the Sun-Earth system, such as ACE, SOHO, WMAP, and Herschel/Planck. The upcoming James Webb Space Telescope and K. E. Davis R. L. Anderson D. J. Scheeres G. H. Born Colorado Center for Astrodynamics Research, University of Colorado at Boulder 429 UCB Boulder, CO Kate.Davis@Colorado.edu

2 2 the Gaia mission will use Sun-Earth LPOs (Gardner, 2003; Lindegren et al., The Artemis mission will send two spacecraft from Earth orbit to the Moon in a transfer that involves numerous lunar approaches, flybys, and low-energy transfer legs in the Sun-Earth system. The trajectories of the two spacecraft culminate with injections into lunar LPOs before a final descent into low-lunar orbit (Broschart et al., LPOs have also been proposed for a wide range of missions, such as lunar navigation and communication relays, or for New Worlds Observer, a space-based observatory that aims to detect and analyze terrestrial extrasolar planets (Farquhar and Kamel, 973; Grebow et al., 2006; Hill et al., 2006; Hamera et al., 2008; Folta and Lowe, As interest in LPOs continues to increase and more missions utilize these orbits, a fuel-efficient method of constructing transfers between unstable periodic orbits will be a powerful mission design tool. The ability to transfer spacecraft between orbits will add more options and flexibility in mission design. Invariant manifolds and an understanding of dynamical systems theory were critical in the Genesis trajectory design and play a vital role in describing the transport of particles within the solar system (Lo et al., 200. An orbit s invariant manifold can be harnessed to construct a wide range of trajectories, from Jovian Moon tours to Earth-Moon transfers (Koon et al., 2002; Parker and Born, 2008; Mingotti et al., Invariant manifolds may also be used to construct transfer trajectories between unstable periodic orbits, and techniques have been developed to transfer between LPOs of the same energy using invariant manifolds (Koon et al., 2000; Gómez and Masdemont, In a previous paper, we proposed a method to construct transfer trajectories between unstable periodic orbits with different energies (Davis et al., 200. A trajectory from within the unstable manifold of the first orbit was connected to a trajectory from within the stable manifold of the second orbit by the execution of two maneuvers. The choice of the individual unstable/stable manifold trajectory combination was dictated by two-body parameters. It was numerically demonstrated that in the vicinity of the secondary, as the two-body parameters of an unstable manifold trajectory more closely matched the two-body parameters of a stable manifold trajectory, the total cost of the transfer decreased. This paper extends our previous research and constructs optimal transfer trajectories in multiple three-body systems. The transfer trajectories presented here will be compared to transfer trajectories that do not employ the use of invariant manifolds. Primer vector theory will be used to optimize the transfer trajectories in this research. Primer vector theory was first developed by Lawden (963, who established the necessary conditions for an optimal impulsive trajectory based on the primer vector and its derivative. Lion and Handelsman (968 extended Lawden s work to develop conditions for improving a reference impulsive trajectory by the addition of interior impulses or by the inclusion of initial and final coasts. Jezewski and Rozendaal (968 devised a method of systematically locating the minimum cost for an N-impulsive trajectory. Jezewski (975 provides a good summary of the contributions of Lawden and Lion and Handelsman to optimal trajectory theory and a good derivation of the equations associated with the primer vector. Many researchers have utilized primer vector theory to locate optimal transfers. For example, D Amario (973 used primer vector theory to compute optimal twoand three-impulse transfer trajectories between the Earth-Moon L 2 point and circular orbits about both the Earth and the Moon. Hiday (992 extended Lawden s work to develop the necessary conditions and transversality conditions for optimal trajectories with coastal arcs and interior impulses in the Elliptic-Restricted Three-Body Problem

3 3 (ERTBP. The necessary conditions for an optimal trajectory in the Circular Restricted Three-Body Problem (CRTBP are the same as those in the ERTBP. A short summary of the necessary conditions for an optimal trajectory and select equations concerning the primer vector will be presented. 2 The Circular Restricted Three-Body Problem The motion of a spacecraft acting under the influence of two massive bodies can be modeled by the equations of motion of the Circular Restricted Three-Body Problem (CRTBP. The two massive bodies are assumed to be in circular orbits about the barycenter of the system. The reference frame rotates at the same rate as the orbital motion of the two bodies and is centered on the barycenter. Henceforth, the larger of the masses will be referred to as the primary, and the smaller of the two will be called the secondary. The x-axis extends from the barycenter through the secondary, the z-axis extends in the direction of the angular momentum of the system, and the y-axis completes the right-hand coordinate frame. The equations describing the motion of the massless third body may be written as ẍ = 2ẏ + x ( µ x + µ R 3 ÿ = 2ẋ + y ( µ y R 3 µ y R2 3 µ x + µ R 3 2 ( z = ( µ z R 3 µ z R2 3, where R and R 2 are equal to the distance from the third body to the primary and secondary, respectively. The parameter µ is the three-body constant used to nondimensionalize the system. It is computed by dividing the mass of the secondary by the total mass in the system. When the system is nondimensionalized, the distance between the primaries is unity, and the locations of the primary and secondary on the rotating x-axis are x = µ and x = µ, respectively. Furthermore, the mean motion of the system is unity and the period of the rotating system is 2π. The reader is directed to Szebehely (967 for a derivation of the equations of motion. An integral of motion emerges in the synodic frame known as the Jacobi constant, C. The Jacobi constant is analogous to energy in the two-body problem, in that the Jacobi constant cannot change unless the spacecraft is perturbed by something other than the two primaries. Note that it is only a function of the position and velocity magnitude expressed in the rotating frame, C = 2Ω V 2, (2 where Ω = 2 (x2 + y 2 + µ R + µ R 2, (3 V 2 = ẋ 2 + ẏ 2 + ż 2. (4

