- Email: [email protected]
Investigation of vehicle reusability for human exploration of Near-Earth Asteroids using Sun–Earth Libration point orbits
Acta Astronautica 90 (2013) 119–128
Contents lists available at SciVerse ScienceDirect
Acta Astronautica journal homepage: www.elsevier.com/locate/actaastro
Investigation of vehicle reusability for human exploration of Near-Earth Asteroids using Sun–Earth Libration point orbits A.K. Zimmer Institute of Space Systems, University of Stuttgart, Pfaffenwaldring 29, 70569 Stuttgart, Germany
a r t i c l e i n f o
abstract
Article history: Received 24 April 2012 Received in revised form 26 August 2012 Accepted 1 October 2012 Available online 13 December 2012
Current plans for human exploration of the solar system envision several missions to Near-Earth Asteroids (NEAs) as stepping stones towards missions to Mars. This research investigates the feasibility of stationing reusable cargo spacecraft, such as habitats, in halo orbits at Sun–Earth Libration points 1 and 2 (L1 and L2 ) between NEA missions in an effort to reduce mission cost and thus overall campaign cost by lowering the mass required to be launched and the amount of new hardware to be built for each mission. Four example missions to the two currently most promising targets of the known NEA population in the 2025–2030 time frame are chosen. In the mission architecture proposed in this study, the crew vehicle directly commutes between Earth and the asteroid in order to keep mission durations for the crew short. The cargo vehicle departs from a halo orbit, rendezvous with the crew vehicle on the outbound trajectory, and returns to a halo orbit after the mission. Manifold trajectories of halo orbits in the northern and southern halo orbit family at L1 and L2 are considered for the transfer of the cargo vehicle to and from the interplanetary trajectory and the total Dv required for this transfer is minimized. This Dv is found to range from a few meters per second to hundreds of meters per second, depending on the specific energy and inclination of the interplanetary trajectory. These results show the great potential of the utilization of Sun–Earth Libration point orbits for enabling vehicle reusability, thus lowering the cost of human exploration missions. & 2012 IAA. Published by Elsevier Ltd. All rights reserved.
Keywords: Mission analysis Reusability Libration point Mission architecture Near-Earth Asteroid Human spaceflight
1. Introduction In April 2010, U.S. President Barack Obama declared missions to Near-Earth Asteroids (NEAs) as the first step in his vision for human exploration of the solar system. Such missions would represent a major milestone as humans set out on their first journey to a destination that is not gravitationally bound to Earth. As financial and technological challenges have delayed both national efforts and international collaborations from committing to a human mission to Mars, it has become apparent that NEAs provide a very effective stepping stone towards this
E-mail address: [email protected]
ultimate goal, both in terms of mission duration and technological complexity. Human missions to NEAs would allow us to acquire deep-space operational experience and test hardware, while at the same time offering extensive and challenging opportunities for space exploration, planetary defense, and scientific discovery. This paper proposes a novel mission architecture which facilitates human exploration of NEAs by enabling vehicle reusability. 2. Motivation and problem description Human exploration missions to NEAs pose a challenging endeavor as mission durations and Dv reach large values resulting in high system and fuel masses, which
0094-5765/$ - see front matter & 2012 IAA. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.actaastro.2012.10.003
120
A.K. Zimmer / Acta Astronautica 90 (2013) 119–128
exceed the limitations of current launch system capabilities and thus require multiple launches per mission. In addition, the spacecraft envisioned for such missions are highly complex systems which are expended at the end of each mission and must be entirely rebuilt and relaunched for the next mission according to current mission architectures [1]. Furthermore, multiple human missions forming a campaign are envisioned. Consequently, the high mass launched per mission as well as the amount of new hardware to be built for each mission multiplied by the number of missions tremendously drive the cost of the overall campaign. Both of these aspects can be addressed by enabling an architecture that allows system elements to be strategically stationed in space and be reused throughout the campaign. 3. Objectives The objective of this investigation is to analyze the feasibility of reusing system elements by stationing them in space between missions. Promising locations include the seven libration points in the Earth’s vicinity: five in the Earth–Moon system and two in the Sun–Earth system. The use of orbits at the Sun–Earth Libration (SEL) points for reusability has previously been proposed [2]. These orbits are of particular interest for this purpose because their locations are nearly time-invariant in the rotating heliocentric frame, i.e., relative to the Sun–Earth line, and therefore do not impose additional phasing restrictions, unlike the Earth–Moon Libration (EML) points. In addition, these libration points represent beneficial gateways for interplanetary missions as they are situated on a high potential energy level with respect to the Earth and are thus barely bound by the Earth’s gravitational attraction. Hence, this study focuses on SEL points. In the scenario proposed in this study, one or several interplanetary transportation elements, such as habitation modules or cargo, are stationed in a libration point orbit. The crew vehicle is launched from Earth into a staging orbit from which it injects directly into an interplanetary trajectory to an NEA in order to keep mission durations for the crew short. After leaving the libration point orbit, the cargo module achieves rendezvous and docks with the crew vehicle on the outbound leg of the interplanetary trajectory before encountering the asteroid. Contingency options in case the docking fails have been investigated in Ref. [3]. Upon return from the NEA to Earth’s vicinity, the crew vehicle undocks from the cargo vehicle to land on Earth while the cargo module returns to a libration point orbit. In order to analyze the feasibility of this scenario, it must be examined whether transfer trajectories connecting SEL point orbits with the interplanetary trajectories of the crew vehicle exist and what the implications of such connections are, e.g., in terms of Dv.
mutual gravitational attraction. The Circular Restricted Three-Body Problem (CR3BP) assumes the mass of one body, e.g., a spacecraft, to be infinitesimally small as compared to the masses of the two other bodies, referred to as the primaries. Furthermore, the motion of the primaries is restricted to circular orbits about their barycenter governed by the Keplerian laws of two-body motion. Since the orbits of most planets in the solar system are nearly circular, the CR3BP has been found to suitably represent the actual dynamics of the motion of objects of negligible mass under the influence of two large gravitational bodies such as the Sun and the Earth. The motion of a spacecraft can then be described using dynamical systems theory. The reference frame chosen for this purpose is centered at and rotates about the barycenter at the same rotation rate as the primaries is thus called the synodic frame. The x-axis points from the larger to the smaller primary, the z-axis is aligned with the direction of the angular momentum of the system, and the y-axis completes the frame as a right-handed coordinate system. All units in the system are scaled using a three-body parameter, m, such that the gravitational parameter, the sum of the mass of the larger body, m1, and the mass of the smaller body, m2, as well as the distance between the primaries equal one and the orbital period equals 2p. Fig. 1 shows an example of a synodic frame of a larger and a smaller body. Note that Fig. 1 represents neither the Sun–Earth nor the Earth–Moon synodic reference frame; but instead a larger value for m is chosen for this example system in order to better depict the location of the libration points. This non-dimensional coordinate system can then easily be applied to any three-body system by just changing m and the equations of motion of the spacecraft can be written as _ mÞ x€ ¼ x þ 2yð1
xþm xð1mÞ m r31 r 32
_ mÞ y€ ¼ y2xð1
y y m 3 r 31 r2
z€ ¼ ð1mÞ
z z m 3 r 31 r2
ð1Þ
4. Astrodynamic background 4.1. The Circular Restricted Three-Body Problem Neglecting non-gravitational forces, the three-body problem describes the motion of three bodies under
Fig. 1. Synodic reference frame of two example bodies and the location of the libration points [4].
A.K. Zimmer / Acta Astronautica 90 (2013) 119–128
X ¼ ½x y z x_ y_ z_ T ; r1 ¼
m¼
X_ ¼ ½x_ y_ z_ x€ y€ z€ T
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx þ mÞ2 þy2 þ z2 ; m2 m1 þ m2
r2 ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx þ m1Þ2 þy2 þ z2
ð2Þ
121
as the derivation of the equations of motion can be found in Ref. [4].
ð3Þ 4.2. Libration points & libration point orbits ð4Þ
where X is the state vector of the spacecraft, X_ is the time derivative of the state, and r1 and r2 are the distances from the spacecraft to the larger and smaller primary, respectively. A comprehensive explanation of the CR3BP as well
There are five equilibrium points in the CR3BP at which the gravitational attraction of the primaries counterbalances the centrifugal force exerted by the rotation of the system. These points are known as libration or Lagrange points and their location is fixed with respect to the primaries, see Fig. 1.
Fig. 2. Representative orbits of the northern L1 and L2 halo orbit families. The Earth is magnified by a factor 10.
