AN APPROACH TO THE DESIGN OF LOW ENERGY INTERPLANETARY TRANSFERS EXPLOITING INVARIANT MANIFOLDS OF THE RESTRICTED THREE-BODY PROBLEM

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1 Paper AAS 4-45 AN APPROACH TO THE DESIGN OF LOW ENERGY INTERPLANETARY TRANSFERS EXPLOITING INVARIANT MANIFOLDS OF THE RESTRICTED THREE-BODY PROBLEM Francesco Topputo, Massimiliano Vasile and Amalia Ercoli Finzi Dipartimento di Ingegneria Aerospaziale, Politecnico di Milano Italy 4 th AAS/AIAA Space Flight Mechanics Conference Maui, Hawaii 8- Feruary 4 AAS 4-45 AAS Pulications Office, P.O. Box 83, San Diego, CA 998

2 AAS 4-45 AN APPROACH TO THE DESIGN OF LOW ENERGY INTERPLANETARY TRANSFERS EXPLOITING INVARIANT MANIFOLDS OF THE RESTRICTED THREE-BODY PROBLEM Francesco Topputo, Massimiliano Vasile and Amalia Ercoli Finzi In this paper a general algorithm for the design of low-energy interplanetary transfers, exploiting invariant manifolds of the periodic orits around liration points, will e presented. The algorithm looks for the est intersection etween invariant manifolds associated to the departure and arrival Sun-Planet-Spacecraft systems. A merit function is associated to the Poincaré sections of the two manifolds, and, after a preliminary systematic search necessary to find a first guess solution, a SQP algorithm is used to optimize the transfer, merging together dynamical system theory and optimization techniques. If an optimal intersection cannot e found, invariant manifolds are linked y solving a Lamert s prolem and the resulting trajectory is then optimized. In order to have a full 3D representation of the transfer, the algorithm has een integrated with an analytical ephemeris model. Some interplanetary trajectories are presented, showing the effectiveness of the proposed approach. INTRODUCTION Traditional patched conic methods are ased on the idea of joining together conic arcs, defined in different reference frames, in order to otain a complete trajectory. These methods take into account only one gravitational attraction at time, solving, in this way, a numer of twoody prolems (BPs). However, in the last ten years, several studies have demonstrated how resorting to a multi-ody model may allow the design of efficient low energy trajectories of practical interest,8,9. More recently Koon, Lo, Marsden and Ross suggested an approach for the design of low energy transfers in the solar system, ased on a division of the full four-ody prolem into two restricted three-ody prolems 9, (R3BPs). Low energy transfers can e otained if the manifolds, associated to the periodic orits around liration points of each three-ody prolem, have a point of intersection in the configuration space. Examples of this kind of transfers can e otained in Planet-Moon systems, to move from one moon to another, or in Sun-Planet systems when gravity constants and orital parameters allow this intersection. However, due to the highly non-linear dynamics, chaotic in nature, characterizing these transfers, their design is not a trivial matter and is extremely sensitive with respect to the sizes of the two orits and the relative phases of the odies involved. A detailed version of the work presented in this paper can e found in the first author s Tesi di Laurea 4. PhD Student, Dipartimento di Ingegneria Aerospaziale, Politecnico di Milano, Milan, Italy, Francesco.Topputo@fastwenet.it Researcher, Dipartimento di Ingegneria Aerospaziale, Politecnico di Milano, Milan, Italy, vasile@aero.polimi.it Full professor, Dipartimento di Ingegneria Aerospaziale, Politecnico di Milano, Milan, Italy, Amalia.Finzi@polimi.it

3 In this paper a computational algorithm, implemented in a Matla tool, called ATOM-C (Astrodynamics Tool for Optimal Manifold Computation), has een developed for the design of such low energy transfers. In this algorithm the full four-ody prolem is split into two R3BPs with manifolds, associated to each three-ody system, computed oth in the synodic and in an asolute inertial reference frame. The latter is then used to look for the est intersection etween the two tues. The manifolds are cut in the inertial frame making oth the sections timedependent. Furthermore, the surface of section is not kept constant and fixed in the space ut it is parametrized allowing a sort of dynamical cutting of the manifolds. A merit function is then associated to each intersection and a systematic search (SS) is used to find the optimal starting and arriving trajectories. After this preliminary search, necessary to find a first guess solution within the required time frame, the algorithm refines each first guess y a SQP method. Since this local optimization method is highly sensitive with respect to the starting point, the use of a refined preliminary systematic search ensures an initial condition inside the asin of attraction of optimal solutions. In this way dynamical system theory and optimization techniques are merged together and all parameters influencing the transfer are optimized. In order to take into account the inclination and eccentricity of the planets, the algorithm has een integrated with an analytical ephemeris model 3 translating the solution of the planar R3BP into a full 3D transfer. After an analysis of some Sun-Planet and Planet-Moon transfers, the approach has een developed further to treat cases in which an optimal intersection could not e found. In this case non-intersecting manifolds are linked solving a Lamert s prolem or through a low thrust arc. In this way the full four ody model is divided into a first three-ody prolem (Sun-Departure Planet-Spacecraft), an intermediate two-ody prolem (Sun-Spacecraft) and a final three-ody prolem (Sun-Arrival Planet-Spacecraft). The proposed patching conic-manifolds method (leading to a multi-urn transfers) exploits the two gravitational attractions of the planets to change the energy level of the spacecraft and to perform a allistic capture and a allistic repulsion. In the first part of the paper, the asic features of the restricted three-ody prolem, including the liration points, the periodic orits around them and the associated invariant manifolds, will e introduced from the dynamical system point of view. Then the algorithm for the intersection of the manifolds is presented, followed y some sample transfers. After that, the fundamental hypothesis, that allows the intermediate conic link etween the manifolds, and the resulting algorithm will e descried. Finally, some trajectory analysis will e presented, showing the effectiveness of the proposed approach, along with some new possile transfers. PROBLEM FORMULATION Dynamic Model There are several ways to derive the equations of motion for a point mass, generically called Spacecraft, under the gravitational attraction of two primaries, generically called Sun and Planet. The method used in this work gives the equations in the second order Lagrangian form written in a synodic reference frame with the following characteristics:. the sum of the mass of the Sun and the Planet is µ S +µ P = with µ P =µ;. the distance etween the Sun and the Planet is normalized to ; 3. the angular velocity of the primaries around their center of mass is normalized to ; where the mass parameter is µ=m P /m S (m P : mass of the Planet; m S : mass of the Sun).