4 4 2. Periodic Orbits There are five equilibrium points in the CRTBP known as libration or Lagrange points. At the libration points, the gravitational acceleration is balanced by the centripetal acceleration. In this work, a prefix is added to denote the system of each libration point by the secondary: EL i denotes the Sun-Earth system, LL i denotes the Earth- Moon system (Lunar libration points, and JEL i denotes the Jupiter-Europa system. There are countless families of periodic and quasi-periodic orbit families that exist under the governing equations of motion for the CRTBP. Particles may be in periodic orbits about the primary, the secondary, or any of the five libration points. Families of periodic orbits have been studied extensively by many researchers: see Darwin (897; Hill (898; Moulton (920; Strömgren (935; Bray and Goudas (967; Broucke (968; Farquhar and Kamel (973; Breakwell and Brown (979; Hénon (2003. There are three common classifications of LPOs: Lissajous, halo, and Lyapunov orbits. A Lissajous orbit is a quasi-periodic orbital trajectory that winds around a torus, but never closes in on itself. Halo orbits are a special case of periodic Lissajous orbits, as the in plane and out of plane frequencies are equal. Halo orbits are three-dimensional; a two-dimensional periodic orbit is termed a Lyapunov orbit. Two common methods for numerically computing LPOs are a Richardson-Cary expansion and a Single-Shooting Algorithm (Richardson and Cary, 975; Howell, Invariant Manifolds Unstable periodic orbits have an associated set of invariant manifolds. A manifold may be defined as an m-dimensional surface embedded in R n which locally possess the structure of R n. The term invariant means that a point on the manifold will remain on the manifold over time. There are two types of invariant manifolds: unstable and stable. The unstable manifold (W U includes the set of all possible trajectories that a particle on a nominal orbit could traverse if it was perturbed in the direction of the orbit s unstable eigenvector. The stable manifold (W S includes the set of all possible trajectories that a particle could take to arrive onto the nominal orbit along the orbit s stable eigenvector. The unstable manifold contains all of the trajectories that exponentially depart the nominal orbit as time moves forward, while the stable manifold contains all of the trajectories that depart the nominal orbit as time moves backwards (i.e., approach the nominal orbit as time moves forward. Each orbit has two associated stable and unstable manifold sets: one corresponding to a positive perturbation, and one corresponding to a negative perturbation. Planar views of the unstable and stable manifolds of an LL 2 halo orbit are shown in Figure. 3 Constructing a Reference Transfer between Orbits Davis et al. (200 developed a method to construct transfers between unstable periodic orbits with invariant manifolds. This section provides a brief description of the method. A bounding sphere is placed about the center of mass of the secondary. Unstable manifold trajectories (W U T of the initial orbit and stable manifold trajectories (WS T of the final orbit are propagated to position intersections with the bounding sphere. Next, each trajectory intersection is integrated inside the bounding sphere such that the

5 5 &"" '"" " / :58;!""" '"" &"".**!" + *,-!'""!&"".**!" + *,- $"" %"" / !%"" " 9:58;!$""!%""!!"""!!"""!"" " "" (**!" + *,- (a Unstable Invariant Manifold!$""!!"""!"" " "" (**!" + *,- (b Stable Invariant Manifold Fig. Planar views of the invariant manifolds of an Earth-Moon L 2 halo orbit. The arrows indicate the direction of motion. position differences of successive integration points are nearly constant. The radius of the bounding sphere is chosen to be approximately one-third of the radius of the sphere of influence of the secondary. Thus, the portions of the trajectories that are contained within the bounding sphere are mainly influenced by the secondary, and the trajectories can be analyzed from a two-body perspective. Each manifold point within the bounding sphere is expressed in terms of two vectors: the normalized angular momentum vector, and the eccentricity vector. The parameter κ is defined as the sum of two quantities: the difference in the normalized angular momentum vectors and eccentricity vectors between a point on a W U T and a point on a WS T. Davis et al. numerically demonstrated that there is an approximate linear relationship between the parameter κ and the total V required for the transfer. As the two-body parameters of a W U T more closely match the two-body parameters of a W S T, the total cost to complete the transfer decreases. The transfers constructed by Davis et al. were not optimized. Only two-impulse transfers were considered. The first maneuver was performed on the W U T and targeted a position on the W S T. After the execution of the first maneuver, a bridging trajectory was propagated to a position intersection with the W S T. The second maneuver was then executed to match the velocity on the bridging trajectory to the velocity on the W S T. In this research, the κ parameter will still be used to determine the unstable/stable manifold trajectory pair for the transfer. However, optimal transfers will be constructed. The number of impulses and their respective locations will be computed using primer vector theory. 4 Primer Vector Theory Formulation Primer vector theory is employed to examine three methods that can lower the cost of a non-optimal two-impulse transfer trajectory: A coastal arc along the initial orbital trajectory (i.e., alter the location of the first maneuver, a coastal arc along the final orbital trajectory (i.e., alter the location of the final maneuver, and the inclusion of interior impulses. Hiday (992 presents a comprehensive derivation of the formulation of the necessary conditions for an optimal trajectory in the ERTBP, as well as the conditions for when coastal arcs or interior impulses will lower the cost of a reference