Fig. 3. Stable and unstable manifolds of a halo orbit at L2 . The Earth is magnified by a factor 10.
122
A.K. Zimmer / Acta Astronautica 90 (2013) 119–128
The system described in the presented approach is the Sun–Earth system, modeling the Sun and the Earth as the primaries and a spacecraft as the third body. Furthermore, only the two collinear libration points which are closest to the Earth, namely Sun–Earth Libration point 1 (L1 ) and Sun–Earth Libration point 2 (L2 ), are addressed.1 There are several families of orbits near libration points. This study focuses on halo orbits, a family of three-dimensional periodic orbits characterized by equal frequencies of in- and out-of-plane motion as well as varying degrees of in- and out-of-plane amplitudes. These orbits are symmetric to the x–z plane but not symmetric to the orbital plane of the primaries. Due to symmetry in the equations of motion of the CR3BP, any existing orbit can be reflected across the x–y plane to obtain another orbit. Consequently, there are two families of halo orbits, a northern and a southern family. A halo orbit intersects the x–z plane in two points, one in the northern and one in the southern hemisphere, called orthogonal axis crossings. By convention, the northern axis crossing is given for northern halos and the southern axis crossing is given for southern halos. In this manner, a halo orbit is uniquely defined by either the xor the z-coordinate of this axis crossing. For the purpose of this paper, halos are identified by the x-coordinate of their axis crossing. See Fig. 2 for the northern L1 and L2 halo orbit families, note the varying degree of out-ofplane amplitudes. In this study, a single shooting algorithm and the continuation method are employed, the application of which to the evolution of the halo orbit families is described in Ref. [5]. The spacecraft state and the state transition matrix are propagated by integrating the equations of motion using a Runge–Kutta–Fehlberg 7(8) method [6]. 4.3. Invariant manifolds There are stable and unstable directions associated with the unstable orbits in these halo families. A small perturbation (E ¼ 106 used in this study) in the unstable direction results in the trajectory exponentially diverging from the halo orbit. Conversely, a trajectory approaching a halo orbit from the stable direction merges onto the orbit. The full set of all such diverging and converging trajectories is referred to as the unstable and stable manifold, respectively. As the perturbation can occur in the positive or negative direction along the vectors of the stable and unstable direction, there are four manifolds associated with each halo orbit, one stable and one unstable manifold on each the near and the far side of the halo as seen from Earth. See Fig. 3 for the invariant manifolds of an example halo orbit. The shape of the manifold is given by the halo orbit, which is defined by its x-axis crossing. A spacecraft traveling along a stable or unstable manifold in this manner can 1 Since this study exclusively investigates the Sun–Earth system, L1 and L2 always refer to Sun–Earth Libration points 1 and 2, respectively.
arrive at or depart from a libration point orbit without any deterministic maneuver. Hence, these manifolds are advantageous for transfer trajectory design. The parameter t is used to specify the position of a spacecraft along the halo orbit. Thus, t defines where along the halo orbit an unstable manifold trajectory begins or where a stable manifold trajectory ends, i.e., t designates a specific trajectory on a given manifold (for a given E). All calculations in this study assume t to take on values from 0 to 1, from first axis crossing until completion of one period. Since x-axis crossing determines a particular halo orbit in a specified family and thus the associated manifold, and t specifies a particular trajectory on the manifold, all possible manifold trajectories can be uniquely defined by the combination of x-axis crossing and t. 4.4. The Oberth effect In a two-body system, the specific orbital energy, e, of a spacecraft with respect to a gravitational body is defined as
e¼
v2 mg 2 r
ð5Þ
where r and v are the position and velocity of the spacecraft, respectively, and mg denotes the gravitational parameter of the body, not to be confused with the three-body parameter m. In order to evaluate the change in specific energy by means of an instantaneous maneuver, i.e., dr¼0, taking the derivative of Eq. (5) gives de ¼ v dv
ð6Þ
which shows that a higher De is obtained for a given Dv where the velocity of the spacecraft along its orbit is highest, i.e., at periapsis. This relation is referred to as the Oberth effect. Since periapsis velocity increases with decreasing periapsis radius, low periapsis radii are desirable for most efficient Oberth maneuvers. This relation also holds for the CR3BP. 5. Approach 5.1. Interplanetary transfer trajectory generation As a first step, the interplanetary transfer trajectories of the crew vehicle are calculated. In order to keep mission durations for the crew short, the crew vehicle commutes directly between Earth and the target asteroid: The transfer trajectory begins in a circular departure orbit around Earth with an altitude similar to that of the International Space Station (ISS), in this case 500 km. A two-impulse transfer is calculated for the outbound leg of the mission trajectory. The first maneuver, i.e., the trans-NEA injection, is performed on the departure orbit and the second maneuver is executed to rendezvous with the asteroid. When leaving the asteroid, a trans-Earth injection maneuver is performed and a direct hyperbolic atmospheric re-entry at Earth is assumed. For these calculations, position and velocity data of the Earth and the respective asteroid are taken from
A.K. Zimmer / Acta Astronautica 90 (2013) 119–128
ephemeris data provided by the Jet Propulsion Laboratory (JPL) [7] and the patched conics approach is used, switching from the Earth-centered coordinate system to the heliocentric coordinate system at the Earth’s Sphere of Influence (SOI), as described in Ref. [8]. The SOI represents a spherical region about a celestial body, in this case the Earth, in which this body is the primary source of gravitational influence on another body, e.g., a spacecraft. Within the SOI of the Earth, the gravitational influence of the Sun and any other celestial bodies is ignored, reducing the dynamics of the system to a twobody problem. The radius of the SOI of one celestial body with respect to another is a function of their respective masses and the distance between them, e.g., radius of the SOI of the Earth is approximately 9:25 105 km. In the CR3BP, it becomes obvious that the concept of patched conics and the SOI is of course only an approximation. The actual gravitational influence of the Earth extends beyond its SOI, which is evident given that the SEL points themselves lie outside the SOI of the Earth, but are by definition dependent on the Earth’s gravitational attraction. However, the SEL points are also caused by the gravitational attraction of the Sun, making the Earth not the body of primary gravitational influence. The target asteroids, launch periods, and interplanetary trajectories of the crew vehicle have been determined in a previous study [3]. From this study, four example missions of different mission duration and Dv values to the two currently most promising targets out of the known NEA population in the 2025–2030 time frame are chosen, see Table 1. The cargo vehicle must also travel along these same trajectories in order to dock with the crew vehicle. Hence, the interplanetary trajectories for the cargo vehicle are the same as those for the crew vehicle. The interplanetary trajectories are defined by their v1 vectors, i.e., the hyperbolic excess velocity vector with respect to the Earth, which is assumed to occur when the outbound trajectory leaves and the inbound trajectory enters the SOI. Finally, the trajectory on which the cargo module leaves and arrives at a halo orbit must be matched to the predetermined v1 vectors on the SOI. For this purpose, these v1 vectors are transformed from the heliocentric ephemeris frame to the CR3BP frame, accepting the inaccuracy caused by the simplifications described above.
123
the outbound trajectory, the cargo vehicle leaves the halo on an unstable manifold. On its trajectory, it performs two maneuvers: one Oberth maneuver at periapsis and a second maneuver when crossing the SOI in order to match the v1 defined by the desired interplanetary transfer. On the inbound trajectory, the cargo vehicle leaves the interplanetary transfer of the crew vehicle by performing a maneuver when entering the SOI. At periapsis, an Oberth maneuver puts the cargo vehicle on a stable manifold, returning it to a halo orbit.