4 The equations of motion for the spacecraft are: && x y& = Ω && y + x& = Ω x y () where ( µ ) x + y µ µ µ Ω = Ω( x, y) = () r r and the suscripts of Ω in Eqs.(), denote the partial derivatives with respect to the coordinates of the spacecraft (x, y). The rotating system in which Eqs. () are written, has the origin in the Sun-Planet center of mass and the x-axis defined y the Sun-Planet line with the Planet on the positive direction (see Figure ). Hence, the distances Sun-Spacecraft and Planet-Spacecraft are respectively:.8 = ( x + µ ) and r ( x + ) + r + y = µ y (3) y L L3 Sun L L x Planet L Figure : Synodic system and liration point representation for µ=. The system has a first integral of motion, called Jacoi integral, which is given y: ( x, y, x&, y& ) = Ω( x, y) ( x& + y& ) C = C (4) and represents a 3-dimensional manifold for the states of the prolem. Once a set of such initial conditions is given, the Jacoi integral defines the foridden and allowed regions of motion for the Spacecraft (Figure ). The zero velocity curves delimiting the foridden regions are called Hill s curves. The energy of the spacecraft and the Jacoi constant are related y: C = E (5) which states that a high value of C is associated to a low energy of the spacecraft. For a low value of the energy the spacecraft is ounded to orit around one of the two primaries. If the energy is increased the allowed regions of motion enlarge and the spacecraft is free to leave the 3

5 primary. Differential system () presents 5 stationary, or liration, points (represented in Figure and called Lagrangian points), three collinear with the primaries (L,L and L3) and two at the vertex of two equilateral triangles (L4 and L5). Periodic Orits around Liration Points L and L In this paper, the possiility to otain free arcs of transfer exploiting the nature of the equilirium points, stationary solution of Eqs. () and -dimensional manifold in the phase space, has een investigated. This goal can e achieved only considering L and L, since L3 has a slow dynamics and a mild instaility 4 while triangular points L4 and L5 are always stale for the Sun-Planet-Spacecraft dynamics in the solar system, therefore they are not suitale for the design of transfers among planets. Furthermore low energy levels, or high values of the Jacoi constant in Eq. (5), are associated to L and L. This means that already for low levels of the energy E, Hill s curves opens at L and L allowing the motion of the spacecraft outside the foridden regions. From the analysis of Eqs. () the linearized dynamics around L and L is that of the product of a saddle point (two real eigenvalues) times a center 7 (periodic motion given y two imaginary eigenvalues). Although some infinitesimal periodic orits (Lyapunov orits) can e constructed taking only the eigenvectors associated to the imaginary eigenvalues, they have not een considered here since only finite size orits are interesting for this study. The existence of periodic orits of finite size is, anyway, guaranteed y Moser s theorem 6 that extends the results associated to the linearized system to the full nonlinear dynamics. y (nondimensional units, Sun Jupiter rotating frame) Foridden Region Sun Jupiter L x (nondimensional units, Sun Jupiter rotating frame) Figure : A periodic L orit in the Sun-Jupiter system. The first guess initial condition (dashed) is then corrected (solid) until the final periodic orit (old) is reached The construction of such periodic orits has een carried out adopting the Lindstedt-Poincaré method 7, truncated at the third order, for the generation of a first guess initial condition (dashed line in Figure ). After this, an algorithm computes the monodromy matrix of the orit and differentially corrects the initial guess 3,4 (solid lines). Once otained one orit (old line), since it is not isolated, a numerical continuation process 4 is used to generate the entire family associated to it. 4