6 6 trajectory. An abbreviated derivation of the necessary conditions in the CRTBP will be presented here. The derivation parallels the work of Jezewski (975 in the two-body problem and the work of Hiday in the ERTBP. The position and velocity of a spacecraft relative to the barycenter in the rotating frame may be expressed as R = [x, y, z] T and V = [ẋ, ẏ, ż] T, respectively. The state equations describing the system can be written as Ṙ = V V = βc m ût + g ṁ = β, (5 where m is the vehicle mass and û T is a unit vector in the thrusting direction with the constraint û T T ût =. The magnitude of the thrust is given by βc, where β is the mass flow rate and c is the effective exhaust velocity. The vector g expresses the equations of motion given in Equation in terms of the pseudo-potentiatl Ω, 2ẏ + Ω x g = 2ẋ + Ω y. (6 Ω z The values of Ω x, Ω y, and Ω z are the first partial derivatives of the pseudo-potential Ω, given in Equation 3, with respect to x, y, and z, respectively. It is important to note that trajectories in the CRTBP are traditionally expressed in rotating coordinates. Thus, the quantity g contains terms concerning the rotation of the synodic frame with respect to the inertial frame, unlike in the two-body problem where g contains only terms from gravitational acceleration. The goal is to minimize a cost function, or performance index, subject to the state equations given in Equation 5. For N-impulsive transfers where the goal is fuel minimization, the most logical choice for the performance index, J, is the fuel expenditure, expressed as N J = V i. (7 i= Adjoint variables are introduced in the form of Lagrange multipliers so the state equations can be treated as differential constraints. The adjoint variables are defined as λ R λ λ V, (8 λ m where λ R and λ V are vectors denoting the adjoints to the spacecraft s position and velocity, respectively, and λ m is a scalar quantity representing the adjoint to the spacecraft s mass. Next, a Hamiltonian, H, for the system can be formulated by multiplying the adjoint variables by their corresponding state equation, given in Equation 5, and summing, ( H = λ T RV + λ T βc V m ût + g λ mβ. (9

7 7 The differential equations for the adjoints are determined by taking the negative partial derivatives of H with respect to each state: λ T R = H R = λt VG R (0 λ T V = H V = λt R λ T VG V ( λ m = H m = βc m 2 λt Vû T. (2 The matrices G R and G V are the partials of g with respect to R and V, respectively, G R = g Ω xx Ω xy Ω xz R = Ω yx Ω yy Ω yz Ω zx Ω zy Ω zz ( , G V = g V = where Ω is given in Equation 3. Note that in the CRTBP, the G V matrix is different than in the ERTBP due to the omission of the eccentricity effects. Henceforth, the adjoint vector to the velocity, λ V, will be called the primer vector, p, and define p = λ V. Equation is expressed in terms of the primer vector and differentiated to obtain p T = λ T R ṗ T G V p T Ġ V. (4 Equation 0 is expressed in terms of p and is substituted into Equation 4 and simplified to result in the following p T = p T ( G R ĠV ṗ T GV. (5 The matrix ĠV is simply a 3x3 matrix of zeros (see Equation 3. Thus Equation 5 simplifies to p T = p T G R ṗ T G V. (6 Also note from Equation 3 that G R = G T R and G V = G T V, so Equation 6 can be transposed to reveal the second order differential equation for the primer vector p = G R p + G V ṗ. (7 This second order differential equation for the primer vector in the CRTBP is identical to the expression for the primer vector in the ERTBP (Hiday-Johnston and Howell, 994. However, due to the formulation in the ERTBP, the matrices G R and G V are not identical. It should also be noted that in Lawden s formulation of the primer vector differential equation in the two-body problem, the matrix G V is a matrix of zeros. The primer vector satisfies Equation 7 with the boundary conditions p(t 0 = V 0 (8 V 0 p(t f = V f V f. (9

8 8 The quantities V 0 and V f denote the impulsive maneuvers performed at the trajectory endpoints. In other words, the primer vector is a unit vector aligned in the velocity direction at the terminal endpoints on the trajectory. The second derivative of the state X, where X = [x, y, z, ẋ, ẏ, ż] T, is governed by the equations of motion given in Equation and can alternatively be written in a similar form as Equation 7 Ẍ = G R X + G V Ẋ. (20 The state transition matrix, Φ, is used to propagate the equations of motion of X. As the equations of motion for the state and the primer vector take on the same form, the value for p can be found at any time, given the initial conditions for p, ṗ and the state transition matrix ] [ p(t ṗ(t = [ Φ (t, t 0 Φ 2 (t, t 0 Φ 2 (t, t 0 Φ 22 (t, t 0 ] [ ] p(t0. (2 ṗ(t 0 The value for ṗ 0 can be found by noting that the primer vector is unity at the endpoints and using the relationship ṗ(t 0 = Φ ( [ 2 tf, t 0 p(tf Φ (t f, t 0 p(t 0 ]. (22 Now, the necessary conditions for an optimal impulsive trajectory can be expressed in terms of the primer vector:. The primer vector is continuous and has a continuous first derivative. 2. The primer vector satisfies the equation: p = G R p + G V ṗ. 3. The primer vector is a unit vector aligned in the optimal thrust direction at an impulse. 4. The magnitude of the primer vector equals unity at an impulse and has a value less than unity at all other instances along the transfer. 5. At all interior impulses ṗ = 0, and the derivative of the primer vector and the primer must be orthogonal due to the continuity of the Hamiltonian at a corner. An adjoint equation will be developed for future analysis of trajectories. The formulated expression will include additional terms not found in the adjoint equation for the two-body problem. Again, the differences are based on the accelerations dependence on velocity. Consider a variation between a reference trajectory and a perturbed trajectory. Let X = [R, V] represent the state on the reference trajectory and let X = [R, V ] represent the state on the perturbed trajectory. Let the variation between the reference and perturbed trajectories be represented as δ, δr = R (t R(t (23 δv = V (t V(t. (24 Assume that both the reference and perturbed trajectories are Null Thrust (NT arcs and that the reference and perturbed trajectories are necessarily close such that a linear