5.3. Parameter space investigation A priori, all northern and southern halo orbits at L1 and L2 as well as all values for t could yield possible trajectory solutions connecting the halo orbit with the interplanetary transfer. Consequently, the parameter space allows the x-coordinate of a halo orbit’s axis crossing to vary throughout the respective halo orbit family and t to take on values from 0 to 1. However, as described by the Oberth effect, it is essential for the manifold trajectories to come relatively close to Earth during their periapsis passages in order to maximize the efficiency of the periapsis maneuver. Consequently, it is investigated how closely manifolds originating at different halo orbits approach Earth, see Fig. 4. In order to generate this plot, one hundred halo orbits from the northern halo family at each L1 and L2 are chosen to represent the both families. For each of these two hundred halo orbits, one thousand trajectories on their unstable manifolds are propagated for two years. Because of the symmetry described in Section 4.2, it is sufficient to analyze the manifolds of either the northern or the southern families for this purpose. The
5.2. Mission architecture for reusability In the proposed mission architecture, cargo vehicles are stationed in northern or southern halo orbits L1 or L2 . For Table 1 Example missions. #
Launch date
Target Asteroid
Total mission duration (days)
Dv (km/s)
1 2 3 4
18 28 13 15
1999 2000 2000 2000
121 161 86 221
7.330 6.968 7.404 6.366
Oct 2025 Mar 2028 Aug 2029 Feb 2030
AO10 SG344 SG344 SG344
Fig. 4. Minimum altitude above the Earth’s surface as a function of xaxis crossing. The red line depicts the minimum altitude encountered along a northern or southern halo orbit of a given x-axis crossing. The blue line shows the minimum flyby altitude of a manifold originating from a northern or southern halo orbit of a given x-axis crossing. The gray line illustrates the altitude of the Earth’s SOI. The Earth is not shown to scale. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
124
A.K. Zimmer / Acta Astronautica 90 (2013) 119–128
Table 2 Minimum matching Dv (km/s) for trajectories matched at their first SOI exit and originating from northern and southern L1 and L2 halo orbits. #
Target NEA
Mission leg
L1 North
L1 South
L2 North
L2 South
1
1999 AO10 2000 SG344 2000 SG344 2000 SG344
Outbound Inbound Outbound Inbound Outbound Inbound Outbound Inbound
0.556 1.302 0.185 0.006 0.851 0.675 0.117 0.538
0.638 0.695 0.112 0.006 0.727 0.668 0.240 0.353
0.687 0.538 0.044 0.002 0.454 0.912 0.245 0.304
0.471 1.010 0.084 0.073 0.460 0.906 0.258 0.401
2 3 4
Table 3 Minimum matching Dv (km/s) for trajectories matched at their second SOI exit and originating from northern and southern L1 and L2 halo orbits. #
Target NEA
Mission leg
L1 North
L1 South
L2 North
L2 South
1
1999 AO10 2000 SG344 2000 SG344 2000 SG344
Outbound Inbound Outbound Inbound Outbound Inbound Outbound Inbound
1.080 0.638 0.122 0.133 0.661 0.450 0.316 0.941
0.441 1.424 0.129 0.101 0.690 0.435 0.186 0.892
0.664 0.685 0.091 0.099 0.543 0.320 0.281 0.516
1.019 1.162 0.076 0.088 0.543 0.328 0.407 0.458
2 3 4
periapsis altitude of the trajectory with the minimum flyby altitude for each halo orbit is recorded. Moreover, the minimum distance from Earth along the halo orbit itself is documented. It can be seen that manifolds originating from halo orbits with lower out-of-plane amplitudes, i.e., halo orbits farthest away from Earth as seen in Fig. 2, have the lowest periapses. The periapsis altitude of the manifolds increases with increasing out-ofplane amplitudes of the halo orbits from which they originate, up to the point where the altitude of the halo orbit and the altitude of the associated unstable manifold hardly differ. With these observations and in order to maximize the efficiency of the maneuvers, this study restricts the parameter space by focusing on the halo orbits whose manifolds have a minimum periapsis altitude less than half of the altitude of the Earth’s SOI. In addition to the increased efficiency of the Oberth maneuver, halo orbits with lower out-of-plane amplitudes provide additional benefits as these orbits are less stable, i.e., less time is required to transit between these halo orbits and Earth, which is an advantage for mission planning.
5.4. Global grid search A global grid search in terms of the x-coordinate of a halo orbit’s axis crossing and t is performed for the
Fig. 5. Examples for outbound and inbound transfer trajectories for the mission to the asteroid 1999 AO10, matched at the first SOI exit. The black line indicates the direction of the v1 vector at the SOI. (a) Outbound trajectory departing from northern L1 halo orbit. (b) Outbound trajectory departing from northern L2 halo orbit. (c) Inbound trajectory arriving at southern L1 halo orbit. (d) Inbound trajectory arriving at southern L2 halo orbit.
A.K. Zimmer / Acta Astronautica 90 (2013) 119–128
northern and southern halo orbit families at L1 and L2 . The values of t are varied from 0 to 1 in steps of 0.01 and the halo orbit family lying within the most promising region as described above is discretized into 40 halo orbits at each L1 and L2 . Then, for each combination of x-axis crossing and t, the unstable manifold trajectory is propagated in the direction of Earth for a flight time up to two years. The minimum altitude recorded during the propagation must be within an altitude range from 500 km to half the altitude of the SOI. If the minimum altitude lies outside this altitude range, the trajectory is discarded. Although trajectories with a minimum altitude below 500 km could be modified to fall within altitude range, such trajectories remain to be investigated in further study. Within the propagation time of two years, some manifold trajectories encounter several Earth flybys and leave and re-enter the SOI repeatedly. For the grid search, up to two SOI exits after entering the altitude range are permitted, and the trajectory is then cut off when leaving the SOI. For the v1 vectors of the interplanetary trajectories, the values of the outbound and inbound trajectories of the four example missions are used. If the target v1 and the velocity vector of a trajectory when leaving the SOI draw an angle of less than 901, the trajectory is retained. All other trajectories are discarded because they would re-enter the SOI once their current v1 vector is adjusted
125
to the target v1 vector and would not arrive at the correct interplanetary trajectory. For each of the remaining manifold trajectories, two maneuvers are introduced in order to match the target v1 vector as described above: a periapsis maneuver in the direction of motion during the last flyby before the SOI exit, and a maneuver at the SOI exit. The sum of the Dv magnitudes associated with these two maneuvers is minimized and the minimum Dv sum along with the respective value of t for each orbit is recorded. In this approach, the transfers originating from the southern halo orbit family are not calculated separately. Since the southern halo orbit family results from the northern halo orbit family by reflection across the x–y plane, matching the southern halo orbit family to a given v1 vector yields the same result as matching the northern halo orbit family to the reflection of the same v1 vector across the x–y plane. The v1 vectors of the inbound interplanetary trajectories must be matched to the stable manifolds of the halo orbit family. However, the symmetry, s, of the equations of motion of the CR3BP as given in Eq. (7) allows the propagation of the stable manifolds backwards in time by means of propagating the unstable manifolds forward in time and reversing the signs of the respective state coordinates. Consequently, by applying this symmetry to the inbound v1 vectors, they can be matched to the
Fig. 6. Examples for outbound and inbound transfer trajectories for the mission to the asteroid 1999 AO10, matched at the second SOI exit. The black line indicates the direction of the v1 vector at the SOI. (a) Outbound trajectory departing from northern L1 halo orbit. (b) Outbound trajectory departing from northern L2 halo orbit. (c) Inbound trajectory arriving at southern L1 halo orbit. (d) Inbound trajectory arriving at southern L2 halo orbit.
126
A.K. Zimmer / Acta Astronautica 90 (2013) 119–128
Fig. 7. Examples for outbound and inbound transfer trajectories for the mission to the asteroid 2000 SG344, matched at the first SOI exit. The black line indicates the direction of the v1 vector at the SOI. (a) Outbound trajectory departing from northern L1 halo orbit. (b) Outbound trajectory departing from northern L2 halo orbit. (c) Inbound trajectory arriving at southern L1 halo orbit. (d) Inbound trajectory arriving at southern L2 halo orbit.