6 Invariant Manifolds Associated to the Periodic Orits There are two manifolds associated to L and L: a stale (W S Li) and an unstale (W U Li) one, oth -dimensional (i=,), The manifolds associated to the periodic orits are centered on these lines. These -dimensional suspaces are here called W S Li,p.o. and W U Li,p.o. (i=,), according to the notation introduced in Ref.. If a Spacecraft is on a stale manifold, its trajectory winds onto the orit and, if it is on the unstale one, it winds off the orit (Figure 3). This aspect is very important for the design of missions to the liration points, for instance, of the Sun-Earth system 6,4. It is important to oserve that, since the Jacoi constant is a 3-dimensional surface, the manifolds are separatrices and they split different regimes of motion. It is not accidental that the transit trajectory shown in Figure 3 passes through the periodic orit and remains within the regions (or tue) delimited y the manifolds. This is due to the initial condition of the transit orit which lies inside the curve associated to the Poincaré section of the manifold 8. y (nondimensional units, Sun Jupiter rotating frame) Foridden Region x= µ Jupiter Sun x= µ Foridden Region Transit Orit Asymptotic Orit x (nondimensional units, Sun Jupiter rotating frame) W s L p.o. W u L p.o. Figure 3: Transit and asymptotic orits for Sun-Jupiter L. Initial condition have een otained according to the Poincaré sections of the manifolds Since the monodromy matrix represents the first order approximation of the flow mapping for a point of the orit x into a point x of an aritrary Poincaré section: ( x ) x a x + M (6) x its eigenvectors gives the direction of the -dimensional manifolds associated to each point of the orit. Hence, if x i is a generic point of the orit, its associated stale manifold can e otained propagating ackward the following initial condition: 5

7 x i i i, S x ± d vs = (7) where v i S is the eigenvector associated to the stale eigenvalue of the monodromy matrix evaluated in x=x i. The parameter d represents the distance etween the point of the orit and the initial condition for the computation of its associated manifold, taken in the direction of the eigenvector. It is clear that, the smaller is the value of d, the etter is the approximation of the manifold that this first order method could yield. The sign ± indicates that there are two different ranches for every manifold. In the same way, the unstale manifold associated to that point can e achieved integrating forward the initial condition taken in the direction of the unstale eigenvector. The two dimensional invariant manifolds associated to the orits have een otained repeating this process for every point of the orit. The stale and unstale dynamics of a liration point Li can e exploited using either the D manifold associated to the point (W Li ) or the D manifolds associated to the periodic orits around that point (W Li,p.o. ). As shown in Figure 3, the latter delimits the regions in which transit orits are contained. Since the intersections among manifolds W Li,p.o occur more frequently than the intersections among manifolds W Li, the transit orits, contained in W Li,p.o, represent the most suitale candidates for the construction of low energy transfers. Once a transit orit is uilt and propagated using Eqs. () in the synodic reference frame, it can e translated at first in the planar sidereal coordinate system and then in the 3D space y the following transformation: as sid X = RX (8) taking into account the real eccentricity and inclination of planetary orits. The first term in (8): as as as { x, y z } as X, = (9) represents the trajectory in the 3D asolute sun-centered reference frame, R=R(i,Ω,ω,θ) is the rotation matrix and X sid is the trajectory in the sidereal system: sid sid { x, y,} sid X = () This transformation is made taking the value of the six orital elements of the planets at a fixed time t=t * introducing, in this way, a small error. Nevertheless this error is expected to affect slightly the accuracy of the transfer ecause the orits of almost all the planets can e assumed circular and, thus, they respect the hypothesis of the R3BP. Repeating the same process for the velocities, it is also possile to otain: V as as as as { x&, y&, z& } = () or the velocity vector in the asolute Sun-centered reference frame. The values of the six orital elements at a given epoch are computed using the analytical ephemeris model in Ref.3. DESIGN APPROACH Intersecting the Manifolds The method of the intersection of the manifolds of two R3BPs, found in the literature 9, is here further developed with some additional and different features:. the surface of section is parametrized, allowing a dynamical cutting of the manifolds; 6

8 . transit orits are chosen according to the minimum value of a merit function; 3. solutions are improved y means of an optimization step; 4. non-intersecting manifolds are linked with a conic arc, solution of a Lamert s prolem. Let a and e respectively the generic departure and arrival planets and, without any loss of generality, let a e inner to. The following description can e easily implemented for the opposite case only y exchanging L with L and vice versa. Let A x,a and A x, e respectively the semiamplitudes of two periodic orits, one around L in the Sun-a system and the other around L in the Sun- system. In order to otain the est intersection etween the manifolds associated to these orits, it has een adopted the sideral plane in which the motion of the planets is circular and coplanar. 5 y (AU, inertial frame) 4 3 Jupiter θ a Surface of Section W u L p.o. (Sun Earth system) Sun W s L p.o. (Sun Jupiter system) Earth θ x (AU, inertial frame) Figure 4: Sideral plane and invariant manifolds for the Earth-Jupiter transfer (a=earth, =Jupiter) On this plane, according to the Figure 4, W U,ext L,p.o. (Sun-a system) and W S,int L,p.o. (Sun- system) must e computed, oth previous otained in the respective synodic frames. The superscripts ext and int means respectively that only the exterior and interior ranch of the manifolds have to e computed on the synodic plane according to Eq. (7). Propagations stop when manifolds develop respectively for an angle of θ a and θ, when angles measurement starts from the corresponding planet (see Figure 4). It has to e oserved that the stale manifold is the result of a ackward integration, so θ grows in opposite direction with respect to θ a and the angular position of is given y θ a +θ. This stoping condition is easy to otain when manifolds are descried in polar coordinates (r, θ) instead of the rectangular ones. Using θ as a parameter, indeed, the surface of section can e easily shifted y varying its value and a sort of fluctuant section can e otained. Let γ a e the curve produced y the section of W U,ext L,p.o. corresponding to the angle θ=θ a and γ the section of W S,int L,p.o. when θ=θ. These curves, represented in Figure 5 for the Earth- 7

