OPTIMIZATION OF PRELIMINARY LOW-THRUST TRAJECTORIES FROM GEO-ENERGY ORBITS TO EARTH-MOON, L 1, LAGRANGE POINT ORBITS USING PARTICLE SWARM OPTIMIZATION

Transcription

1 AAS OPTIMIZATION OF PRELIMINARY LOW-THRUST TRAJECTORIES FROM GEO-ENERGY ORBITS TO EARTH-MOON, L 1, LAGRANGE POINT ORBITS USING PARTICLE SWARM OPTIMIZATION Andrew J. Abraham, David B. Spencer, and Terry J. Hart INTRODUCTION A technique for the preliminary global optimization of a low-thrust transfer trajectory from Earth orbit to a nominal, collinear Lagrange point orbit in the Earth-moon system is presented. The initial Earth orbit has a Jacobi energy equal to that of a geosynchronous Earth orbit (GEO-energy). Particle swarm optimization (PSO) is utilized to quickly locate the globally optimal patch-point where the low-thrust trajectory is terminated and the spacecraft begins to ballistically coast along the stable manifold. The results of the PSO algorithm are compared with that of a stochastic Monte Carlo algorithm. The Earth-Moon three-body environment contains unique topography that has only recently been utilized by spacecraft mission planners. 1, 2 Missions such as ARTEMIS have orbited the L 1 and L 2 Lagrange points of the Earth-Moon system in order to study the distant regions of Earth s magnetotail. Past studies of these Lagrange point orbits have focused on applications such as: communications, 3 observation and surveillance, 4 navigation, and human operations 5, 6 and may utilize Lyapunov, Lissajous, or various types of Halo orbits to accomplish mission objectives. While many studies have focused on impulsive maneuvers for insertion into a Lagrange point orbit, few focus on finite, low-thrust transfers. Mingotti et al 7 focused on low-thrust transfers from a Geosynchronous Transfer Orbit (GTO) to Earth-Moon halo orbits of L 1 and L 2 via stable manifolds. Mingotti used direct transcription and collocation to develop his trajectory beginning with an initial guess of a simple two-body, low-thrust spiral. Unfortunately, this work was not primarily concerned with identifying the globally optimal patch point of the stable manifold. A patch point is defined as the point on a stable manifold, where the low-thrust trajectory arc ends and ballistic motion along the manifold begins. This paper presents a method of identifying a near-globally-optimal patch point which would likely be a better initial-guess solution for Mingotti s original collocation problem. 7, 8 The problem is framed in terms of a non-convex optimization problem and solved using an evolutionary algorithm known as the Particle Swarm Optimization (PSO) method with the given search-space defined as the entire stable manifold itself. It is hypothesized that an evolutionary method would be able to avoid local minima in search of the globally-optimal solution. 9 The PSO method is also directly compared with a brute-force and stochastic Monte Carlo method to demonstrate its ability to identify large numbers of usable patch points given a modest number of function evaluations. PROBLEM STATEMENT While low-thrust trajectories have been studied extensively in low Earth orbits such trajectories are less studied (although references 7, 8, 14, 15 begin to address the problem) in regions above Geosynchronous Earth Orbit (GEO) where the three-body, perturbative effects of the Moon become increasingly dominant. The ratio of the Moon s gravitational force to the Earth s gravitational force is around 10 4 in GEO and Ph.D. Candidate, Mechanical Engineering, Lehigh University, 19 Memorial Dr. W., Bethlehem, PA, Professor of Aerospace Engineering, Penn State University, 229 Hammond Building, University Park, PA, Professor of Practice, Mechanical Engineering, Lehigh University, 19 Memorial Dr. W., Bethlehem, PA,

2 rapidly increases above that altitude. An optimal trajectory is desired that takes a spacecraft from GEO (or an orbit with equivalent energy) to a desired Earth-Moon, Lagrange point orbit via a low-thrust arc that terminates on the invariant stable manifold of the nominal (i.e. desired) Lagrange point orbit. Since the main objective of this study is to focus on a regime of motion where three-body effects are significant, any low-thrust trajectory less than the orbital energy of GEO will not be considered. Instead, the spacecraft s initial Earth orbit will begin with a Jacobi energy roughly equal to that of a GEO orbit and will increase as a function of time and thrust. It is assumed that the spacecraft can be placed into this initial Earth orbit via either: 1. a finite thrust maneuver or a series of finite thrusts which are considered to be high-thrust 2. a low-thrust spiral from a Low Earth Orbit (LEO) to the initial GEO-energy orbit. Of course the method of raising the orbit (either method 1 or 2) would have a significant influence on the type of GEO-energy orbit allowed. A high-thrust maneuver would offer a great deal of flexibility while an efficient low-thrust maneuver would likely be restricted to a GEO-orbit of relatively low eccentricity. This is because a low-thrust orbit that originates in LEO must have a very small eccentricity or risk impacting the Earth s surface. It is well known that the most time and fuel-efficient continuous low-thrust control law in two-body 7, 14 dynamics is that of tangential thrust. Since continuous tangential low-thrust will not significantly alter the eccentricity of a circular orbit, it is assumed that the eccentricity of the initial GEO-energy Earth orbit should, ideally, be low if not zero. This goal is incorporated into the fitness function described in later sections of this paper. The concept of utilizing the stable manifold is not arbitrary as a spacecraft that is placed on the manifold will be ballistically delivered to the nominal Lagrange point orbit after some finite amount of time. Unfortunately, the stable manifolds in the Earth-moon system never come very close to the Earth (i.e. LEO nor GEO, as is evidenced by Figure 3). It, therefore, becomes unavoidable for a low-thrust mission to operate an appreciable amount of time in a regime that includes three-body effects. System Dynamics The Circular Restricted Three Body Problem (CR3BP) is used to model the motion of a third body of negligible mass (the spacecraft) under the simultaneous gravitational acceleration of two massive primaries. Refering to Figure 1, which uses a synodic reference frame with its origin located at the barycenter of the two primaries, the equations of motion can be expressed as ẍ 2ẏ = Ω x + u x (t), ÿ + 2ẋ = Ω y + u y (t), z = Ω z + u z (t), ṁ (t) = T (t) I spg o (1) with the subscripts of the pseudopotential Ω (x, y, z) = 1 2 ( x 2 + y 2) + 1 µ r 1 + µ r 2 (2) indicating partial derivatives with respect to the sub-scripted variable. The mass parameter µ = m 2 m 1 + m 2 (3) represents the ratio of the mass of the smaller primary divided by the total mass of the system. In the case of the Earth-Moon system, this mass parameter is µ = The distances r 1 = (x + µ) 2 + y 2 + z 2 and r 2 = (x [1 µ]) 2 + y 2 + z 2 represent the distance between the spacecraft and the smaller primary (Moon) and larger primary (Earth), respectively. The vector u (t) = [u x (t), u y (t), u z (t)] T = [ ] T Tx(t) m(t), Ty(t) m(t), Tz(t) m(t) represents the acceleration of the third body due to the low-thrust engine in the synodic 2