unstable manifolds yielding the same results as matching the original inbound v1 vectors to the stable manifolds. _ y, _ z_ Þ-ðt,x,y,z,x, _ y, _ z_ Þ s : ðt,x,y,z, x,
ð7Þ
In this manner, the propagation of only the unstable manifolds of the northern L1 and L2 halo orbit families are required and the results for the southern halo orbit families as well as the stable manifolds of all halo orbits can efficiently be deduced. 5.5. Initial guess selection & local optimization A local optimum is computed for the northern and southern halo orbit families at each L1 and L2 for both one and two SOI exits, giving a total of eight trajectories to be optimized for each outbound and inbound v1 vector. The cost function is formulated in terms of matching Dv as a function of x-axis crossing, i.e., halo orbit, and t. For each of the eight trajectories, the most Dv-efficient combination of halo orbit and t from the global grid search is used as an initial guess for further optimization. The lower and upper bounds for the x-axis crossing are given by the two halo orbits adjacent to the initial guess halo orbit. The lower and upper bounds for t are given by the best initial guess values for t along those adjacent halos, since a continuous behavior of t from one halo orbit to the next is observed in most cases when comparing
values of t of the most Dv-efficient transfers originating from adjacent halo orbits. The initial guess for the x-axis crossing and t along with their respective lower and upper bounds are fed into a simple optimizer based on a finite-difference algorithm. 6. Results Tables 2 and 3 summarize the results from the optimization of trajectories being matched to the v1 vectors at their first and second SOI exit, respectively. Note that these Dv values, termed matching Dv, represent the maneuvers required to connect halo orbits to the desired interplanetary transfer using the architecture described above. They do not include maneuvers required to rendezvous with or depart from the asteroid. Example matching trajectories between the libration point orbits and the SOI for the first two missions are shown in Figs. 5–8. In addition to the 3D trajectory, projections on all three coordinate planes are plotted. Since some trajectories have a very low perigee altitude at Earth, not plotting the Earth to scale would result in the trajectory seemingly intersecting the Earths’s surface. Hence, the Earth is plotted to scale at ½1m,0,0T and consequently it is barely visible. Large differences in Dv are observed for the same interplanetary trajectory, e.g., ranging from 0.638 km/s
A.K. Zimmer / Acta Astronautica 90 (2013) 119–128
127
Fig. 8. Examples for outbound and inbound transfer trajectories for the mission to the asteroid 2000 SG344, matched at the second SOI exit. The black line indicates the direction of the v1 vector at the SOI. (a) Outbound trajectory departing from northern L1 halo orbit. (b) Outbound trajectory departing from northern L2 halo orbit. (c) Inbound trajectory arriving at southern L1 halo orbit. (d) Inbound trajectory arriving at southern L2 halo orbit.
to 1.424 km/s for the inbound trajectory leg of the mission to asteroid 1999 AO10, depending on whether the cargo trajectory originates at a northern or southern L1 or L2 orbit and whether it is matched to the crew trajectory at the first or second SOI exit. In addition, the matching Dv required for a certain transfer depends strongly on the heliocentric specific energy and inclination of the transfer trajectory to the target NEAs given by the v1 vectors as well as the periapsis altitude of the Oberth maneuver. As expected, the matching Dv increases with increasing specific energy of the interplanetary trajectory since the difference between this energy level and that of the manifold transfer must be bridged. Also not surprisingly, the increase in matching Dv can be counterbalanced by decreasing the periapsis altitude of the Oberth maneuver. Although the halo orbit family contains orbits with very large out-of-plane amplitudes, the manifolds of these orbits do not approach Earth quickly enough to be of value in this study. However, halo orbits with small outof-plane amplitudes translate to manifolds with small out-of-plane amplitudes and do not by themselves reach high heliocentric inclinations. Hence, a maneuver raising the inclination is inevitable for higher inclinations of the heliocentric transfer. Consequently, these two factors should be taken into consideration when choosing the heliocentric transfer trajectory to the target NEAs, giving
priority to transfer trajectories with lower specific energies and inclinations. If the heliocentric trajectory is of low energy level and inclination, ‘‘free’’ connections to halo orbits can be found as is the case in the first mission to asteroid 2000 SG344.2 Especially since the connection between the interplanetary transfer and the manifold trajectory must be made twice for vehicle reusability, i.e., before and after each mission, matching Dv penalties are inflicted twice for high inclinations and specific energies. The x-axis crossings of the optimum halo orbits for stationing a vehicle in the proposed scenario vary within a certain range for each L1 and L2 . However, since heteroclinic, i.e., free transfers exist between halo orbits at L1 and L2 and hopping from one halo orbit to another orbit in the same family can be accomplished with few tens of m/s of Dv [10,11], it is not necessary to choose a single orbit for parking the spacecraft. It is observed during the optimization that small changes in t and x-axis crossing of the halo orbit yield great savings in matching Dv. Conversely, this demonstrates
2 This seems particularly promising if this scenario is combined with a chemical-electric hybrid-propulsion architecture as described in Ref. [9], as the interplanetary trajectories in such architectures initially are of lower specific energies and inclinations than is the case for architectures relying entirely on chemical propulsion.
128
A.K. Zimmer / Acta Astronautica 90 (2013) 119–128
how sensitive these trajectories are to errors in orbit control and manifold injection. Stationing unmanned vehicles in halo orbits at L1 or L2 for reuse in subsequent missions is advantageous if the cost of building and launching these vehicles anew for each mission is higher than the cost of building and launching the fuel stages required for the maneuvers of this architecture. Hence, the effectivity of vehicle reusability in the proposed scenario depends on the actual mission and campaign and is to be evaluated once these costs are determined.
Acknowledgments The research described in this paper was carried out at the Jet Propulsion Laboratory, California Institute of Technology and at the Institute of Space Systems (IRS), University of Stuttgart. The author wishes to thank Professor Dr. Ernst Messerschmid, chair of the Astronautics and Space Stations Group at the IRS, as well as Dr. Roby Wilson, supervisor of JPL’s Inner Planet Mission Analysis Group, for their dedicated support and sharing their comprehensive knowledge.
7. Conclusion & outlook References This study presents a mission architecture enabling the reuse of cargo modules, such as habitation modules or laboratories, throughout a campaign of several missions to NEAs, thus lowering campaign cost. For this purpose, the feasibility of parking such cargo modules in SEL point orbits at L1 and L2 and connecting them to interplanetary trajectories is shown. For four example missions, the manifolds of northern and southern L1 and L2 halo orbits are matched to these interplanetary trajectories by performing a maneuver at periapsis and when crossing the SOI. It is demonstrated that these connections are always possible, the required Dv ranging from a few meters per second to hundreds of meters per second, depending on the heliocentric specific energy, inclination, and periapsis altitude of the manifold trajectory. Lower Dv values for missions to the same targets can be expected if transfer trajectories of lower specific energy and inclination are chosen. In addition, chemicalelectric hybrid-propulsion architectures promise a further decrease in the Dv required to match halo orbits to interplanetary trajectory as the specific energy and inclination of such interplanetary trajectories are significantly lower when crossing the SOI. Furthermore, the use of an additional maneuver at halo orbit departure and its effects on launch period as well as periapsis altitude and thus, total Dv is to be investigated. Moreover, the study can be extended to address other interplanetary targets such as additional NEAs or Mars and its moons. As the total Dv is very sensitive to errors in orbit control and manifold injection, contingency scenarios counteracting these errors must be investigated in future work. Finally, all trajectories remain to be calculated in a full ephemeris model.
[1] R. Landis, P. Abell, D. Korsmeyer, T. Jones, D. Adamo, Piloted operations at a near-earth object (NEO), Acta Astronautica, 2009. [2] W. Huntress, D. Stetson, R. Farquhar, J. Zimmerman, B. Clark, W. O’Neil, R. Bourke, and B. Foing, The Next Steps in Exploring Deep Space, Universities Press, 2004. [3] A. Zimmer, E. Messerschmid, Going Beyond: target selection and mission analysis of human exploration missions to near-earth asteroids, Acta Astronautica 69 (11-12) (2011) 1096–1109. [4] W. Koon, M. Lo, J. Marsden, S. Ross, Dynamical Systems, the ThreeBody Problem and Space Mission Design. California Institute of Technology, 2006. [5] K. Howell, Three-dimensional, periodic, ‘halo’ orbits, Celestial Mechanics, 32 (1) (1984) 53. [6] E. Fehlberg, Classical Fifth, Sixth, Seventh and Eighth Order RungeKutta Formulas with Stepsize Control, NASA Technical Report TR R-287, 1968. [7] Jet Propulsion Laboratory, Solar System Dynamics, /http://ssd.jpl. nasa.gov/?horizonsS, 2012. [8] R. Bate, D. Mueller, J. White, Fundamentals of astrodynamics, Dover, 1971. [9] D. Landau, N. Strange, Human Exploration of Near-Earth Asteroids via Solar Electric Propulsion, AAS/AIAA Space Flight Mechanics Meeting, New Orleans, Louisiana, February 2011, pp. 13-17. [10] W. Koon, M. Lo, J. Marsden, S. Ross, Heteroclinic connections between periodic orbits and resonance transitions in celestial mechanics, Chaos, 2000. [11] K. Davis, Locally Optimal Transfer Trajectories Between Libration Point Orbits Using Invariant Manifolds, Dissertation, University of Colorado, 2009. Aline K. Zimmer received a M.Sc. in Aerospace Engineering from the Georgia Institute of Technology and a Dr.-Ing. (German Ph.D. equivalent in Aerospace Engineering) from the University of Stuttgart, Germany. Her research interests include astrodynamics, mission analysis, asteroid exploration, and human spaceflight.