9 Jupiter case, are diffeomorphic to ellipses and are oth time-dependent ecause they are plotted in the sidereal system, when there are no constant of motion. Hence, every point of these curves is characterized y a value of r and r& so, with the angular coordinate and the Jacoi constant associated to the departure periodic orit, it defines uniquely a trajectory. Now, let the ojective function e: D = r I i a a, r& = min i I a, j I i a γ ; a ( i j r a r ) + w( r& i a r& j ) r j, r& { i : i =,.., N }; I = { j : j =,.., N } a j γ where N a and N is an aritrary numer of points respectively on γ a and γ and w is a parameter (set to e-3 in this implementation) weighting the importance of r& (expressed in m/s) with respect to r (measured in AU) (see Figure 5). The ojective function is used to select the closest point of the two curves, since, if the manifolds do not intersect each other, the algorithm works in order to move one manifold closer to the other and vice versa. The states of the prolem are defined as: { A A, ϑ,, d d } x, a, x, a ϑ a () X =, (3) and, due to the nonlinearities of the system and the feature of the manifolds section, a complex relation D=D(X) exists. The last two parameters in Eq.(3), d a and d are the distances descried in the Eq. (7) and adopted in each three-ody prolem for the computation of the manifolds. At this point, a systematic search (SS) is required in order to find an initial guess for the set of the six states (3). The prolem can e formulated in the following way: suject to the oundary constraints: These constraints are: X min D ( X ) (4) LB UB < X X (5) LB UB X = {,,,,,} and X { A x, a, Ax,,π,π, d, d} = (6) If the definition of the lower ound X LB is ovious, the value of the upper ound is not a trivial matter. The parameter d directly influences the computation of the manifold of each point of the orit and, due to the strong nonlinear nature of the prolem, its value must remain small to ensure a good level of approximation of the manifolds. Furthermore, the upper ound for the semiamplitude A of the orits is necessary ecause oth the orits and their manifolds are otained y means of a first order method ased on the availaility of the monodromy matrix. This scheme is founded on the linearization of the flow near the liration points L and L, and, when the semiamplitude rise, this approximation loses its validity. Local Improvement of the Intersection of the Manifolds The systematic search provides a set of first guess parameters: FG FG FG FG FG FG { A A, ϑ,, d d } FG X x, a, x, a ϑ a, = (7) 8

10 with the associated initial value of the ojective function D FG =D(X FG ). At this point an optimization process is started in order to reduce further the value of the ojective D(X). Prolem (4) is solved once more starting from the initial value X =X FG. 5 Sections of W s L p.o. (Sun Jupiter system) r (m/s, inertial frame) Sections of W u L p.o. (Sun Earth system) A D FG D OPT B r (AU, inertial frame) Figure 5: First guess and optimum Poincaré section of the manifolds y (nondimensional units, Sun Earth rotating frame) L Earth L Foridden Region W u L p.o. Starting Trajectory x (nondimensional units, Sun Earth rotating frame) Figure 6: Departure trajectory in the Sun- Earth synodic frame y (nondimensional units, Sun Jupiter rotating frame) W s L p.o. Arriving Trajectory L Jupiter L Foridden Region x (nondimensional units, Sun Jupiter rotating frame) Figure 7: Arrival trajectory in the Sun- Jupiter synodic frame Eqs. (4) and (5) represent a standard optimization prolem, which can e efficiently solved y a SQP algorithm. When the process terminates, the states assume the values X OPT and the ojective function is D OPT =D(X OPT ) (see Figure 6). Selection of the Departure and Arrival Trajectories The two points of minimum distance D OPT of γ a or γ are: ( r a, OPT, r& a, OPT ) and ( r, OPT, r&, OPT ) (8) 9