3 frame. Finally, T represents the thrust of the low-thrust engine, I sp is the specific impulse, m the spacecraft s instantaneous mass, and g o is the Earth s gravitational acceleration at sea level. Note that the Eq. 1 are written in dimensionless units, that is to say that: the angular velocity of the larger and smaller primary (ω), the sum of the masses of each primary, and the distance between each primary are all intentionally set to unity. The unit of distance, [du], is the distance from the larger primary to the smaller primary, while the unit of time, [tu], is equal to the synodic period of the primaries divided by 2π. The velocity unit, [vu], is simply equal to [du] /[tu]. Setting u = T = 0 defines a ballistic trajectory in the CR3BP. Under ballistic conditions, exactly one constant of motion can be defined 16 for Eq. 1 C (x, y, z, ẋ, ẏ, ż) = 2Ω (x, y, z) ( ẋ 2 + ẏ 2 + ż 2) = constant, (4) and is known as the Jacobi energy. For a given spacecraft state, X = [x, y, z, ẋ, ẏ, ż] T, there exists exactly one Jacobi energy in the ballistic CR3BP. This energy defines the allowable and forbidden regions of motion for the spacecraft. It is also possible for the spacecraft to locate itself at an equilibrium point of Eq. 1. It is widely known that there are five such equilibrium points under ballistic dynamics; three collinear points that are aligned with both primaries (L 1, L 2, and L 3 ) and two triangular points ( L 4 and L 5 ) which are located at the vertex of two equilateral triangles formed by each primary and the respective Lagrange point. It is possible to define complex, periodic trajectories in the vicinity of each collinear Lagrange point. This is due to the fact that a linear analysis of the collinear points reveals a saddle center center nature to the dynamics of a spacecraft placed near those points. Due to the presence of the center part, it is possible to define two-dimensional Lyapunov orbits, three-dimensional halo orbits, and quasi-periodic Lissajous trajectories. Figure 1. Coordinate System of CR3BP Invariant Stable Manifolds A Lagrange point orbit can be defined either via tabulated values 16 or a numeric computation that solves a two-point boundary value problem. 17 The Lagrange point orbit is then defined by an arbitrary state vector (anywhere along the orbit), X orbit, and the period of the orbit, P. Under controlled CR3BP dynamics, the state vector for the spacecraft is expressed in terms of position, velocity, and spacecraft mass as X = x y z ẋ ẏ ż m (5) 3