9 9 analysis can be used. Then, the variations for position and velocity on the NT arc can be written as δṙ = δv (25 δ V = δ R = δg = g g δr + R V δv δ R = G R δr + G V δv. (26 Premultiplying Equation 26 by p T and Equation 7 by δr T and subtracting gives p T δ R δr T p = p T G R δr + p T G V δv δr T G R p δr T G V ṗ. (27 In the CRTBP, G R = G T R and G V = G T V. Rearrange and simplify Equation 27 to obtain p T δ R δr T p p T G V δv δr T G T Vṗ = 0. (28 The quantity ṗ T δv is added and subtracted to Equation 28 and the result is rearranged to render ( p T δ V ( ( + ṗ T δv p T δr + ṗ T δṙ p T G V δr + ṗ T G V δr = 0. (29 Equation 29 may be expressed as a total time derivative ( d p T δv ṗ T δr p T G dt V δr = 0. (30 Equation 30 is integrated to obtain the adjoint equation p T δv (ṗ T + p T G V δr = Constant. (3 The adjoint equation is valid over any interval between two impulses (i.e., on an NT arc. The adjoint equation presented here differs from the two-body adjoint equation formulated by Jezewski (975. If the matrix G V were reduced to a matrix of zeros (i.e., no second order dependence on velocity, Equation 3 would reduce to the two-body adjoint equation. 4. Coastal Arcs Coastal arcs are employed to lower the cost of a reference two-impulse transfer trajectory. A coastal arc will move a maneuver in time, and subsequently, in position along its corresponding orbit. In this research, the initial maneuver is moved along the W U T of the first orbit, and the final maneuver is moved along the W S T of the second orbit. Let Γ be some reference two-impulse transfer trajectory and let Γ denote a perturbed trajectory that has been allowed to coast on the initial or final orbit. The difference in cost between the two trajectories can be written as δj = J Γ J Γ = ( V 0 + V f ( V0 + V f. (32 Hiday-Johnston and Howell (996 examined Equation 32 twice: Once for an initial coast and once for a final coast. They used relationships between the noncontemporaneous and contemporaneous variations in velocity immediately before and after the

10 0 impulse in conjunction with the adjoint equation to express Equation 32 in terms of the primer vector, δj Initial Coast = ṗ T 0 (dr 0 V 0 dt 0, (33 δj Final Coast = ṗ T ( f drf V f dt f. (34 If the perturbed trajectory Γ includes initial and final coastal arcs, Equations 33 and 34 can be combined and simplified to obtain δj = ṗ 0 V 0 dt 0 ṗ f Vf dtf. (35 A perturbed trajectory will have a lower cost than the reference trajectory if the quantity δj is negative, hence ṗ 0 V 0 dt 0 ṗ f Vf dtf < 0. (36 For a given trajectory, the values of ṗ 0 and ṗ f are known at the terminals. Consequently, these values dictate the signs of the coastal periods at the initial and final impulses. The following four cases determine the signs for the initial and final coastal arcs: ṗ 0 > 0 and ṗ f > 0 dt 0 > 0 and dt f > 0 Initial Coast/Late Arrival. ṗ 0 > 0 and ṗ f < 0 dt 0 > 0 and dt f < 0 Initial Coast/Final Coast. ṗ 0 < 0 and ṗ f > 0 dt 0 < 0 and dt f > 0 Early Departure/Late Arrival. ṗ 0 < 0 and ṗ f < 0 dt 0 < 0 and dt f < 0 Early Departure/Final Coast. The signs of the primer magnitude slopes at the terminals (ṗ 0 and ṗ f are used to adjust the times of the coastal arcs accordingly (dt 0 and dt f. The magnitude of the adjustment to the coastal arc time is proportional to the value of the terminal slope. A larger absolute value of the slope corresponds to a larger adjustment to the time of the initial or final coastal arc. After the coastal arcs have been adjusted, the slopes of the primer magnitude at the initial and final impulses are again computed and the process of adjusting the coastal arcs is repeated until the slopes at the terminals are close to zero, within a desired tolerance. This process can be implemented in an iterative fashion to quickly determine the coastal arcs that minimize the overall cost of the two-impulse transfer. The plot of the primer magnitude for a minimum two-impulse transfer would show primer magnitude values of one and slopes of zero at the terminal endpoints. A more detailed derivation of Equations may be found in Hiday-Johnston and Howell (996, Hiday (992, or Davis ( Addition of Interior Impulses If the cost of a two-impulse transfer has been minimized such that the slopes are zero at the terminals, but the primer exceeds unity at some point along the trajectory, the transfer is not optimal. The effect of an interior impulse is investigated by comparing a reference two-impulse trajectory to a neighboring trajectory that contains an interior impulse. The difference in cost between the two trajectories can be expressed as δj = J Γ J Γ = ( V 0 + V m + V f ( V0 + V f. (37