11 and in the est case correspond to the point of intersection of the two curves. Anyway the coordinates in Eq. (8) do not identify the departing and arriving legs of the transfer, ecause, integrating such initial conditions, the corresponding trajectories would lie on the manifolds. As shown in Figure 3 this kind of trajectories wind onto (on the stale manifold) and wind off (on the unstale manifold) from the periodic orit. Therefore the values of r and r& in Eq. (8) correspond to a link etween the orits around liration points and not etween the planets. A connection etween the planets is possile only y means of trajectories flowing inside the manifolds ecause, as shown earlier, this is the only circumstance when a transit orit passes through the gateway left open y the Hill s curves and approaches the planet. On the other hand, the selection of points inside the curves γ a or γ, raises the energy level of the transfer. For these reasons, the points on the Poincaré section which correspond to the starting and arriving trajectories must e inside the two curves γ a and γ and as close as possile to their intersection points. To this aim, the points A and B in Figure 5 are modified as follows: A : ( ra OPT ra r& a OPT r&, ± δ,, ± δ a ) (9) B : r ± δr, r& ± δr& ( ), OPT, OPT with δ r a, and δ r & a, oth positive and aritrary small, for example: δr δr& a, a, =. =. ( ra, (max) ra, (min) ) ( r& r& ) a, (max) a, (min) The sign amiguity is solved with the imposition that A and B must e inner to the curves. The two transit trajectories plotted in Figures 6 and 7 have een otained integrating forward the state vector specified y the variation of the point B and ackward the one given y the variation of point A. The integrations was stopped at the closest approach to each one of the planets. Jupiter-Saturn Transfer The approach for the design of low energy transfers descried aove has een applied at first to a known case that can e found in the literature. By simply considering the manifolds associated to the points, as in Ref., a low energy transfer from Jupiter to Saturn has een otained y linking, in the sidereal plane, W U L of the Sun-Jupiter system with W S L of the Sun- Saturn system. In Tale the results otained in this work are compared to those found in Ref. and to the cost of a traditional Hohmann transfer while Figure 8 illustrates the corresponding low energy transfer. As can e seen, the proposed systematic search, assures a V FG lower than the V in Ref. and almost one fourth of the Hohmann s one. This V FG is then improved y the SQP algorithm which reduces the total cost further down to 93 m/s in case of solution and to 9 m/s in case of solution. It is remarkale how the solution, found with the comination of SS and SQP, is aout days faster than the one in Ref. and just 38 days slower than the Hohmann one. Although, in comparison to a direct Hohmann transfer, the saving in v otained exploiting the stale and unstale manifolds is remarkale, these sort of trajectories have a longer time of flight. In addition the asymptotic nature of stale manifolds makes the approach to the target planet even slower reducing significantly the transport rate (i.e. at a given cost, the ratio etween the total mass transferred and the required time for the transfer). ()

12 Figure 8: Jupiter-Saturn transfers optimization using invariant manifolds associated to the liration points Tale.Comparison etween Hohmann, Ref. and current work for a Jupiter-Saturn Transfer V FG (m/s) V OPT (m/s) t (day) Hohmann Lo and Ross SS+SQP Sol SS+SQP Sol Tour of the Uranian Moons A further validation of the proposed technique has een performed designing a tour of the Uranian moons. The odies involved in these transfers, as in the Petit Grand Tour of the Jovian moons 9, are the four largest moons of Uranus: Ariel, Umriel, Titania and Oeron. In fact, as already suggested in previous works 7, the Uranian system is a downscaled version of the Jovian system. This means, that from a dynamical point of view, the relative large mass parameter of the Uranus-Moon systems, is expected to allow an extension of the manifolds far from their source orits with a consequent easy intersection among them. Furthermore, the small eccentricity and the low inclination characterizing the orits of all the moons (see Tale ) guarantee that the physical 3BP is well approximated y the theoretical circular R3BPs model employed in this study. The optimized solution for the Titania-Oeron transfers, shown in Figure, is characterized y v=4 m/s and t=.6 days. The Umriel-to-Titania transfer costs 97 m/s and is 9.3 days long. Finally the Ariel-Umriel trajectory, requires a v=35 m/s at the patching point and a 4 days transfer. These trajectories are very low energy transfers ecause, as shown in Figure 9 for the Titania-Oeron case, oth the manifolds and their Poincaré sections intersect.

13 Tale. Orital and mass parameters for the Uranian system Moon a (km) e i(deg) µ Ariel e-5 Umriel e-5 Titania e-5 Oeron e-5 5 W u Lp.o (Oeron) 4 x 3 3 Oeron Titania r (m/s) 5 y (AU) v 5 W s Lp.o (Titania) r (AU) x 3 Figure 9: Poincarè sections for the Titania- Oeron transfer x (AU) x 3 Figure : Titania-Oeron transfer Tale 3. Results for two Uranian moon transfers Transfer V (m/s) t(day) Ariel-Umriel Umriel-Titania Titania-Oeron Multi-urn Transfers and the Fundamental Hypothesis for a Conic Link An interplanetary transfer exploiting the invariant manifold method of the R3BP requires that the unstale manifolds, computed in the starting system, and the stale one, that allows the capture y the arriving planet, have an intersection in the configuration space. In the solar system this intersection occurs frequently for outer planets (as already oserved in Ref. ) thanks to their high mass parameter µ which allows an extension of the manifolds far from their points of origin. On the other hand for inner planets (Mercury, Venus, Earth and