4 with the controlled CR3BP expressed in terms of a first-order differential equation as Ẋ = ẋ ẏ ż ẍ ÿ z ṁ = v x v y v z 2v y + Ω x + u x 2v x + Ω y + u y Ω z + u z T I spg o. (6) Numeric integration of Eq. 1 (with ballistic assumptions of u = T = 0 and m = constant) as well as the State Transition Matrix (STM) Φ (t,t0) = X(X0,t) Xo, for one period, P, with the initial condition X 0 = X orbit enables the calculation of the monodromy matrix, M = Φ (t=p,t0) = X(X0,t=P ) Xo associated with the initial state, X orbit. The stable eigenvector, ν, of the monodromy matrix is multiplied by a very small number, ɛ, where ɛ = Two perturbed initial states, X pert = X orbit ± ɛν, are then integrated backward in time using the ballistic version of Eq. 1. In the case of L 1 and L 2 orbits, the integration is terminated when the spacecraft crosses the yz-plane from the negative x-directon. This defines two trajectories that are members of the stable manifold of the nominal Lagrange point orbit. The nominal Lagrange point orbit can be discretized into N states defined as X (k) orbit with k [ 1, N ]. The process is then repeated for other values of X (k) orbit which, in turn, constructs a stable manifold of 2N trajectories. If a spacecraft s state lies along a trajectory within this manifold then the ballistic flow forward in time will take it to the state X (k) orbit and the spacecraft ( will be inserted into the nominal Lagrange point orbit. In this way, any state within the manifold, X s.m. τs.m., k ), can be expressed via two parameters: 1. An integer k [ 1, N ] that corresponds to a state on the nominal orbit X (k) total number of states that represent a discretization of the orbit. orbit with N being the 2. A time parameter τ s.m. that represents the time remaining for a ballistic flow of Eq. 1 to reach the state X (k) orbit. Suboptimal Trajectory Generation Using Dynamical Systems Theory (DST), the invariant stable manifold of the nominal Earth-Moon Lagrange point orbit is constructed as described in the preceding section. Next, a particular trajectory segment, k (±), is randomly selected from the stable manifold, W s(±), with the positive sign indicating a manifold constructed by the positive perturbation, +ɛ, and the negative sign indicating a manifold constructed by the ) negative perturbation, ɛ. From this trajectory segment a random state, X (k(±) orbit, is chosen to be a patch point. A low-thrust trajectory is then propagated backwards in time, according to Eq. 1, from this patch point until the Jacobi energy of the spacecraft is equal to the Jacobi energy of a Geosynchronous Earth Orbit (GEO); then the integrator stops. The control law for the low-thrust acceleration during this propagation is quite straightforward: u (t) = Tmax m(t) (7) û (t) = ˆv (t) with T max being the magnitude of the maximum thrust of the engine, ˆv (t) being a unit vector oriented in the same direction as the instantaneous, three-body velocity of the spacecraft. This control law was successfully used by Mingotti et al 7 and was shown to be the most fuel and time efficient control law possible. A summary of this method for an are arbitrarily chosen trajectory and patch point is described below: 1. Create an invariant stable manifold, W s(±), based on the nominal L-point orbit by using DST and backwards integrating the Eq. 1 of the CR3BP 4

5 ( 2. Arbitrarily select a trajectory, k (±), from this manifold and then arbitrarily select a patch point, X s.m. τs.m. k (±) ), from within this trajectory 3. Propagate the low-thrust trajectory from this patch point backwards in time until the Jacobian matches that of GEO. It is now possible to describe an efficient and continuous trajectory from a GEO-energy orbit to a Lagrange point orbit given a patch point, X s.m.. It, therefore becomes necessary to optimize this trajectory by selecting the optimal patch point, X s.m.. Thus, the search space becomes all the possible points within the stable manifold. ALGORITHM Fitness Function In order for the optimal patch point to be located, one must be able to evaluate the cost or fitness function of an arbitrary patch point within the search space. The fitness function used in this study can be expressed as: J (X s.m. ) = c 1 e GEO (X s.m. ) e desired + c 2 m (X s.m. ) + c 3 T (X s.m. ) (8) where e GEO is the eccentricity of the GEO-energy orbit, e desired is the desired eccentricity of the GEOenergy orbit, m is the fuel used by the low-thrust maneuver, and T is the total Time of Flight (TOF) from the initial Earth orbit to the nominal Lagrange point orbit. The constants c 1, c 2, c 3 are all weighting constants controlled by the researcher. The goal of this paper is to minimize the eccentricity of the original orbit and make it as circular as possible; therefore e desired = 0. The motivation for this goal becomes clear when one considers the fact that any low-thrust trajectory that originates from Low Earth Orbit (LEO) must have an eccentricity near zero; if it does not then it will impact the surface of the Earth. It is well known 7 that the most efficient control law for constant thrust, Eq. 7, is the tangential thrust strategy, which will increase the specific energy of the spacecraft while leaving its eccentricity virtually unchanged (i.e. a nearly circular orbit). Thus, it is important that the target GEO-energy orbit also be circular if a mission planner desires to utilize a low-thrust transfer from LEO as well. The choice of the constants c 1, c 2, c 3 is of critical importance. No one term should dominate the others (unless the researcher wants it to). The constants c 2 and c 3 should, therefore, be chosen such that the ratio c 2 m (X s.m. ) c 3 T (X s.m. ) (9) is reasonable. For example, if m 10 1, T 10 2 and the desired ratio of Eq. 9 was roughly 10 : 1 an appropriate choice of constants would be c 2 = 10 3, c 3 = This selection method does not require a priori knowledge of the value of m or T, but rather an a priori estimate of their relative magnitudes. Once c 2 has been determined, it is possible to set a value for c 1 by setting some target threshold for the first term in Eq. 8. Values below this threshold would be overwhelmed by the other terms in Eq. 8 while values above this threshold would dominate Eq. 8. In this study, a threshold value of 1% seems reasonable since this represents a relatively small error in eccentricity. Thus the ratio c 2 m (X s.m. ) c 1 e GEO (X s.m. ) e desired (10) should be roughly 1 : 1 assuming c 2 = 10 3, m 10 1, and the threshold value of e GEO (X s.m. ) e desired is roughly This gives a value of c 1 = 1. In summary, following the procedure outlined in this example yields the values c 1 = 1, c 2 = 10 3, c 3 = 10 4 for the weighting constants. This setup ensures convergence on an optimal solution that rigidly enforces the eccentricity constraint while primarily minimizing the fuel usage and secondarily minimizing the time of flight. 5