11 Equation 37 is manipulated using the contemporaneous variations in velocity immediately before and after the interior impulse and by applying the adjoint equation on the two NT arcs that are separated by the interior impulse on Γ. The resulting cost variation is found as ( δj = C p T m ˆη, (38 where C is the magnitude of the interior impulse, p m is the primer vector at the interior impulse, and ˆη is a unit vector in the direction of the interior impulse. If δj can be made negative, then the perturbed trajectory, Γ, has a cost improvement over the reference trajectory, Γ. Examining Equation 38 shows that δj can be made negative if and only if the magnitude of the primer on the reference trajectory at the time of the interior impulse is greater than unity ( p m >. Equation 38 also shows that the greatest improvement in cost, to the first order, can be realized by executing an interior impulse at the time when the magnitude of the primer is at a maximum. Furthermore, the maneuver should be in the direction of the primer vector. Hiday-Johnston and Howell (994 provide a detailed derivation of Equation 38 as well as a description of the equations required to compute the location and magnitude of the interior impulse. They also describe how to move the interior impulse, if the resulting three-impulse trajectory does not meet some of the criteria for an optimal trajectory. 5 Optimal Transfer Trajectories Optimal transfer trajectories will now be computed in three different three-body systems. First, a reference two-impulse trajectory will be computed using the method described by Davis et al. (200. Next, coastal arcs will be included to find the best two-impulse trajectory. If the two-impulse trajectory is non-optimal, interior impulses will be included to produce an optimal trajectory. 5. Transfers in the Sun-Earth System Optimal transfers were constructed for five different orbit pairs about EL. The z- amplitude of the initial halo orbit, A z, varied from 60,000 km to 240,000 km, and the z-amplitude of the final halo orbit, A z2, was held constant at 0,000 km. The reference trajectory and the corresponding primer vector history for the orbits A z = 60,000 km and A z2 = 0,000 km are shown in Figure 2. There is very little motion in the z-direction of this transfer, so only a planar projection is presented. The arrows in Figure 2 and all subsequent figures denote the direction of motion of the trajectory. Figure 2(b clearly shows a non-optimal primer history: the slopes at the terminals are not zero. The total V for the reference transfer is 3.30 m/s. The maneuver locations will be varied by employing initial and final coastal arcs to determine the best two-impulse transfer. The results are shown in Figure 3. The total V for the best two-impulse transfer is 20.6 m/s. The coastal arcs along the unstable and stable manifold trajectories that minimized the V were computed to be days and.74 days, respectively. Figure 3(b shows that this two-impulse transfer is optimal. The primer history never exceeds unity, and the slope is zero for a very small amount of time immediately after the initial impulse, as shown in the

12 2.*+!$ % +,- ' " ( $!(!"!'!!V2.*+!( % +,- ($% (!($%!!!V Earth Flybys!""!""$%!"%!"%$%!"&!"&$%!"'!"'$% *+!( % +,-!"$!"%!"&$!"&%!%$$!%$%!%!$ *+!$ % +,- (a Reference Two-Impulse Trajectory Time (Days past!v (b Primer Vector Magnitude History Fig. 2 A planar projection of the reference two-impulse transfer trajectory and corresponding primer vector magnitude history for the orbits A z = 60,000 km and A z2 = 0,000 km. The transfer is not optimal. The total V = 3.30 m/s. /*+!$, +-. ' " ( $!(!"!'!!V2 /*+!(, +-. ($% (!($%!!!V Earth!""!""$%!"%!"%$%!"&!"&$%!"' *+!(, +-.!V!"$!"%!"&$!"&%!%$$!%$%!%!$ *+!$, +-. (a Optimal Two-Impulse Trajectory Time (Seconds past!v Time (Days past!v Zoomed in near!v (b Primer Vector Magnitude History Fig. 3 A planar projection of the optimal two-impulse transfer trajectory and corresponding primer vector magnitude history for the orbits A z = 60,000 km and A z2 = 0,000 km. The total V = 20.6 m/s. inset of Figure 3(b. After the short duration of zero slope, the primer magnitude drops quickly. Compare the maneuver locations in Figure 2(a and Figure 3(a. The first maneuver has been moved to a location just prior to the closest approach to the secondary. The time between V and the closest approach represents the short time where the primer magnitude slope is zero. However, the spacecraft is traveling with a very high velocity at this point, and the effect of the flyby causes the precipitous drop in the primer magnitude right after the flyby. Interestingly, when the manifolds were propagated for a longer duration, a solution was located which retraced the middle portion of the trajectory. The reference two-impulse transfer is shown in Figure 4 and is denoted as a two-loop transfer. The corresponding primer vector magnitude history does not have zero slopes at the trajectory endpoints and is therefore non-optimal. The cost of this reference two-impulse two-loop solution is 3.4 m/s, an improvement of 7 m/s from the one-loop solution. The maneuver locations were altered by coastal arcs to find the minimum two-impulse solution. The best two-impulse transfer trajectory and primer vector history are presented in Figure 5.