Contents lists available at SciVerse ScienceDirect
Acta Astronautica journal homepage: www.elsevier.com/locate/actaastro
Investigation of vehicle reusability for human exploration of Near-Earth Asteroids using Sun–Earth Libration point orbits A.K. Zimmer Institute of Space Systems, University of Stuttgart, Pfaffenwaldring 29, 70569 Stuttgart, Germany
a r t i c l e i n f o
abstract
Article history: Received 24 April 2012 Received in revised form 26 August 2012 Accepted 1 October 2012 Available online 13 December 2012
Current plans for human exploration of the solar system envision several missions to Near-Earth Asteroids (NEAs) as stepping stones towards missions to Mars. This research investigates the feasibility of stationing reusable cargo spacecraft, such as habitats, in halo orbits at Sun–Earth Libration points 1 and 2 (L1 and L2 ) between NEA missions in an effort to reduce mission cost and thus overall campaign cost by lowering the mass required to be launched and the amount of new hardware to be built for each mission. Four example missions to the two currently most promising targets of the known NEA population in the 2025–2030 time frame are chosen. In the mission architecture proposed in this study, the crew vehicle directly commutes between Earth and the asteroid in order to keep mission durations for the crew short. The cargo vehicle departs from a halo orbit, rendezvous with the crew vehicle on the outbound trajectory, and returns to a halo orbit after the mission. Manifold trajectories of halo orbits in the northern and southern halo orbit family at L1 and L2 are considered for the transfer of the cargo vehicle to and from the interplanetary trajectory and the total Dv required for this transfer is minimized. This Dv is found to range from a few meters per second to hundreds of meters per second, depending on the specific energy and inclination of the interplanetary trajectory. These results show the great potential of the utilization of Sun–Earth Libration point orbits for enabling vehicle reusability, thus lowering the cost of human exploration missions. & 2012 IAA. Published by Elsevier Ltd. All rights reserved.
Keywords: Mission analysis Reusability Libration point Mission architecture Near-Earth Asteroid Human spaceflight
1. Introduction In April 2010, U.S. President Barack Obama declared missions to Near-Earth Asteroids (NEAs) as the first step in his vision for human exploration of the solar system. Such missions would represent a major milestone as humans set out on their first journey to a destination that is not gravitationally bound to Earth. As financial and technological challenges have delayed both national efforts and international collaborations from committing to a human mission to Mars, it has become apparent that NEAs provide a very effective stepping stone towards this
E-mail address: [email protected]
ultimate goal, both in terms of mission duration and technological complexity. Human missions to NEAs would allow us to acquire deep-space operational experience and test hardware, while at the same time offering extensive and challenging opportunities for space exploration, planetary defense, and scientific discovery. This paper proposes a novel mission architecture which facilitates human exploration of NEAs by enabling vehicle reusability. 2. Motivation and problem description Human exploration missions to NEAs pose a challenging endeavor as mission durations and Dv reach large values resulting in high system and fuel masses, which
0094-5765/$ - see front matter & 2012 IAA. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.actaastro.2012.10.003
120
A.K. Zimmer / Acta Astronautica 90 (2013) 119–128
exceed the limitations of current launch system capabilities and thus require multiple launches per mission. In addition, the spacecraft envisioned for such missions are highly complex systems which are expended at the end of each mission and must be entirely rebuilt and relaunched for the next mission according to current mission architectures [1]. Furthermore, multiple human missions forming a campaign are envisioned. Consequently, the high mass launched per mission as well as the amount of new hardware to be built for each mission multiplied by the number of missions tremendously drive the cost of the overall campaign. Both of these aspects can be addressed by enabling an architecture that allows system elements to be strategically stationed in space and be reused throughout the campaign. 3. Objectives The objective of this investigation is to analyze the feasibility of reusing system elements by stationing them in space between missions. Promising locations include the seven libration points in the Earth’s vicinity: five in the Earth–Moon system and two in the Sun–Earth system. The use of orbits at the Sun–Earth Libration (SEL) points for reusability has previously been proposed [2]. These orbits are of particular interest for this purpose because their locations are nearly time-invariant in the rotating heliocentric frame, i.e., relative to the Sun–Earth line, and therefore do not impose additional phasing restrictions, unlike the Earth–Moon Libration (EML) points. In addition, these libration points represent beneficial gateways for interplanetary missions as they are situated on a high potential energy level with respect to the Earth and are thus barely bound by the Earth’s gravitational attraction. Hence, this study focuses on SEL points. In the scenario proposed in this study, one or several interplanetary transportation elements, such as habitation modules or cargo, are stationed in a libration point orbit. The crew vehicle is launched from Earth into a staging orbit from which it injects directly into an interplanetary trajectory to an NEA in order to keep mission durations for the crew short. After leaving the libration point orbit, the cargo module achieves rendezvous and docks with the crew vehicle on the outbound leg of the interplanetary trajectory before encountering the asteroid. Contingency options in case the docking fails have been investigated in Ref. [3]. Upon return from the NEA to Earth’s vicinity, the crew vehicle undocks from the cargo vehicle to land on Earth while the cargo module returns to a libration point orbit. In order to analyze the feasibility of this scenario, it must be examined whether transfer trajectories connecting SEL point orbits with the interplanetary trajectories of the crew vehicle exist and what the implications of such connections are, e.g., in terms of Dv.
mutual gravitational attraction. The Circular Restricted Three-Body Problem (CR3BP) assumes the mass of one body, e.g., a spacecraft, to be infinitesimally small as compared to the masses of the two other bodies, referred to as the primaries. Furthermore, the motion of the primaries is restricted to circular orbits about their barycenter governed by the Keplerian laws of two-body motion. Since the orbits of most planets in the solar system are nearly circular, the CR3BP has been found to suitably represent the actual dynamics of the motion of objects of negligible mass under the influence of two large gravitational bodies such as the Sun and the Earth. The motion of a spacecraft can then be described using dynamical systems theory. The reference frame chosen for this purpose is centered at and rotates about the barycenter at the same rotation rate as the primaries is thus called the synodic frame. The x-axis points from the larger to the smaller primary, the z-axis is aligned with the direction of the angular momentum of the system, and the y-axis completes the frame as a right-handed coordinate system. All units in the system are scaled using a three-body parameter, m, such that the gravitational parameter, the sum of the mass of the larger body, m1, and the mass of the smaller body, m2, as well as the distance between the primaries equal one and the orbital period equals 2p. Fig. 1 shows an example of a synodic frame of a larger and a smaller body. Note that Fig. 1 represents neither the Sun–Earth nor the Earth–Moon synodic reference frame; but instead a larger value for m is chosen for this example system in order to better depict the location of the libration points. This non-dimensional coordinate system can then easily be applied to any three-body system by just changing m and the equations of motion of the spacecraft can be written as _ mÞ x€ ¼ x þ 2yð1
xþm xð1mÞ m r31 r 32
_ mÞ y€ ¼ y2xð1
y y m 3 r 31 r2
z€ ¼ ð1mÞ
z z m 3 r 31 r2
ð1Þ
4. Astrodynamic background 4.1. The Circular Restricted Three-Body Problem Neglecting non-gravitational forces, the three-body problem describes the motion of three bodies under
Fig. 1. Synodic reference frame of two example bodies and the location of the libration points [4].
A.K. Zimmer / Acta Astronautica 90 (2013) 119–128
X ¼ ½x y z x_ y_ z_ T ; r1 ¼
m¼
X_ ¼ ½x_ y_ z_ x€ y€ z€ T
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx þ mÞ2 þy2 þ z2 ; m2 m1 þ m2
r2 ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx þ m1Þ2 þy2 þ z2
ð2Þ
121
as the derivation of the equations of motion can be found in Ref. [4].
ð3Þ 4.2. Libration points & libration point orbits ð4Þ
where X is the state vector of the spacecraft, X_ is the time derivative of the state, and r1 and r2 are the distances from the spacecraft to the larger and smaller primary, respectively. A comprehensive explanation of the CR3BP as well
There are five equilibrium points in the CR3BP at which the gravitational attraction of the primaries counterbalances the centrifugal force exerted by the rotation of the system. These points are known as libration or Lagrange points and their location is fixed with respect to the primaries, see Fig. 1.
Fig. 2. Representative orbits of the northern L1 and L2 halo orbit families. The Earth is magnified by a factor 10.