6 Parametrization of the Search Space To locate the global optimal patch point of the manifold one must select a suitable optimization technique. Sampling of the search space has demonstrated a lack of convexity and concavity in the fitness function. Due to this complexity, an evolutionary optimization algorithm seems well suited to the task. This paper focuses on a Particle Swarm Optimization (PSO) algorithm. To use the PSO algorithm one must first parametrize the search space. Here, a very simple parametrization of the search space is accomplished by holding fixed a particular trajectory of the stable manifold, k (±) fixed W s(±) given a particular sign (±). Each trajectory can be parametrized by the time it takes for a spacecraft to flow, backwards in time, from X (k(±) ) ( orbit,) on the nominal Lagrange point orbit, to an arbitrary state on the trajectory k fixed, X s.m. τs.m., k fixed. This time, τ s.m., is known as manifold time. Assuming this state is within the search space, it can be considered a patch point and has a cost associated with that point that is defined by Eq. 8. The PSO algorithm will attempt to fly through this search space and locate the patch point with the globally minimal cost. This process can be repeated for other values of k fixed within W s(±). Once a representative number of trajectories have been addressed with the PSO algorithm, the results of each trajectory can be compared and the global optimal patch point for all of W s+, W s, and/or W s(±) can be identified. It is important to place bounds on the parameter τ s.m. since this defines the search space for the PSO algorithm. The bounds are expressed as τ L.B. τ s.m. τ U.B. with: τ L.B. being the time that trajectory k crosses the yz-plane from the positive x direction τ U.B. being the time that trajectory k crosses the yz-plane located at x = L 1. The τ U.B. bound was chosen to ensure that the fitness function (Eq. 8) can be evaluated at any point within the search space. If, for example, a patch point was located on the other side of L 1 the control law Eq. 7 would cause the spacecraft to flow towards a lunar orbit instead of an Earth orbit; thus invalidating Eq. 8. The opposite bound, τ L.B. was chosen as a matter of convenience and practicality. While it is true that the trajectories of W s continue to flow for an infinite amount of time, one needs to cut off this flow after a finite amount of time due to the limitations of computing power. Since the run times of the PSO method can become quite large, a smaller search space is needed to adequately cover the space with a limited number of function evaluations (i.e. patch points). While this limited search space does preclude the possibility of lunar phasing maneuvers, such as those used by Mingotti et all, 7 it is more than adequate to address a fitness function that attempts to minimize time of flight, such as Eq. 8. Particle Swarm Optimization The PSO algorithm is quite simple in principle. It starts by randomly distributing N p particles in the search space of a single trajectory. Particle i has a position, χ (j) (i), (parametrized on trajectory k by time parameter τ s.m., ) with χ = τ s.m. for a given k fixed and j = 1 being the initial iteration. Note that χ ( τ s.m. k fixed ) maps directly to the state X s.m. ( τs.m., k fixed ) via the parameters τs.m. and k fixed. Particle i has an initial velocity, ω (j) (i), that is also randomly selected at j = 1. The fitness function, Eq. 8, is then evaluated for each particle and two values are recorded. The first value is the global best, Y (j), and it represents the best known value of the cost function recorded by any particle in the swarm. The second value is the personal best, ψ (j) (i), and it represents the personal best value of particle i given all the patch points that this particle has visited in the past. Once these values have been recorded, the particles are allowed to move to a new location by the rules: ω (j+1) (i) = C i (1 + R 1 (0, 1)) ω (j) (i) + C c R 2 (0, 1) ( ψ (j) (i) χ (j) (i) ) + C s R 3 (0, 1) ( Y (j) χ (j) (i) ) (11) 6

7 χ (j+1) (i) = χ (j) (i) + ω (j+1) (i) (12) with R 1, R 2, R 3 being random numbers that are uniformly distributed between zero and one. The constant C i stands for the inertial weighting coefficient and controls the particle according to Newtonian mechanics. The constant C c is the cognitive weighting constant and allows the particle to remember the best location that it has visited so far. The final constant, C s, is the social weighting constant and allows for the particles to communicate with each other and share information about the global best location found so far. The constants used in this study have been inspired by the work of Pontani and Conway 18 and are as follows: C i = 0.15 C c = 1.5 C s = 1.5. DATA AND RESULTS Data for three studies is presented in this section. Each study represents a different form of the fitness function found in Eq. 8. All three studies share a common manifold and nominal Lagrange point orbit in the Earth-Moon system. All numeric integration was performed using MATLAB s ODE113 solver; a variable order Adams-Bashforth-Moulton PECE solver with a relative tolerance of and an absolute tolerance of This method was chosen because it is relatively efficient when using stringent tolerances as well as being efficient with computationally intensive functions. Destination Orbit and its Associated Manifold Data using the PSO technique has been collected for a northern halo orbit of the Earth-moon L 1 point. This destination orbit has a period of P = [tu] and an initial state of X 0 = expressed in [du] and [vu]. The orbit is displayed in Figure 2 with the orbit being discretized into N = 791 points with each point corresponding to two different trajectories on the invariant stable manifold. Note that every 10 th point, X (k) orbit, is displayed such that only 10% of the points are shown in Figure 2 because they would be indistinguishable when displayed in the plot otherwise. 7

8 Figure 2. The Nominal Earth-Moon L 1 Northern Halo Orbit. The orbit is broken into N = 791 points with every 10 th point displayed in this figure. This orbit was chosen arbitrarily as an example of the PSO technique on a highly three dimensional and fairly complicated nominal orbit. According to Dynamical Systems Theory (DST) the invariant stable manifold of this northern halo orbit may be generated by the following procedure: 1. Select a random point on the orbit and integrate the State Transition Matrix (STM) forward in time for one period. 2. Calculate the eigenvectors associated with the direction of the stable manifold at this point. 3. Multiply this eigenvector by a small number, ɛ, and add (green trajectories in Figure 3) or subtract (blue trajectories in Figure 3) this small perturbation to the point. 4. Propagate this state vector backward in time until it crosses the x-axis from the x direction. This procedure can be repeated numerous times; once for each point on the nominal orbit. The trajectories generated by all points on the nominal orbit comprise the invariant stable manifold, W s(±), of the nominal orbit. Figure 3 illustrates the invariant stable manifold of the Earth-moon, L 1, northern halo orbit. 8