13 3.*+!$ % +,- ' " ( $!(!"!'!!/ (!/!!"$!"%!"&$!"&%!%$$!%$%!%!$ *+!$ % +,- (a Reference Two-Impulse Trajectory, Two Loop Solution Time (Days past!v (b Primer Vector Magnitude History, Two-Loop Solution Fig. 4 A planar projection of the reference two-impulse transfer trajectory with two-loops and corresponding primer vector magnitude history for the orbits A z = 60,000 km and A z2 = 0,000 km. The transfer is not optimal. The total V = 3.4 m/s..*+!$ % +,- ' " ( $!(!"!'!!/ (!/!!"$!"%!"&$!"&%!%$$!%$%!%!$ *+!$ % +,- (a Best Two-Impulse Trajectory, Two Loop Solution Time (Days past!v (b Primer Vector Magnitude History, Two-Loop Solution Fig. 5 A planar projection of the best two-impulse transfer trajectory with two-loops and corresponding primer vector history for the orbits A z = 60,000 km and A z2 = 0,000 km. The transfer is not optimal. The total V = 0.7 m/s. By allowing coastal arcs, the total V for the two-impulse transfer has been lowered to 0.7 m/s. Obviously, this is not an optimal transfer, as the magnitude of the primer vector exceeds unity around 3.7 days after the first maneuver as seen in Figure 5(b. However, Figure 5 does represent the best two-maneuver transfer. The term best should not be confused with the term optimal. The word best will refer to the lowest possible V for a transfer with a set number of maneuvers. The best two-maneuver transfer will exhibit zero slopes at the terminal points, but the primer magnitude may exceed unity. On the contrary, an optimal trajectory will meet all of Lawden s necessary conditions for optimality. An optimal trajectory with N-impulses will always be the best N-impulse trajectory, but the converse is not always true. An interior impulse was added to the two-loop transfer to produce a lower total V. The resulting trajectory and primer vector magnitude history are shown in Figure 6. An optimal transfer is shown. The primer vector is unity at the impulse locations (initial, interior, and final and is less than unity at all other locations. The inset in

14 4 Figure 6(b zooms in near the interior impulse to show that there is no cusp in the primer magnitude, but rather a region of zero slope in the primer magnitude at the interior impulse. Thus, all of the criteria for an optimal trajectory are satisfied. The inclusion of an interior impulse ( V Int decreased the total V from 0.68 m/s to 0.52 m/s, a cost savings of 4.6 mm/s. In a real mission design scenario, it is highly unlikely that the interior impulse would be performed, as a noticeable benefit is not realized. However, this is a good demonstration that an optimal trajectory can be computed..*+!$ % +,- ' " ( $!(!"!'!!V2!V!VInt Earth!VInt!"$!"%!"&$!"&%!%$$!%$% 0.992!%!$ *+!$ % +, (a Optimal Transfer Trajectory, Two Loop Solution Zoomed in near!vint!vint Time (Days past!v (b Primer Vector Magnitude History, Two-Loop Solution Time (Days past!v Fig. 6 The optimal transfer trajectory and primer vector magnitude history for the two-loop transfer for the orbits A z = 60,000 km and A z2 = 0,000 km. The yellow dots denote the impulse locations. The total V = 0.5 m/s. The cost of the two-loop trajectory is approximately one-half the cost of the oneloop transfer. The κ parameter for the two-loop transfer was also much smaller than the κ parameter associated with the one-loop transfer. The idea that three-body trajectories can retrace their paths has been explored by Lo and Parker (2005 and Parker et al. (200 in their work on chaining periodic orbits. The concept of chaining implies that the orbital trajectory shown here could theoretically retrace its path an infinite number of times. However, the time constraints on mission designs, and the likely need for large amounts of station-keeping renders such an idea as practically unfeasible. Transfers were constructed and optimized for the other four orbit pairs in the Sun- Earth system. A summary of the transfer costs and times of flight are presented in Table. One and two-loop solutions were found for each orbit pair. All of the transfers required either two or three impulses for optimality. Table also includes the costs and times of flight for the transfers constructed by Howell and Hiday-Johnston (994 which did not employ invariant manifolds. There are several pieces of information to note in Table. First, the transfers constructed using invariant manifolds show a drastic decrease in the total V. The transfer costs decreased by as much as 36% for the one-loop case and 72% for the twoloop case when compared to the transfers constructed without invariant manifolds. Note that the one-loop transfer results are nearly linear, as are the results from Howell and Hiday-Johnston. Gómez et al. (998 stated that the transfer cost between orbits should be nearly proportional to the z-amplitude difference between the orbits. However, this may not necessarily hold true for transfers constructed using invariant manifolds that

15 5 Table Transfer trajectory costs (m/s and times of flight (days for optimal transfers in the Sun-Earth system. A z A z A z A z A z Initial Cost (m/s Howell Davis Davis z-amplitude TOF (days Solution -Loop 2-Loop = 60,000 km = 80,000 km = 200,000 km = 220,000 km = 240,000 km V TOF V TOF V TOF V TOF V TOF have a limited time of flight. Consider the two-loop solution transfers for Case A, the transfer with the initial z-amplitude of 60,000 km, and Case B, the transfer with the initial z-amplitude of 80,000 km. Although the energy difference is higher in Case B, the transfer cost is smaller (3.37 m/s for Case B vs. 0.5 m/s for Case A. A quick investigation of the κ values showed that the κ parameter was almost 50% lower for Case B than for Case A. Given infinite propagation time for the manifolds, it is likely that the transfer for Case A will have a lower cost than the transfer for Case B. Finally, the clear drawback to the decrease in the fuel cost of the transfers using invariant manifolds is a seven-fold and eleven-fold increase in total time of flight for the one-loop and two-loop transfers, respectively. This increase in transfer time suggests that the methods presented here would be more applicable in a three-body system with smaller time scales, such as the Earth-Moon system, or the Jupiter-Europa system. 5.2 Transfers in the Earth-Moon System Two orbits about LL 2 were selected for a transfer in the Earth-Moon system. The initial orbit was chosen as a Southern halo orbit with Jacobi constant of C = and a period of 7.5 days. The final orbit was selected to be a Southern halo orbit with a Jacobi constant of 3.06 and a period of 3.6 days. Both orbits were proposed as part of a lunar navigation and communication relay; Hamera et al. (2008 designed the initial orbit for South Pole coverage, and the final orbit was designed by Hill et al. (2006 to have excellent far side coverage. First, a transfer trajectory will be constructed between the orbits that does not use invariant manifolds. It will later be compared to the transfer trajectory that does use invariant manifolds. The impulse locations on the initial and final orbits were arbitrarily selected as the maximum z-amplitudes of each halo orbit. The first maneuver targeted the position vector on the final orbit corresponding to the location of the second maneuver. The magnitude and components of the first impulse were computed using a Level differential corrector, which is described by Wilson (2003. The second maneuver corrected the velocity to match the velocity on the second orbit at the position intersection. The bridging trajectory was propagated under the equations of motion of the CRTBP. The