Fig. 3. Stable and unstable manifolds of a halo orbit at L2 . The Earth is magnified by a factor 10.
122
A.K. Zimmer / Acta Astronautica 90 (2013) 119–128
The system described in the presented approach is the Sun–Earth system, modeling the Sun and the Earth as the primaries and a spacecraft as the third body. Furthermore, only the two collinear libration points which are closest to the Earth, namely Sun–Earth Libration point 1 (L1 ) and Sun–Earth Libration point 2 (L2 ), are addressed.1 There are several families of orbits near libration points. This study focuses on halo orbits, a family of three-dimensional periodic orbits characterized by equal frequencies of in- and out-of-plane motion as well as varying degrees of in- and out-of-plane amplitudes. These orbits are symmetric to the x–z plane but not symmetric to the orbital plane of the primaries. Due to symmetry in the equations of motion of the CR3BP, any existing orbit can be reflected across the x–y plane to obtain another orbit. Consequently, there are two families of halo orbits, a northern and a southern family. A halo orbit intersects the x–z plane in two points, one in the northern and one in the southern hemisphere, called orthogonal axis crossings. By convention, the northern axis crossing is given for northern halos and the southern axis crossing is given for southern halos. In this manner, a halo orbit is uniquely defined by either the xor the z-coordinate of this axis crossing. For the purpose of this paper, halos are identified by the x-coordinate of their axis crossing. See Fig. 2 for the northern L1 and L2 halo orbit families, note the varying degree of out-ofplane amplitudes. In this study, a single shooting algorithm and the continuation method are employed, the application of which to the evolution of the halo orbit families is described in Ref. [5]. The spacecraft state and the state transition matrix are propagated by integrating the equations of motion using a Runge–Kutta–Fehlberg 7(8) method [6]. 4.3. Invariant manifolds There are stable and unstable directions associated with the unstable orbits in these halo families. A small perturbation (E ¼ 106 used in this study) in the unstable direction results in the trajectory exponentially diverging from the halo orbit. Conversely, a trajectory approaching a halo orbit from the stable direction merges onto the orbit. The full set of all such diverging and converging trajectories is referred to as the unstable and stable manifold, respectively. As the perturbation can occur in the positive or negative direction along the vectors of the stable and unstable direction, there are four manifolds associated with each halo orbit, one stable and one unstable manifold on each the near and the far side of the halo as seen from Earth. See Fig. 3 for the invariant manifolds of an example halo orbit. The shape of the manifold is given by the halo orbit, which is defined by its x-axis crossing. A spacecraft traveling along a stable or unstable manifold in this manner can 1 Since this study exclusively investigates the Sun–Earth system, L1 and L2 always refer to Sun–Earth Libration points 1 and 2, respectively.
arrive at or depart from a libration point orbit without any deterministic maneuver. Hence, these manifolds are advantageous for transfer trajectory design. The parameter t is used to specify the position of a spacecraft along the halo orbit. Thus, t defines where along the halo orbit an unstable manifold trajectory begins or where a stable manifold trajectory ends, i.e., t designates a specific trajectory on a given manifold (for a given E). All calculations in this study assume t to take on values from 0 to 1, from first axis crossing until completion of one period. Since x-axis crossing determines a particular halo orbit in a specified family and thus the associated manifold, and t specifies a particular trajectory on the manifold, all possible manifold trajectories can be uniquely defined by the combination of x-axis crossing and t. 4.4. The Oberth effect In a two-body system, the specific orbital energy, e, of a spacecraft with respect to a gravitational body is defined as
e¼
v2 mg 2 r
ð5Þ
where r and v are the position and velocity of the spacecraft, respectively, and mg denotes the gravitational parameter of the body, not to be confused with the three-body parameter m. In order to evaluate the change in specific energy by means of an instantaneous maneuver, i.e., dr¼0, taking the derivative of Eq. (5) gives de ¼ v dv
ð6Þ
which shows that a higher De is obtained for a given Dv where the velocity of the spacecraft along its orbit is highest, i.e., at periapsis. This relation is referred to as the Oberth effect. Since periapsis velocity increases with decreasing periapsis radius, low periapsis radii are desirable for most efficient Oberth maneuvers. This relation also holds for the CR3BP. 5. Approach 5.1. Interplanetary transfer trajectory generation As a first step, the interplanetary transfer trajectories of the crew vehicle are calculated. In order to keep mission durations for the crew short, the crew vehicle commutes directly between Earth and the target asteroid: The transfer trajectory begins in a circular departure orbit around Earth with an altitude similar to that of the International Space Station (ISS), in this case 500 km. A two-impulse transfer is calculated for the outbound leg of the mission trajectory. The first maneuver, i.e., the trans-NEA injection, is performed on the departure orbit and the second maneuver is executed to rendezvous with the asteroid. When leaving the asteroid, a trans-Earth injection maneuver is performed and a direct hyperbolic atmospheric re-entry at Earth is assumed. For these calculations, position and velocity data of the Earth and the respective asteroid are taken from
A.K. Zimmer / Acta Astronautica 90 (2013) 119–128
ephemeris data provided by the Jet Propulsion Laboratory (JPL) [7] and the patched conics approach is used, switching from the Earth-centered coordinate system to the heliocentric coordinate system at the Earth’s Sphere of Influence (SOI), as described in Ref. [8]. The SOI represents a spherical region about a celestial body, in this case the Earth, in which this body is the primary source of gravitational influence on another body, e.g., a spacecraft. Within the SOI of the Earth, the gravitational influence of the Sun and any other celestial bodies is ignored, reducing the dynamics of the system to a twobody problem. The radius of the SOI of one celestial body with respect to another is a function of their respective masses and the distance between them, e.g., radius of the SOI of the Earth is approximately 9:25 105 km. In the CR3BP, it becomes obvious that the concept of patched conics and the SOI is of course only an approximation. The actual gravitational influence of the Earth extends beyond its SOI, which is evident given that the SEL points themselves lie outside the SOI of the Earth, but are by definition dependent on the Earth’s gravitational attraction. However, the SEL points are also caused by the gravitational attraction of the Sun, making the Earth not the body of primary gravitational influence. The target asteroids, launch periods, and interplanetary trajectories of the crew vehicle have been determined in a previous study [3]. From this study, four example missions of different mission duration and Dv values to the two currently most promising targets out of the known NEA population in the 2025–2030 time frame are chosen, see Table 1. The cargo vehicle must also travel along these same trajectories in order to dock with the crew vehicle. Hence, the interplanetary trajectories for the cargo vehicle are the same as those for the crew vehicle. The interplanetary trajectories are defined by their v1 vectors, i.e., the hyperbolic excess velocity vector with respect to the Earth, which is assumed to occur when the outbound trajectory leaves and the inbound trajectory enters the SOI. Finally, the trajectory on which the cargo module leaves and arrives at a halo orbit must be matched to the predetermined v1 vectors on the SOI. For this purpose, these v1 vectors are transformed from the heliocentric ephemeris frame to the CR3BP frame, accepting the inaccuracy caused by the simplifications described above.
123
the outbound trajectory, the cargo vehicle leaves the halo on an unstable manifold. On its trajectory, it performs two maneuvers: one Oberth maneuver at periapsis and a second maneuver when crossing the SOI in order to match the v1 defined by the desired interplanetary transfer. On the inbound trajectory, the cargo vehicle leaves the interplanetary transfer of the crew vehicle by performing a maneuver when entering the SOI. At periapsis, an Oberth maneuver puts the cargo vehicle on a stable manifold, returning it to a halo orbit.