9 Figure 3. Invariant stable manifold of the nominal, L 1 northern halo orbit. Green trajectories represent s +, while blue trajectories represent s. Note that the manifold never approaches the vicinity of near Earth orbit. Study A: Eccentricity-Only Fitness Function Fitness Function: In this study, a very simple fitness function was chosen in such a way as to focus the goal of the PSO algorithm on obtaining a circular GEO-energy orbit. This was accomplished by setting e desired = 0 and c 1 = 1, c 2 = c 3 = 0 in Eq. 8. Therefore, the fitness function for Study A becomes J (χ) = e GEO (χ). (13) A circular GEO orbit is the most likely orbit attainable by low-thrust technology for its given specific orbital energy. Study A focuses on the exploration of the solution space in an effort to identify all GEO-energy orbits that have a low eccentricity (e GEO 0.01). It is important to prove that the PSO method is able to find a large number of circular GEO-energy orbits because any usable fuel-optimal solutions must be a subset of the acceptable solution space found in Study A. Data: Numerous trials were conducted using the cost function found in Eq. 13 with different amounts of particles, iterations, and function evaluations per trajectory k fixed. In total, four trials were conducted and directly compared to the stochastic Monte Carlo method. The results are summarized in Table 1 and Table 2. Table 1 compares the PSO method with the Monte Carlo method via function evaluations. Note that the +/ signs correspond to W s+ and W s, respectively. Function evaluations are defined as the number of patch points visited per trajectory k fixed. Note that an entire low-thrust trajectory is computed for each patch point that is visited by either method. For example, the 300 Samples + trial of the Monte Carlo method will randomly sample 300 patch points on each trajectory of W s+ for a total of 300 function evaluations per k fixed. The table then reports the optimal value of the cost function Eq. 13, the τ s.m. and k of that value, and the amount of time it took for the computer to complete the trial. Likewise, the same can be said for the PSO method. The 30 Particles, 10 Iterations + trial samples a total of 300 patch points on each trajectory of W s+ for a total of 300 function evaluations per k fixed. The remaining columns are analogous to the Monte Carlo method. Note that it is fair to directly compare the PSO method to the Monte Carlo method given the same number of function evaluations. Looking at Table 1, one may note that the PSO method usually finds a better optimal solution than the Monte Carlo method regardless of the number of function evaluations used by both methods. 9

10 Table 1. Top: Results from eight Monte Carlo trials. Both the negative perturbation (-) and positive perturbation (+) are investigated. Bottom: Results from eight PSO trials. Both perturbations are also investigated. The optimal case is shown in bold. Monte Carlo Trial Optimal J = e GEO τ s.m. [tu] k (±) CPU Time (hrs.) 300 Samples Samples Samples Samples Samples Samples Samples Samples PSO Trial Optimal J = e GEO τ s.m. [tu] k (±) CPU Time (hrs.) 30 Particles, 10 Iterations Particles, 10 Iterations Particles, 10 Iterations Particles, 10 Iterations Particles, 10 Iterations Particles, 10 Iterations Particles, 5 Iterations Particles, 5 Iterations Table 2. Comparison of PSO vs. Monte Carlo (MC) methods. Both the negative perturbation (-) and positive perturbation (+) are investigated. The PSO and MC column indicates the number of solutions found with an eccentricity less than 0.01 using the Particle Swarm Optimization and Monte Carlo methods, respectively. Function Evaluations PSO MC PSO/MC % Dominated by PSO % Dominated by MC

11 To further illustrate this point refer to Table 2. The second and third column of this table identifies the number of patch points with e GEO 0.01 for the PSO and Monte Carlo method, respectively. Note that for a given number of function evaluations and perturbation (i.e. + / ) the PSO method significantly outperforms the Monte Carlo method in the number of useful patch points that it has identified. The fourth column indicates the ratio of the values of column two to column three; or the ratio of useful patch points identified by the PSO vs. that of the Monte Carlo method. Finally, the fifth column identifies the percentage of trajectories within W s(±) that obtained a better optimum value using the PSO method vs. the Monte Carlo method. The sixth column is simply the opposite of the fifth. Note that the PSO method significantly outperformed the Monte Carlo method in terms of the number of suitable patch points it was able to identify according to both measures. Figure 4. PSO vs. Monte Carlo. Percentage of function evaluations that yield an eccentricity less than PSO (diamond and squares) clearly outperformed the Monte Carlo (triangles and X s) method. Logarithmic curve fit (PSO) and linear curve fit (MC) are drawn for reference. Figure 4 illustrates the effectiveness of the PSO method very clearly. The figure plots the percentage of patch points in W s with e GEO 0.01 (as identified by the PSO and Monte Carlo methods) given a set amount of function evaluations. The figure shows that the percentage of suitable patch points identified by the Monte Carlo method was fairly uniform with regard to the number of function evaluations per k fixed. The percentage of suitable patch points identified by the Monte Carlo method is on the order of 0.001% which is fairly low and is to be expected from a random sampling of the search space. In contrast, the percentage of suitable patch points identified by the PSO method varies between 0.02% and 0.44% and generally increases with more function evaluations. This provides convincing evidence that the PSO method quickly and efficiently identifies suitable patch points far better than random chance would allow. Additionally, it would seem that all data originating from W s+ contains more suitable patch points than that from W s although no clear explanation for this phenomena currently exists. Overall, the optimal trajectory found has an eccentricity of e GEO = and is located at time τ s.m. = [tu] on trajectory number k = of W s+ and was identified by both the PSO and Monte Carlo methods. Figure 5 displays the cost of each patch point visited by the PSO algorithm on trajectory Note that there are three local minima and two local maxima. The global minimum of k = occurs at time [tu] and is a rather sharp minimum, with the fitness function increasing rapidly from both the left and right sides. Indeed, the approach towards this minimum is so abrupt that traditional, gradient-based solvers may never reach it; instead becoming trapped within the other two local minima. This 11