16 6 reference two-impulse trajectory and corresponding primer vector magnitude history are presented in Figure 7. %!0 % /*+! *,-!%!$!' &.*+! *,-! Orbit!&!" Orbit 2 $ $% (*+! *,- (a Reference Two-Impulse Transfer Trajectory Time (Days past!v (b Primer Vector Magnitude History Fig. 7 The reference two-impulse transfer trajectory and primer vector magnitude history for the Earth-Moon system halo-to-halo orbit transfer with no invariant manifolds. The transfer is not optimal. The total cost of the transfer is 220 m/s. The primer history shows a transfer that is clearly non-optimal. The slopes are non-zero at the terminals and the primer exceeds unity. Furthermore, the trajectory itself, Figure 7(a, shows a transfer where the first maneuver of m/s acts almost perpendicularly to the velocity direction of the initial orbit to target the position on the final orbit. The final impulse of m/s also acts in a nearly normal direction to the velocity of the final orbit. Coastal arcs of.07 days and 4.5 days were added to the initial and final orbits, respectively, to reduce the cost of the reference transfer. The best two-impulse trajectory and primer vector history are shown in Figure 8. The trajectory given in Figure 8(a is a vast improvement over the reference trajectory. The initial and final coastal arcs substantially lowered the total cost from over 200 m/s down to 7.0 m/s. The time of flight also increased; the time of flight for the reference transfer was approximately 3 days while the best two-impulse transfer has a time of flight of 2.68 days. The best two-impulse transfer is non-optimal as the primer vector exceeds unity. The distance to the secondary has been superimposed over the primer history in Figure 8(b to show that the primer magnitude peaks very near the closest approach to the secondary. Recall that the largest cost improvement can be realized when an interior impulse is added at the location where the primer magnitude has its maximum value. Thus, similar to the transfer seen in Figure 6(a, the interior impulse will be located very close to the flyby of the secondary. The optimal transfer and primer vector history are presented in Figure 9. The primer history satisfies all of the criteria for an optimal trajectory. The primer and its derivative are continuous at the interior impulse and the primer magnitude is less than or equal to unity at all instances. The addition of the interior impulse decreased the total cost of the transfer from 7.0 m/s to 3.20 m/s.

17 7 /*+! *,- %!%!$!' & 02.*+! *,-!3 +!3 %!&!" $ $% (*+! *,- (a Best Two-Impulse Transfer Trajectory Minimum Distance to the Secondary Time (Days past!v (b Primer Vector Magnitude History and Distance to the Secondary Distance to the Center of the Secondary, 0 3 km Fig. 8 The best two-impulse transfer trajectory and primer vector magnitude history for the Earth-Moon system halo-to-halo orbit transfer with no invariant manifolds. The transfer is not optimal. The total V = 7.0 m/s. /*+! *,- %!%!$!'! !0!0 % + & $% $.*+! *,-!&!" (*+! *,- (a Optimal Transfer Trajectory Time (Days past!v (b Optimal Primer 0.36Vector Magnitude Time (Days past!v History!VInt Zoomed in near!vint Fig. 9 The optimal transfer trajectory and primer vector magnitude history for the Earth- Moon system halo-to-halo orbit transfer with no invariant manifolds. The interior impulse is denoted by the circle. The total V = 3.20 m/s. Now the transfer constructed without manifolds will be compared to a transfer trajectory that has been constructed using manifolds. The unstable and stable manifold trajectories with the smallest κ value were selected and a reference two-impulse trajectory was created. The initial and final maneuver locations were selected to be approximately eight hours before and eight hours after the closest approach in position between the manifolds. The reference two-impulse transfer trajectory and corresponding primer vector magnitude history are shown in Figure 0. The total V for this transfer is m/s. Although this transfer has not been optimized, it is a drastic

18 8 decrease from the transfer that was constructed without manifolds (total V = 3.20 m/s. z, 0 3 km !0!20!30!40!50!60!70!V Moon Moon!V 2!V2! y, 0 3 km (a Reference Two-Impulse Transfer Trajectory Time (Days past!v (b Primer Vector Magnitude History Fig. 0 The reference two-impulse transfer trajectory and primer vector magnitude history for the Earth-Moon system halo-to-halo orbit transfer. The total V = m/s. The transfer is clearly non-optimal as the primer exceeds unity and does not display zero slopes at both terminals. Initial and final coastal arcs will be employed to locate the best two-impulse transfer trajectory. The best two-impulse trajectory and corresponding primer vector magnitude history are presented in Figure *+(" & +,- '" (" "!("!'"!&"!%"!!"!$"!"!2 ( /00!2 '!V2!!" "!" *+(" & +,- (a Best Two-Impulse Transfer Trajectory Minimum Distance to the Secondary Time (Days past!v (b Primer Vector Magnitude History and Distance to the Secondary 50 Distance to the Center of the Secondary, 0 3 km Fig. The best two-impulse transfer trajectory and corresponding primer vector magnitude history for the halo-to-halo orbit transfer in the Earth-Moon system. The transfer is not optimal. The total V is m/s. Figure (b shows a non-optimal primer vector magnitude history. Although the slopes at the endpoints are zero, the primer exceeds unity, violating the necessary