5.3. Parameter space investigation A priori, all northern and southern halo orbits at L1 and L2 as well as all values for t could yield possible trajectory solutions connecting the halo orbit with the interplanetary transfer. Consequently, the parameter space allows the x-coordinate of a halo orbit’s axis crossing to vary throughout the respective halo orbit family and t to take on values from 0 to 1. However, as described by the Oberth effect, it is essential for the manifold trajectories to come relatively close to Earth during their periapsis passages in order to maximize the efficiency of the periapsis maneuver. Consequently, it is investigated how closely manifolds originating at different halo orbits approach Earth, see Fig. 4. In order to generate this plot, one hundred halo orbits from the northern halo family at each L1 and L2 are chosen to represent the both families. For each of these two hundred halo orbits, one thousand trajectories on their unstable manifolds are propagated for two years. Because of the symmetry described in Section 4.2, it is sufficient to analyze the manifolds of either the northern or the southern families for this purpose. The
5.2. Mission architecture for reusability In the proposed mission architecture, cargo vehicles are stationed in northern or southern halo orbits L1 or L2 . For Table 1 Example missions. #
Launch date
Target Asteroid
Total mission duration (days)
Dv (km/s)
1 2 3 4
18 28 13 15
1999 2000 2000 2000
121 161 86 221
7.330 6.968 7.404 6.366
Oct 2025 Mar 2028 Aug 2029 Feb 2030
AO10 SG344 SG344 SG344
Fig. 4. Minimum altitude above the Earth’s surface as a function of xaxis crossing. The red line depicts the minimum altitude encountered along a northern or southern halo orbit of a given x-axis crossing. The blue line shows the minimum flyby altitude of a manifold originating from a northern or southern halo orbit of a given x-axis crossing. The gray line illustrates the altitude of the Earth’s SOI. The Earth is not shown to scale. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
124
A.K. Zimmer / Acta Astronautica 90 (2013) 119–128
Table 2 Minimum matching Dv (km/s) for trajectories matched at their first SOI exit and originating from northern and southern L1 and L2 halo orbits. #
Target NEA
Mission leg
L1 North
L1 South
L2 North
L2 South
1
1999 AO10 2000 SG344 2000 SG344 2000 SG344
Outbound Inbound Outbound Inbound Outbound Inbound Outbound Inbound
0.556 1.302 0.185 0.006 0.851 0.675 0.117 0.538
0.638 0.695 0.112 0.006 0.727 0.668 0.240 0.353
0.687 0.538 0.044 0.002 0.454 0.912 0.245 0.304
0.471 1.010 0.084 0.073 0.460 0.906 0.258 0.401
2 3 4
Table 3 Minimum matching Dv (km/s) for trajectories matched at their second SOI exit and originating from northern and southern L1 and L2 halo orbits. #
Target NEA
Mission leg
L1 North
L1 South
L2 North
L2 South
1
1999 AO10 2000 SG344 2000 SG344 2000 SG344
Outbound Inbound Outbound Inbound Outbound Inbound Outbound Inbound
1.080 0.638 0.122 0.133 0.661 0.450 0.316 0.941
0.441 1.424 0.129 0.101 0.690 0.435 0.186 0.892
0.664 0.685 0.091 0.099 0.543 0.320 0.281 0.516
1.019 1.162 0.076 0.088 0.543 0.328 0.407 0.458
2 3 4
periapsis altitude of the trajectory with the minimum flyby altitude for each halo orbit is recorded. Moreover, the minimum distance from Earth along the halo orbit itself is documented. It can be seen that manifolds originating from halo orbits with lower out-of-plane amplitudes, i.e., halo orbits farthest away from Earth as seen in Fig. 2, have the lowest periapses. The periapsis altitude of the manifolds increases with increasing out-ofplane amplitudes of the halo orbits from which they originate, up to the point where the altitude of the halo orbit and the altitude of the associated unstable manifold hardly differ. With these observations and in order to maximize the efficiency of the maneuvers, this study restricts the parameter space by focusing on the halo orbits whose manifolds have a minimum periapsis altitude less than half of the altitude of the Earth’s SOI. In addition to the increased efficiency of the Oberth maneuver, halo orbits with lower out-of-plane amplitudes provide additional benefits as these orbits are less stable, i.e., less time is required to transit between these halo orbits and Earth, which is an advantage for mission planning.
5.4. Global grid search A global grid search in terms of the x-coordinate of a halo orbit’s axis crossing and t is performed for the
Fig. 5. Examples for outbound and inbound transfer trajectories for the mission to the asteroid 1999 AO10, matched at the first SOI exit. The black line indicates the direction of the v1 vector at the SOI. (a) Outbound trajectory departing from northern L1 halo orbit. (b) Outbound trajectory departing from northern L2 halo orbit. (c) Inbound trajectory arriving at southern L1 halo orbit. (d) Inbound trajectory arriving at southern L2 halo orbit.
A.K. Zimmer / Acta Astronautica 90 (2013) 119–128
northern and southern halo orbit families at L1 and L2 . The values of t are varied from 0 to 1 in steps of 0.01 and the halo orbit family lying within the most promising region as described above is discretized into 40 halo orbits at each L1 and L2 . Then, for each combination of x-axis crossing and t, the unstable manifold trajectory is propagated in the direction of Earth for a flight time up to two years. The minimum altitude recorded during the propagation must be within an altitude range from 500 km to half the altitude of the SOI. If the minimum altitude lies outside this altitude range, the trajectory is discarded. Although trajectories with a minimum altitude below 500 km could be modified to fall within altitude range, such trajectories remain to be investigated in further study. Within the propagation time of two years, some manifold trajectories encounter several Earth flybys and leave and re-enter the SOI repeatedly. For the grid search, up to two SOI exits after entering the altitude range are permitted, and the trajectory is then cut off when leaving the SOI. For the v1 vectors of the interplanetary trajectories, the values of the outbound and inbound trajectories of the four example missions are used. If the target v1 and the velocity vector of a trajectory when leaving the SOI draw an angle of less than 901, the trajectory is retained. All other trajectories are discarded because they would re-enter the SOI once their current v1 vector is adjusted
125
to the target v1 vector and would not arrive at the correct interplanetary trajectory. For each of the remaining manifold trajectories, two maneuvers are introduced in order to match the target v1 vector as described above: a periapsis maneuver in the direction of motion during the last flyby before the SOI exit, and a maneuver at the SOI exit. The sum of the Dv magnitudes associated with these two maneuvers is minimized and the minimum Dv sum along with the respective value of t for each orbit is recorded. In this approach, the transfers originating from the southern halo orbit family are not calculated separately. Since the southern halo orbit family results from the northern halo orbit family by reflection across the x–y plane, matching the southern halo orbit family to a given v1 vector yields the same result as matching the northern halo orbit family to the reflection of the same v1 vector across the x–y plane. The v1 vectors of the inbound interplanetary trajectories must be matched to the stable manifolds of the halo orbit family. However, the symmetry, s, of the equations of motion of the CR3BP as given in Eq. (7) allows the propagation of the stable manifolds backwards in time by means of propagating the unstable manifolds forward in time and reversing the signs of the respective state coordinates. Consequently, by applying this symmetry to the inbound v1 vectors, they can be matched to the
Fig. 6. Examples for outbound and inbound transfer trajectories for the mission to the asteroid 1999 AO10, matched at the second SOI exit. The black line indicates the direction of the v1 vector at the SOI. (a) Outbound trajectory departing from northern L1 halo orbit. (b) Outbound trajectory departing from northern L2 halo orbit. (c) Inbound trajectory arriving at southern L1 halo orbit. (d) Inbound trajectory arriving at southern L2 halo orbit.
126
A.K. Zimmer / Acta Astronautica 90 (2013) 119–128
Fig. 7. Examples for outbound and inbound transfer trajectories for the mission to the asteroid 2000 SG344, matched at the first SOI exit. The black line indicates the direction of the v1 vector at the SOI. (a) Outbound trajectory departing from northern L1 halo orbit. (b) Outbound trajectory departing from northern L2 halo orbit. (c) Inbound trajectory arriving at southern L1 halo orbit. (d) Inbound trajectory arriving at southern L2 halo orbit.