12 is not the case, however, with the evolutionary PSO technique. Note that the blue points in Figure 5 are the initial guess solutions of the PSO algorithm, while the red points are the solutions of the intermediate iterations, and the green points are the solutions of the final iteration. While the initial points are randomly distributed throughout the search space, the PSO algorithm quickly converges to the global minimum of trajectory k = The majority of the function evaluations (i.e. patch points) are located near the global minimum which is a much more efficient use of computational resources as opposed to a random distribution with the Monte Carlo method. Figure 5. Fitness vs. Manifold Time: Fitness (eccentricity) of different locations along trajectory k = of the positively perturbed stable manifold of the nominal halo orbit. A blue point indicates the location and cost of an initial particle, red points indicate that of a particle during the optimization process, and green points indicate that of a particle at the end of the PSO algorithm. Figure 6 shows two different views of the optimal low-thrust trajectory (red) and its associated trajectory k = W s+ shown in green. A gray sphere surrounds the Earth (small sphere at the center) and indicates the radius of a circular GEO-energy orbit while a blue trajectory indicates the GEO-energy orbit that the spacecraft has originated from. The blue dots on trajectory k indicate the location of the initial patch points of the PSO algorithm (as in Figure 5) while the red dots indicate the location of the final patch points of the PSO algorithm (which correspond to green points in Figure 5). The spacecraft began with an initial mass of m 0 = [kg] and has a final mass of m f = [kg] with a mass fraction of m f m 0 = at the optimal patch point. The controlled flight took [tu] while the total time of flight took [tu] or approximately 122 [days]. Optimization of an Arbitrary k fixed : Table 3 illustrates the optimization of nine independent trajectories of W s+, namely k fixed = through Note that many trajectories have relatively high values of e GEO as well as high values of the optimal e GEO for that trajectory. 12

13 Figure 6. Optimal Low-Thrust Trajectory with e GEO = Shown with parent trajectory k = as well as the nominal halo orbit. Three separate views of this trajectory are given in order to show perspective. Table 3. Optimal patch points found for nine trajectories ( ) using PSO. k fixed Optimal τ s.m. [tu] Optimal J = e GEO

14 Figure 7 represents the fitness/eccentricity generated from each patch point visited by the PSO algorithm for k fixed = The figure is broken into six plots; one for each trajectory. As before, a blue point indicates the location and cost of an initial particle, red points indicate the location and cost of a particle during the optimization process, and green points represent the final location and cost of a particle at the end of the PSO algorithm. Note that the structure of the cost functions changes significantly between successive values of k fixed. Some functions have large fluctuations with low minima and high maxima while others, such as have far less variability. All trajectories display a semi-cyclic nature with periodic extrema that occur in a semi-periodic manner. While the magnitude of each extrema can vary significantly, some can have very similar magnitudes such as the local minima found in trajectory Also note that this simple fitness function yielded a highly non-convex (nor concave) cost function that makes it difficult, if not impossible, for more traditional convex optimization methods to be used. Figure 7. Cost Map of Trajectory ; 228-Upper Left; 233-Lower Right. A blue point indicates the location and cost of an initial particle, red points indicate that of a particle during the optimization process, and green points indicate that of a particle at the end of the PSO algorithm. Note how the result of the fitness function varies dramatically between adjacent trajectories (a) - (f). This demonstrates the discontinuous nature of the cost function if viewed as a function of trajectory number. 14

15 Study B: Eccentricity and Fuel Optimizing Fitness Function Fitness Function: The fitness function in Study B becomes slightly more complex with the addition of the fuel-optimization constraint c 2 0. Therefore the fitness function becomes J (X s.m. ) = c 1 e GEO (X s.m. ) + c 2 m (X s.m. ) (14) with the values c 1 = 1 and c 2 = 10 3 in accordance with Equation 10. This choice of weighting constants ensures convergence on trajectories with very low eccentricity values for their initial orbits while simultaneously minimizing fuel consumption. Data: Numerous trials were conducted using the cost function found in Eq. 14 with different amounts of particles, iterations, and function evaluations per trajectory k fixed. In total, four trials were conducted and directly compared to the stochastic Monte Carlo method. The results are summarized in Table 4 and Table 5. Table 4. Top: Results from eight Monte Carlo trials. Both the negative perturbation (-) and positive perturbation (+) are investigated. Bottom: Results from eight PSO trials. Both perturbations are also investigated. The optimal case is shown in bold. Monte Carlo Trial Opt. e GEO Opt. m [Kg] Opt. J τ s.m. [tu] k (±) CPU Time [hrs.] 300 Samples Samples Samples Samples Samples Samples Samples Samples PSO Trial Opt. e GEO Opt. m [Kg] Opt. J τ s.m. [tu] k (±) CPU Time [hrs.] 30 Particles, 10 Iterations Particles, 10 Iterations Particles, 10 Iterations Particles, 10 Iterations Particles, 10 Iterations Particles, 10 Iterations Particles, 5 Iterations Particles, 5 Iterations