14 Mars) manifolds do not extend far enough in order to intersect each other. Then the prolem ecomes to design an interplanetary transfer, exploiting invariant manifolds of the restricted three-ody prolem, without any intersection in the sidereal frame. Figure shows the external leg of the L unstale manifold (W U,ext L) in the Sun-Earth system integrated for five years. It should e noticed that, after an initial evolution, the trajectory ecomes invariant and quasi-periodic, which suggests that, in general, in systems with a low mass parameter and far from the smallest primary, a spacecraft is only suject to the gravitational attraction of the largest primary. This can e verified calculating the Sun-Spacecraft two-ody energy without considering the presence of the Earth. It is well known, indeed, that in the two-ody prolem the energy remains constant therefore for any point of W U,ext L it has een calculated: k S E = v () r where r and v are respectively the modulus of the position and the velocity vectors, oth expressed in the asolute system; k S is the gravitational constant of the Sun. As can e see in Figure, the two-ody energy E=E(t) after an initial growth, when the spacecraft is close to the Earth, remains almost constant and the motion of the spacecraft is mainly governed y the Sun. Therefore in the two-ody representation of the dynamics the spacecraft exploits the gravitational fields of the planets in order to change its energy level with a propulsive effect, at departure, and with a raking effect at arrival. On the other hand when the spacecraft is far from oth planets, the energy associated to the Sun-spacecraft two-ody prolem stailizes aout a constant value and the end point of the two legs of the manifolds in the configuration space can e considered suject only to the gravitational attraction of the Sun. If no third-ody effects can e considered the two terminal points can e linked with a conic arc, solution of a Lamert s two-ody prolem x y (AU, inertial frame) Earth Orit Sun W u L Earth E (m /s ) x (AU, inertial frame) Figure : Unstale manifold of the Sun- Earth L in the asolute system t (days) Figure : Manifold s two-ody energy versus time Therefore the original method of the manifolds 9 can e extended to treat cases in which no intersection can e found decomposing the full four-ody prolem as follows: 3

15 . an initial R3BP with primaries Sun and departure Planet. The unstale manifolds associated to the periodic orits around L or L are computed in this system;. an intermediate BP Sun-Spacecraft in which a conic arc links the extremes (or terminal points) of the two manifolds; 3. a final R3BP with primaries Sun and arrival Planet. The stale manifolds of periodic orits around L or L are calculated in this second reference system. Two intermediate deep space maneuvers are then required to realize to link the conic arc in the phase space with the stale and unstale manifolds. The total v associated to the conic link can e tuned changing the dimensions of the sources periodic orits with a consequent change of the energy associated to the transit orits: the higher the energy of the transit orit (low values of the Jacoi constant) the closer the intersection points on the Poincaré sections are. Solution of the Lamert s Prolem and Selection of the launch window After the departure and arrival transit trajectories have een generated they are placed on the respective orital planes, with the correct phase, using the analytical ephemeris as shown in Figure 3. A z (AU).. Jupiter Earth Sun. B 5. y (AU) x (AU) 4 6 Figure 3: Departure and arrival trajectories in the Sun-centered inertial frame Let V A and V B e respectively the velocities corresponding to the points A and B in Figure 3 and let V and V e the two velocities, at A and B respectively, computed solving the associated Lamert s prolem. The total cost of the link is: A B v = v + v = V V + V V () which is a function v = v t L, T ) (3) ( s of the departure epoch T s from planet a and of the time t L required to join the two points with the conic arc. If t a and t denote respectively the time lengths of the departure and arrival transit trajectories, the optimization prolem ecomes: min v( t L, Ts ) (4) suject to: 4

16 T e,min T s s,min T + t T T a s + t L s,max + t T e,max where (T s,min, T s,max ) and (T e,min, T e,max ) are the launch and arrival dates. The prolem stated in Eqs. (4) and (5) has een solved using an SQP algorithm and the results will e presented in the next section. It is important to notice that if no conic link was required and an intersection etween the manifolds occured spontaneously, as for outer planets, the algorithm would set automatically one of the two v s to zero. RESULTS FOR THE MULTI-BURN CASE In this section the patched manifold approach descried in the previous chapter is applied to the design of some representative transfers from the Earth to Venus and Mars. The results are compared in terms of total v and time of flight to the corresponding classical icircular Hohmann transfers. The prolem has een solved considering two different cases: a powered capture into a circular orit around the target planet and a allistic capture around the target planet. In the former case two additional maneuvers, v S and v E, must e considered. In particular in the following, v S and v E indicate respectively the cost necessary to depart from a circular orit of radius r S around planet a and the cost for the insertion into a circular orit of radius r E around planet. The values r S and r E are not imposed a priori ut computed simply propagating forward and ackward in time the starting and arriving legs of the transfer, which are uniquely specified y Eq. (9), and taking the closest point to the corresponding planet. The costs of the multi-urn and of the Hohmann transfers are then calculated as the total v necessary to transfer the spacecraft etween the two circular orits of radius r S and r E. Earth to Venus Direct Transfer The first test case is a direct transfer from the Earth to Venus. The systematic search is focused on a 3 days launch window at the end of 6. The results otained for three different transfers are reported in Tale 4 and compared to the corresponding Hohmann transfer. Among the three in the tale solution is the one that offers the lowest total v and has een represented in Figures 4 and 5. For solution, the initial orit around the Earth has a radius r S = km and the destination orit around Venus is circular with radius r E =6 km. An initial v S =66 m/s is required to insert the spacecraft into a trans-venus trajectory. A first deep space maneuver of 48 m/s is performed after 8 deg to decrease the altitude of the aphelion down to the terminal point of the stale manifold of the Venus-Sun system. A second urn is the required to insert the spacecraft into the transit orit reaching Venus and a final urn v E =369 m/s injects the spacecraft in the circular orit around Venus. The total cost of the transfer is 3949 m/s, aout m/s less than the Hohmann transfer ut the travel time is more than three times larger Tale 4 Solutions for an Earth-Venus transfer T S (MJD) v (m/s) v (m/s) v (m/s) t(day) Hohmann Solution Solution Solution (5) 5