19 9 conditions for an optimal trajectory. The distance to the secondary has been overlaid on top of the primer vector magnitude history to show that the peak in the primer vector corresponds to a flyby of the Moon. The inclusion of the coastal arcs drastically reduced the total V of the transfer. The best two-impulse transfer cost is m/s, an improvement of over 33 m/s. An interior impulse was added at the location where the primer magnitude achieved its maximum value. The resulting trajectory and primer history are plotted in Figure 2. The inset of Figure 2(b zooms in near the interior impulse to verify that there are no cusps or discontinuities in the primer vector, which would invalidate the optimality of the solution. The primer history is smooth, continuous, and never greater than unity. Therefore, the transfer is optimal. The interior impulse was necessary to render an optimal trajectory. However, it had a very small effect on the total transfer cost, as the V decreased by only 9 mm/s.. %!!VInt !VInt 0+,!,-.!!%!!!$!&!V!!V2! %!(!' &!&!" $ $% /+,!,-. *+,!,-. (a Optimal Transfer Trajectory Zoomed in near!vint Time (Days past!v (b Primer Vector Magnitude History Time (Days past!v Fig. 2 The optimal transfer trajectory and primer vector magnitude history for the Earth- Moon system halo-to-halo orbit transfer. The total V is m/s. The cost of the optimal trajectory using invariant manifolds is over three times less than the cost of the transfer trajectory that did not use manifolds. With the decreased cost came an increase in the time of flight. The time of flight for the optimal transfer with manifolds is 04 days. However, as shown in Figure 2, a large portion of the time of flight is spent asymptotically departing the initial orbit. Recall that the initial and final halo orbits were originally designed for a lunar navigation and communication relay and were selected for their lunar coverage characteristics. As the majority of the trajectory stays on the far side of the Moon in the vicinity of the initial orbit, this path provides excellent coverage of the South Pole and far side of the Moon during the transfer. Hence, the longer time of flight is not a hindrance to mission design. Despite the fact that this transfer satisfies the conditions for an optimal trajectory, it may not represent the global minimum in terms of the total V cost. The transfers illustrated in Figures 9 and 2 both satisfied the necessary conditions for an optimal trajectory. However, the cost of the transfer that does employ invariant manifolds is only m/s, over three times less than the transfer that did not use invariant manifolds. A transfer trajectory with a lower V may be achieved by propagating the

20 20 manifolds for a longer duration, as was seen in the two-loop solutions for the transfers in the Sun-Earth system. Hence, the trajectories should be called locally optimal. Now consider the reverse transfer, i.e., the path from the larger halo orbit to the smaller halo orbit. If a transfer has already been constructed to connect Orbit A to Orbit B using invariant manifolds, finding the transfer from Orbit B to Orbit A can be quite simple. There is an inherent symmetry of the invariant manifolds in the CRTBP about the x-axis. This symmetry can be seen by examining the invariant manifolds in Figures (a and (b. If a transfer from Orbit A to Orbit B is known, the transfer from Orbit B to Orbit A can be found by reflecting the original transfer about the x-axis. Let the parameter τ describe the position of a particle on a periodic orbit, where the value of τ ranges from 0 2π radians, increasing in the direction of orbital motion. τ is defined to be zero at the orbit s intersection with the x-z plane in the +ẏ direction and π radians at the intersection with the x-z plane in the ẏ direction. The parameter τ is used to define the starting locations of the manifold trajectories along their respective orbits. To reflect the transfer about the x-axis, the τ values are altered as follows τ UB to A = 2π τ SA to B (39 τ SB to A = 2π τ UA to B. (40 In a similar fashion, the new times to propagate the manifolds can be found as t UB to A = t SA to B (4 t SB to A = t UA to B. (42 It should be noted that Equations 4 and 42 give a first guess at the times to propagate the manifolds. It is likely that the new times will have to be optimized using primer vector theory. A trajectory was constructed to transfer the spacecraft back from the larger halo orbit to the smaller halo orbit. The optimal transfer trajectory and primer vector magnitude history are shown in Figure 3. Three maneuvers were required for an optimal transfer. The interior impulse was executed near the closest approach to the secondary. The trajectories given in Figures 2(a and 3(a are symmetric about the x-axis. It was also noted that the primer history shown in Figure 3(b is a reflection of the primer history given in Figure 2(b. The transfers costs differ by only 4 mm/s. 5.3 Transfers in the Jupiter Europa System Many different types of unstable periodic orbits and their associated invariant manifolds exist within the various Jupiter-Jovian Moon three-body systems (Anderson and Lo, Recently, Russell (2006 located families of unstable periodic orbits about Europa. Therefore, in the Jupiter-Europa system, it is possible to construct transfers that connect an LPO to an unstable periodic orbit about Europa. The initial orbit was selected as a JEL 2 halo orbit with Jacobi constant C = The final orbit was chosen to be an unstable periodic orbit about Europa with Jacobi constant C = The initial and final orbits are shown in Figure 4. The unstable/stable manifold pair with the smallest κ parameter was selected for the transfer. The initial and final maneuver locations for the reference two-impulse transfer trajectory were chosen to be approximately 2 hours before and after the closest position approach of the manifold trajectories. The primer vector magnitude history