16 Table 5. Comparison of PSO vs. Monte Carlo (MC) methods. Both the negative perturbation (-) and positive perturbation (+) are investigated. The PSO and MC column indicates the number of solutions found with a fitness function less than J = 0.1 using the Particle Swarm Optimization and Monte Carlo methods, respectively. Function Evaluations PSO MC PSO/MC % Dominated by PSO % Dominated by MC Table 4 compares the PSO method with the Monte Carlo method via function evaluations and is organized in much the same way as Table 1 of Study A. The only new addition is the thrid column corresponding to the optimal amount of fuel used during the low-thrust maneuver. Based on the results of Table 4 it can be clearly seen that the PSO method matched or outperformed the Monte Carlo method in every case. Note that the PSO method clearly dominates the Monte Carlo method for a low number of function evaluations; such as 40 Monte Carlo samples vs. eight particles and five iterations via the PSO method. Table 5 further demonstrates the superiority of the PSO method. The table records the number of patch points with J 0.1 which are considered to be suitable candidates for further study. Note that the PSO method significantly outperformed the Monte Carlo method just as was observed in Study A. The globally optimal patch point found in this study occurred on trajectory k = with a total fuel consumption of m = kg, an eccentricity of e GEO = , and a fitness value of J = Coincidentally, this also happens to be the globally optimal patch point found in Study A and is graphed in Figure 6. Figure 8 plots the fuel consumption verses eccentricity of the patch points used in Study B. Note that the maximum fuel used by any point is around 74 kg and is roughly 15% more fuel than the globally optimal value found in Study B. The figure also suggests that the globally minimal fuel usage does not correspond to an eccentricity of zero but closer to e GEO = 0.25 which may lower the fuel used to only 63.7 kg. Finally, at nearly zero eccentricty the range of feasable fuel consumption is very narrow; differing by less than a kilogram. This suggests that optimization with respect to fuel may not be the most fluid parameter, given a zero eccentricity restriction, and optimization with respect to Time of Flight (TOF) should be given more consideration. 16

17 Figure 8. Plot of Fuel Consumption vs. Eccentricity for patch points generated in Study B. Study C: Eccentricity, Fuel, and Time of Flight Optimizing Fitness Function Fitness Function: The fitness function in Study C becomes slightly more complex with the addition of the Time-of-Flight-optimization constraint c 3 0. Therefore the fitness function becomes J (X s.m. ) = c 1 e GEO (X s.m. ) + c 2 m (X s.m. ) + c 3 T (X s.m. ) (15) with the values c 1 = 1, c 2 = 10 3, and c 3 = 10 4 in accordance with Equation 10 and 9. This choice of weighting constants ensures convergence on trajectories with very low eccentricity values for their initial orbits while simultaneously minimizing fuel consumption and TOF. Data: Numerous trials were conducted using the fitness function found in Eq. 15 with different amounts of particles, iterations, and function evaluations per trajectory k fixed. In total, four trials were conducted and directly compared to the stochastic Monte Carlo method. The results are summarized in Table 6 and Table 7. Table 6 compares the PSO method with the Monte Carlo method via function evaluations and is organized in much the same way as Table 1 of Studies A and B. The only new addition is a column corresponding to the optimal spacecraft Time of Flight (TOF). Based on the results of Table 6 it can be clearly seen that the PSO method outperformed the Monte Carlo method in nearly every case. Table 7 further demonstrates the superiority of the PSO method. The table records the number of patch points with J 0.1 which are considered to be suitable candidates for further study. Note that the PSO method significantly outperformed the Monte Carlo method just as was observed in Studies A and B. The globally optimal patch point found in this study occurred on trajectory k = with a total fuel consumption of m = kg, an eccentricity of e GEO = , TOF of tu, and a fitness value of J = Coincidentally, this also happens to be the globally optimal patch point found in Studies A and B and is graphed in Figure 6. Figure 9 plots the TOF verses eccentricity of the patch points used in Study C. Note the cyclical nature of TOF at zero eccentricity. This suggests that the TOF is in great need of optimization and has, indeed, been accomplished in Study C. 17

18 Table 6. Top: Results from eight Monte Carlo trials. Both the negative perturbation (-) and positive perturbation (+) are investigated. Bottom: Results from eight PSO trials. Both perturbations are also investigated. The optimal case is shown in bold. Monte Carlo Trial Opt. e GEO Opt. m [Kg] Opt. TOF [tu] Opt. J τ s.m. [tu] k (±) CPU Time [hrs.] 300 Samples Samples Samples Samples Samples Samples Samples Samples PSO Trial Opt. e GEO Opt. m [Kg] Opt. TOF [tu] Opt. J τ s.m. [tu] k (±) CPU Time [hrs.] 30 Particles, 10 Iterations Particles, 10 Iterations Particles, 10 Iterations Particles, 10 Iterations Particles, 10 Iterations Particles, 10 Iterations Particles, 5 Iterations Particles, 5 Iterations Figure 9. Plot of Fuel Consumption vs. Eccentricity for patch points generated in Study C. 18