17 y(au) Venus Earth.5 v Sun v x (AU).5 z (AU) Figure 4: Earth-Venus transfer projected on the ecliptic plane.5.5 Conic Link (Sun) Arriving (Sun Venus) Starting (Sun Earth) x (AU).5.5 y (AU).5 Figure 5: 3D Earth-Venus transfer in the Sun-centered inertial frame.5 Earth to Mars Direct Transfer For the Earth-Mars case, the systematic searched has een restricted to a 3 days launch window at the eginning of 9.The results otained for three different transfers are reported in Tale 5 and the trajectory with the lowest total v (solution ) is represented in Figures 6 and 7. Also in this case the total time of transfer increase as for the Earth-Venus transfer ut in the explored short launch windows no transfer etter than the Hohmann could e found. In fact, for the est solution, the initial trans-mars maneuver performed from a circular orit around the Earth with radius r S =48 km, is v S =46 m/s and the final circularization maneuver around Mars is v E =864 m/s at radial distance r E = km. These two maneuvers added to the two required deep space maneuvers yield a total cost of 5469 m/s. Therefore this transfer appears to e less convenient than the Homann one, nevertheless it should e noticed that it provides a free allistic capture at Mars. In fact if no raking maneuver at destination is performed the spacecraft remains temporary captured y the read planet and stays within the Hill regions for a period of aout 3 days efore exiting through the gateway open at L (see Figure 8). If a small 6 m/s maneuver is performed at the pericenter r E of the incoming trajectory (see Figure 9) the gateway closes down and the spacecraft remains permanently captured within the limits of Hill s curves at a total cost, for the transfer, of 473 m/s. Furthermore it is expected that a etter exploration of the solution space, enlarging the launch window, will provide etter solutions as for the Venus case. Tale 5 Solutions for an Earth-Mars transfer T S (MJD) v (m/s) v (m/s) v (m/s) t(day) Hohmann Solution Solution Solution

18 y (AU).5.5 Earth z (AU) v v.5 Sun. Arriving (Sun Mars). Starting (Sun Earth) Conic Link (Sun).5 Mars x (AU) Figure 6: Earth-Mars transfer projected on the ecliptic plane.5 x (AU) y (AU) Figure 7: 3D Earth-Mars transfer in the Sun-centered inertial frame.5.5 y (nondimensional units, Sun Mars rotating frame) x 3 L Foridden Region (efore maneuver) Mars x (nondimensional units, Sun Mars rotating frame) Figure 8: Earth-Mars temporary capture y (nondimensional units, Sun Mars rotating frame) x 3 L Arrival Trajectory Foridden Region (after maneuver) V Modified Trajectory Mars x (nondimensional units, Sun Mars rotating frame) Figure 9: Earth-Mars: permanent capture L For high v transfers as the one in Figures 6 and 7, a high I sp thrust system could represent an interesting alternative. The conic arc is sustituted with an electric-propulsion arc. In Figure a 8kg spacecraft is equipped with a cluster of electric engines delivering a maximum of.5 N of thrust at an Isp=3s. Bold lines represent thrust arcs while dotted lines represent coast arcs. The C3 at departure is 46 m/s as in the previous case and the total propellant consumption is 75 kg. The same concept can e applied to a small 5 kg spacecraft equipped with a single low-thrust engine delivering.4n at an Isp=3s. In this case the travel time is 4 days longer ut the total propellant consumptions is aout 75kg (see Figure). 7