unstable manifolds yielding the same results as matching the original inbound v1 vectors to the stable manifolds. _ y, _ z_ Þ-ðt,x,y,z,x, _ y, _ z_ Þ s : ðt,x,y,z, x,
ð7Þ
In this manner, the propagation of only the unstable manifolds of the northern L1 and L2 halo orbit families are required and the results for the southern halo orbit families as well as the stable manifolds of all halo orbits can efficiently be deduced. 5.5. Initial guess selection & local optimization A local optimum is computed for the northern and southern halo orbit families at each L1 and L2 for both one and two SOI exits, giving a total of eight trajectories to be optimized for each outbound and inbound v1 vector. The cost function is formulated in terms of matching Dv as a function of x-axis crossing, i.e., halo orbit, and t. For each of the eight trajectories, the most Dv-efficient combination of halo orbit and t from the global grid search is used as an initial guess for further optimization. The lower and upper bounds for the x-axis crossing are given by the two halo orbits adjacent to the initial guess halo orbit. The lower and upper bounds for t are given by the best initial guess values for t along those adjacent halos, since a continuous behavior of t from one halo orbit to the next is observed in most cases when comparing
values of t of the most Dv-efficient transfers originating from adjacent halo orbits. The initial guess for the x-axis crossing and t along with their respective lower and upper bounds are fed into a simple optimizer based on a finite-difference algorithm. 6. Results Tables 2 and 3 summarize the results from the optimization of trajectories being matched to the v1 vectors at their first and second SOI exit, respectively. Note that these Dv values, termed matching Dv, represent the maneuvers required to connect halo orbits to the desired interplanetary transfer using the architecture described above. They do not include maneuvers required to rendezvous with or depart from the asteroid. Example matching trajectories between the libration point orbits and the SOI for the first two missions are shown in Figs. 5–8. In addition to the 3D trajectory, projections on all three coordinate planes are plotted. Since some trajectories have a very low perigee altitude at Earth, not plotting the Earth to scale would result in the trajectory seemingly intersecting the Earths’s surface. Hence, the Earth is plotted to scale at ½1m,0,0T and consequently it is barely visible. Large differences in Dv are observed for the same interplanetary trajectory, e.g., ranging from 0.638 km/s
A.K. Zimmer / Acta Astronautica 90 (2013) 119–128
127
Fig. 8. Examples for outbound and inbound transfer trajectories for the mission to the asteroid 2000 SG344, matched at the second SOI exit. The black line indicates the direction of the v1 vector at the SOI. (a) Outbound trajectory departing from northern L1 halo orbit. (b) Outbound trajectory departing from northern L2 halo orbit. (c) Inbound trajectory arriving at southern L1 halo orbit. (d) Inbound trajectory arriving at southern L2 halo orbit.
to 1.424 km/s for the inbound trajectory leg of the mission to asteroid 1999 AO10, depending on whether the cargo trajectory originates at a northern or southern L1 or L2 orbit and whether it is matched to the crew trajectory at the first or second SOI exit. In addition, the matching Dv required for a certain transfer depends strongly on the heliocentric specific energy and inclination of the transfer trajectory to the target NEAs given by the v1 vectors as well as the periapsis altitude of the Oberth maneuver. As expected, the matching Dv increases with increasing specific energy of the interplanetary trajectory since the difference between this energy level and that of the manifold transfer must be bridged. Also not surprisingly, the increase in matching Dv can be counterbalanced by decreasing the periapsis altitude of the Oberth maneuver. Although the halo orbit family contains orbits with very large out-of-plane amplitudes, the manifolds of these orbits do not approach Earth quickly enough to be of value in this study. However, halo orbits with small outof-plane amplitudes translate to manifolds with small out-of-plane amplitudes and do not by themselves reach high heliocentric inclinations. Hence, a maneuver raising the inclination is inevitable for higher inclinations of the heliocentric transfer. Consequently, these two factors should be taken into consideration when choosing the heliocentric transfer trajectory to the target NEAs, giving
priority to transfer trajectories with lower specific energies and inclinations. If the heliocentric trajectory is of low energy level and inclination, ‘‘free’’ connections to halo orbits can be found as is the case in the first mission to asteroid 2000 SG344.2 Especially since the connection between the interplanetary transfer and the manifold trajectory must be made twice for vehicle reusability, i.e., before and after each mission, matching Dv penalties are inflicted twice for high inclinations and specific energies. The x-axis crossings of the optimum halo orbits for stationing a vehicle in the proposed scenario vary within a certain range for each L1 and L2 . However, since heteroclinic, i.e., free transfers exist between halo orbits at L1 and L2 and hopping from one halo orbit to another orbit in the same family can be accomplished with few tens of m/s of Dv [10,11], it is not necessary to choose a single orbit for parking the spacecraft. It is observed during the optimization that small changes in t and x-axis crossing of the halo orbit yield great savings in matching Dv. Conversely, this demonstrates
2 This seems particularly promising if this scenario is combined with a chemical-electric hybrid-propulsion architecture as described in Ref. [9], as the interplanetary trajectories in such architectures initially are of lower specific energies and inclinations than is the case for architectures relying entirely on chemical propulsion.
128
A.K. Zimmer / Acta Astronautica 90 (2013) 119–128
how sensitive these trajectories are to errors in orbit control and manifold injection. Stationing unmanned vehicles in halo orbits at L1 or L2 for reuse in subsequent missions is advantageous if the cost of building and launching these vehicles anew for each mission is higher than the cost of building and launching the fuel stages required for the maneuvers of this architecture. Hence, the effectivity of vehicle reusability in the proposed scenario depends on the actual mission and campaign and is to be evaluated once these costs are determined.
Acknowledgments The research described in this paper was carried out at the Jet Propulsion Laboratory, California Institute of Technology and at the Institute of Space Systems (IRS), University of Stuttgart. The author wishes to thank Professor Dr. Ernst Messerschmid, chair of the Astronautics and Space Stations Group at the IRS, as well as Dr. Roby Wilson, supervisor of JPL’s Inner Planet Mission Analysis Group, for their dedicated support and sharing their comprehensive knowledge.
7. Conclusion & outlook References This study presents a mission architecture enabling the reuse of cargo modules, such as habitation modules or laboratories, throughout a campaign of several missions to NEAs, thus lowering campaign cost. For this purpose, the feasibility of parking such cargo modules in SEL point orbits at L1 and L2 and connecting them to interplanetary trajectories is shown. For four example missions, the manifolds of northern and southern L1 and L2 halo orbits are matched to these interplanetary trajectories by performing a maneuver at periapsis and when crossing the SOI. It is demonstrated that these connections are always possible, the required Dv ranging from a few meters per second to hundreds of meters per second, depending on the heliocentric specific energy, inclination, and periapsis altitude of the manifold trajectory. Lower Dv values for missions to the same targets can be expected if transfer trajectories of lower specific energy and inclination are chosen. In addition, chemicalelectric hybrid-propulsion architectures promise a further decrease in the Dv required to match halo orbits to interplanetary trajectory as the specific energy and inclination of such interplanetary trajectories are significantly lower when crossing the SOI. Furthermore, the use of an additional maneuver at halo orbit departure and its effects on launch period as well as periapsis altitude and thus, total Dv is to be investigated. Moreover, the study can be extended to address other interplanetary targets such as additional NEAs or Mars and its moons. As the total Dv is very sensitive to errors in orbit control and manifold injection, contingency scenarios counteracting these errors must be investigated in future work. Finally, all trajectories remain to be calculated in a full ephemeris model.
[1] R. Landis, P. Abell, D. Korsmeyer, T. Jones, D. Adamo, Piloted operations at a near-earth object (NEO), Acta Astronautica, 2009. [2] W. Huntress, D. Stetson, R. Farquhar, J. Zimmerman, B. Clark, W. O’Neil, R. Bourke, and B. Foing, The Next Steps in Exploring Deep Space, Universities Press, 2004. [3] A. Zimmer, E. Messerschmid, Going Beyond: target selection and mission analysis of human exploration missions to near-earth asteroids, Acta Astronautica 69 (11-12) (2011) 1096–1109. [4] W. Koon, M. Lo, J. Marsden, S. Ross, Dynamical Systems, the ThreeBody Problem and Space Mission Design. California Institute of Technology, 2006. [5] K. Howell, Three-dimensional, periodic, ‘halo’ orbits, Celestial Mechanics, 32 (1) (1984) 53. [6] E. Fehlberg, Classical Fifth, Sixth, Seventh and Eighth Order RungeKutta Formulas with Stepsize Control, NASA Technical Report TR R-287, 1968. [7] Jet Propulsion Laboratory, Solar System Dynamics, /http://ssd.jpl. nasa.gov/?horizonsS, 2012. [8] R. Bate, D. Mueller, J. White, Fundamentals of astrodynamics, Dover, 1971. [9] D. Landau, N. Strange, Human Exploration of Near-Earth Asteroids via Solar Electric Propulsion, AAS/AIAA Space Flight Mechanics Meeting, New Orleans, Louisiana, February 2011, pp. 13-17. [10] W. Koon, M. Lo, J. Marsden, S. Ross, Heteroclinic connections between periodic orbits and resonance transitions in celestial mechanics, Chaos, 2000. [11] K. Davis, Locally Optimal Transfer Trajectories Between Libration Point Orbits Using Invariant Manifolds, Dissertation, University of Colorado, 2009. Aline K. Zimmer received a M.Sc. in Aerospace Engineering from the Georgia Institute of Technology and a Dr.-Ing. (German Ph.D. equivalent in Aerospace Engineering) from the University of Stuttgart, Germany. Her research interests include astrodynamics, mission analysis, asteroid exploration, and human spaceflight.