19 Table 7. Comparison of PSO vs. Monte Carlo (MC) methods. Both the negative perturbation (-) and positive perturbation (+) are investigated. The PSO and MC column indicates the number of solutions found with a fitness function less than J = 0.1 using the Particle Swarm Optimization and Monte Carlo methods, respectively. Function Evaluations PSO MC PSO/MC % Dominated by PSO % Dominated by MC CONCLUSIONS This paper shows that preliminary optimization of a low-thrust transfer trajectory from a GEO-energy orbit to a nominal Earth-Moon Lagrange point orbit is indeed possible using Particle Swarm Optimization. Since the search space is often non-convex, traditional convex optimization methods do not apply. Additionally, gradient based optimization algorithms would likely have a great deal of difficulty avoiding a local minima. Instead, an evolutionary PSO algorithm has proven to be a highly useful optimization technique that significantly outperforms random chance algorithms, such as the Monte Carlo technique, as evidenced by Figure 4. While a large number of function evaluations is important, the percentage of candidate trajectories identified by the PSO method increases only logarithmically with increasing function evaluations. Since run time increases proportionately with function evaluations it seems reasonable to identify 160 function evaluations as a compromise between robust candidate identification and CPU run time. It is also interesting to discover that the positive perturbation of the stable manifold, W s+ normally contains more candidate patch points than its counterpart, W s. It is unknown if this is the result of generalized system dynamics or simply a phenomina unique to the given nominal orbit. The optimization of the low-thrust trajectory with respect to fuel consumed and TOF was sucessfully demonstrated in this paper. The PSO algorithum is, in general, much more sucessful in optimizing the trajectory than random chance would allow. Convergence on the discreate, global minimum, as observed in Figure 9, demonstrate the robustness of the PSO method; especally compaired to gradient based algorithms used by Mingotti et all 7 which can become trapped in a local minima. Although the PSO method can not guarantee convergence upon the globally optimal solution it does offer the benifits of a wide coverage of the searchspace and would be an ideal choice for preliminary trajectory optimization. REFERENCES 1 V. Angelopoulos, The ARTEMIS Mission, Space Science Review, 2010, pp. 1 23, /s T. H. Sweetser, S. B. Broschart, V. Angelopoulos, G. J. Whiffen, D. C. Folta, M.-K. Chung, S. J. Hatch, and M. A.Woodard, ARTEMIS Mission Design, Space Science Review, Vol. 165, March 2012, pp , /s R. W. Farquahr, A Halo-Orbit Lunar Station, Astronautics & Aeronautics, Vol. 10, 1972, pp E. Davis, Transfers to the Earth-Moon L3 Halo Orbits, AAS/AIAA Astrodynamics Specialist Conference, Minneapolis, MN, August 2012, /

20 5 Lockheed Martin, Early Human L2-Farside Missions, tech. rep., Lockheed Martin, 2010, Final.pdf. 6 T. A. Heppenheimer, Steps Toward Space Colonization: Colony Location and Transfer Trajectories, Journal of Spacecraft and Rockets, Vol. 15, September - October 1978, pp , / G. Mingotti, F. Topputo, and F. Bernelli-Zazzera, Combined Optimal Low-Thrust and Stable-Manifold Trajectories to the Earth-Moon Halo Orbits, AIP Conference Proceedings (E. Belbruno, ed.), American Institute of Physics, 2007, / G. Mingotti, F. Topputo, and F. Bernelli-Zazzera, Transfers to distant periodic orbits around the Moon via their invariant manifolds, Acta Astronautica, Vol. 79, October-November 2012, pp , /j.actaastro B. A. Conway, ed., Spacecraft Trajectory Optimization. Cambridge, W. A. Scheel and B. A. Conway, Optimization of Very-Low-Thrust, Many Revolution Spacecraft Trajectories, Journal of Guidance, Control, and Dynamics, Vol. 17, November-December 1994, / A. L. Herman and D. B. Spencer, Optimal, Low-Thrust Earth-Orbit Transfers Using Higher-Order Collocation Methods, Journal of Guidance, Control, and Dynamics, Vol. 25, January-February D. C. Redding and J. V. Breakwell, Optimal Low-Thrust Transfers to Synchronous Orbit, Journal of Guidance, Control, and Dynamics, Vol. 7, No. 2, 1984, / D. C. Redding, Highly Efficent, Very Low-Thrust Transfer to Geosynchronous Orbit: Exact and Approximate Solutions, Journal of Guidance, Control, and Dynamics, Vol. 7, 1984, / G. Mingotti, F. Topputo, and F. Bernelli-Zazzera, Numerical Methods to Design Low-Energy, Low-Thrust Sun-Perturbed Transfers to the Moon, Proceedings of 49th Isreal Annual Conference on Aerospace Sciences, Tel Aviv-Haifa, Isreal, 2009, pp J. Senet and C. Ocampo, Low-Thrust Variable-Specific-Impulse Transfers and Guidance to Unstable Periodic Orbits, Journal of Guidance, Control, and Dynamics, Vol. 28, March-April V. Szebehely, Theory of Orbits: The Restricted Problem of Three Bodies. New York, NY: Academic Press Inc., T. A. Pavlak, Mission Design Applications In The Earth-Moon System: Transfer Trajectories and Stationkeeping, Master s thesis, School of Aeronautics and Astronautics, Purdue University, West Lafayette, Indiana, May M. Pontani and B. A. Conway, Particle Swarm Optimization Applied to Space Trajectories, Journal of Guidance, Control, and Dynamics, Vol. 33, September-October 2010, /