19 z [AU].6.4. Earth. Mars.4.6 x [AU] Figure : High-thrust Earth-Mars transfer y [AU] z [AU] x [AU] Earth Mars Figure :Low-thrust Earth-Mars transfer y [AU] FINAL REMARKS AND FUTURE WORK In this paper an approach to the design of low energy interplanetary transfers has een presented. This approach, ased on an effective lend of dynamical system theory and optimization technique, provides good results oth for the preliminary and for the advanced analysis of the interplanetary trajectories. Applied to the case of a Jupiter-Saturn transfer improved the results reported in Ref.. Despite the gain in v demonstrated y this case, the time of flight increases excessively making these transfer of questionale practical interest. A second intriguing example has shown how a low-energy tour of the Uranian moons (analogous to the Petite-Grand Tour of the Jovian moons 9 ) was possile exploiting the manifolds associated to the Uranus-Spacecraft-Moon R3BP. Furthermore an extension of the technique of the manifolds is proposed in order to treat transfers for which no intersections among manifolds 5,9, can e found. Here the non intersecting manifolds have een linked together either solving a Lamert s prolem or using a low-thrust arc. This extended algorithm has applied to the design of low-energy transfers from the Earth to Venus and Mars. If, in the former case, a clear advantage in terms of v could e found, in the latter case a real gain is achieved only when no circularization is performed at destination. However a promising improvement of the algorithm is currently in progress: the search for good first guess solutions has een extended using a comination of evolutionary algorithms (EA) and SQP 5 in order to perform a etter exploration of the solution space. A preliminary result can e seen in Figure where the prolem of a transfer from the Earth to Mars, presented aove, has een solved again minimizing the difference etween the total V of the multiurn trajectory and the total V H of the corresponding Hohmann transfer. In this case the total cost of the transfer plus the insertion into a stale circular orit is some m/s lower than the one of the correspondent Hohamnn transfer. In future works an extension of the R3BP to the spatial elliptic restricted three-ody prolem (SER3BP) will e considered as well. 8

20 45 4 Ojective function ( V V H ) GA SQP Numer of function evaluations Figure.. Convergence history for an optimal Earth-Mars transfer REFERENCES [] BATTIN, R. H. An Introduction to the Mathematics and Methods of Astrodynamics, AIAA, New York, 987 [] BELBRUNO, E. A. and MILLER, J. K. Sun-Perturated Earth-to-Moon Transfers with Ballistic Capture, Journal of Guidance, Control and Dynamics, Vol. 6, No. 3, July-August 993, p.p [3] DYSLI, P. Analytical Ephemeris for Planets, Dysli, 977 [4] GÒMEZ, G. and MONDELO, J. M. The Dynamics Around the Collinear Equilirium Points of the RTBP, Physica D 57,, p.p [5] GÒMEZ, G., KOON, W. S., LO, M. W., MARSDEN, J. E., MASDEMONT, J. and ROSS, S. D. Invariant Manifolds, the Spatial Three-Body Prolem and Space Mission Design, AAS/AIAA Specialist Meeting, Queec City, Canada, August, Paper AAS -3 [6] HOWELL, K. C., BARDEN, B. T. and LO, M. W. Application of Dynamical System Theory to Trajectory Design for a Liration Point Mission, The Journal of Astronautical Sciences, Vol. 45, No., April-June 997, p.p [7] JORBA, Á. and MASDEMONT, J. Dynamics in the Center Manifold of the Collinear Points of the Restricted Three Body Prolem, Physica D 3, 999, p.p [8] KOON, W. S., LO, M. W., MARSDEN, J. E. and ROSS, S. D. Heteroclinic Connections etween Periodic Orits and Resonance Transition in Celestial Mechanics, Chaos, Vol., No., June, p.p [9] KOON, W. S., LO, M. W., MARSDEN, J. E. and ROSS, S. D. Constructing a Low Energy Transfer etween Jovian Moons, Proceedings of the International Conference on Celestial Mechanics, Northwestern University, Chicago, Illinois, 5-9 Decemer, 999 9

21 [] LLIBRE, J., MARTÌNEZ, R. and SIMÒ, C. Transversality of the Invariant Manifolds associated to the Periodic Orits near L in the Restricted Three-Body Prolem, Journal of Differential Equations, No. 58, 985, p.p [] LO, M. W. and ROSS, S. D. Low Energy Interplanetary Transfers using the Invariant Manifolds of L, L and Halo Orits, AAS/AIAA Space Flight Mechanics Meeting, Monterey, California, 9- Feruary, 998 [] SZEBEHELY, V. Theory of Orits: the Restricted Prolem of Three Bodies, Academic Press inc., New York, 967 [3] THURMAN, R. and WORFOLK, P. A. The Geometry of Halo Orits in the Circular Restricted Three-Body Prolem, Geometry Center Research Report, GCG95, University of Minnesota, 996 [4] TOPPUTO, F. Le Varietà Invarianti del Prolema dei Tre Corpi Ristretto: Uno Strumento Innovativo per Trasferimenti Interplanetari a Basso Costo, Tesi di Laurea, Politecnico di Milano, A. A. -3, in Italian. [5] VASILE, M. and FINZI ERCOLI, A. Comining Evolutionary Programs and Gradient Methods for WSB Transfer Optimization, XVI Congresso Nazionale AIDAA, Palermo, 4-8 Settemre, [6] WIGGINS, S. Introduction to Applied Nonlinear Dynamical System and Chaos, Springer, New York, 99 [7] HEATON, A. F. and LONGUSKI, J. M. The Feasiility of a Galileo-Style Tour of the Uranian Satellites. AAS -464 AAS/AIAA Astrodynamics Specialist Conference, Queec City, Queec,Canada July 3-August,. [8] BROUCKE, R. Traveling Between the Lagrangian Points and the Moon. Journal of Guidance and Control, Vol.,No.4,979,pp [9] KOON, W. S., LO, M. W., MARSDEN, J. E., ROSS, S. D. Shoot the Moon, AAS/AIAA Astrodynamic Specialist Conference, Florida, 3-6 January,.